-
’ y"..£’-11~~--:~u . ...‘......., -1.“ '7 ‘
.‘r '
O

A STUDY OF- PRECISION III

‘ ’ spzcmopnommnmc TECHNIQUES

 

 

- - , ‘ A missefiaticn‘ for the Gegree‘ bf Ph ‘D
' I I ,MIGAIGAN STATE UNIVERSITY
" v 7 ‘ h ‘ " LESLIE SAVID ROTI‘III‘AN
' 1974 '

   

J

III III IIIIIIIIII

31293010 ;LIBRARY Q,

Michigan State
Universit)’

This is to certify that the

thesis entitled . ~
A Study of Precision in
Spectrophotometric Techniques
presented by
Leslie David Rothman
has been accepted towards fulfillment

of the requirements for

Ph.D. degree in Chemistry

fl4/4/4/ gm

Major professor a;
J I 1" I i
D ngulx 22! 1974 5

 

0-7639

 

 

 

 

PLACE lN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

fi- .1 ‘

$32816;

 

 

 

 

 

 

 

 

 

___,_______-————-‘—..- A... __.__—_——-———
44#______ .___._———-—-'—A

 

 

 

 

 

 

2/05 cfi- RafioDUQJndd-p. 15

ABSTRACT

A STUDY OF PRECISION IN SPECTROPHOTOMETRIC TECHNIQUES

By

Leslie David Rothman

A previous theoretical investigation (1) of the precision to be
expected in molecular absorption spectrophotometry has revealed that
optimum measurement precision may not occur at 36.8% T. This treat-
ment proposes the effect of three limiting cases on measurement pre-
cision; limited readout resolution or dark current noise, photocurrent
shot noise, and source flicker noise.

An experimental study is presented which compares the observed
measurement precision to that predicted by theory for the three proposed
limiting cases. An addition to the theory is presented which predicts
the effect of sample cell positioning imprecision on measurement pre-
cision. An experimental study of the effect of sample cell positioning
imprecision is also presented. The four cases studied showed measure-
ment precision was optimized near 33% T for the dark current noise limit,
l0% T for the photocurrent shot noise limit, 2% T for the source flicker
limit and was still improving at l% T for the sample cell positioning
limit. The best measurement precision observed in this study was
approximately i0.0I%.

An instrument for photometric titrations is described which

determines the sample concentration by a curve-fitting technique.

€3\ Leslie David Rothman

o“

(if) A minicomputer controls the instrument and fits the theoretically
predicted titration curve to that observed experimentally. Titrations
performed with this instrument were typically precise to i0.3-0.l%.

A device is described which prepares reagent solutions automatically
under computer control. It is hoped that this device may be applied
to fundamental chemical investigation requiring large numbers of ac-
curately and precisely prepared solutions. It is projected that the
present device will be able to prepare reagent solutions from a
single stock solution over a concentration range ratio of 3.6 x lo4 with

all concentrations precise to within 0.1%.

A STUDY OF PRECISION IN SPECTROPHOTOMETRIC TECHNIQUES

By

Leslie David Rothman

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirement

for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemistry

1974

TO ANN, MOM AND DAD

ii

ACKNOWLEDGMENTS

This author wishes to express his deep gratitude to Dr. S. R.
Crouch for his guidance, encouragement and friendship during the
course of this investigation.

Thanks are also due to Dr. C. G. Enke for serving as second reader
and for providing valuable comments.

This author wishes to thank Dr. J. D. Ingle, Jr. for his original
proposal of the sampling theory and for his time in discussion of the
relationship between his theories and this author's experiments.

The following institution and businesses deserve special mention:
Digital Equipment Corp., for providing the author with an opportunity
to learn computer repair; Mother Nature, for providing the author
with a blood pressure directly proportional to the distance separating
the author and the local draft board; and the American Society for
Medical Technology, for giving the author the "inside dope" on hospital
laboratories.

The fellow members of the author's research group deserve special
mention for their comments, suggestions, criticisms and incredible
capacity for food and drink.

The author expresses his thanks to Michigan State University for
providing aid in the form of assistantships, and for providing excellent
facilities for the pursuance of this research. Ron Haas of the depart-
ment electronics shop and Russ Geyer of the machine shop are acknowledged
for their aid.

Finally, this author wishes to thank his parents for their support,

his wife, Ann, for keeping him fed, clothed, inspired and happy, and

Pippin, for waiting up for him at night and being purrfectly charm-

ing.

iv

TABLE OF CONTENTS

Page
LIST OF TABLES ....................... ix
LIST OF FIGURES ....................... x
CHAPTER I - Introduction .................. l
CHAPTER II - A Theoretical and Experimental Investigation
of Factors Affecting Precision in Molecular
Absorption Spectrophotometry .......... 4
A. Introduction .................... 4
B. Theory ....................... 5
1. Precision in the Absence of Sampling
Error ..................... 6
2. Errors due to Sampling Imprecision ....... 12
3. Calculation of Theoretical Curves ....... 18
C. Experimental .................... 19
l. Spectrophotometer ............... l9
2. Reagents .................... 21
3. Procedure ................... 21
D. Results and Discussion ............... 25
1. Introduction .................. 25
2. Theoretical Predictions ............ 25
a. Introduction ................ 25
b. Fundamental Noise Sources ......... 25
c. Excess Noise ................ 26
d. Calculation of Theoretical Relative
Measurement Error ............. 27

3. Case I - Variance Independent of Photocurrent . 31

4. Case 11 - Variance Proportional to
Photocurrent .................. 35

V

Chapter

5. Case III - Variance Proportional to

Photocurrent Squared ..............

6. Case IV - Sampling Imprecision .........

7. Effect of Limited Readout Resolution ......

CHAPTER III - Photometric Titrations ............

A. Introduction ....................

B. Historical .....................

1. Titrant Delivery Systems ............

2. End Point Detection ..............

3. Automated Photometric Titrators ........

4. Computerized Analysis of Titration Curves . . .

C. Theory .......................

0. End Point Detection ................
E. Least Squares Method for Titration

Curve Analysis ...................

1. Introduction ..................

2. Simplest Least Squares Approach ........

3. Extension to a Real System ...........

CHAPTER IV - Photometric Titrations-Experimental ......

A. Introduction ....................

B. Instrumentation ..................

1. Spectrophotometer ...............

2. Buret .....................

3. Computer System ................

a. General ..................

b. Buret Interface ..............

C. Analog-to-Digital Converter ............

vi

Page

38
39
46

49
49
49
51
52
53
55
51

65
65
65
67
7O
7O

7O
75
75

76
78

Chapter Page

D. Software ..................... 82

E. Determination of Instrument Linearity ...... 82

F. Experimental Results ............... 88

1. Chemical System ............... 88

2. Investigation of Signal Modifiers ...... 88

a. Precision of Results ........... 88

3. Accuracy of Results ............. 91

G. Test of Converging Ability of Software ...... 92

H. Future Prospectives ............... 92

CHAPTER V - An Automated, High Precision Reagent

Preparation System .............. 95

A. Introduction and Historical ........... 95

1. The Deming-Pardue System ........... 96

2. The Megargle-Marshall System ......... 97

8. Proposed System ................. 98

1. Introduction ................. 98

2. Stepping Motor-Syringe Assembly ....... 98

3. The Dilution Chamber ............. 103

C. Experimental Evaluation ............. 107

0. Proposed Applications .............. 109

BIBLIOGRAPHY ....................... 111
APPENDIX A - Description and Documentation of Curve-

Fitting Program for Photometric

Titrations .................. 114

A. Introduction ...... . . . .......... 114

8. Description of Software Operation ........ 114

C. Definition of Symbols .............. 117

PROGRAM LRTITR.FT . . . . . ............. 120

vii

Chapter Page
APPENDIX B - Transferring Data to and from the

RK8E Disc .................. 124
PROGRAM LRPUT.FT ................... 129
PROGRAM LRAD.FT ................... I33
PROGRAM LRGET.FT ................... 134

viii

III
IV

VI

VII

VIII-A

VIII—B

IX

XI‘AsB

LIST OF TABLES

Page
Definition of Terms ................ 9
Specified Noise Appearing in the Readout Due
to the Keithley I-V Converter ........... 28
Noise Appearing in the Readout Due to Dark
Current Shot Noise ................. 28
Instrumental Parameters .............. 30
Experimental Data ................. 34
Comparison of Readout Quantizing Error to
Errors Due to Shot Noise and Source Flicker
Noise ....................... 47
Crosstalk Levels in the A/D Converter
System ....................... 83
Modifications to BASIC/RT to Allow Use
of Datel A/D Converter ............... 84
Comparisons of Precision of Results
Obtained With Different Signa1 Modifiers ...... 90
Comparison of Accuracy of Results Obtained
with Different Signal Modifiers .......... 90
Test of Software Converging Ability ........ 93
Description of Valving for Figure 17 ........ 102

. Data from Experimental Evaluation of

Solution Preparation System ............ 108

ix

LIST OF FIGURES

Figure Page
1 Case I Study .................... 32
2 Case II Study ................... 36
3 Case III Study ................... 40
4 Case IV Study ................... 42
5 Comparison of Theory and Experiment from

all Four Studies .................. 45
6 Various Possible Photometric Titration

Curve Shapes .................... 56
7 Dependence of Titration Curve Shape of

the Complex Formation Constant ........... 58
8 Dependence of Titration Curve Shape on

Ratio of Titrant-to-Sample Concentration

([LJO/[M]o) .................... 60
9 Signals Derived from Derivative Endpoint

Detection Circuit ................. 63
10 Titration Cell ................... 72
11 Motor-Driven Buret Interface ............ 77
12 A/D Converter Interface .............. 80
13 Typical Computer-Operator Dialog for

Photometric Titration Program ........... 85
14 Flowchart for Photometric Titrator Program ..... 86
15 Determination of System's Adherence to

Beer's Law ..... . ............... 87
16 Diagram of Automatic Solution Preparation

Device ....................... 99
17 Stepping Motor Interface ............ ‘. . 101
18 Circuit Diagram for Optical Sensor 8 ........ 104
19 Flowchart for Operation of Automatic

Solution Preparation Device ............ 106

CHAPTER I

Introduction

Molecular absorption spectrophotometry is probably the most widely
applied instrumental technique for chemical analysis. This technique
owes its popularity to its simplicity, wide potential range of applica-
tion and the simple relationship between the signal from the output
transducer and the concentration of the chemical species being analyzed.
Often, however, the precision inherent in high quality spectrophotometric
instruments is not realized due to improper use of the equipment or
problems related to sample handling. This thesis describes three
studies which were conducted to identify the limits of spectrophotometric
measurement precision and to minimize measurement errors.

In the first study a theoretical and experimental investigation
of measurement precision in absorption spectrophotometry is described.
Random errors arising both from instrumental factors (fundamental and
excess noise in the photomultiplier, current-to-voltage converter and
readout device and flicker noise in the light source) and from sampling
procedures are considered. This study includes both a critical examina-
tion of previously proposed theories (1,2) and the presentation of
additions to this theory to explain the effect of imprecision in sample
cell positioning on measurement precision.

In the second study the development of a computerized system for
photometric titrations is described. Photometric titrations have long
been used for precision measurements, but it is important to note that

while spectrophotometric instrumentation has improved in quality in

the last 50 years, the precision of photometric titrations has not
undergone a concurrent improvement. In 1928, the precision of single
spectrophotometric measurements was on the order of il%, while photometric
titrations were often precise to 0.3 - 0.5%. Today, single spectro-
photometric measurements may be made to a precision of 0.01% (as shown
in this thesis), but the precision of photometric titrations has not
improved. The photometric titration system described here utilizes
an on-line minicomputer to collect and process titration data. Chemical
concentration is obtained via a curve-fitting procedure which is highly
precise. The insensitivity of the observed relative precision to sample
concentration indicated that the precision of concentration determina-
tions is better than the volumetric precision of sample and titrant
addition to the titration cell. The obvious volumetric imprecision
clearly reveals the importance of the third study reported in this
thesis.

In the third study the conception and initial development of a
high precision computer-controlled reagent preparation system is des-
cribed. rThe system is applicable to any analytical procedure involving
the precise control of solution concentrations and volumes. While
partially patterned after existing systems, this reagent preparation
system includes a novel method for the addition of diluent to the
reagent stock solutions, making it possible to prepare solutions over
a concentration range ratio of 104 from a single stock solution. Portions
of this device may also be applied to sample and titrant addition in
high precision titration experiments or simultaneous addition of two
or more solutions to a sample cell for reaction-rate methods of analysis.

This solution preparation system has been proposed in the expectation

that its application to all analytical methods in this laboratory,

both spectrophotometric and other methods, will eliminate or greatly
reduce errors due to imprecision in the preparation of sample and re-
agent solutions. In addition, because the system is totally automated,
solution preparation time is greatly reduced for high sample through-
put rate analytical instruments, such as a stopped flow system developed

in this laboratory.

CHAPTER II
A Theoretical and Experimental Investigation
of Factors Affecting Precision in Molecular

Absorption Spectrophotometry

A. Introduction

 

A recent treatment (1,2) of the precision to be expected in molecu-
lar absorption spectrophotometric measurements has revealed that the
assumption that optimum measurement precision occurs near 37% T may be
grossly in error under certain conditions. In this treatment, a unique
expression was derived for the signal-to-noise ratio (S/N) of measure-
ments in molecular absorption spectrophotometry. This expression was
evaluated for three different types of instrumental noise sources and
two different measurement techniques. Consideration was given to the
effect of these noise sources and measurement techniques on both over-
all measurement precision and the transmittance at which optimum measure-
ment precision is obtained.

Many modern instruments permit the operator sufficient latitude
in the choice of instrumental operating conditions so that any of the
noise sources previously described could become predominant under various
conditions. It is therefore important to understand how the various
instrumental noise sources contribute to overall measurement error.

In particular, a study of the relation between the theoretically
predicted relative measurement standard deviation and that attainable
in practice would give some important insights into how the theory
relates to experiment.

In this study, a single beam spectrophotometer with a readout

4

linear in transmittance was used to determine the relative measurement
standard deviation as a function of transmittance in the presence of
varying amounts and types of instrumental noise. The results of the
experimental study are compared to the relative measurement standard
deviation as a function of absorbance as predicted by theory.

In the first part of this chapter,a unique form of the theoretical
relation is developed. In addition to treating the noise sources pre-
viously considered (1,2), this development further considers the contribu-
tion of imprecision in cell positioning to the overall relative measure-
ment error. The second section describes the measurement system and
techniques used in the experimental study. The third section contains
a comparison of the experimental standard deviation of spectrophotometric
measurements to the theoretically predicted values. The contributions
of various individual noise sources to the total measurement error are
discussed. Studies were performed with measurement precision limited
by (1) amplifier-readout noise, dark current shot noise and excess
noise,(2) photocurrent shot noise, (3) source flicker noise and (4)
imprecision in sample cell positioning. The final section compares
the errors introduced by limited readout resolution to the errors from
other instrumental noise sources. This comparison clearly demonstrates
the constraints which limited readout resolution may place upon the

overall measurement precision.

8. Theory

Measurement precision in molecular absorption spectrophotometry
is dependent upon a number of factors. In order to understand how

these factors affect precision, it is first necessary to understand

what these factors are and how they relate to the measurement process.
The most commonly followed procedure for a spectrophotometric measure-
ment involves: (l) the measurement of the reference photocurrent
(100%T), (2) measurement of the dark current (0%T), (3) introduction
of the sample into the optical path, and (4) measurement of the sample
photocurrent (%T of the sample). Overall measurement precision is
influenced by imprecision in each step. The measurements in steps 1,2
and 4 are imprecise due to noise in the amplifier-readout system, and
shot noise and excess noise in the dark current. The measurements in
steps 1 and 4 are also imprecise due to shot noise in the photocurrent,
fluctuations in light source intensity and irreproducibility in per-
forming step 3. The contribution of each source of imprecision, except
cell positioning imprecision, has been previously treated (1,2). Hence,
the effects of noise sources not related to the sampling step will be
reviewed briefly followed by detailed treatment of errors due to sample

cell positioning.

1. Precision in the Absence of Sampling Error

 

Numerous treatments have been presented of the precision to be
expected in molecular absorption spectrophotometry (see 8-5) and references
in these papers). Many of these treatments only consider errors due
to the uncertainty in reading a linear scale. When such treatments are
applied to a spectrophotometer with a linear transmittance readout,
the result is the familiar prediction that absorbance measurement
precision is optimized at 36.8%1. Many modern spectrophotometers either
are or may be equipped with low noise, high resolution readout devices.

The precision of absorbance measurements performed with modern instruments

may not be limited by the amplifier-readout system, and in such cases,
the measurement precision will rarely be optimized at 36.8%T. Careful
consideration must therefore be given to the effect of the predominant
noise source.

The actual goal of any theoretical treatment of precision in
spectrophotometry is to predict the magnitude of the relative error
in a measurement of the concentration of some chemical species. The
relationship between the relative concentration error (ac/C) and the
relative absorbance error (oA/A) may be easily established. When Beer's

Law holds,
A=ebC (2.1)

where A is the solution absorbance, e is the molar absorptivity, b is
the sample cell path length and C is the concentration of the absorbing
species. The standard deviation in absorbance, 0A, is related to the
standard deviation in the measured concentration, cc, as shown in

Equation 2.2.
CA = Eb CC (2.2)

The relation between the relative absorbance error, oA/A, and the relative

concentration error, oC/C, may be written

CA - Choc CC
’A" ch t“ . (2'3)

 

As Equation 2.3 indicates, the relative errors in absorbance and

concentration are equal when Beer's Law holds. The theory presented
here expresses the relative measurement error as the relative absorbance
error, oA/A.

The following equations have been applied to a description of the
relative precision of absorbance measurements performed on an instrument
with a linear transmittance readout (1,2). It is assumed that the
transmittance measurement is performed by measuring a voltage, Ert’
corresponding to 100% transmission, a voltage, Ed, corresponding to
0% T, and a voltage, Est’ corresponding to the sample transmittance.
Variance in each measurement is treated. Equations 2.4-2.7 express the

transmittance, T, the standard deviation in the transmittance, oT, and

the relative absorbance error, oA/A. All terms are defined in Table I.

E - E
T . _____Est _ Ed (2.4)
rt d
2 2 1/2
_ T c‘st 2 (l-T) 2
GT—E;[(T)+Ort+(T)Odt] (2'5)
0 0
A _ r
T “ :‘r—rrr—r' (2'5)
0 1/2
A _ l -l 2 2 2 -2 -l
‘A— - W [KmeEr (1+r ) + 2Er g] +2(KmR1.lZd+o‘W )(l+T -T )] (2.7)

This derivation assumes that all noise sources are statistically in-
dependent, that there is negligible unidirectional instrumental drift
between the measurements of 100% T, 0% T and the sample transmittance,

and that sample cell positioning errors are negligible. Similar

Table I

Definition of Terms

Ert = Er + Ed (2.9)
ES = signal due to source radiation passed by sample cell containing
sample, V.

E = signal due to source radiation passed by sample cell containing
reference, V.

Ed = signal due to dark current, V.

CT = standard deviation in transmittance measurement, dimensionless.

ost = standard deviation in Est’ V.
2 2 2 1 2
= (Odt + (as) q+s + (as) f) / (2.10)
art = standard deviation in Ert’ V.
2 2 2 l 2
= (0dr; + (0,.) cm + (or) f) / (2.11)
Gdt = standard deviation in Ed, V.
2 2 1/2
- (o q+s + 0 ar) (2.12)
Oar = amplifier-readout standard deviation, V.
°q+s = dark current shot noise, V.
= RfKEd (2.13)

m = average current gain of photomultiplier, dimensionless.
Rf = feedback resistance in OA current-to-voltage converter, 0.

K = 2e Af (l + a) (2.14)

10

e = charge of an electron, C.

Af = noise equivalent bandwidth of amplifier-readout systems, Hz.

a = secondary emission factor, dimensionless.

(cir,.)q,.S

(OrIf

E] = source flicker factor, dimensionless.

(Us)q+s

(Us)f =

rms shot noise in Er’ V.

JmeKEr
rms flicker noise in Er’ V.

5l Er

rms shot noise in Es’ V.
= J'meKES = J'meKTEr
rms flicker noise in ES, V.

51 E5 = 51 TEr

(2.15)

(2.16)

(2.17)

(2.18)

ll

equations have been derived for cases involving different numbers of

0%T measurements (1), a linear absorbance readout (l) or chemical scale

expansion (6). The only significant difference occurs when the relative
precision of a linear absorbance readout instrument is limited by read-

out resolution (1).

Three limiting cases of Equation 2.7 have been identified (1).
These cases correspond to the situations in which the first (Case 11),
the second (Case III), or the third (Case I) term in Equation 2.7 is
predominant. The Case I term includes errors due to amplifier-readout
noise, dark current shot noise and excess noise. Errors included in
the Case I term are independent of photocurrent. When the Case I term
predominates, the relative standard deviation in absorbance, oA/A,
is predicted to decrease from 100%T to 38.8%, then rise rapidly as the
transmittance decreases further. If the voltage, Ed’ due to the dark
signal is measured twice, once for the term Ed in the numerator of
Equation 2.4 and a second time, independently, to determine the term Ed
in the denominator, the form of Equation 2.7 is changed slightly and
the relative absorbance error is predicted to reach a minimum at 33.1%T
(1). Commonly, the dark current is only measured once, and that is
the situation considered in the theory presented here. The Case I
term includes the sources of imprecision in the classical treatments
and predicts nearly the same result.

The Case II term describes the measurement variance due to shot
noise in the reference and sample photocurrent. The variance due to
these terms is proportional to photocurrent. When the Case II term
predominates, the relative absorbance error is predicted to decrease

from 100%T to 10.9%T and to increase with further decreases in %T.

12

If the noise in the reference photocurrent measurement is ignored in
the theoretical treatment, the relative measurement error is predicted
to reach a minimum at 13.5%T.

The Case III term describes the measurement variance due to varia-
tions in the light source intensity, referred to here as source flicker
noise. The variance due to source flicker is proportional to photo-
current squared. When the Case III term predominates, the relative
measurement error is predicted to decrease continuously with transmittance.
No optimum transmittance is predicted for Case III, but Case I terms begin
to become important at very low transmittances and limit the measurement
precision. Further, stray light will cause Beer's Law deviations, and
invalidate the assumption that oA/A = oC/C. The most precise mode of
instrument operation is that which causes the Case III term to pre-
dominate in the overall measurement error. Source flicker is excess
noise, as is the excess noise in the dark current, and therefore the
magnitude of the error due to source flicker may not be predicted from
fundamental theory. The magnitude of the excess noise terms may be

decreased by improvements in instrumentation.

2. Errors due to Sampling Imprecision

 

The theory presented above considered the effect of electronic
instrumental factors on measurement precision. In practice, these
limits are not always observed since all real measurements are also
affected by imprecision in the preparation of the sample and positioning
of the sample and reference cells. It is therefore important to under-
stand how sampling errors affect relative measurement precision.

Sampling errors may arise either from sample preparation or from

l3

imprecision in the positioning of the cells in the spectrophotometer.
The treatment of errors due to sample preparation is straightforward
and independent of any spectrophotometric factor and will therefore not
be discussed here. Imprecision due to cell positioning may arise from
irreproducibility in placing a sample cell in a cell holder, imprecision
in a mechanical cell positioning apparatus or from changes in the posi-
tion of a fixed cell due to the operations of filling and emptying the
cell, thermal expansion and vibration. These variations in cell position
can cause a variance in both the reference and sample photocurrent.
Sample positioning imprecision may cause a measurement error by at
least two different mechanisms. Variations in cell position will lead
to changes in the reflective losses at the four interfaces in the sample
cell (two air-glass, and two glass-solution), and to changes in the cell
path length. First, consider the effect of random variations in cell
position on the reflective losses. Systematic errors due to reflective
losses have been previously treated (9). The reflective losses at the
air-glass interfaces are generally larger than those at the glass solu-
tion interface and are the only losses which will be considered in this
treatment. A reflective loss, R], may be identified at the cell wall
through which the source radiation enters the sample cell and a second
loss, R2, may be identified at the exit point. These reflective losses
may be related to the radiant power incident on the cell, Po, as shown
in Equations 2.19 and 2.20.
R

= P f (2.19)

R2 = (PO-Pof1)Tf2 (2.20)

14

The terms f] and f2 represent the fraction of the incident light re-
flected and T is the transmittance of the solution within the cell.

The total radiant power transmitted, P], is, therefore,
P1 = (Po-R])T - R2 (2.21)

The variance in the total radiant power transmitted, oP 2, can be

1
calculated by propagation of error mathematics, assuming of? and ofzz

to be the only sources of variance, and is shown in Equation 2.22.

 

2 _ 2 2 2 2 2 2 2 2
. . . 2 2 -
DlVldlng through by P0 T , one obtains
op 2
212 = (f 2-f +1) 2 + (f 2-f +l)o 2 (2 23)
POT. ,2 2 0r1 1 1 f2 °

The terms f] and f2 are generally small (m0.05), and Equation 2.23
may be approximated by Equation 2.24, if the first and second-order
terms in f1 and f2 are neglected.

2 o 2
f2 (f1 = f2<<l) (2.24)

 

2...

If it is assumed that of1 = Uf229 and that PC = KEr, Equation 3.24 may

be rewritten.

2 z 2 2 2 2
op1 - 20f1 T K Er (2.25)

15

It may readily be seen that Equation 2.25 is of the same form as
Equation 2.18 describing errors due to source flicker. When T = 1.0,
Equation 2.25 reduces to ZoflzlgErz. The relative standard deviation in
reflective losses, of], may be treated like source flicker, and the
product /2 Kof] will be referred to here as the cell transmission flicker

factor, E2. The total flicker noise in the reference signal may there-

fore be written as

. = (512+a22) Er2 (2.26)

and the total flicker noise in the sample signal may be written as
(asifz = (:12 + 522) TZEr2 (2.27)

The expression derived for the cell transmission flicker factor should
be accurate to within ~3%, given the assumptions used in the derivation.
From Beer's Law, it is clear that the absorbance of a given solu—
tion varies proportionally with molar absorptivity, cell path length
and concentration. Since variations in cell position may affect the
cell path length, it is important to consider the effect that such
variations will have upon measurement precision. A sampling flicker
factor, E3, may be identified which represents the relative standard
deviation in absorbance due to the dependence of cell path length on
cell position.

The effect of E3 on ES may be derived as follows:

r = io'EPC (2.28)

l6

°r2 = (-eC io'ebc)2 obz (2.29)
2 10 T log T 2 2
o = ——9——-10 o (2.30)
r . b b
_ log T log T _ T log T '
O O 0'
A - __;E_.= _ ___2__.=
TA" -TlnT 2.3035 g3 (2°32)

As Equation 2.32 indicates, the relative absorbance error due to
sampling flicker is a constant, independent of absorbance. If measure-
ment precision were limited by variations in cell path length, a constant
error would be predicted, regardless of photocurrent. A factor,
53*, may be identified as the relative standard deviation in Es due
to sampling flicker.
* OT
53 = 2.303 53A = :T' (2 33)
The total flicker noise in ES is given by Equation 2.34 where both cell
positioning effects are taken into account.

2 2 * 2 2 2 * 2 2 2
(05).: = [E] + £22 + (£3 )2] E5 = [E] + £2 + (£3 ) ]T Er (2°34)

*

It may be noted that the magnitude of :3 in Equation 2.34 increases
as transmittance decreases. The sampling flicker factor will eventually

limit measurement precision if not limited by other factors.

If the reference solution absorbs light at the analytical wavelength,

l7

Er will also be affected by path length changes. This may be incor-
porated into 53.

Sampling imprecision may be incorporated into Equation 2.7 by
substituting Equations 2.26 for 2.16 and 2.34 for 2.17 to obtain the

result shown in Equation 2.35.

2 2

35-- (-E l r)‘2 [KmR E 1+T'1) + ( + 2)2E +
A "{ r " r r( 51 52 r

1/2
2(i+r'2-r")(i<meEd + oar2)] + 532} (2.35)

Comparison of Equation 2.35 to 2.7 indicates how sampling imprecision
may affect the relative precision of an absorbance measurement. If
Case I or II conditions predominate, the effect of sampling imprecision
will not be observed. Case 111 conditions may prevail due to source
flicker or variations in the reflective losses at the air-glass
interfaces. The predicted effect of either variance is increased
measurement precision as absorbance increases. If the sampling
flicker factor becomes the limiting factor at high absorbance,
oA/A = 53, and the relative measurement error will become a constant
independent of absorbance. As absorbance continues to increase,
the relative measurement error will eventually also begin to increase,
due to the increasing significance of signal and dark current shot
noise and amplifier-readout noise.

The equations presented have been discussed assuming that only

one of the error terms in Equation 2.35 is significant at any given

l8

absorbance. In practice, the measurement variance is the sum of
variances due to all noise sources and, often more than one of these
noise sources will be significant. In such a case, the observed
relative measurement error as a function of absorbance will be inter-
mediate between the error functions of the individual noise sources.
If the values of all the parameters in Equation 2.35 are known, the
relative standard deviation in absorbance as a function of absorbance
may be predicted for any instrument.

3. Calculation of Theoretical Curves

 

The parameters in Equation 2.35 were evaluated for the instrument
used in this study. The constant K from Equation 2.35 had the value

1.6 x 10"9

, since the bandwidth of the readout device was 0.5 Hz.

The effect of secondary emission noise was ignored when calculating K.

The source flicker factor, E], was measured directly by measuring the
standard deviation in Er under source flicker limited conditions. The
cell transmission flicker factor, g2, was evaluated by placing a reference
solution in each of two sample cells in a sample cell positioner. Er
was set with cell A in position, then cell 8 was moved into the optical
path and the transmittance was measured. This procedure was repeated

50 times to establish a variance in the measurement of the transmittance
of sample cell B. The dark current voltage signal, Ed, and the variance
in Ed, Odtz’ were readily measured. The gain of the photomultiplier
transducer (PMT) was determined by measuring the photoanodic current of
the PMT and comparing that current to the photoanodic current of a type
935 vacuum photodiode (RCA), when both the PMT and the vacuum photo-

diode received the same incident radiant power. Since the photocathodes

of both devices have the same spectral response (S-5) and since the

l9

photodiode has no internal gain, the ratio of the observed photoanodic
currents represents the approximate internal gain of the PMT. No value
was determined for £3, for reasons which will be discussed later. Once
determined, these parameters were used to calculate predicted values of
oA/A as a function of absorbance for the four different instrumental
cases studied in this work. Two theoretical curves were calculated

for each case. First, a curve was calculated considering only amplifier
noise and shot noise as dark current noise sources. This curve repre-
sents the limit of precision which may be approached if all excess noise
in the dark current is eliminated. A second curve was calculated con-
sidering the experimentally measured dark current variance as the dark
current noise. This curve represents the predicted limit of precision
under the experimental conditions studied and should agree well in shape

and magnitude with the experimental data.

C. Experimental

 

l. Spectrophotometer

 

The instrument used in this study was a single beam molecular
absorption spectrophotometer (EU-701A, GCA McPherson Corp. Acton, MA),
with a high quality current-to-voltage (I-V) converter and high resolu-
tion, linear transmittance readout. The monochromator slit width,photo-
multiplier supply voltage and I-V converter gain were varied to change
the contributions of Case I, II and III noise sources to the relative
measurement error. The modular construction of the spectrophotometer
also allowed the use of two different sample cell positioning modules
in addition to a module with a rigidly positioned sample cell.

The tungsten light source power supply was operated in the optical

20

feedback mode for maximum stability. A Corning number 5-57 bandpass
filter was placed at the entrance slit of the monochromator to reduce
stray light. The monochromator was set to 450.0 nm, and the slit width
was varied to vary the source spectral radiance entering the monochromator.
The lP28A photomultiplier transducer (PMT) was operated at power supply
voltages from -320 VDC to -650 VDC. At 450.0 nm, the cross-correlation

of the tungsten lamp source spectral radiance and the PMT spectral
sensitivity is nearly maximized, which allows maximum flexibility in

the control of instrumental parameters.

The sample cell used in the study of Cases I, II and III was a 2.5
cm path length glass cell held firmly in position by a plexiglass base.
Sample cells used in the study of cell positioning errors were 1 cm
quartz cells.

The current-to-voltage (I-V) converter (Model 427, Keithley Instru-
ments, Ind., Cleveland, OH) used in this study provided a variable range
of transfer functions (104 - 1010 V/A) and a wide range of current
suppression (10'3 - 10‘13A), which was useful in nulling out dark current.
Noise from the I-V converter and current suppression circuitry was
insignificant in nearly all cases. The rise time of the I-V converter
was switch selectable from 10 microseconds to 300 milliseconds. A l
msec rise time was selected for this work to prevent the amplifier from
limiting the noise equivalent bandwidth of the amplifier-readout system.

The readout device used in this study was a 6-digit digital voltmeter
(DVM) (EU-805A, Heath Co.). This DVM consists of a voltage-to-frequency
converter and a gated counter. A one second gating time was used in
all studies (noise equivalent bandwidth of 0.5 Hz), with one volt giving

a full scale reading. In order to prevent errors due to limited readout

21

resolution, the I-V converter gain was adjusted to maintain an output
signal within the range of 0.1 V to 1.0 V for all sample and reference
photocurrents. The conservatively rated noise specifications of the I-V
converter indicated that changes in gain did not significantly affect
the measured standard deviation in absorbance.

In order to assure maximum freedom from instrumental drift, all
electronics were permitted a minimum of 24 hours warmup time. The
measurements for each limiting case were performed in the minimum time
permitted by the measurement technique used in order to reduce irregulari-
ties in the data due to long term instrumental drifts. One factor
limiting the measurement rate was a fatiguing effect in the PMT, which
gave a short term (m20 seconds) drift in the photoanodic current follow-
ing a large change in the incident radiant power. All measurements were

performed after settling of the photocurrent.

2. Reagents

All analyte solutions were prepared by dilution of a potassium
dichromate stock solution (5000 ppm K20r207 in 0.1 N sulfuric acid)
with 0.1 N H2304. Analyte solution concentrations were chosen spanning
the absorbance range of greatest interest for the limiting case being
studied. Distilled, deionized water was used as the reference solution.
All solutions were filtered prior to use to reduce errors due to particu-

late matter in the solutions.

3. Procedure
The rigidly positioned sample cell was used in the studies of
Cases I, II and III to eliminate effects due to cell positioning im-

precision. A preliminary set of data indicated that the best cell

22

positioning module available for the spectrophotometer introduced a
sampling error that was very significant in the Case II and III studies.
The dark signal voltage, Ed, was nulled out with the current suppression
control of the I-V converter. The cell was then filled with the
reference solution and the reference signal voltage, Er’ was set to
1.0000 V by adjustment of the PMT power supply voltage. The reference
solution was aspirated from the cell, the cell was rinsed with the analyte
solution, aspirated dry and then refilled with the reference solution.
After equilibration of the photocurrent, 10 consecutive measurements
of the sample signal voltage, Es’ were recorded at a rate of one measure-
ment per second. This process was repeated to collect a total of 40
measurements of Es' The individual variances in E5 of the 10 point data
sets were used to calculate the variance of the 40 point group. Sets
of data with excessively high variance were rejected. The occurrence
of such highly imprecise data sets was predictable during data collec-
tion when mechanical perturbations were inadvertently introduced (doors
slammed, laboratory bench bumped or leaned upon) or when transient
signals appeared on the AC line voltage due to the use of other electrical
equipment in the laboratory. Ten percent of the data were rejected for
gross disagreement with other measurements. Ninety percent of the
deviant data sets were predicted during the data collection process.

The method of sampling used in the Case I, II and III studies were
chosen on the basis of a previous study (8) and a study performed in
this laboratory. Nelson and Hawes compared the relative absorbance
error of two sampling procedures for solutions with A = 1.0. One
procedure involved the removal and reinsertion of the sample cell

prior to each measurement, while the second method involved the filling

23

and emptying of a fixed cell. They observed that measurement precision
for the procedure using the fixed cell was as much as 18 times better
than that for the procedure in which the cell was reinserted before
each measurement.

A second study, performed in this laboratory, also compared the
relative absorbance error of two sampling techniques. The first
technique involved filling a fixed sample cell with the reference solu-
tion, adjustment of Er’ rinsing and filling with the analyte solution

and measurement of Es’ The reference voltage, E , was set prior to

r
each measurement of Es' At an absorbance of approximately 1, the relative
absorbance error was 20.036%. This method was compared to the procedure
of setting Er once and then performing 10 successive measurements of E5,
the method subsequently used in this investigation. This technique
gave a relative absorbance error of 0.01% at the same absorbance. Ten
measurements were performed by each method. The F test may be applied
to determine if the variances of the two sets of data are significantly
different. With nine degrees of freedom in each data set, the ratio
of the variances must exceed 10.8 to be significant at the 99.9% confi-
dence level. The ratio of the variances is (.036/.Ol)2 = 12.86. The
difference in precision for the two methods is therefore highly signi-
ficant.

Since the investigation of Cases 1, II and III must be made with
the assumption that sampling error is insignificant, the sampling
method used in these investigations must minimize these errors. Since
the studies just discussed show that the minimum sampling error is

observed when the sample cell is not moved or emptied between measure-

ments of Es’ this method was used in the study of Cases I, II and III.

24

The variance in the data sets collected using this measurement
technique is treated differently than the variance of an alternate
reference, dark, sample voltage measurement technique in determining
oA/A. The data treatment will be discussed later. This sampling
method will yield reliable results if unidirectional instrumental drift
is low during the period between the first and tenth measurements of
sample transmittance. Long term drift in this instrument was measured
and found to be typically 0.002 - 0.0005% during the measurement period,
which was insignificant in all cases with respect to random instru-
mental noise.

A sample cell positioning module (model EU-70l-ll, GCA McPherson)
with a manually controlled four position sample cell holder was sub-
stituted for the module containing the fixed sample cell in the
study of sample cell positioning imprecision. A similar, but less
comprehensive, study was performed using a sample cell module with a
motor driven cell positioner (EU-7ll-ll, GCA McPherson). The procedure
followed in this study was: (1) nulling out the dark current, (2)
measurement of the reference photocurrent, (3) positioning of the sample
cell in the optical path and (4) measurement of the sample photocurrent.
This is the procedure for which Equation 2.35 describes the relative
measurement error.

During the sampling imprecision study, the instrument was operated
under conditions similar to those used in the source flicker (Case III)
study, thereby minimizing all noise due to sources other than sampling
imprecision. The data clearly illustrate the constraints which sampling

imprecision may place upon overall measurement precision.

25

0. Results and Discussion

 

1. Introduction

 

In this section, results of experimental measurements of spectro-
photometric precision are compared to results of theoretical calcula-
tions. In order to compare results and to predict the absorbance at
which best measurement precision is obtained, plots of the relative
absorbance error, oA/A (directly proportional to the relative concentra-
tion error, oC/C) were made as a function of absorbance for the limiting

cases discussed in the previous section.

2. Theoretical Predictions

 

a. Introduction - The total relative measurement error may be

 

predicted if the magnitudes of the individual instrumental errors may
be measured experimentally, calculated from theory or derived from
existing instrument specifications. It is necessary to consider the

instrumental noise sources described in the following sections.

b. Fundamental Noise Sources - Fundamental noise is generated by

 

motion of discrete charges in circuits, has a flat power density
spectrum, and cannot be completely eliminated (9). The two most
important types of fundamental noise are Johnson, or thermal noise,
and shot noise. Johnson noise is produced by the random motion of
electrons in a resistive element, caused by thermal agitation. The
rms noise voltage due to Johnson noise, vrmsu’ contained in bandwidth
Af may be expressed as

- 1/2
17mSJ - (4kTRAf) (2.36)

26

wherel< is the Boltzmann constant, T is the temperature of the resistor
in °K, and R is the resistance of the element. In spectrophotometry,
Johnson noise is present in the voltage signal at the output of an I-V
converter, and arises from the internal feedback resistance, Rf, and the
dark current suppression resistor, Rs’

Shot noise results from the random motion of discrete charge
carriers across a junction. In spectr0photometry, the main source of
shot noise is the photomultiplier transducer (PMT), due to the discrete
nature of the electrons ejected from the photocathode (quantum noise)
and from the dynode stages (secondary emission noise). The rms shot
noise current, Irmss’ may be expressed in terms of the average current
i, by the Shottky equation

' - Qei I/2
1rmsS ' _E_ (2-37)

Qe is the charge on a single electron and t is the integration time of
the measurement device. For an integrating measurement device, the
noise equivalent bandwidth, Af, is equal to l/(2t), so the Schottky
equation may be written as

1/2

1 = (ZQeiAf)

(2.38)
rmsS

The amount of noise added by secondary emission noise increases Arms
5

by only 10% to 40% and is often ignored.

c. Excess Noise - Excess noise is due to imperfect instrumentation

 

and may be reduced or eliminated when instrumentation is improved.

27

Excess noise is almost always frequency dependent. The most common
sources of excess noise in spectrophotometry are variations in light
source intensity (source flicker), variations in PMT gain due to power
supply fluctuations (photomultiplier flicker) and flicker noise in

the semiconducting elements in the I-V converter.

d. Calculation of Theoretical Relative Measurement Error - The
relative measurement error due to Johnson noise in the amplifier-
readout system may be calculated if the resistances Rf and Rs are known.
Noise in the amplifier-readout system is generally expressed as the rms

noise current at the amplifier input. This may be converted to the

standard deviation in the voltage observed in the readout, Oar’ using
Equation 2.39,
_ 2 . 2 1/2
Oar - (Rf (Arms) Af) (2.39)

109’

where Af is the bandwidth of the readout system. When Rf = 10
Car for the Keithly amplifier used in this study is specified to be

10'15 A/unit bandwidth. The standard deviation in the amplifier out-
put voltage (assuming Af - 0.5 Hz) may be calculated to be 7 x 10"6 V.
In addition, the noise due to the current suppression resistor, R

when it is assumed RS = 10100, may be calculated to be 6.6 x 10'6

S,
V.

Both noise levels are insignificant with respect to the overall noise
voltage. Various amplifier-readout noise levels are listed in Table
IIA.

The rms shot noise in the dark current, 0 , may be calculated

0+5
from Equation 2.13. Various dark current shot noise levels are listed

in Table IIB.

28

Table II-A

Specified Noise Appearing in the Readout Due to

the Keithley I-V Converter

 

 

 

 

 

 

 

 

Rf,0 irms’ per unit bandwidth, A Oar’ V
106 2 x 10"3 1 4 x 10‘7
107 2 x 10"4 1 4 x 10‘7
108 4 x 10"]5 2 8 x 10‘7
109 1 x 10“5 7 x 10‘7
1010 1 x 10'15 7 x 10'6
10H 8 x 10"6 5 7 x 10'5
Table II-B
Noise Appearing in the Readout due to Dark
Current Shot Noise

Rf, Q m Ed,V oq+s,V
106 560 8 x 10"5 8.5 x 10'8
107 560 8 x 10‘4 8.5 x 10"7
1010 560 0.8 8.5 x 10'4
106 3 x 105 2.6 x 10’4 3.5 x 10'6
109 3 x 105 0.26 3.5 x 10‘3
1010 2520 1.0 2.0 x 10'3

 

 

29

The rms shot noise in the photocurrent, (as) , may be calculated

9+5

from Equation 2.15(<5 = (KmeTEr)1/2).

s)q+s

The rms source flicker noise may be calculated from equation 2.18
((os)f = ngEr)' The source flicker factor, 5], must be determined
empirically, since flicker noise is excess noise, and is not theoretically
calculable. This is also true for the sampling flicker factors, 52
and E3.

The theoretical plots presented in this work are derived in two
manners. First, the theoretical limit of the relative measurement error
was calculated from the measured source flicker factor, the specified
amplifier-readout noise, and the fundamental shot noise terms. A second
theoretical limit was calculated from the same factors with the measured
dark current variance substituted for the variance due to shot noise.
The first set of calculations therefore represents the theoretical limit
of precision for the instrument used in this study if all excess noise
in the dark current may be eliminated. The second set of calculations
represents the theoretical limit of precision predicted for this study
under actual experimental conditions. These theoretical calculations
are plotted as the smooth curves in Figures 1 through 5. These plots
represent the predicted relative absorbance error as a function of
absorbance for the experimental conditions listed in Table III.

The experimental data presented in this section were collected
in two different forms. Due to the measurement technique used, the
variances in the sets of data collected for the Case I, II and III
studies represent the variance in the sample signal voltage measure-‘
ment, Cstz’ not the variance in the transmittance measurement, 0T2. In

order to calculate 0T2, it was necessary to determine the total variance

30

II
N
LU‘

 

 

 

 

 

 

 

mmoo.o
.AP vv on o» vmsammm 8v mmLmQEm mp-o_ x o._ u m< mm u g
~ooo.o u _8
so. u Em .;o_8n 8cm .o.o u e
.1
“com A Ev Am_-o_ x mu.PA AoF-o~ x No.NAV =o_m_oata
OOON omm N2 8OF op-o_ x m.o 8-op mcw_asmm
Aomm u as AN.-oP x m~._4v A_,-o_ x No.~AV
omw_ omm .mo_ Ac_ op-o_ x 8.8 N.2 HHH ammo
“888.com u as A~,-o_ x m¢._AV Ao_-o_ x «As
ocm omo N2 mo_ o_-op x 8.~ 8-0— HH ammo
Aommm u as Am_-op x F.NAV Am_-op x m.NAV
_N omm P_o_ opofi o_-op o_-o_ H 8448
.ACOLULE .=_mm 8:8 .Uo> Fo.o-P.oue P.o-_ue .< .coeamwsao .< .=84882>8o
.588_z “Pym .88m8_o> #28 L nemecaam 8:8 etaucmom 8:8
copmsoccoocoz a . m < .pcmggau xcmo < .ucmcgaoouozm

mocmcmmmm

 

 

mgmumsmcma .muc052cum:H

HHH mpnmh

31

2

in the reference and dark signal measurements, 0 t2 and Odt and calculate

r
0T2 from Equation 2.5. The relative standard deviation in absorbance
may then be calculated from Equation 2.6. The variances in the sets

of data collected for the study of sampling imprecision represent the
variances in the transmittance measurements directly, due to the measure-
ment procedure used. The experimental values of oA/A calculated from

Equation 2.6 are plotted as the individual points in Figures 1 through

5.

3. Case I - Variance Independent of Photocurrent

 

The instrumental conditions selected from the study of Case I

are summarized in Table III. The reference photocurrent and the dark

"ADA. The PMT supply voltage was

-lO

current were both adjusted to 10
adjusted with the PMT in darkness to give a dark current of 10 A,
which was then nulled out with the current suppression circuitry of

the I-V converter. The PMT was then illuminated by the source radiation
passing through a reference solution, and the monochromator slit width
was adjusted to obtain the desired photocurrent. The relative standard
deviation in the dark current and reference photocurrent were measured
and determined to be 0.21 and 0.291% respectively. It may be calculated
that the dark current shot noise and the specified amplifier noise
individually give rise to relative standard deviations in the dark
current of 0.2% and 5 x 10'4%, respectively. The remaining noise was
excess noise in the PMT due to the low operating voltage. The measured
standard deviation in the dark current agreed so well with that pre-

dicted due to shot noise that the two theoretical curves could not be

resolved.

 

32

1.2 I:

1.8 —

2.0 '-

 

 

 

22 I I I I I I I
0 .2 .4 .6 .8 1.0 1.2 1.4

ABSORBANCE

Figure 1. Case I Study.
0 - experimental data

- - theory

33

The experimental data are plotted in Figure l as the individual
points and are listed in Table IV. Theory predicts that the best
relative measurement precision will be 0.8% at T = 0.335 (A = 0.475)
if excess noise is ignored and 0.805% at T = 0.339 (A = 0.470) when the
total measured dark current noise is considered in the theoretical
calculations. The difference between the predicted transmittance at
optimum measurement precision for this study, T = .339 and that for
pure Case I, T = .388 is due to the magnitude of the photocurrent shot
noise term. At T = 0.40, the dark current shot noise is only 1.6
times as large as the photocurrent shot noise. The measurement pre-
cision was therefore limited by a mixture of Case I and II, with a
predictably lower optimum transmittance than that for pure Case I.

The best relative standard deviation in absorbance observed experimentally
was 0.9% at T = 0.378 (A = 0.423).

Case I behavior is typically observed in an instrument with limited
readout resolution (analog meter or 2 1/2 or 3 1/2 digit digital panel
meter (DPM)), excessive amplifier noise, or very low photocurrent such
as might be observed near the limits of PMT sensitivity. Case I
generally represents the mode of instrument operation yielding the
poorest measurement precision. The most common cause of Case I con-
ditions is the use of a readout device with limited resolution. If
noise cannot be observed in the readout device, measurement precision
may nearly always be improved by using a readout device with higher
resolution. Amplifier noise may be reduced to near the Johnson noise
limit by the use of a high quality I-V converter. The noise due to
dark current shot noise may be reduced by reducing the equivalent noise

bandwidth of the readout device, by cooling the PMT to reduce dark

Experimental Data

34

Table IV

 

 

 

 

CASE I CASE II
A oA/A A oA/A
.127 0.0238 .256 0.00155
.192 0.0175 .335 0.00114
.257 0.0114 .421 0.000765
.354 0.0099 .492 0.000650
.423 0.0090 .650 0.000514
.549 0.00925 .851 0.000417
.759 0.0122 1.015 0.000360
1.205 0.0170 1.152 0.000408
1.360 0.000463
2.020 0.000484
CASE III CASE IV
A oA/A A oA/A
.298 0.000513 .144 0.0427
.678 0.000380 .326 0.0174
1.215 0.000153 .422 0.0129
1.800 0.000121 .542 0.0098
2.57 0.000204 .718 0.0071
3.18 0.000532 1.013 0.0049
1.34 0.0049
1.84 0.0029

 

 

35

current due to thermionic emission or by using a PMT Specially selected
for low dark current, since the magnitude of dark current shot noise
seen in the readout is proportional to the square root of both the read-

out bandwidth and the average dark current.

4. Case 11 - Variance Proportional to Photocurrent

 

In this study, an attempt was made to isolate Case II by making
photocurrent shot noise predominant over a wide range of transmittances.
As Indicated in Table III, the radiant power incident on the PMT was
increased by increasing the monochromator slit width to make the reference
photocurrent much larger than the dark current. The PMT supply voltage
was adjusted to give a high value for the average PMT gain, m, to assure
that the reference photocurrent shot noise would be greater than the
source flicker noise. The reference photocurrent and dark current were
10'6A and 2.6 x lO'IOA, respectively. The measured relative standard
deviation of the reference photocurrent was 0.04%, and that of the dark
current was 0.5%. In this study, the theoretically predicted noise
due to dark current shot noise agreed well with the dark current noise
observed experimentally. At a PMT supply voltage of 650V, the principal
source of dark current is thermionic emission, and the principal source
of dark current noise is shot noise, which has a predictable magnitude.
The two theoretical curves in Figure 2 agreed so well that they could
not be resolved. At the predicted optimum transmittance for Case II
(T = 0.109), the calculated dark current shot noise is 0.046% of the
calculated photocurrent shot noise, while source flicker noise is 16%
of the photocurrent shot noise. This indicates that these measurements
will be somewhat source flicker limited, and that the optimum trans-

mittance will be somewhat lower than T = 0.109. Examination of Figure 2

36

 

37 1 ‘
1.0 20

 

ABSORBANCE

Figure 2. Case II Study.

a - experimental data

- - theory

37

reveals that the predicted transmittance optimum is T = 0.086 (A =
1.065), where it is predicted that the relative measurement precision
will be 0.034%. The most precise measurement obtained in practice was
0.036% at T = 0.0965 (A = 1.015).

The data obtained in the Case II study show a significant improve-
ment in precision over the Case I data. The conditions selected for
the Case II study are quite common. Many analytical procedures involve
the monitoring of an absorption line with an instrument having a suf-
ficiently narrow spectral bandpass to exclude interferences from nearby
absorption lines. Since reduction of the spectral bandpass also re-
duces the radiant power incident on the PMT, the PMT gain, m, is
increased to produce an easily measurable photocurrent. The increase
in gain increases the relative measurement error due to shot noise.

It is not common to find a conventional spectrophotometer in a pure
Case II limit. In this study, source flicker noise was also signifi-
cant, and reductions in the incident light level to reduce the effect
of source flicker would increase the fraction of the total noise due

to dark current (Case I) noise. A fast amplifier system may cause
measurements to be shot noise limited due to the large noise equivalent
bandwidth of the amplifier-readout system (Af = l/(4RC)).

Malmstadt, Franklin and Horlick (11) studied the limits of measure-
ment precision in a photon counting spectrophotometer. These investi-
gators found that measurement precision was optimized at T = 0.11,
which corresponds quite favorably with the predicted optimum transmit-
tance, T = 0.109, for the pure shot noise limit. Shot noise is expected
to predominate in photon counting measurements because the radiant power

must be relatively low to avoid pulse pileup.

38

If measurement precision is limited by shot noise, precision may
be improved by several techniques. Measurement precision in an instru-
ment using a fast amplifier - A/D converter system may be improved by
a factor of JN, where N is the number of A/D conversions averaged per
measurement. Relative measurement precision in photon counting systems
may be improved by JN, where N is the number of pulses counted per
measurement. Increasing the counting time will increase measurement
precision. In conventional spectrophotometers, measurement precision
may be improved by increasing the radiant power incident on the PMT
and reducing the PMT gain. Each solution may have undesirable side
effects, however, due to increased measurement time in the fast
amplifier-AID converter system and photon counting system or increased
monochromator spectral bandpass and subsequent Beer's Law deviations

for the conventional spectrophotometer.

5. Case III - Variance Proportional to Photocurrent Squared

An attempt was made to place the instrument in the source flicker
limited condition in order to investigate Case III. The instrumental
conditions selected for this study are listed in Table III. For this
study, the source radiant power incident on the PMT was maximized by
opening the monochromator slits. The reference photocurrent was
10'7A and the dark current was 8 x lO'IIA. The relative standard
deviation in the reference photocurrent and the dark current were
found to be 0.02% and 0.022% respectively. Under these conditions,
source flicker noise should predominate over a wide transmittance
range. The value of the source flicker factor, E], was found, experi-

4

mentally, to be 2 x 10' . Theory predicts that under Case III conditions,

the relative measurement precision will continue to increase with

39

absorbance until other noise sources become important. At T = 0.1,

dark current noise and photocurrent shot noise are 8.2% and 35% of the
source flicker noise. These measurements are, therefore, somewhat shot
noise limited, and it is predicted that the best relative measurement
precision will be 0.0096% at T = 0.026 (A = 1.685). Examination of the
experimental results plotted in Figure 3 and listed in Table IV re-
veals that the best relative measurement precision obtained experi-
mentally was 0.012% at T = 0.016 (A = 1.8). This represents nearly the -
most precise measurement possible with the instrument used in this study.
Further significant improvements would require the use of a light

source which was both more intense and more stable or the use of a PMT
that had higher sensitivity and quantum efficiency. It should be noted
that measurements taken at the source flicker limit are by far the most
precise measurements in the entire study, regardless of the transmit-

tance.

6. Case IV - Sampling Imprecision

 

It is somewhat misleading to designate sampling imprecision as
a single limiting case of instrumental noise, since theory shows that
cell positioning errors may be due to two flicker factors, 52 and E3,
and that these two factors contribute to the total instrument noise
in entirely different manners. Two limiting forms of Case IV are
therefore proposed. If 52, the cell transmission flicker factor,
predominates, the relative measUrement precision is predicted to in-
crease with absorbance. If 53, the sampling flicker factor, predominates,
the relative measurement error is predicted to be a constant, independent

of absorbance. Obviously, a mixture of these factors may be observed

40

24 —

 

 

 

ABSORBANCE

Figure 3. Case III Study.
. - experimental data

2 2
- - theory, OA+S + Ohr measured

---- - theory, OD+S2 + OBFZ pFEdICCEd by theory

41

in practice.

An approximate value for E3 was calculated from geometric considera-
tions. The manual sample cell positioning module used in this study
has a specified lateral positioning error of 10.002 inches and an
angular positioning error of 15 minutes of angle. Assuming that the 1
cm cells have a path length uniform to 0.0001 inches (as measured in
this laboratory) or 2.5 x 10"4 cm, the standard deviation in path
length due to lateral positioning errors would be approximately 8 x 10'5
cm. The standard deviation in path length due to angular positioning
imprecision would be approximately 1 cm/cos (15') = 10'5 cm. The total
relative standard deviation due to changes in path length, 53, would
therefore be approximately 8.06 x 10'3%. Examination of the experimental
data in Figure 4 and Table IV reveals that the observed errors due to
sampling are significantly larger than 0.008%, and that 53 is probably
insignificant with respect to $2.

In order to minimize any contribution to the total measurement
error by noise sources not related to sampling, the instrument was
operated at conditions similar to those used in the Case III study.

Under these conditions, the basic instrumental noise was principally
due to source flicker and was quite low.

The data in Figure 4 and Table IV reveal the extent to which sampling
imprecision may degrade measurement precision in spectrophotometry. The
best relative measurement standard deviation obtained in this study
was 0.16% at T = 0.015 (A = 1.823). This represents over an order of
magnitude more error than that predicted due to source flicker or
sampling flicker.

The data for this study strongly indicate the importance of

42

1.01-

1.4

 

 

 

1.8
-Logg_A.
1A

2.2

246

3.0 ' '

O 1.0 21)
ABSORBANCE

Figure 4. Case IV Study.
GD -experimental data

- - theory

43

maintaining good quality optical surfaces on the cuvets used in
spectrophotometry. Scratches, dust and oil on the optical surfaces

of the cell tend to increase reflective and scattering losses, and
possibly the magnitude of the variances in these losses. A procedure
presently recommended for cleaning the optical surfaces of cuvets
prior to use calls for the use of a lens tissue soaked in spectrograde
methanol and held by forceps. This tissue is drawn across the optical
faces of the cuvet and the cuvet is air-dried. This procedure reduces
the possibility of transferring oil from the skin to the cuvet, while
removing dust and oil from the cell surface (11).

The manner in which the cuvet is placed in the sample cell holder
was also found to be important. Experiments performed in this labora-
tory indicate that simply removing the sample cell from the cell holder
and reinserting it may, depending on cell holder design, change the
measured transmittance by 10% or more. Additionally, if some care is
not exercised in the manner in which cells are placed in the cell holder,
the cell transmission flicker factor may be drastically increased.
Experiments indicate that improper attention to positioning the cell
squarely in the holder may increase E2 by a factor of 2 to 4.

A comparison study was performed using a motor-driven sample cell
positioning module. Despite the fact that this module is specified to
position the sample cell with the same reproducibility as the manually
controlled module, the relative measurement error observed at T = 0.08
(A = 1.1) was 0.038%, near the source flicker limit. The cell trans-
mission flicker factor calculated for this module was 5 x 10-4, as
compared to 6.7 x 10'3 calculated for the manually controlled module.

Continued improvement in sample cell positioning precision is occurring

44

in commercial spectrophotometers (11,12) and such improvements should
eventually lead to realization of the maximum precision inherent in
the commercial instruments.

Figure 5 displays the experimental data and theoretically pre—
dicted curves (considering only dark current shot noise in the dark
current noise expression) for the four cases discussed previously.

It is interesting to note that, at low absorbance, sampling imprecision
is greater even than the imprecision observed in the Case I study.

The data clearly show that as noise due to sources other than sample
positioning is reduced, the observed optimum transmittance shifts

to higher absorbance. This implies that, whether limited by sampling
imprecision or electronic errors, when the instrumental parameters

are optimized, the best measurement precision will be obtained at
relatively high absorbance, with a low PMT gain and a high source
intensity. The absorbance at which optimum measurement precision is
obtained may be predicted from the theory presented in this work, and
the reliability of that prediction is clearly high, as indicated by

the agreement between theory and experiment shown in Figure 5. Further
improvement in the agreement may be achieved by determining the quantity
of excess noise in the dark current and the secondary emission factor,
01.

Most obvious from Figure 5 is the importance of sampling in
spectrophotometry. The relative measurement precision observed when
samples were positioned by the manual cell positioning module did
not approach the maximum precision possible at the source flicker
limit (Case III), and the plots of the data for these two cases

clearly shows the extent to which sampling imprecision limited the

Figure 5.

 

45

 

 

ABSORBANCE

Comparison of Theory and Experimental from all Four Studies.

0 - Case I

X - Case II

A - Case III
I - Case IV

46

measurement precision. The best precision obtained under Case IV
limited conditions was poorer than that measured under Case III limited
conditions by a factor of thirty. It is obvious that some attention
should be directed toward improving sampling precision in spectrophoto-

metric measurements.

7. Effect of Limited Readout Resolution

 

Many modern spectrophotometers are capable of performing absor-
bance measurements with high precision. Some of these units, and many
older models, are equipped with readout devices with limited resolu-
tion. The most commonly used readout devices are the D'Arsonval analog
meter, the 3 1/2 digit digital panel meter (0PM) and the 4 1/2 digit
0PM. It is instructive to compare the imprecision due to quantizing
errors resulting from the use of such meters to the imprecision due to
the noise sources studied in this work. Consider an analog meter
readable to 0.5% of full scale (analogous to a 2 1/2 digit 0PM) and a
3 1/2 and 4 1/2 digit 0PM, accurate to 21/2 the quantizing level.

Table V lists the quantizing error of these meters, assuming that no
other source of noise is significant, that these readouts are linear

in transmittance with no scale expansion and that 100% T is set to 1
volt. Also listed are the experimentally measured values of the relative
measurement error under Case II and III limited conditions. The
magnitudes of these quantizing errors and experimental errors are
compared at the observed optimum transmittances for the Case I, II and
III studies presented here. The error in reading the analog meter is
treated as a quantizing error, since reader bias generally results in

a preponderance of 0's and 5's in the last significant figure of readings

Comparison of Readout Quantizing Error to Errors

47

Table V

Due to Shot Noise and Source Flicker Noise

 

 

Noise Source

Quantizing Error, q, in percent of reading, or

% Relative Standard Deviation

 

Readout Error
2 1/2 digit DPM
or analog meter
3 1/2 digit 0PM
4 1/2 digit DPM
Shot Noise

Source Flicker

 

or = 0.339 r = 0.086 r = 0.016
q = 1.4% q = 2.4% q = -5.8%
+4.8%
q = 0.14% q = 0.24% q = 0.53%
q = 0.014% q = 0.024% q = 0.053%
10.05% :0.034% :0.039%
20.028% :0.012% 40.0095%

 

 

48

from such meters.

Table V reveals that the use of an analog meter or a 3 1/2 digit
0PM may severely limit measurement precision in a good quality spectro-
photometer. In the worst case, the reading error for the analog meter
is five hundred times greater than the measured source flicker limited
instrumental noise at T = 0.016. It is therefore obvious that the use
of analog meters should be restricted to instruments not used for
precision measurements. The 3 1/2 digit 0PM significantly reduces
readout error, but obviously does not approach the level of precision
necessary for truly precise measurements. Even the 4 1/2 digit 0PM
has an inherent imprecision which is significant with respect to other
instrumental noise sources. It should be noted that a recent theoretical
treatment of the effect of quantizing errors (13) has shown that as
the signal noise approaches the quantizing level, q, the standard
deviation due to quantizing noise reduces to q/«T2 or 0.29q. Since
the signal noise observed experimentally does approach the quantizing
level of the 4 1/2 digit 0PM, the observed standard deviation due to
quantizing noise would be 0.004%, 0.007% and 0.016% for the 4 1/2
digit DPM in the three cases listed in Table V. It may readily be
seen that the quantizing noise is still larger than instrumental noise
at high absorbance under source flicker limited conditions. The effect
of quantizing noise may be reduced by scale expansion or by the use of
a higher resolution readout device. It is obvious that 4 1/2 digits
is the minimum acceptable resolution for a readout device used in con-
junction with a high quality spectrophotometer if the readout is linear

in transmittance and electronic scale expansion is not employed.

CHAPTER III

Photometric Titrations

A. Introduction

 

The term photometric titration refers to an analytical technique
in which photometric instrumentation is used to determine the end-
point in a titration. A considerable number of reactions may be utilized
for such titrations, since the only requirement is that the reaction
include some light-absorbing reactant or product, or be coupled to a
reaction involving such species. The instrumentation required for
such titrations is quite simple. The requirements are: a means of
delivering accurately known quantities of the titrant to the reaction
vessel, an instrument designed to determine the absorbance of the solu-
tion in the reaction vessel, and some means of interpreting the data.
Some degree of automation has been implemented in each area.

Photometric titrations are commonly employed when precise con-
centration data are needed. Most applications of automation have been
performed in an attempt to improve the precision of the results. This

subject is covered in several books (28,56,57).

B. Historical

 

1. Titrant Delivery Systems

In 1926, Field and Bass-becking (l4) objectively measured the color
change of the starch-iodine reaction in order to determine the endpoint
of a titration. The first application of automation to the technique
they developed was reported in 1928 by MUller and Partridge (15). The

MUller-Partridge titrator utilized a standard buret with a length of

49

50

rubber tubing connecting it to the titration cell. A solenoid-controlled
pinch Clamp placed on the rubber tubing was used to stop the flow of
titrant into the cell when the photocurrent of the detection system
exceeded a pre-set level. This was the first reported automatic photo-
metric titrator with automatic "end point" detection, and it served

as a model for several systems which will be discussed later. In 1933,
Hickman and Sanford (16) reported a slightly more complex version of

the same instrument, adding a chart recorder to the output of the photo-
detector and automating the introduction of the sample solution to the
titration cell.

In 1948, Lingane (17) designed a motor-driven syringe for titrant
delivery. The drive system consisted of an electric motor and reduction
gears, with a mechanical revolution counter to indicate volume delivered.
Each revolution of the drive screw delivered 0.4088 ml 20.03%. The
counter had a resolution of 0.006 ml/count. The advantages of this
buret are obvious. It would deliver 50 ml of titrant, resolved to
0.0012%,with good precision. This style of motor buret is the model
for several commercial units and has been applied with good results in
photometric titrations (18).

In 1957, Malmstadt (19) reported a device for automatically
twisting buret stopcocks under remote control using two rotary solenoids.
Eyewitnesses (20) report that this was an awe-inspiring device.

More recently, digital-type burets have been developed. Many
workers have used stepping motor - syringe combinations for reagent
delivery (21-23). The digital nature of stepping motors and the small
step sizes available (l.8°) allow precise control over reagents with

digital circuitry. Such systems have been reported with a volume

51

resolution of 0.6-270 n1 33% (noncumulative). Stepping rates of up to
500 or more steps per second allow flexible control over the rate of
titrant delivery.

Another system, reported by Hieftje (24), consists of a droplet
generator which can deliver the titrant in the form of 0.3 ul droplets
at rates up to 600 Hz. This device will deliver 2893 droplets/ml,
10.1%. This may be compared to the system described by Megargle and
Marshall (22) which could deliver 1 m1 10.01%. The Megargle-Marshall
system made use of computer-controlled stepping motor-micrometer
syringe assemblies. This system had the unique advantage that computer
control could simply consist of loading an up-down counter with the
number of steps desired and a second register with a control word to
select the stepping rate. The computer was free to perform other tasks

while a hardware controller operated the stepping motors.

2. End Point Detection

 

Prior to 1954, endpoints of titrations were generally determined
by graphical methods (25) or by arbitrarily selecting a detector voltage
and halting the titration at that voltage (15,16,26). Commercial instru-
ments available at that time generally made use of the pre-set voltage
detection technique (27-29). Since this technique is only useful for
sigmoid-type titration curves, complexation titrations yielding curves
with the form of two intersecting lines could only be treated by the
graphical method of extrapolating the linear portions to their inter-
section (18.30-34).

In 1954, Malmstadt and Fett (35) reported the development of an
automatic differential potentiometric titrator, which took the second

derivative of the titration curve electronically. The second derivative

52

of the sigmoid shape titration curves crosses zero at the inflection
point of the original titration curve. Upon detection of this zero
crossing, the instrument operated a solenoid-controlled pinch valve
of the Muller type (15). Photometric titration curves which are sigmoid
may also be treated by this method (36,37).

In 1956, Malmstadt and Roberts (36) reported the development of
a detection system for the intersecting-line complexation-type titrations.
This unit took the third derivative of the titration curve electronically.
The third derivative of this type of curve has a zero crossing at the
break in the original curve. The disadvantage of this type of system
is the noise sensitivity, requiring careful circuit design and the
requirement that the titrant be added rapidly to maximize the third
derivative signal. The second disadvantage prohibits the use of this
type of detection system when titrating species which do not equilibrate
rapidly. Application of this technique to fast redox reactions has been

performed with success (36,37).

3. Automated Photometric Titrators

In addition to the automated systems already mentioned (15,16,
21-23,27-29,35,36), many other automated instruments have been reported.
Wise, Gilles and Reynolds (38) developed an automatic coulometric
titrator with photometric detection of the endpoint. Malmstadt and
Gohrbandt (18) modified a Cary spectrophotometer to allow it to plot
directly the titration curve of the copper-EDTA system, and the curve
for the titration of vanadium with coulometrically generated titanous
ion (26). Marple and Hume (39) similarly modified a Beckman model B

spectrophotometer. Malmstadt and Vasallo (40) constructed an automatic

53

spectrophotometric titrator from a commercial unit, the Sargent-
Malmstadt potentiometric titrator (41) and applied this instrument to
several analyses (42-44). The SargentFWMImstadt Spectro-Electro
Titrator is the commercial version of this instrument.

Thoburn, Jankowski and Reynolds (45) described the development of
a commercial automatic titrator, the CENCO. While similar to the
Sargent-Pblmstadt titrator, end-point detection was accomplished by
shutting off the buret at a pre-set level of photocurrent.

Jagner (46) has described an elaborate system capable of 45
simultaneous titrations. All operations are sequenced by a hardware
controller. Data are collected and punched on paper tape by an ASR-33
teletype. The data are subsequently read into an IBM 360/65 computer
for treatment by the curve-fitting program, LETAGROP. Both potentiometric
and photometric titrations have been carried out with this instrument.

MUeller and Burke (21) described a computer-controlled reagent
addition system used for photometric titrations. Instrumentation
included a Bausch and Lomb Spectronic 20, Hewlett-Packard 2115A mini-
computer and stepper motor - micrometer syringe delivery systems. End-
point detection was implemented by having the computer search for the
portion of the titration curve which exceeded a pre-set level.

Slanina, et al., (23) described an automatic titrator with pre-set
level-type endpoint detection. This instrument was used in the deter-

mination of calcium on the 100 ng to 160 pg level.

4. Computerized Analysis of Titration Curves
In recent years, a considerable effort has been expended in the
development of computer programs for the analysis of titration curves.

By far, the greatest portion of this effort has been directed toward

54

the analysis of curves derived from monitoring pH potentiometrically.
There is also considerable variation in the complexity of the programs.

An example of a simple titration curve analysis program is that
described by Hunter, Sinnamon and Hieftje (47). The primary function
of the computer interfaced to their instrument was control of the drop-
let generator used as a buret. Data analysis consists of monitoring
the output of a Malmstadt-type second-derivative endpoint detector (35).
Upon receiving a signal denoting the endpoint, the computer is programmed
to calculate the sample concentration. In the same work, a somewhat
more sophisticated end-point detection routine was implemented. This
involved the collection of all data points by the computer, followed
by smoothing and differentiation for endpoint detection. Comparisons to
a hardware controller were discussed. The precision of the hardware
controlled system was 0.2% - 0.4%, while the computer controlled system
was said to be precise to 0.05% - 0.3%.

The computer system described by Jagner (46) is similar. The data,
collected by a hardware controller, are punched on paper tape and read
into an IBM 360/65.

A number of investigators have applied curve-fitting techniques
to the analysis of titrations curves. Liteanu and Cormos (48) determined
the endpoint of a complex-formation titration by finding the best-fit
straight lines for the two branches of the titration curves. The inter-
section of these lines was designated as the endpoint. Jandera,

Kolda and Kotrly (49) and Vrestal and Kotrly (50) published an extensive
analysis of the errors involved in such a technique.

A number of investigators have devised least-squares fitting

programs for formation constant calculation (51-54). These programs

55

make use of pH titration (potentiometric) data. All are intended for
use on large computer systems. LEAST (54) is capable of treating only
systems of one metal ion and one ligand. MINIQUAD (53) is capable of
treating far more complex systems. LETAGROP (51) and SCOGS (52) are
programs of intermediate power with documented flaws (53,55), but,at
present, wide acceptance.

This thesis presents a novel automated photometric titrator using
an on-line minicomputer for data analysis. The data analysis consists
of carrying out a least-squares fit of the theoretical curve shape to
the data generated during the titration. This end-point detection
technique is shown to be quite precise, and the advantages and dis-

advantages of this method are discussed.

C. Theory

The shape of titration curves derived from photometric monitoring
may fall into either of two general classes. For a few types of photo-
metric titrations, the titration curves are similar to those observed
when titrations are monitored potentiometrically. Since such titrations
were not treated experimentally in this work, they will not be further
considered.

A second class of titration curves observed photometrically
includes those resulting from complex formation reactions. In the most
ideal form (high formation constant, negligible dilution) such titration
curves have the form of two intersecting straight lines as illustrated
in Figure 6. The endpoint of such titrations is indicated by the inter-
section of the line segments. In practice, the shape of such titration

curves is less ideal.

56

 

 

 

 

 

 

 

 

 

A A
VOL VOL
(A) (B)
A A
VOL VOL
(c) (D)

Figure 6. Various Possible Photometric Titration Curve Shapes.

57

For the sake of simplicity, this treatment will initially assume
that the titration curves observed are the result of the formation of
a 1:1 metal-ligand complex. It is further assumed that, of all species
present in solution, only the complex has an appreciable molar absorptivity.
The effects of dilution and the complex formation constant on the titra-
tion curve shape and end point detection technique will be considered.
Figure 7 illustrates the effect of the complex formation constant
on the titration curve shape. It is assumed that the titrant concentra-
tion is one hundred times greater than the sample concentration, and
that dilution effects are therefore minimal. It is further assumed
that the sample solution contains the metal ion, M, at initial concentra-
tion [M]o, and that the titrant solution contains the ligand, L, at

initial concentration [L]o. If the reaction in solution is
M + L +-ML (3.1)
the complex formation constant, Kf may be written

”L (3.2)

where activity coefficient corrections have been neglected. During
the course of a titration involving the addition of discrete volumes
of ligand solution, the concentrations [M] and [L] after the ith
addition may be rewritten as:

V [M]

[M]. = 11,79 {ML}, (3.3)
i

58

.ucaumcou covumsgo. xm_aeou on» yo mamzm m>c=u covpmguFP mo mocmucmgmo .n «campm

 

 

 

 

. . 86> x ..>
0.. 8.. c._ u. o.. 0.0 8.0 To «.0 o
H _ d u q . q u .q

l 0.0
O J
E

7:

1 o._

 

 

59

and

vL.[L]o
[L].- = -—\',-—- [ML]. (3.4)
Ti

where Vm is the initial sample volume, VL is the total volume of titrant

i
added at the ith addition, and Vri = vm + vL,. Substituting 3.3 and
1

3.4 into 3.2 and solving for the complex concentration after the ith

addition of titrant, [ML]i,

 

 

v [M] VL.[L]0
m 0 I
[MLJI = Kf[ VT. '1’ r] 'I‘ I -
1 1
Kf VT '1’ VT + 1 - 4Kf TEMJOU-JO (2Kf) (3.5)
l 1 T

1

Figure 7 illustrates the result obtained by setting [L]o = 100[M]o
and substituting various values of Kf into Equation 3.5. Figure 8 is
plotted as the ratio of the complex concentration, [C] to the initial
metal ion concentration versus the ratio of the volume of titrant
added to that volume necessary for equivalence.

It may readily be seen that when Kf is large, the titration curve
shape is nearly ideal with a sharp break at the endpoint. However, as
Kf decreases, the endpoint of the titration becomes less well-defined.
The effect that this lack of definition has on endpoint detection methods
will be discussed later in this thesis.

The shape of the titration curve is also dependent upon the volume

of titrant which must be added to the sample solution to achieve equivalence,

60

.AomzH\oH.uv :owumgpcoocou upmsmmroprpcncuph we ovumz co mamgm a>g=o :owpmcpwh eo wocmucmawa .m oczmwm

 

 

 

 

co
> \ .>
a. o. v; a. o. .6 0.0 to «6
< q d — u - u u d O
_.O
\.
L00
.\\
\\\\\
0.1 l
\ 13..
\ 3
/ \\
/\\
O.
0121. l o.—
oo_. 1L
3.

 

 

61

as may be seen by examination of Equation 3.5. Figure 8 illustrates

the manner in which the shape of a titration curve is dependent upon

the concentration ratio of titrant to sample [LJO/[M]o. The curves

shown in Figure 8 assume a complex formation constant of 1010. It

may be seen that when the titrant is far more concentrated than the
sample, and the volume of titrant added to reach equivalence is there-
fore small with respect to the initial volume of sample, the break in

the titration curve at the endpoint is readily seen. However, as the
concentration ratio decreases, the break at the endpoint is less clearly
defined. In addition, the titration curves no longer resemble the inter-
section of two straight lines. The effect of dilution on endpoint detec-
tion methods will also be discussed later in this thesis.

It is easy to speculate on further changes in the titration curve
shape caused by the combination of a low formation constant and a large
dilution effect. The observed titration curve would have the appearance
of a nearly straight line with near-zero slope. A curve such as this
would be nearly impossible to analyze by any endpoint detection method.

Needless to say, chemical systems which might give this type of titra-

tion curve are avoided in practice.

0. End Point Detection

 

Two popular methods are presently used for the detection of end-
points in photometric titration curves. The classical method involves
extrapolation, generally by a human estimate of the extrapolations of
the straight line portions of the titration curve to the intersection
of the two straight lines. At least one individual has computer-

generated the best least squares fit to the line segments (48) for

 

62

endpoint detection. The extrapolation method has inherent advantages.
It may be performed by hand on manually plotted data, the titration
curve may be plotted automatically (26,31,39,58) and manually analyzed,
or the data may be computer-collected and analyzed (46,47,52,53,55,59).
The results obtained by this method are not strongly influenced by a
constant background absorbance in the solution. Therefore, the extrapola-
tion method may be applied to any of the curve shapes illustrated in
Figure 6, as well as others.

The disadvantages of this particular endpoint detection method may
easily be understood by examining the titration curves in Figures 7
and 8. As the complex formation constant decreases and/or the dilution
effect increases, the portions of the titration curve representing the
straight lines to be extrapolated become increasingly difficult to
define. The dilution effect is especially troublesome, since it causes
pronounced curvature in opposite directions in the two limbs of the
titration curve, making accurate extrapolation exceedingly difficult.
It is therefore necessary to maintain a relatively high titrant-to-
sample concentration ratio to reduce the dilution effect. This may be
done by either diluting the sample (and subsequently lowering both
observed absorbance changes and the measurement precision) or by using
small volumes of a very concentrated titrant, with the attendant in-
creased volumetric imprecision in titrant addition. Conversely, the
effect of a low formation constant is to induce curvature at the end-
point, and examination of Figure 7 will reveal that somewhat useful
straight line portions of the titration curve exist at Kf = 103, but

2

not at Kf = 10 . Use of the extrapolation technique is probably most

successful when dilution effects are minimized, even at low values of Kf.

63

 

 

 

 

 

A
B
s
o
R
B
A
N
c
E
o.
dA I
_ 04
dt
dzA
_ 0
A. V
63A
__ 0
dt

 

 

TIME (Titrant added continuously)

Figure 9. Signals Derived from Derivative Endpoint Detection
Circuit.

64

A second method for endpoint detection, developed by Malmstadt
et al. (36), involves electronic differentiation of the titration curve.
Malmstadt's technique involves the continuous addition of titrant with
continuous monitoring of solution absorbance, giving the derivative with
respect to time. A third derivative circuit will yield the results shown
in Figure 9 for a titration of the type previously discussed, with only
the complex absorbing. The third derivative of the titration curve
crosses zero at the break in the titration curve, i.e. the endpoint.

Like the extrapolation technique, this method is not subject to inter-
ferences due to a constant background absorbance and may be applied to
titration curves of nearly any type as illustrated in Figure 6.

Examination of Figures 7 and 8 reveals the effect of Kf and dilution
effects on the third derivative method. As Kf decreases, the slope
of the titration curve changes less rapidly near the endpoint, and
the observed detector signal is subsequently reduced. However, dilu-
tion effects in a system with a large Kf do not totally obscure the
sharp break in the titration curve at the endpoint. It would there-
fore appear that the endpoint detection precision of the third deriva-
tive method would be less dependent on dilution effects than the pre-
cision of the extrapolation method, but more dependent on the magnitude
of Kf. A choice between these methods would be dictated by the instru-
mental and chemical systems to be used.

The principal disadvantages of this method limit the types of
titrations to which it may be applied. One disadvantage is that the
third derivative signal due to noise in the instrument signal is generally
larger than the third derivative of the titration curve. As a result,

some care must be taken to minimize both instrumental noise and the

65

noise sensitivity of the detection circuitry.

In order to maximize the third derivative signal, titrant must be
added somewhat rapidly. This can cause inaccuracy and possibly im-
precision if mixing in the titration vessel is slow and/or the observed
reaction is slow. This limits the applicability of this technique to

titrations involving fast reactions.

E. Least Squares Method for Titration Curve Analysis

1. Introduction

 

The criteria for least squares fitting of equations to experimental
data have been well established (60,61). Reliability in the application
of this technique depends upon the experimental error being random
and Gaussian. Least squares methods have been applied by many indi-
viduals to the analysis of titration curves, particularly potentiometric
titration curves (51-54). In general, the computing systems used in
these studies have been large batch processing (IBM 360/65, CDC 6500,
etc.). It is possible to perform similar calculations on a laboratory
minicomputer, albeit on a less complicated level. The following sec-
tion describes the least squares equation which may be solved for a

single unknown, the sample concentration.

2. Simplest Least Squares Approach

 

The simplest case which may be treated is that of the previously
described chemical system, where only the complex formed duringthe
titration has an appreciable molar absorptivity. If it is assumed
that the complex concentration can be directly monitored at each of n
points in a titration curve, combining Equations 3.1, 3.3 and 3.4

th

gives, at the 1 point

66

 

v M
[MLJi = Kf 1} 1° - [ML].
T1 i

Due to the complexity of the equation resulting from this derivation,

several symbols will be redefined as shown below:

V1 = Vm/Vri V2 = VLi/VTi

(3.7)
M = [M]o L = [L]0 C = [MLJi
Equation 3.6 may be rewritten
C = KfEVIM-C][v2L-C] (3.8)
The sum of the residuals, S, may be evaluated.
" 2
s = :2: {C-Kf[V]M-C][V2L-C]} (3.9)

i=1

If the initial metal ion concentration, M (or [M]0) is unknown, Kf,

Vi’ V2 and L may be specified or calculated and C can be measured,

then the partial derivative of S with respect to M may be set equal

to 0.

n
as _ _ 2 2 2 2 2 2 2
.3.“ -0 _ Z 2mf (v1 v2 L M+v1 MC +2V1V2LC

i=1

2 2 2 2

3

2 .

67

solving 3.10 for M,

 

 

2
n V V LC V C
2 2 2 3 1 2 1
Z zvlszc -v2 VIL c-v]c — Kf + .5.
i=1
M = (3.11)
n
2 2 2 2 2 2

If experimental values of C are substituted in Equation 3.11 and known
values for L, Kf, V], and V2 are used, evaluating the summation will

give the initial concentration of the sample.

3. Extension to a Real System

 

Equation 3.11 applies to any titration involving the formation
of a 1:1 complex. This equation can only be evaluated properly if
the complex concentration at each point in the titration can be ac-
curately determined. A problem arises when the absorbance signal from
the spectrophotometer contains some information not directly related
to the complex concentration as shown in Equation 3.12

A (3.12)

Ameasured = complex + Asample + Atitrant + Abackground

In this case, the measured absorbance signal contains information
related to all species present in the observed reaction, plus a back-
ground signal which may or may not vary during the titration. The
titration curves in Figure 6 illustrate the curve shapes which may be
obtained if more than one of the absorbance terms in Equation 3.12 is
significant during the titration. Curve A might be observed if both

the sample and the titrant had appreciable molar absorptivities. Curves

68

B and C could be observed if both the complex and the ligand absorbed.
Curve 0 might result if the metal ion and ligand absorbed only in the
uncomplexed state.

In the particular case investigated in this study, both the sample
and the complex had significant molar absorptivities. In this case,
the complex concentration could only be calculated by correcting the
observed absorbance for the absorbance due to the metal ion. Since
the initial metal ion concentration is unknown, the proper corrections
and metal ion concentrations can only be calculated by an iterative
process which will be described later.

The disadvantages of this particular technique are generally
related to the fact that different data correction routines must be
used for each different case of sample and/or titrant and/or complex
absorption. Additionally, corrections for background absorbance
interference must be either simultaneously fitted or ignored. These
disadvantages can be overcome by expanding the curve fitting routine
to include more parameters, but at the expense of increased calculational
time. Whether or not the increase in calculational time will be
significant depends upon the software and hardware in use.

The particular advantage of a curve-fitting technique for titra-
tion endpoint detection is the ease with which dilution effects may
be treated. Data may be corrected for the dilution effect when col-
lected or the dilution effect may be incorporated in the theoretical
equation used in the least-squares fit. Likewise, a low formation
constant need not cause problems, since the shape of the titration
curves is easily predicted and fitted if the correct formation constant

is known but_the lower the formation constant, the more accurately it

 

69

mg§t_be_knggn, Bias is eliminated in both the collection and analysis
of data, since neither of these steps involves any human operations.
In the case of titrations in very dilute systems with small changes
in a small absorbance, computer programming could allow the separation

of the absorbance signal from the background noise by averaging tech-

niques.

CHAPTER IV

Photometric Titrations-Experimental

A. Introduction

 

In this chapter, a new endpoint detection technique for use in
photometric titrations is described and evaluated. Other endpoint
detection methods have some limitations due to the effects that dilu-
tion and low formation constants have upon the shapes of the complex-
formation titration curves. The method reported in this chapter has
been developed to circumvent these limitations, which are discussed
in more detail in the previous chapter. The least squares technique
described in Chapter III was applied to the analysis of titration curves
obtained by spectrophotometric monitoring of the titration of copper
(II) with EDTA (ethylenediaminetetracetic acid) in 0.1 M acetate buffer
(pH 4.7). In acetate buffer, the product of molar absorptivity and
path length for the copper ion was found to be 72. The same product
for the copper-EDTA complex was measured and found to be 230. The
significant c0pper absorbance prior to the endpoint necessitates cor-
rection of the experimental data, as discussed in Chapter III, Part
E, Section 3. The software used for the data correction and least

squares fit is discussed in this chapter.

8. Instrumentation

 

l. Spectrophotometer

 

A single beam molecular absorption spectrophotometer (EU-701A,

GCA McPherson, Acton, MA) was used in this study. The sample cell

70

71

compartment was modified to accept a special 20 ml sample cell, shown
in Figure 10. The cell was cylindrical, with a 2.54 cm path length.
The cell was carefully baffled to limit the entrance and exit light
paths of the cell to an area 6 mm high by 8 mm wide. This baffling

was necessary to provide an optical path which did not include either
the vortex due to mixing or the meniscus of the solution in the titra-
tion cell. Including either of these in the optical path would obviously
cause significant inaccuracy in absorbance measurements, both of a
random and systematic nature. Relatively rapid mixing was accomplished
by the use of a small magnetic stirring bar (Bel-Art, Pequannock, NJ)
placed inside the cell and activated by a compressed air-driven mag-
netic stirrer (G. F. Smith Co., Columbus, OH). Selection of the proper
stirrer speed was critical. Operation at a too-slow speed failed to
provide good mixing, while operation at high speed caused the vortex
and air bubbles to enter the optical path.

Due to the shape of the titration cell, cell positioning was also
critical. A lateral displacement of as little as two mm changed the
reference photocurrent by 90%. The cell was initially positioned
such that the empty cell transmittance was approximately 20% of the
reference. A plexiglass base held the cell securely in place, to
prevent displacement due to vibrations of the magnetic stirrer.

The cover of the cell was perforated to contain the glass funnel
used for the introduction of the reference and sample solutions, the
plastic tubing used for titrant delivery and removal of waste solu-
tions and an air vent. The plastic tubing was terminated with stain-
less steel hypodermic needles, both of which were positioned near the

bottom of the sample cell, but outside the optical path defined by

72

SAMPLE

I
N /

‘

ro WASTE ,r”rnou auner

 

 

 

 

 

 

 

 

 

14 _J
. I
PLEMGLASS
BASE \~»., . AIR DRlVEN
\ C33 —1 MAGNETIC
SHRRER

 

 

 

Figure 10. Titration Cell.

73

the baffles. The use of a fixed aspirator attachment was necessary
due to the requirement that the cell position be absolutely constant.
Manual insertion of an aspirator into the cell inevitably resulted in
some cell displacement, regardless of the care exercised by the ex-
perimenter.

The physical size of the titration cell and the number of tubing
connections to it made it necessary to operate the instrument with the
lid to the sample cell compartment removed. In order to reduce the
resulting stray light level, the sample cell module was placed between
the light source module and the monochromator instead of in the con-
ventional position between the monochromator and the photomultiplier
transducer (PMT). Stray light causes negative deviations from Beer's
Law, with a resulting loss of both photometric accuracy and precision,
and should therefore be minimized to realize optimum instrument per-
formance.

The tungsten light source was operated in the optical feedback
mode for optimum stability. A Corning 2-64 cutoff filter was used
to reduce stray light.

The monochromator was set to 750.0 nm with a slit width of 2000
microns (4 nm bandpass). The lP28A PMT was typically operated at 650 V.
Under these conditions, measurement precision was probably limited by
photocurrent shot noise, since the low sensitivity of the PMT to light
at 750 nm necessitated the use of a high PMT gain to obtain a reference
photoanodic current of one microampere. More precise measurements
should be possible with a red-sensitive PMT.

Two different types of signal modifiers were evaluated. Prelimin-

ary measurements of the molar absorptivities of all chemical species

74

were performed using a one volt per decade logarithmic amplifier (EU-
703-31, McPherson) with a 3 1/2 digit panel meter readout, (EU-700-62,
Weston). The errors caused by using such a signal modifier-readout
combination will be discussed later. The logarithmic amplifier (log
amp) and a current-to-voltage converter (Model 427, Keithley Instruments,
Inc., Cleveland, OH) were both used as signal modifiers, and the results
obtained with each will be compared later in this text. The reference
voltage, Er’ was set to 0.000 V, by adjustment of the PMT supply voltage,
when the log amp was used. When the current-to-voltage (I-V) converter
was employed, Er was set to +5.000 V. The computer collected the data
as a voltage equal to absorbance when the log amp was used or as a
voltage equal to the transmittance when the I-V converter was used.

The log amp data were stored directly after scaling and averaging.

The I-V data were also scaled and averaged, but were converted to
absorbance by a calculational routine prior to storage. The advantage
of using the I-V converter is obvious. Since data were input to the
computer via a :5 V, 12 bit A/D converter, the I-V converter made use

of half the dynamic range of the A/D converter (0-+5 volts). From 0

to l absorbance unit, the I-V converter output changes by 4.5 volts,
while that of the log amp changes by only 1 volt. Additionally, when
the solution absorbance is low, the I-V converter output voltage is

near full-scale for the A/D converter, and measurement quantizing errors
are therefore minimized. In the case of the log amp, the output will

be near 0 volts at 0 absorbance, and measurement quantizing errors are
maximized. Assuming that the computer can accurately calculate log
values, one would expect measurement precision (and hence titration

precision) to be poorer for the log amp than for the I-V at low solution

75

absorbance. By a reciprocal argument, one would expect the opposite
at high solution absorbance. Experimental results verify the first

conclusion (at low absorbance) and show the second to be unimportant.

2.829:

The device used for the delivery of titrant was a motor driven
buret with a 20 m1 capacity (Metrohm AG). The buret had two modes
of delivery, stepping and continuous. Short-circuiting two control
leads while the motor selector switch is set to either of the two
"deliver"positionsstarts the motor. In the stepping mode, shorting
the control leads for 0.5 to 1.5 seconds starts the buret on a fixed
cycle which terminates after the delivery of 0.2 ml of titrant. In
the continuous mode, the buret would deliver as long as the control
leads are shorted. This permits both variable and finer control over
titrant addition. Since some signal is necessary to indicate the buret
plunger position, a lO-turn potentiometer was mechanically coupled
to the buret gear system. A 5 volt signal supplied by a voltage
reference source was dropped across the potentiometer. The poten-
tiometer was used as a variable voltage divider, providing a voltage
signal proportional to the buret plunger position and related to the

volume of titrant delivered.

3. Computer System

 

a. General - The computer used in this study was a PDP-8/e 12
bit minicomputer with 8192 words of core memory, a programmable real
time clock, a 1.6 million word disc with Operating system (OS/8)
and poSitive I/O bus interface. The computer was interfaced to the

instrument through a minicomputer interface system (Heath model EU-801E).

76

Input information was sent through this interface directly to the com-
puter or through an interfaced analog-to-digital (A/D) converter (Datel
OAS-l6). The operator issues commands, sets initial values and receives

output through an ASR-33 Teletype (Teletype Corp.)

b. Buret Interface

 

i. Stepping Mode. Control over the buret was simple in this

 

mode. It was only necessary to short the control leads together for
0.5-1.5 seconds to deliver 0.2 ml of titrant. The interface is shown
in Figure 11(A). The device select code is 54. When a 6541 instruc-
tion is performed, the buret control leads are shorted by relay #2,
initiating delivery. After a computer-controlled delay, a 6542 in-
struction de—energizes relays #1 and 2, opening the buret control
leads. Switch A provides an independent manual control over the buret
drive. Switch C allows the operator to change the state of the flip-
flop. Two relays were used in tandem, since switching transients
generated in the internal circuits of the buret often fed back and
changed the state of FFl. This essentially Pee-ran the buret. There-
fore, the relay 1 was used to switch relay 2 which had an independent
power supply. This provided a sufficient buffer against unwanted
noise. These same switching transients also strobed the monochromator
control circuitry, if energized. Conseqently the monochromator wave-
length control circuit was de-energized (unplugged) after being used
to select the proper wavelength.

ii. Continuous Mode. In this mode, the same control circuit,

 

shown in Figure 11(A) was used. The 100 K0 potentiometer, Figure 11(8),

was used as a volume encoder. The computer started titrant delivery

77

 

 

 

 

 

 

 

 

0.5.54
109 1
5v
SET- “NC NC
0
swétch T RELAY l
5»
CLEAR

~o

‘ to
as 54 bum
IOP2 switch NC

3 RELAY 2
(A)
t
/_ __ __ __ o buret
/ motor
/
” :0
+5 V
'00 K to A/D converter

 

O
I)

Figure 11. Motor-Driven Buret Interface.

78

and monitored the output of the potentiometer until a preselected

voltage change was detected. At that point, the "stop" signal (6542)

was generated and the voltage across the potentiometer contacts was
carefully measured (average of 100 A/D conversions), to determine the
exact buret position. Theoretical volume resolution is 1 part in

2047 (approximately 0.01 ml). Practical considerations limited actual
control over the minimum volume deliverable per addition to approximately
0.1 ml, but reproducibility was poor (220%). This irreproducibility

was due to the excessive noise in the signal derived from the volume
encoder (100 K0 voltage divider) due to electromagnetic interference
(EMI) from the buret motor. This irreproducibility necessitated storage
of both absorbance and volume readings in core. Subsequent studies
revealed that no significant improvement in titrator precision could

be realized by continuous buret operation as compared to the stepping
mode for this buret, due to nonlinearities in the buret bore itself.
Thus, the continuous mode of operation was not employed in the succeeding

studies.

C. Analogeto-Digital Converter

 

The analog-to-digital (A/D) converter used in this study was the
Datel OAS-16. This is a 12-bit, 5 volt full scale, bipolar, successive
approximations A/D converter with 8 independent, multiplexed input
channels (numbered 0-7). The output of the converter is in two's
complement form (+5V = 37778, -5V = 40008), directly compatible with
the two's complement arithmetic used in the minicomputer. The conver-
sion time from initial selection of the multiplex channel to comple-

tion of the conversion is 20 microseconds. The interface designed to

79

operate the A/D converter under computer control is shown in Figure 12.
The selected multiplexer channel is determined by the states of bits
9-11 in the accumulator when a 6521 instruction is performed by the
computer. Input timing considerations dictate that a 5 microsecond
delay follow the instruction to allow the input lines to settle.
Therefore, the 6521 instruction should be followed by two NOP's. This
allows sufficient time, since the multiplexer channel is selected at
the beginning of a 2.6 microsecond IOT and the NOP's operate in 1.2
microseconds apiece. A 6531 instruction simultaneously starts con-
version and clears the "end of conversion" flag. The 6532 instruction
clears the accumulator and checks the "end of conversion" flag. When
the flag goes high, the 6532 instruction generates a skip pulse. The
6534 instruction activates a 12-bit gated driver (EU-800-JL, Heath Co.),
driving the data from the A/D converter into the accumulator through
an I/O patch card (EU—801-21, Heath Co.). The suggested program to

perform one A/D conversion is as follows:

CLA /clear accumulator

TAD MUX /MUX = A/D converter channel #, 0-7

6521 /select channel

NOP /delay

NOP /delay

6531 /begin conversion and clear flag
WAIT,6532 /clear AC and skip on flag

JMP WAIT

6534 /drive data into AC

The interface exists in hard-wired form. It includes the gated driver

80

.oocecmpcm Lougm>cou n\< .NF mcamwm

 

fl
¢<m40
._.
O

 

 

 

 

 

 

 

 

 

 

E

.83..

 

 

 

   

K u>.¢o
cubic
:
bou4uw unocho

 

 

       
     

32.2....
E...
oxw ion.

(40

 

 

 

“to. .nodd

 

 

 

 

_GO_.Nm ma

 

 

 

.ao..nn.m.o

 

81

and I/O patch card mentioned previously, plus a custom designed de-
coder board. The A/D converter power supplies are designed for over-
voltage and overcurrent shutdown. The interface is powered by an
independent +5 volt supply.

Three difficulties have been observed in use with the A/D converter
system. The first conversion performed after a change in the input
voltage level often gives a result equal to the results obtained in
conversions previous to the voltage change. Second and subsequent
conversions after the voltage level change are always correct. No
reason for this problem could be detected in the interface circuitry.
Data may be collected properly if the first data point after a voltage
level change is discarded, but this problem will greatly reduce the
utility of the A/D converter for following rapidly changing voltage
signals, such as those in stopped-flow spectrophotometry. The author
suggests that individuals desiring to use the A/D converter for such
applications consider trouble-shooting and/or redesigning the inter-
face. A mode of multiplexer operation in which the multiplexer channels
are sequentially scanned automatically is available in the converter,
but not directly provided by the interface. Channels may be scanned
in any desired order under software control.

The 7476 integrated circuit flip-flop used in the flag circuit
is prone to early failure. During one year of operation it failed
twice. When only the six most significant bits are transferred out of
the A/D converter on an A/D conversion cycle, failure of this 10 should
be suspected.

Crosstalk has been observed between the multiplexer channels,

due to ohmic leakage in the multiplexer FET switches. Table VI

82

shows the crosstalk levels recorded with a 1 volt signal on the re-
ceiving channel and a 5 volt signal on the broadcasting channel.
Ambitious individuals are invited to rewire the input section with
properly shielded cable. The most desirable channel(s) for any applica-
tion may be selected after examination of Table VI.

The A/D converter may be operated by BASIC/RT, if the BASIC/RT
A/D converter handler is slightly modified. It is only necessary to
change five locations in memory field 1 to the instructions listed in

Table VII.

0. Software

All programs used in this study were written in OS/8 FORTRAN.
All input-output instructions and clock control commands were written
in SABR assembler language. The program used in this study is listed
and explained in Appendix A. The program dialog and flowchart are

shown in Figures 13 and 14.

E. Determination of Instrument Linearity

 

Proper operation of this endpoint detection method depends upon
systematic adherence to Beer's Law. Deviations from Beer's Law may
be expected due to the quantity of room light entering the sample cell
module and the resultant possible stray light level at the PMT. An
investigation of the stray light level was performed. Varying con-
centrations of potassium dichromate were prepared in 0.1 M sulfuric
acid by dilution of a 1000 ppm stock solution. A Beer's law plot at
450 nm was produced and the result is shown in Figure 15. It may be

seen that, at 450 nm, no significant deviation due to stray light is

83

Table VI

Crosstalk Levels in the A/D Converter System

 

 

Percentage of 5 volt Signal Appearing in Receiving

 

Channel
Receiving Channel Broadcasting Channel

0 l 2 3 4 5 6 7
0 ---- mlOO 0 0 O 0 0 0
l mlOO --- 0 O 0 0 0 0
2 9.4 0.3 -- O 0 0 0 0
3 mlOO 0.3 0 --- 0.2 0 0 0
4 9 4 0 3 0 0.2 --- 0 0 0
5 9 0 0.3 0 0 2 0 -- 0 0
6 9 0 0.3 0 0 2 0 0 -- 0
7 mlOO 0.3 0 0 2 0 0 0 ~-

 

 

84

Table

VII

Modifications to BASIC/RT to Allow Use of

Datel A/D Converter

 

 

 

Location Old Contents Change to
10214 6531 6521
10216 6536 7200
10365 6532 6531
10366 6534 6532
10373 6533 6534

 

 

This A/D converter may be operated at rates up to 5 KHz when controlled

by Basic/RT. External A/D start (e.g. by real time clock) and A/D

DONE Interrupt are disabled.

85

ENTER TITEANT CONCENTRATION) MOLAP. UNITS.

0.031

ENTER THE COMPLEX FORMATION CONSTANT.
IOGE+8

ENTER MOLAR ABSORPTIVITY OF THE COMPLEX.

23G.

ENTER SAMPLE MOLAR ABSORPTIVITY.

72.

SPECIFY SIGNAL MODIFIER IN USE. OPTIONS ARE:
I VOLT PER DECADE LOG CONVERTER. TYPE I
2 VOLT PER DECADE LOG CONVERTER: TYPE 2
5 VOLT FULL SCALE I TO V: TYPE 3
I VOLT FULL SCALE I TO V: TYPE 11

[I

INI TI AL. ESTIMATE
I TERATION NUMBER

= 0.287307E-B‘2
I
ITERATION NUMBER 2

O. 29136513-012
0. 29203815-02

THE SAMPLE CONCENTRATION IS
0.2920422-02 MOLAR

ANY CHANGES? TYPE 1 FOR YES: Z FOR N0.
0

DO YOU DESIRE A LISTING OF THE DATA?

0

PRESS CONTINUE FOR NEXT EXPERIMENT.

4 Figure 13. Typical Computer-Operator Dialog for Photometric Titration
Program.

536

,——-o—

( START /)
I

 

 

 

 

I INITIALIZE I _ .._—_
I
l - I L-
IAVERAOE
,___. IOOO A/O . __
~ CONVERSIONS I—. '-fi
__ __J I
a ”"“’l“'”““ I
PULSE I
I BURET I I
I |
__I
I I
‘ __._L-_.-L__
DELAY I I SET BY _—_I
I LOOP I ----- 1 -------------------------- 4--1 USER
I I i
I 1K
r SOth
I POINT? \\\59———-J
I
I
. lYES
I A
I USE FIRST
. POINT TO I
I
I

IESTIMATE [M]

 

It

I ICORRECT OATA,
I ICALCULATEO

 

SUMMATIONS

 

 

I _.-
////ESTIMATEW USE RESULT I

0F CALCULATIONS

 

I

 

I CORRECT? ‘ AS NEw
\\ ESIIfiflIE
YES
IPRINT
ISAMPLE I
CONCENTRATION;
__.._.I
--L_IL_____. ,/J\\

333,3: Inc ,4," \ m
r——~ -/ CHANGES? "-——*—-"'—'”
J ‘.

 

 

\
\
\

Figure l4. Flowchart for Photometric Titrator Program.

 

87

.3m4 m.gmmm op wucmgmcu< m.soumxm we cowum:_scmumo .mp mgsmwm

saa.zo_»<m»zmuzou

000 00» 00m 000 00¢ OOn com 00. O

u a I W 1 I J u

mmd

()
N

 

 

88

observed at 2.55 absorbance units. A larger effect due to stray light
may be expected when the monochromator is set to 750 nm, due to in-
creased PMT gain, but it is felt that the instrument response will
still be linear to A = 1.0, all that proved to be necessary for this

work.

F. Experimental Results

 

l. Chemical System

 

Titration curves were generated by recording absorbance during
the titration of copper (II) ion with EDTA (ethylenediaminetetra-
acetate). The copper solutions were prepared by dilution of a 0.1 M
cupric chloride solution, in 0.l M acetic acid-sodium acetate buffer
(pH = 4.7), with acetate buffer. EDTA solutions were prepared in
acetate buffer from the disodium salt. Concentrations of 0.0l M and
0.001 M EDTA were prepared.

The copper-EDTA complex has a conditional formation constant
of approximately 108 at pH 4.7. In fact, the value of the formation
constant is unimportant if greater than l06, due to truncation errors

in calculations performed by the 8/e.

2. Investigation of Signal Modifiers.

 

a. Precision of Results - Two signal modifiers were evaluated,

 

a one volt per decade log amplifier and a current-to-voltage converter
(I-V). The 100% T reference voltage from the I-V was set to 5.000
volts by adjusting the PMT supply voltage, while 0% T was set at 0.000
volts, using the amplifier current suppression circuitry to suppress
dark current. The zero absorbance signal was set to zero volts from

the log amplifier. The precision of titrations performed with each

89

signal modifier was evaluated at various concentration levels. The
results are shown in Table VIIIA. There is no statistically significant
difference between the precisions of results obtained with the two
signal modifiers at Cu+2 concentrations greater than 0.5 mM. Below
that, the precisions are statistically different at the 99.5% confidence
level, as determined by the F test. Two points deserve mention. First,
at the 8 mM Cu+2 concentration level, the calculation routine failed

to converge to an answer. Second, as the overall change in absorbance
during the titration decreased, the precision of titrations using the
log amp as a signal modifier decreased faster than that for titrations
in which the I-V was used as the signal modifier. The reasons for

this are simple to see. When absorbance measurements are made with a
one volt per decade log amplifier and a 12 bit, :5 volt A/D converter
at absorbances very close to A = 0, the instrument is essentially read-
out limited, since a change of l bit in the A/D converter represents

a change of 0.0025 absorbance units. 0n the other hand, at 90% trans-
mission, using the I-V converter as the signal modifier, a change of

1 bit in the A/D converter represents a change of 0.00024 absorbance
units - a factor of l0 better resolution. For the titration of 0.0002
M Cu+2 (molar absorptivity x pathlength = 72 in this work) with 0.001
EDTA (complex molar absorptivity x pathlength = 230), the solution
absorbance changes from 0.00l4l initially to 0.0383 at the endpoint.

The limited resolution of the log amplifier-A/D converter (1 part in

15, maximum) system at this low signal level causes obvious quantizing
errors. 0n the other hand, since the I-V converter is operating at

peak resolution (approximately 1 part in 2000) quantizing errors are

minimal. One would expect the opposite effect at high solution

90

Table VIII A
Comparisons of Precision of Results Obtained With

Different Signal Modifiers

 

 

 

Cu+2 Concentration, mM Relative Standard Deviation
(Titrant, 0.0] M EDTA) Log amp 1-v
8 --- ---
5 0.25 0.08
3 0.35 0.16
l 0.35 0.425
0.5* 0.97 0.l5
0.2* 0.9 0.30

 

 

*Titrant - 0.00l M EDTA

Table VIII 8
Comparison of Accuracy of Results Obtained With

Different Signal Modifiers

 

 

2

 

Cu+ Concentration, mM Error, %
(Titrant, 0.0l M EDTA) Log amp 1-v
3 ---- ----
5 +0.75 -4.9
3 -l.25 -2.9
1 +0.30 -3.4
0.5* -l.8 -5.l

0.2 -3.0 +6.2

 

 

91

absorbance, but two considerations must be mentioned. First, at high
sample concentrations, the calculations did not converge. Second, the
greater interest lies in the direction of lowering the concentration
limit of good precision. Over-concentrated solutions may always be
diluted. It would therefore seem that the I—V converter would be the
Signal modifier of choice for this work. In routine use on approxi—
mately 100 individual titrations, the instrument previously described,
using the I-V converter as the signal modifier, was typically precise
to 10.15 to 10.3% relative standard deviation over a copper concentra-
tion range of 0.5-5.0 mM. It may be seen by examination of Table
VIIIA, and was routinely observed in practice, that the endpoint de-
tection method used in this study was not the factor limiting precision.
In general, imprecision was more attributable to the volumetric work
of the author (10.05% - 10.2%) and to the motor driven buret (10.l% -

10.5%) than to the endpoint detection technique.

3. Accuracy of Results

 

The accuracy of the titrator was tested over a range of Cu+2

concentrations, using both signal modifiers. Accuracy is dependent
upon a number of factors upon which precision does not depend, i.e.
proper values for molar absorptivities and absolute accuracy in all
volumetric work. The accuracy of this instrument was found to be
highly influenced by these factors. Table VIIIB shows the results

of the accuracy study. Titrations performed using the log amplifier
were more accurate in all cases, due to the fact that the log amplifier
was used in the determination of all molar absorptivities. Non-

linearities in both the log amplifier and the A/D converter gave rise

92

to the scatter of errors on both sides of zero.

G. Test of Converging Ability of Software

 

This study was performed to determine the effect that the initial
estimate of sample concentration had upon the final answer derived

2 with 0.01 M EDTA

from the calculations. A titration of 3.0 mM Cu+
was performed and all data points were stored in a data file on the l.6
million word disc. These data were then used in the calculational
routine. Initial estimates of the sample concentration, entered from
the teletype keyboard,ranged from 0 mM to 1000 mM. The results are
listed in Table IX. In all cases, the fitted result was 0.3T7214E-02
(3.l72l4 mM), constant to six significant figures. The number of
iterations needed to arrive at the final answer was rarely greater

than three, unless the initial estimate exceeded the actual concentra-

tion by a factor of 50 or more. The program used for storing data on

the disc and retrieving the data for analysis are listed in Appendix B.

H. Future Prospectives

 

The method of photometric titration endpoint detection just
described has been shown to be applicable to one of the simplest
complexometric titrations. To be truly useful, this method must be
shown to be applicable to the titration of systems with low formation
constants, systems with a background interference and those in which
the ligand may also have an appreciable molar absorptivity. The prin-
cipal pitfall observed by the author was that expansion of the soft-
ware to allow more flexible treatment of the data also involved a
far greater expansion in the time required to process the data.

Ideally, the program should, given all the molar absorptivities, fit

93

Table IX

Test of Software Converging Ability

 

 

 

Initial Estimate, mM Number of Iterations Final Results, mM
0 3 .317214
.01 3 .3l7214
.l 3 .317214
3 3 .3l7214
1.0 3 .3l7214
3.0 2 .3l7215
10.0 4 .3l7214
30.0 4 .3l7214
100.0 4 .317214
1000.0 4 .3l7214

 

 

94

the formation constant, a background term and the sample concentra-
tion. Practically, the software required to perform this task would
not operate on the present minicomputer, as it is now configured,
in a reasonable time. One iteration of the present routine requires
approximately 15 seconds of computer time. However, as access to
faster, more sophisticated processors and software becomes available,
the multiparameter fit for minicomputer controlled photometric titra-
tions will become more feasible. More complicated systems are already
treated by large computer systems, as was discussed in Chapter III.
Further improvement in instrumental precision will require the
development of better volumetric equipment for the delivery of the
sample and titrant to the titration cell. Such a system is presently
under development and promises to yield volumetric accuracy and precision

to 10.01% for diluting and delivering solutions.

95

CHAPTER V

An Automated, High Precision Reagent Preparation System

A. Introduction and Historical

 

Perhaps the single greatest handicap to the speed and efficiency
of automated analytical instrumentation is the solution preparation
step preceding the analysis. This step is generally performed manually
using volumetric glassware, and may often introduce more imprecision
into a measurement than all instrumental noise sources combined. In
addition, many studies of chemical systems require the preparation of
large numbers of solutions. It is not uncommon in such cases to find
that, of the three major steps in such studies, preparation of samples,
analysis of the samples and analysis of the data, the sample prepara-
tion step may be the most time-consuming. Certainly, this is the case
for a computer-controlled stopped-flow system developed in this labora-
tory (62). Many commercial automated instruments achieve good precision
in dilutions at fixed ratios of sample and diluent, but none exist
which allow precise, computer-controlled dilutions at widely varying
ratios. The application of a computer to the control of an automated
dilution system could be advantageous in several ways. Provided with
a few stock solutions, the proper peripherals, and a list of the solu-
tions to be prepared, the computer-dilutor system could: (l) prepare
solutions quickly; (2) calculate activity corrections prior to solution
preparation; (3) prepare solutions at constant pH or ionic strength;
(4) maintain optimum precision by selection of the proper dilution
equipment; (5) prepare unstable solutions immediately before use; (6)

accurately control the times at which various reagents are added to a

96

solution for reaction rate studies, etc. The utility of such a device
cannot be overestimated.

The very fact that the problem of solution preparation is widely
recognized is evidenced by the papers that have recently appeared
describing computer-controlled systems for solution preparation
(21,22,63). Although no large number of papers have been published,
those which have appeared reflect a genuine concern with the problems

faced by analytical chemists with regard to solution preparation.

1. The DemingePardue System

 

Deming and Pardue (63) developed an automated system for the
characterization of chemical systems. The general concept of this
system, that is, the use of an automated instrument for the investiga-
tion and optimization of experimental conditions in an analytical pro-
cedure, is of general interest, but of particular interest is the manner
by which reagents were handled. By means of micrometer syringes driven
by stepping motors, multicomponent reagent solutions were prepared
from Single component stock solutions, with varying ratios of the
various components, all under computer control. The volumetric system
consisted of 2.5 ml syringes (0.l microliter precision) and stepping
motors with a step angle of 22.5° (13.0%, noncumulative). The micrometer
head of the syringes underwent 50 revolutions from limit to limit (2.5
ml delivery). This resulted in a volume resolution of 1 part in 800
or 3.125 microliters (10.14 microliters or 14.5%) per step of the
stepping motors (assuming that the only sources of imprecision are
the syringe and stepping motor). Two drawbacks of moderate importance
may be anticipated with this system. First, due to the type of syringe

used, the maximum dilution which may be performed with 0.1% precision

97

is approximately 1:18. This means that four or five stock solutions
might have to be prepared if a study were carried out over a four or
five order of magnitude concentration range. Dilution of a single
solution over this concentration range would probably result in more
internally consistent data, but is not possible with this system if
good precision is to be maintained. The second fault lies in the
manner in which the stepping motors are controlled. Each step of a
motor is initiated by a pulse from a minicomputer. During the dilution
operation, the computer must be dedicated to control of the stepping
motors, and is not available for the performance of other functions.
Nonetheless, this system represents an important step toward computer

control of the reagent preparation step in chemical analyses.

2. The Megargle-Marshall System

 

Megargle and Marshall (22) described a stepping motor-micrometer
syringe system for volumetric delivery with a Significant improvement
over the Deming-Pardue system. The stepping motors were operated by
a hardware controller. The computer program they developed set the
number of steps, the direction of motor rotation and the stepping rate
in hardware registers, using one to three computer instructions. The
hardware controller then generated all the pulses necessary to perform
the number of steps specified by the computer. During the time required
for these operations, the computer was free to perform other tasks.
However, the use of micrometer syringes for metering all solutions will
limit the useful range of dilution much as it did in the Deming—Pardue

system.

98

B. Proposed System

 

1. Introduction

 

Because the system proposed here has been under development for
only a short period, this section of Chapter V will serve as a
speculative guide to further work on the system, and as a brief presenta-
tion of the data obtained to date.

The basic proposal for this system is that addition of the diluent
solution to the reagent be controlled differently than the manner in
which the reagent stock solution is dispensed in order to provide a
wide range of dilution ratios with good precision. The reagent stock
solutions are dispensed by stepping motor-micrometer syringe assemblies.
These solutions are dispensed into a tube, through which the diluent
solution flows on its way into a set of glass chambers, as illustrated
in Figure 16. Sensors at points A, B and C sense the presence of solu-
tion in the glass tubing connecting the chambers and may shut off the
flow of diluent by closing valve #3. The solution contained in the
glass chambers is then dispensed into a receiving vessel. The glass
chamber assembly is constructed so that the total solution volume to
sensor C is approximately 10 ml, to sensor B approximately 100 ml and

to A, 1000 ml.

2. Stepping Motor-Syringe Assembly_

 

The micrometer syringes used for this device are 2.5 and 0.25 ml
capacity syringes, precise to 10.004% of full scale (#7870 and 7874,
Gilmont ultra-precision micrometer syringe, Cole-Palmer Instrument Co.,
Chicago, IL). These are mechanically coupled to the stepping motors
via a precisely aligned mechanical linkage. The stepping motors used

in this study provide a 9 degree rotation per step (#1820130-40-F75,

99

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5
;
diluent venl I5
N
L; 2
L--sensor A
I”
<—-———sensor B
4———sensor C
QL fl 4
6
acu m
‘—E[:>> I é§=v "
7 dehvery

 

 

 

 

 

 

 

 

 

2

stepping Inolor-
micromeier syringe
assembly

Figure 16. Diagram of Automatic Solution Preparation Device.

100

Sigma Instruments, Inc., Braintree, MA). The stepping error is 13%,
noncumulative. This assembly will deliver the full syringe volume
with 50 revolutions of the stepping motor. The minimum quantity of
solution which may be delivered is 1.25 microliters from the 2.5 ml
syringe and 0.125 microliters from the 0.25 ml syringe, 18.5%. The
range of syringe volume over which delivery is precise to within 0.1%
is from 4 to 100 percent of the syringe volume.

The stepping motors are controlled by a hardware control unit.
This unit contains a register which may be set by a minicomputer or
by the use of manual switches. This register contains the number of
steps for which the motor is to be driven and allows computer-specified
solution preparations to proceed while the computer is free to operate
instruments or analyze data. A block diagram of the controller is
shown in Figure 17. The micrometer syringes are driven from limit to
limit by 2000 steps of the stepping motor. The 11 bit down counter
may be set to 2047 counts, sufficient to drive the system from limit
to limit. When a count is set in the presettable down counter, the
"COUNT = 0" gate goes to logic "1", which allows the clock pulses to
activate simultaneously the motor control circuitry and enter counts
to the down counter. The direction of motor rotation is controlled
by BAC bit 0, if operated under computer control, or a switch, if
operated in the manual mode. The syringe may be filled or emptied in
10 seconds. A three-way valve (number 1 or 2) connects the syringe
either to a reservoir of reagent stock solution for refilling the
syringe or to a delivery tube connected to the diluent reservoir.
The functions of all the valves shown in Figure 16 are listed in

Table X. It should be noted that it may be possible to substitute

.mommgwpcn Lopez mcwnnmum .mp «Czar;

 

 

 

 

 

101

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A312. 4mz<a
Azaoo :ptsm
mopoz
oEaauhm
dompzoo a eupzaoo
3:.
mokoz .asw kIIJI mmpzaoo
ozaamhm zsoo
cozoozu m4m<ppummma
L #5-: :10
10040
N: oom
zotzm
4<32<2

 

 

 

 

s nvq.m

 

102

Table X

Description of Valving for Figure 16

 

 

 

Valve # Function
De-Energized Energized
l Syringe to reservoir Syringe to diluent
2 Same as 1 Same as l
3 Closed Open
4 Reservoir to diluent Reservoir to receiving vessel
5 Reservoir to atmosphere Reservoir to nitrogen tank
6 Closed Open
7 Closed Open

 

 

103

check valves for the three-way valves used to fill and empty the

syringes.

3. The Dilution Chamber

 

The dilution chamber pictured in Figure 16 consists of three
glass globes of approximately 10, 100 and 1000 ml volume connected
by glass tubing. Optical level sensors are used to detect the solution
rising past the top of each chamber. These sensors consist of an optical
fiber light guide, illuminated by a high intensity tungsten-halogen
lamp, and a photocell (NSL-364) positioned on opposite sides of the
glass tube. Typical cell resistance when the glass tube is empty is
20 K0 and when the tube is filled with water, 10 MO. The change in
cell resistance as the solution meniscus rises through the optical
path may be used as a signal to turn off valve #3. This arrangement
allows the accumulation of a precisely controlled quantity of solution
in the chamber assembly.

The Optical sensor circuitry is diagrammed in Figure 18. Only
the sensor circuit for sensor 8 is shown. The photocell forms part of
a variable voltage divider. The voltage across the other resistor in
the divider is applied to one comparator input and a variable reference
voltage is applied to the other input. The variable reference is
adjusted so that the TTL compatible output of the comparator is at
logic level 0 when solution is present in the glass tube and logic
level 1 to 0 pulse applied to the SET input of the ENABLE flip-flop.
This sets 0 of the flip-flop to a l, which allows information from
the comparator to pass through gate 1 to gate 2. All inputs to gate

2 are at logic level 1 until an enabled comparator is at logic level

104

  
     

 

 

.Illll%

 

0 3...... _ uk<o

< 3...... _ mh<o

   

h a
I... 33528 IDI

.m gomcom pauppao sop Eagmavo «waugwu

.m_ wgzmpm

 

W as:

 

 

 

 

 

 

 

 

 

 

x no.

xx .m.a

105

which sets the relay to state 1 and applies power to the solenoid
control unit of valve number 2. When the solution level rises past

the enabled sensor, the comparator and gate 2 outputs go low, the relay
breaks the solenoid energizing circuit,and the monostable applies a
pulse to clear the flip flop and disable the comparator.

The process of performing a dilution therefore consists of inject-
ing the desired quantity of reagent stock solution into a tube and
washing this reagent into the dilution chamber where the flow of diluent
may be stopped at a preset total solution volume. Diluent flow may
be implemented with a variable speed peristaltic pump or by gravity
flow from an elevated reservoir.‘ The quantity of reagent solution
added and the optical sensor enabled are determined by the desired
solution volume and concentration. The quantity of reagent and the
optical sensor to be energized may be calculated manually or by the
computer, depending upon the desired mode of operation. A flow chart
describing the dilution operation is shown in FigurelQ. After the
solution has been prepared, it is delivered, by energizing valve 4,
to a receiving vessel where more thorough mixing may take place.

In order to maintain high delivery precision, dry nitrogen at low
pressure (0.5 psi) is used to expel the solution clinging to the chamber
walls and tubing. Valve 5 controls the flow of nitrogen. Valve 6

or 7 is operated to either discard the solution or allow it to be

used by an instrument. Alternatively, various receiving vessels may

be mechanically positioned under the delivery tube for solution collec-

tion.

106

 

.1
[LOAD DOWN COUNTER . I

LENABLE AN OPTICAL SENSOIEI

 

 

[DELIVER REAGENT .J
[ERIN VALVE 3. REFILL SYRINGES.]

l

—'

[CLOSE VALVE 3 WHEN CHAMBER 13 FULL]

+ l

ENERGIZE VALVE 4. DELIVER SOLUTION]

[ENERGIZE VALVE 5. BLOW CHAMBER DRY.|

E-ENERGIZE VALVES .

]

 

 

 

 

 

 

 

 

 

 

 

 

RESET AND WAIT-
FOR NEW
INST UCTIONS. V

 

 

 

 

 

 

[IS THIS A WASTE SOLUTION?} YESL{0#PEN VALVE 6.}
l N0
Ig’EN VALVE 7.l

 

Figure 19. Flowchart for Operation of Automatic Solution Preparation

Device.

107

C. Experimental Evaluation

 

The only portion of the automatic solution preparation system
available for evaluation at the time this thesis was written was the
dilution chamber. Tests were performed to determine the volumes of
the three compartments and the precision with which the chambers could
be filled and emptied. The volume of the assembly to the first optical
sensor was 7.5 ml, to the second sensor, 86.5 ml and to the third,
approximately llOO ml. The filling and delivery precision was tested
by energizing the second optical sensor, allowing distilled water to
flow into the chamber from an elevated reservoir and dispensing the
water into a weighing cup. The weight of water delivered was determined
with a top-loading balance. The volume delivered is tabulated in
Table XIA and XIB, assuming the density of water at 25°C to be
0.997044 g/ml. Table XIA shows the results of simple delivery using
only gravity flow. Note that reproducibility is not good (t0.22%).
Table XIB is a study performed using low pressure nitrogen to expel
clinging drops of solution from the chambers and tubing. Ignoring
data point number lO, which is aberrant due to mechanical failure of
the nitrogen supply, a mean value of 86.53 ml of solution was delivered
with a relative standard deviation of 0.022%. Some of this imprecision
is due to errors in reading the solution weight from the top-loading
balance.

Mechanical problems prevented testing of the other two bulbs.

If it can be assumed that they may be relied upon to fill and deliver
with the same precision as the bulb tested, the range over which dilu-
tions may be performed with O.l% precision may be calculated. Since

two micrometer syringes, 2.5 ml and 0.25 ml, and three chambers, 7.5,

108
Table XI A
Data from Experimental Evaluation of

Solution Preparation System

 

 

Sample # Volume Delivered, ml

 

Oxoooxnosmth-a
CD
Ch
(A)
(A)

—-l

 

 

Mean = 86.33
Relative Standard Deviation - 0.22%

Table XI 8

 

 

Sample # Volume Delivered, ml

 

omoowmmpwm—a
CD
0‘
U1
(A)

—l

 

 

109

86.5 and 1100 ml, are available, a wide usable range is anticipated.
The maximum allowable syringe imprecision, assuming a desired total

imprecision of O.l% and a diluent imprecision of 0.022%, is

 

== ‘[(.001)2-(.00022)2 = 0.0976% (5.1)

0' .
5 r1 n
‘y gemax

For each syringe, this means that delivery of 4.1% to 100% of the syringe
volume followed by dilution can probably be accomplished with 0.1%
precision. The smallest dilution factor available (100% of the volume
of the 2.5 ml syringe diluted to 7.5 m1) is 1:3 and the largest (4.1%
of the volume of the 0.25 ml syringe diluted to 1100 m1) is l 1.0 x 105.
This means that this system can perform dilutions of a single stock
solution to a precision of 0.1% over a relative concentration range

of approximately 36,000, a factor of 2000 greater than the Deming-
Pardue system. Two stock solutions of a reagent of the proper concen-
trations may be used to produce a set of solutions, diluted to 0.1%,
spanning a relative concentration range of 1.3 x 109, enough for most

studies. Practical considerations will limit this range to some extent.

0. Proposed Applications

 

The first application of this device should be to solution prepara-
tion for the stopped-flow devices used in this laboratory. These
devices have been used in studies of chemical systems over relative
concentration ranges of as many as three orders of magnitude (64).

Large numbers of solutions are used in such studies and a significant
quantity of time is consumed in solution preparation. A dilution
device such as this one, operating next to the stopped-flow instrument,

could prepare solutions rapidly, on demand, and would be essential for

110

the design of a truly interactive computerized stopped-flow system.
Such a system could be used to investigate a few representative sample
concentrations, analyze the data to determine the concentration range
of greatest interest and prepare solutions within that range in sequence
with subsequent analyses.

This dilutor could also be used as a modular device, perhaps

operated like the Deming-Pardue model, for less critical applications.

BIBLIOGRAPHY

(A)

\JO‘UW

CD

10.

11.
12.
13.
14.
15.
16.

17.
18.
19.
20.
21.
22.

BIBLIOGRAPHY

J. D. Ingle, Jr. and S. R. Crouch, Anal. Chem., 4% 1375 (1972).
. D. Ingle, Jr., Ph.D. Thesis, Michigan State University, 1971.

w. T. Gridgeman, Anal. Chem., 3%, 445 (1952).

C. M. Crawford, Anal. Chem., 343 343 (1959).

H. K. Hughes, Appl. Opt., 2, 937 (1963).

J. D. Ingle, Jr., Anal. Chem., 45, 861 (1973).

R. W. Burnett, Anal. Chem., 45, 383 (1973).

D. C. Nelson and R. C. Hawes, "A Technique for Measuring Photometric
Precision and Accuracy", Pittsburgh Conference for Applied Spectroscopy
and Analytical Chemistry, March, 1964.

H. V. Malmstadt, C. G. Enke and S. R. Crouch, "Electronic Measurements
for Scientists", w. A. Benjamin, Menlo Park, CA, 1974, pp. 727-753.

H. V. Malmstadt, M. L. Franklin and Gary Horlick, Anal. Chem., 44,
63A (1972).

Terry Searles, Varian Inc., personal communication.

Howard Schwartz, GCA McPherson, personal communication.

. C. Kelly and Gary Horlick, Anal. Chem., 45, 518 (1973).

. Field and L. G. M. Baas-Decking, J. Gen. Physiol., 9, 445 (1926).

. H. Mfieller and H. M. Partridge, Ind. and Eng. Chem., 29, 423 (1928).

. Hickman and C. R. Sanford, Ind. and Eng. Chem., Anal. Ed., 5,
1933 .

P

J

R

K

6 (

J. J. Lingane, Ana1. Chem., 20, 285 (1948).
H. V. Malmstadt and E. C. Gohrbandt, Anal. Chem., 26, 442 (1954).
H. V. Malmstadt, Anal. Chem., 29, 1901 (1957).
c. G. Enke, Michigan State University, personal communication.
K. A. Mneller and M. F. Burke, Anal. Chem., 43, 641 (1971).
R. Megargle and J. Marshall, J. Chem. Instru.4, 29 (1972).

111

23.

24.
25.
26.
27.
28.

29.
30.
31.
32.
33.
34.
35.
36.
37.
38.

39.
40.

41.
42.
43.

44.
45.

46.

112

J. Slanina, P. Vermeer, G. Mook, H. F. R. Reinders and J. Agter-
denbos, 2. Anal. Chem., 260, 354 (1972).

G. M. Hieftje and B. M. Mandarano, Anal. Chem., 4%, 1616 (1972).
H. A. Robinson, Trans. Electrochem. Soc., 22, 445 (1947).
H. V. Malmstadt and C. B. Roberts, Anal. Chem., 27, 741 (1955).
Beckman Instruments, Inc., Pasadena, CA, Bull, 239A(1951).

J. J. Lingane, "Electroanalytical Chemistry", New York, Interscience,
1953.

Precision Scientific Co., Chicago, IL., Bull, 9588 (1949).

R. F. Goddu and D. N. Hume, Anal. Chem., 22, 1314 (1950).

M. L. Nicols and B. H. Kindt, Anal. Chem., 22, 785 (1950).

. Sweetster and P. B. Bricker, Anal. Chem., 24, 409 (1952).
. Underwood, Anal. Chem., 25, 1910 (1953).

. Reilley and B. Schweizer, Anal. Chem., 26, 1124 (1954).

. Malmstadt and E. R. Fett, Anal. Chem., 26, 1348 (1954).

IIO>O

. Malmstadt and C. B. Roberts, Anal. Chem., 28, 1408 (1956).

H. . Malmstadt and C. B. Roberts, Anal. Chem., 28, 1412 (1956).

Z < < < Z l— m

E. . Wise, P. N. Gilles and C. A. Reynolds, Anal. Chem., 25,
1344 (1953).

T. L. Marple and D. N. Hume, Anal. Chem., 28, 1116 (1956).

H. V.)Ma1mstadt and D. A. Vassallo, Anal. Chim. Acta.,,LQ, 455
1957 .

E. H. Sargeant Co., Sci. Apparatus and Methods, 7, 2 (1955).
H. V. Malmstadt and D. A. Vassallo, Anal. Chem., 313 862 (1959).

H. V.)Ma1mstadt and T. P. Hadjiioannou, Jour. A.w.N.A., 51, 411
1959 .

H. V. Malmstadt and D. A. Vassallo, Anal. Chem., 31, 206 (1959).

J. M. Thoburn, C. M. Jankowski and M. S. Reynolds, Anal. Chem.,
31, 124 (1959).

D. Jagner, Anal. Chim. Acta., 50, 15 (1970).

47.

48.
49.
50.
51.
52.
53.
54.
55.
56.

57.

58.
59.
60.
61.

62.
63.
64.

113

T. H. Hunter, J. T. Sinnamon and G. M. Hieftje, Pittsburgh Conference
on Analytical Chemistry and Applied Spectroscopy, Cleveland, OH,
March, 1974, Abstract No. 255.

C. Liteanu and D. Cdrmvs, Talanta, z, 18 (1960).

P. Jandera, S. Kolda and S. Kotrly, Talanta,,LZ, 443 (1970).

J. Vrestal and S. Kotrly, Talanta,,LZ, 151 (1970).

N. Ingri and L. G. Sillen, Arkiv. Kemi.,2g, 47 (1965).

I. G. Sayce, Talanta, 15, 1397 (1968).

A. Sabatini, A. Vacca and P. Gans, Talanta, 24, 53 (1974).
A.Sabatini and A. Vacca, J. Chem. Soc., (Dalton), 1693 (1972).
P. Gans and A. Vacca, Talanta, 21, 45 (1974).

J. B. Headridge, "Photometric Titrations", Pergamon, New York,
NY, 1961.

A. L. Underwood, in "Advances in Analytical Chemistry and Instrumentation"
Vol. 3, C. N. Reilley, Ed., Interscience, New York, NY, 1964, p.31.

E. 0. Olsen and C. C. Foreback, J. Chem. Ed., 42, 206 (1972).
J. Kragten and M. Nijzenbeck, Z. Anal. Chem.,‘ggg, 7 (1972).
W. E. Wentworth, J. Chem. Ed., 42, 96, 162 (1965).

N. J. Youden, ”Statistical Methods for Chemists", John Wiley and
Sons, Inc., New York, 1951.

P. M. Beckwith and S. R. Crouch, Anal. Chem., 44. 221 (1972).
S. N. Deming and H. L. Pardue. Anal. Chem. 43, 192 (1971).
P. M. Beckwith, Ph.D. Thesis, Michigan State University, 1972.

APPENDIX A

APPENDIX A
Description and Documentation of Curve-Fitting Program

for Photometric Titrations

A. Introduction

 

This program determines the endpoint of a photometric titration
by a curve-fitting technique. It is intended for application to chemical
systems in which the product of the titration reaction absorbs light
at the analytical wavelength, the sample may or may not absorb light
at that wavelength and the titrant does not absorb light to any appreciable
extent. It is assumed that a 10.0 ml sample is used in the titration cell
and that the buret adds 0.2 m1 of titrant to the cell on each command
from the computer. A total of 10.0 ml of solution is added during each
titration. Each titration curve is therefore composed of 50 data
points. Instructions are written into the program to control the
buret and the A/D converter described in Chapter V. This program permits
the use of any of four signal modifiers. The signal modifier used may
be a one volt per decade logarithmic amplifier, a two volt per decade
logarithmic amplifier or an I-V converter set to give either a one
volt or five volt signal for the reference photocurrent. The voltage
signal is connected to the A/D converter multiplexer input number 4.
Input 4 was selected due to its low sensitivity to crosstalk. This

program is written to operate on a PDP-8/e 05/8 system.

8. Description of Software Operation
All symbols used in the computer program are defined in Section
C of this appendix. The program is resident on the 1.6 million word
disc and is put into operation by typing the following instructions to
114

115

the 05/8 system keyboard monitor.
.R FORT
*LRTITR.FT/G

After being complied, assembled and loaded, the program enters an
initialization routine in which the operator enters via the teletype
keyboard the values of titrant concentration, the complex formation
constant, the molar absorptivity-pathlength product for the complex
and the sample and the amount of time that will be allowed for the
reagents to reach equilibrium after the titrant has been added to the
titration cell. The program then requests information about the type
of signal modifier used. After the signal modifier is specified, the
program enters a loop which begins by averaging 1000 A/D conversions.
This average is stored in an array for further treatment. The buret
is pulsed by performing the START instruction, 6541, delaying for
approximately one second under control of the real-time clock, then
performing the STOP instruction, 6542. A second real-time clock con-
trolled delay is performed. The length of this delay is set by the
user in the initialization routine. This time delay allows the reagents
to equilibrate after the titrant addition. After this delay, the
program returns to the beginning of the loop, where 1000 A/D conversions
are again averaged. This loop is performed 50 times. At the end of
the 50th pass, the program enters the data treatment routine. The
program computes an initial estimate of the sample concentration and
prints the computed value on the teletype. The initial sample concen-
tration is calculated by dividing the first average absorbance value

stored in the array by the molar absorptivity-pathlength product of

116

the sample. A loop is then entered in which the estimated complex
concentration at each data point is calculated from Equation 3.5
assuming M is equal to the initial estimate calculated earlier. A

value is calculated equal to the estimated complex concentration at each
data point, using the specified values of the two molar absorptivity-
pathlength products, the measured value of the solution absorbance,

and the calculated estimates of the initial sample concentration and

the complex concentration at each data point (the estimated complex
concentrations used in the calculation are estimates based upon the
first absorbance measurement). The complex concentrations calculated
for each data point by this portion of the program are estimates based
upon the individual absorbance measurement at that point. The estimated
values of complex concentration are stored in an array. In the same
loop, the summation terms of Equation 3.11 are evaluated.

At the completion of 50 passes through this loop the quotient of
the summation terms is calculated. This is the new estimate of the
sample concentration. The new estimate is compared to the initial
estimate. If the two estimates are more than 0.01% different, the new
estimate of sample concentration is printed, then used as a new initial
estimate in the calculational loop. Calculations and comparisons
continue until the estimate calculated by the calculational loop agrees
with the estimated sample concentration used in the calculations to
within 0.01%. When sufficient agreement is reached, the sample concentra-
tion is printed on the teletype. The operator then has the option of
listing the measured absorbance, corrected values of complex concentra-
tion and values of complex concentration calculated from the sample

concentration for each data point, returning to the initialization

117

routine or recalculating the sample concentration using a different
value for the formation constant. The program then halts to allow
the operator sufficient time to prepare the instrument for another

titration.

C. Definition of Symbols

 

 

Symbol Name Use(§)
A Titrant concentration, moles/liter.
AB Product of complex molar absorptivity and

cell path length, cm liters/mole.

ABU Product of sample molar absorptivity and

cell path length, cm liters/mole.

8 Calculated least squares value of sample

concentration, moles/liter.

80 Estimated value of sample concentration,

either initial or iterated, moles/liter.

C Calculated value of complex concentration,
used in correction of experimental data

and in listing option, moles/liter.

D(N) Dimensioned array of all data corrected

for sample absorbance.

DATA Summation 1000 A/D conversions collected
before each addition of titrant. Scratch

pad.

FK

ICHA

IDATA

IFORM

IT

ITIME

IYN

OF

118

Complex formation constant, liters/mole.

Flag set by user. If set to 1, program
reenters the initilization routine.

Fixed point.

Scratch pad storage for individual A/D
conversions prior to floating and summa-

tion. Fixed point.

Flag set by user. If set to 1, allows
user to enter new value of the formation
constant, then refits data from last

titration. Fixed point.

Register to store number of iterations

performed. Fixed point.

Register used to set clock buffer-preset

register. Fixed point.

Flag set by user. When set to 1, programs
list raw data, corrected data and complex

concentration for each data point.

Flag set by user. Specifies which type
of signal modifier is used, directs
selection of proper data treatment state-

ments.

Scratch pad. Used in calculation of D(N).

TIME

V1

V2

V0

VT

VTI

X1

119

Dimensioned array of average absorbance

values measured for each data point.

Time allowed for solution equilibration
after titrant delivery and prior to

data acquisition, seconds.

Same as R.

Fraction of total solution volume in cell

due to initial sample volume.

Fraction of total solution volume in cell

due to titrant additions.

Initial sample volume. 10 m1.

Total volume of titrant added, m1.

Volume of titrant added per addition, m1.

Scratch pad used in calculation of 8.

Same as X.

Same as X.

 

—()ncwrin‘7n‘T"°

460

25

ryntfi

HP“:

(“018(005M1n01M

mgwf1fit5

120

PROGRAM LRTITR.FT. F1GE 1o
PHOTOMETRIC TITRATION PROGRAM FOE THE SPC A/D.

TITRATION DETERMINATIONS DY CURVE FITTING.

THIS :ROGRAM ASSUMES THAT: COMPLEX ABSOPRS: SAMPLE MAY (1R
MAY NOT ABSORB AND TITRANT DOES NOT ABSORB. IT FPDTHER
ASSUMIS THAT THE INPUT '5 THRU THE DATEL 12 H11 A/P VIA
CHANNEL FOUR: SIGNAL MIDIFIEF MAY BE OF FOU! TYPES.

CONTINUE
DIMENIION D(SO):S(SO)

WRITE «1:29)

FOIMAT ('ENTER TITRANT
VTI=0.2

READ (1:41) A

FORMAT (E10:8)

V0-1¢.O

ZONCENTRATION:MOLAR UNITso')

WRITE (1:42“)

FORMAT ('ENTEFT
REAE (1:43) FK
FORMAT (Elflo8)

THE.COMPLEX FORMATION CONSTANT.')

WRITE (1:436)

FORMAT ('ENTER MOLAE ABSORPTIVITY OF THE COMPLExo')
REAE (1:44) AB

FORMAT (E10.8)

WRITE (1:446)

FORMAT ('ENTER METAL ION MOLAP ABSORPTIVITY.')
READ (1:45) ABU

FORMAT (E14o8)

WRITE (1:21)

FORMAT ('ENTER TIME DELAY FOR ECUIL IN sscowrs.')
READ (1:22) TIME

FORMAT (E10.4)

TIME=~100.0*TIME
ITIMEIIFIX(TIME)

SELECT PROPER DATA TREATMENT ROUTINE.

WRITE (1:450) ‘
FORMAT('SPECIFY SIGNAL MODIFIER IN USE.
1 1 VOLT PER DECADE LOG CONVERTER: TYPE
2 2 VOLT PER DECADE LOG CONVERTER: TYPE
3 5 VOLT FULL SCALE I TO V: TYPE. (1':/:'
4 I VOLT FULL SCALE I TO V: TYPE 4':/:)
READ (1:469) OP

FORMAT (E16.4)

OPTIONS APEI'1/o'
I':/a'
2.3/3'

DATA'O
IT-O
IDATA-G

LOOP TO PERFORM 50 TITRANT DELIVERIES AND MEASUREMENT CYCLES.

DO 60 J'1:SE:1
CLA CLL
TAD (4
6521

NOP

NOP

6531
6532
JMP JMP4
6534

CLA CLL

ISELECT A/D CHANNEL 4.

ICONVERT.

ISKIP ON FLAG.

IREAD A/D BUFFER.

THE ABOVE ROUTINE EXCERCISES THE A/D TO CLEAR THE SAMPLE

AND HOLD. WITHOUT THIS PRECAUTION: THE FIRST DATA
POINT OF TfFH SF" " '“CORRFC’

121
c PROGRAM LRTITR.FT. Fast 2.
DATA-D

LOOP TO TAKE 1000 DATA POINTS.

GOO

DO 50 I=1:1000:1
CLA CLL
TAD (4
6521 ISELECT AID CHAJNEL A.
NOP
NOP
6531 ICONVERT.
MP1: 6532 ISKIP ON FLAG.
JMP JMPI
6534 IREAE A/D BUFFER.
DCA \IDATA /PUT DATA IN 1 SAFE PLACE.
CLA CLL

U)
'32
4
D
C)
s

GET DATA FROM SAFE PLACE AND FLOAT AND SUM IT.

(“00101.“L‘1MEMUIU'1U1LA

DATA-(DATA+FLOAT(IDATA))
50 CONTINUE

C GOTO PROPER DATA TREATMENT P'WTINE.
C

IF ‘20-'01”) 471:472:473
4"1 IF (3.-OP) 47‘14751475
.2 S(J)"(DATA)/(2.*4R9.4#IBUG.)

ASSUMES -1 VOLT= 1 AESORBANC? UNIT

GOTO 479
473 S(J)=-DATA/(409.4I1000.)

C ASSUMES -2 VOLTS=1 ABSORBANCE MVIT

GOTO 479
7 S(J)--(ALOG(DATA/4.694E+S))/2.30258

A
C
C ASSUMES 1 VOLTII 100% T.
C

GOTO 479
475 S(J)=-(ALOG(DATA/2.047E+6))/2.303
479 CONTINUE

ASSUMES S VOLTS-IOBXT.
TIMING ROUTINE FOR THE BURFT.

CLA CLL
6541 lI-O COMMAND TO BURET-- COMMENCE DELIVERY.
CLA CLL IDELAY LOOP - PERMITS BURET SUFFICIENT
TAD (-15@O / TIME TO INITIATE FIXED 0.2 ML CYCLE.
6133
CLA CLL
TAD (5362
6132
”Pa: 6131
JMP JMP2
6135
CLA CMA
6130 [STOP CLOCK. PREPARE TO ALTER PRE-SET BL‘FER.
CLA CLL
6542 ISTOF BURET.
CLA CLL
TAD \ITIME IPROGRAMMEL DELA’ LOOP TO ALLOW F P
6133 IEOLILIBRIUM IH SAVPLF CELL. ITINE SET 1' USER.
CLA CLL
TAD (5222
6132
MP3: 6131
JMP JMP3
6135
CLA CMA
6130
CLA CLL
CONTIHIE

gmmmmmmmmmmmmrmmmmmmvmnonnn

(.‘L'Il‘UlIAU'!

R

“TC

61

103

71

rvn

n,,nr)3 gtwr,nc1n

4‘!
0“.)

0

ON
U"

)oc1cancfirancwczg;3rvn<5

122

PROGRAM LRTITR.FT. IAGE 3.
CALCULATE AN INITIAL EETIMATF FOL THE SAMFLE CONCINTRATION.

B0=S(1)/ALU
R80
U80
C80

WRITE (1.103) 80
FORMAT ('INITIAL ESTIMITE =':2X.F14.8)

X80
X180
Y=0
VTI-VTI

INITIAL ESTIMATE OF METAL ION CONCENTRATION.

DO 70 Kal.su.l

D(K)=S(K)

VT-VT+VTI

VI=VT/(VT+UO)

v2-v01<v1+v0)
R8FK¥<V1tA+V2tBO>+I.
UaSORT(Rt:2-4.tFKtt2tvItVZ*AtPe)
C-(R-U)/(2.DtFK)
D(K)=(D(K)-ABU¢<U2#B@-C))/AB

ABOVE CALCULATES ESTIMATED COMPLEX CONCENTRATION AT EACH
DATA POINT. BELOU CALCULATES METAL ION CONC. FROM
ESTIMATES AND OBSERVED DATA.

X8X+2.¥V18V2¢A*D(K)882-V1882*U28Att28D(K)-V2#D(K)#83
Xl-Xl-UltU2tAtD(K>/FK0V2:D(K)#:2/FK
Y-Y+VI*t2tv2tt2tAtt2+v2:t2tD(K)tt2-2.tv2¢VItU2¢AtD(K)
CONTINUE

ROUTINE TO CHECK IF ESTIMATED SAMPLE CONCENTRATION AGREES
TO WITHIN 0.012 WITH CALCULATED VALUE.

Y=-Y

B8(X+XI)/Y

VTt-VTI

IF (8-80-80/10000.00) 73:79.76
IF (5-80+B0/10000.0) 76a79a79
8088

ITIIT+1

WRITE (1.215) IT.BB
FORMAT ('ITERATION NUMBER '.2x.12.'-‘.2X.E|a.8)

GOTO 71
OUTPUT SAMPLE CONCENTRATION.

WRITE (1:80) B
FORMAT (‘THE SAMPLE CONCENTRATION IS'.2X.E14.8)

ASK IF LISTING DESIRED.

(‘C‘fififififfc

230
220

109
110

120

190
140

123

PROGRAM LRTITR.FT. IAGF 4.

URITF (I:400)

FORMAT ('LISTING? 1=YIS:0=NO.')
READ (1:410) IYN

FORMAT (I5)

IF (1-IYN)109:101:109

VT=-VTI
PRINT READINGS FOR LISTING.

WRITE (1:358)
FORMAT ('ABSORBANCE DATA':2X:'MEASURED ICI':2X:

DO 229 L-I:50:I

VT-VTOVTI

VlaVT/(VT+UO)

U2-VO/(UT+VO)
R-FKt(VItA+V288)+I.
u-SQRT(Rt*2-a.tFKtt2tVlcvztAta)
C=(R-U)/(2.GtFK)

WRITE (1:230) S(L):D(L).C
FORMAT (3E17.8)
CONTINUE

ASK IF ANY PARAMETERS ARE TO BE CHANGED.

WRITE (1:110)

FORMAT (‘ANY CHANGES?')
READ (1:120) ICHA
FORMAT (I5)

IF (I-ICKA) 190:1:190

WRITE (1:140)

FORMAT ('PRESS CONTINUE FOR NEXT RUN.’)
PAUSE

GOTO 25

DVD

'CALCULATED [(1')

APPENDIX B

 

APPENDIX B

Transferring Data to and from the RK8E Disc

Many experiments generate data far more quickly than they may be
treated mathematically. The classic approach to such experiments has
been the use of some device to store the data until analysis is possible.
The availability of laboratory minicomputers and mass storage devices
has greatly simplified data storage and treatments. Proper use of mass
storage devices requires some understanding of the basic hardware,
although the required level of comprehension is greatly reduced when
these mass storage devices are controlled by software operating systems
such as the 03/8 operating system used in this laboratory, with a
PDP-8/e minicomputer and RK8E 1.6 million word disc (Digital Equipment
Corp., Maynard, MA). The version of FORTRAN II included in the 05/8
system may directly read data from or write data onto the disc in
file-structured format. Since this greatly increases the available
storage space for data, it is important to be familiar with the simple
programming steps necessary to perform the read-write operations and
to understand the limitations on the use of the disc as a mass storage
device.

The information necessary for writing FORTRAN programs to read
data from and write data onto the disc is presented in Programming
Languages, 1972 edition, pp. l3-37 to l3-39, and in the 03/8 System
Reference Manual, p. 105 (published by Digital Equipment Corp.). This
Appendix will present examples of such data transfers to serve as further
elucidation for the existing instructions.

Data files on any storage device controlled by the 08/8 system may

124

125

be created by the use of the instruction, CALL OOPEN. The format for

the instruction is:

CALL OOPEN ('DEVICE', FILE NAME)

The device name is always enclosed in single quotes, e.g. 'RKBD',
'DATD', ‘PTP', but the file name may be either an absolute file name
or a variable representing a file name. If the file DATADl is to be
created,the file name may be specified in the CALL OOPEN command in
either of two manners. If an absolute file name is specified, it must
be enclosed in single quotes, e.g. 'DATADl', but a variable file name

may be created as shown in the following set of instructions:

WRITE (1,10)

1D FORMAT ('ENTER FILE NAME')
READ (1,20) FILE

2% FORMAT (A6)
CALL OOPEN ('RKBD', FILE)

The variable representing the file name is not enclosed in quotes. In
operation, this program will print the message on the teletype asking
for the file name, in response to which the user must enter from the
teletype a six character file name which may consist of alphabetic or
numeric characters, e.g. FILENl, LGZTDY. The file name must consist

of six characters. If shorter file names are desired, the remaining,
non-used characters must be entered by typing SHIFT/P(@), e.g. to enter
the file name LDT, type LDT@@@ in response to the message. Subsequent
WRITE statements specifying device code 4 will write information into

the file on the storage device specified in the CALL OOPEN statement.

126

When the process of writing data is complete, the instruction CALL
OCLOSE should be performed, since the data file will not otherwise
exist. All data files created in this manner are given the extension.
.DA

Data may be written into these files in any format (12, A6, El0.6,
etc.), but the most efficient format for writing numerical data is A6
format for floating point numbers and A2 format for integers. Attempts
to list such data files with PIP will create unintelligible printout.
Transfers of such files using PIP should always be performed in core
image mode (/I option).

Program LRPUT.FT, listed at the end of this appendix,demonstrates
the creation of data files in a real program used in data collection.
The steps used in the creation of these files are circled. This program
is similar to the data collection portion of program LRTITR.RT, described
in Appendix A. This program requests values for parameters, A, FK,

AB and ABU, and a name for the file to be created, then performs the
titration and stores the data in an array. At the conclusion of the
titration, the CALL OOPEN instruction is performed, the values of A, FK,
AB, and ABU are written into the specified file, followed by the ab-
sorbance data, S(K), from the dimensioned array in core memory. The
file is closed and the program loops back to ask for new values for A,
FK, AB and ABU (optional) and a new file name (not optional). A typical
copy of the printout follows the program listing, where the specified
file name for the first data set is DATADl. 0n the disc directory,

this file will appear as DATADl. The file extension (.DA) is not
specified by the user.

Once present on the disc, data files may be called up for use by

127

the program by the use of the instruction CALL IOPEN, followed by a
READ statement specifying device code 4. The format for the CALL IOPEN

command is similar to that for CALL OOPEN:

CALL IOPEN ('DEVICE', FILE NAME)

The device and file name are specified in the same manner as is
the CALL OOPEN command. File names may be specified by a variable,
but must exist on the specified device. It is not necessary to close
input files. An example program which will create a data file contain-
ing N (where N is any integer) A/D conversions with a subsequent program
to read the data file and store the data in an array is listed at the
end of this Appendix as program LRAD.FT.

This program stores both the A/D conversions and the number of
conversions, N. It is important to note that the format of the READ
statement used to read a data file must be the same as the format of
the WRITE statement used to create the file.

It is possible to store file names in a dimensioned array for use
in the program. Program LRGET.FT, listed at the end of this Appendix,
uses this method to allow the program user to specify which data files
are to be analyzed. This program is the data analysis section of LRTITR.FT,
listed at the end of Appendix A. The data files to be analyzed by this
program are created by LRPUT.FT (previously described), and each file
contains the titrant concentration, the complex formation constant,
the molar absorptivities of complex and sample and the 50 data points
collected during program operation. Program LRGET.FT sequentially
reads in and analyzes each data set until the entire specified number

of data files is analyzed. A copy of a typical dialog between program

128

and operator follows the program listing.

When storing information on the disc, it is important to under-
stand the limits to the rate at which data may be stored. A program
was written to collect an A/D conversion and write the individual result
on the disc as a floating point number. This operation was performed
lO5 times during program operation. Total execution time was 9 3/4
minutes, or an average time of 6 msec to perform and record each A/D
conversion. Since the converter converts in 20 sec, most of the time
is taken up by software. Storage of data in an array and transfer
of the array does not substantially reduce the total data transfer
rate, but allows much faster acquisition of the data stored in the
array. The implication is that when data are acquired slowly, the
method used to transfer data to the disc is unimportant. However, if
high data acquisition rates are required, all the data should be
collected and stored in core, then after data acquisition is complete,
the data file may be created on the disc. If each data point is written
into the file when it is collected, the maximum acquisition rate is
one data point per 6 msec, or approximately l70 Hz. If all the data
are stored in core prior to being written onto the disc, the maximum
acquisition rate is approximately 40 KHz. Obviously, the quantity
of data which may be collected by such a procedure is limited by the

available core space.

129.

C PROGIAM LRPUT.FT. PA3F 1.
c
c PHOTOMETRIC TITRATION FZOGRAM FOF THF SRC A/D.
c THIS PROGRAM COLLFCTs TIE TITRATION DATA IN THE FORM OF
c ABSORBANCE AND STORES TIE DATA ON THE DISC IN A6 FORMAT.
C EACH DATA SET IS PRECEEED BY THE FOLLOVINC DATAI
C THE TITRANT CONCENTRAT11N. THF COMPLEX FORMATION CONSTANT.
C THE COMPLEX MOLAR ABSOFPTIVITY. THF SAMPLE MOLAR ABSORPTIVITY.
C
C
1 CONTINUE
TVMENSION D(SO).S(SO)
C
IO VRITF (1.22)
22 FORMAT ('ENTER TITRANT :ONCENTRATION.MOLAR UNITS.')
VTX:002
READ (1.41) A
u WWMT(U0£)
V0810.0
C
VRITF (1.420)
422 FORMAT (/:'ENT 2 THE COIPLEX FORMATION CONSTANT.')
READ (1.43) FR
4: FORMAT (E10.8)
(
WRITE (1.436)
422 FORMAT (/:'ENTER MOLAR ABSORPTIVITY OF THE COMPLFx.-)
READ (1.44) A5
44 FORMAT (210.8)
VRITE (1.44O)
442 FORMAT (/:'ENTER SAMPLE MOLAR ABSOFPTIVITY.°)
READ (1.45) ABU
4S FORMAT (214.8)
c
VPITF (1.21)
21 FORMAT (/.'ENTER TIME [ELAY FDR EQUIL IN SECONDS.')
READ (1.22) TIME
22 FORMAT (210.4)
T1M£--IOO.O.TIME
LItInfisijXIJIMjaL _
C
VRITE (1.4525'
452 FORMAT(I:'SPECIFY SIGNAL MODIFIE! 1N CS2. I :IONS APE:':/:'
1 1 VOLT PER DECADE LOG CONVERTEf: TYPE 1'./.'
2 2 VOLT PER DECADE LOG CONVERTEZ: TYPE 2'./.'
3 s VOLT FULL SCALE I To V. TYPE 3'./.'
4 1 VOLT FLLL SCALE I To V. TYPE 4-./.)
READ (1:460) OP
46C FORMAT (£1o.4)
C
461 WRITE (1.46)
46 FORMAT(/.'SPECIFY SIx LETTER FIL; STORAGE NuME')
PEA: (1.47) FILE
47 FORMAT (A6)
25 DATA=0
IT=o
IDATA=O
C
C LOOP To FERFORM so TITRANT DELIV'RIES AND MEASUEFNENT CYCLES.
C
Do 60 u=1.so.1
s CLA CLL
s CLA CLL
s TAE (4
s 6521 ISELECT CHANNEL 4.
s NOP
s NOP
s 6531 ICONVERT.
SJIP4. 6532 ISKIP ON FLAG.
s JMP JMP4
s 6534 /RFAD A/D BUFFER.
s CLA CLL
c
C THE ABOVE ROUTINE EXCFRCISFS THE A/D TO CLLAF THF SAMPLF
c ANT HOLT. VITHODT THIS PPECAUTIlN. THF FIFST DATA
C POINT OF EACH SLT IS InconnFCT.
c

DATA‘V:

.g o‘

(3(3’1

SATOE:

annlw')

r1nrvm
Q

b'b
Q‘Q
‘OU‘

MP2;

nr)(10th",mtnUIMECUIm1nUTm¢nU1mrfififi

130

PEOGFAV LRPUT.FT. VICE 2.
LOOF TO TAKE 1390 EATA POINTS.

DO 50 181:IOOO.1

CLA CLL

CLA CLL

TAD (4

6521 ISELECT CHANNEI 4.
NOF

NOP

6531 ICONVERT.

6532 ISKIP ON FLAG.
JMP JMP1

6534 /READ AID BUFFIR.
DCA \IDATA [STORE A/D BUFFER DATA.
CLA CLL

FLOAT A/D CONVERSION AND SUM.

DATAI(DATA+FLOAT(IDATA))
CONTINUE

SELECT PROPER DATA TREITMENT ROUTINE.

SF (Zn-OP) 471:0721473
IF (Bo-OP) “7414750475
S(J)3'(DATA)/(20*409oat1900-)

ASSUMES -1 VOLT= 1 ABSORBANCE UNIT

GOTO 479
S(J)B-DATA/(h09.4#1000.)

ASSUMES -2 VOLT581 ABSORBANCE UNIT

GOTO 479
S(J)=-(ALOG(CATA/Q.O941+5))/2.38258

ASSUMES 1 VOLT= 1001 T.

GOTO 479
S(J)‘-(ALOG(DATA/2.647E+6))/2.303
CONTINUE

ASSUMES 5 VOLTS=IOOZT.

CLA CLL

6541 [START EURET. THIS TIMING LOOP ALLOWS SUFFICIENT
CLA CLL ITIME FOR THE BURET TO BEGIN A FIXED 6.2 ML.
TAD {-1500 IDELIVERY CYCLE.

6133

CLA CLL

TAD ($366

6132

6131

JMP JMP2

6135

CLA CMA

6130 ISTOP CLOCK. PREPARE TO RESET FEE-SET BUFFER.
CLA CLL

6542 ISTOP BURET.

rmmmmmnn'finr‘fr‘

MFB.

finfi111n'1th1htn
1‘)

o honor)
1:

non

66
(5

122

161

131

PROGRAM LRPUT.FT. PIGE 3.

CLA CLL

TAD \ITIME lPROGRIMMED DELAY LOOP TO ALLOW FOR
6133 IEQUILIDRIUM IN SAMPLE CELL. ITIME SET BY USER.
CLA CLL

TAD (5280

6132

6131

JMP JMP3

6135

CLA CMA

6130 ISTOP CLOCK.

CLA CLL

CONTINUE

ROUTINE TO STORE DATA CN DISC.
CALL OOPHV ('RKBO'aFILE)

STORE SPECIFIED PARAMETERS. TITPANT CONCENTRATION.
FORMATION CONSTANT.COMPLEX MOLAR ABSORPTIVITY AND SAMPLE
MOLAR ABSORPTIVITY.

WRITE (4:64) A: FKJABI ABU
FORMAT (4A6)
00 65 K-I.SG

STORE DATA FROM EXPERIMENT.

URITE (4.66) S(K)
FORMAT (A6)
CONTINUE

CALL OCLOSE

URITE (1:67)
FORMAT (la/.Ia'TITRATION COMPLETE!‘)

VRITE (1.100)

FORMAT ('ARE THERE ANY CHANGES IN THE INPUT FARAMETERS'./.
1 'TYPE 1 FOR YES AND 2 FOR N0.‘

READ (1.101) I

FORMAT (11)

IF (1") 461:1:461

BUD

132

.R FORT
uLFPUT.FT/O/G

FNTER TITRANT CONCBVTRATIONaMOLAR UNITS.
0.91

ENTEP THE COMPLEX FORMATION CONSTANT.
I. 9E§8

ENTER MOLAR ABSORPTIVITY OF THE COMPLEX.
230.

ENTER SAMPLE MOLAR ABSORPTIVITY.
72.

ENTER TIME DELAY FOF EOUIL IN SECONLS.
6.61

SPECIFY SIGNAL MODIFIER IN USE. OPTIONS ARE:
1 VOLT PER DECADE LOG CONVERTEW. TYPE 1

2 VOLT PER DECADE LOG CONVERTER. TYPE 2

S VOLT FULL SCALE I TO V. TYPE 3

1 VOLT FULL SCALE I TO V: TYPE A

3

SPECIFY SIX LETTER FILE STORAGE NAME
LRDATA

TITRATION COMPLETE!

ARE THERE ANY CHANGES IN THE INPUT PARAMETERS
TYPE 1 FOR YES AND 2 FOR NO.

2

SPECIFY SIX LETTER FILE STORAGE NAME
'C

nann

F308 OOH (50010

f .

OOGMIOVDEMMMMUI
3
v
.-
5

fifinbU‘
GS

fifififi

133

LRAD.FT

PROGRAM TO STORE DATA ON THE RKS-F DISC AND RETRIEVE IT

ALL UNDER FORTRAN CONTR(..

KRITE (1.10)

FORMAT ('ENTER N')

READ (1.20) N

FORMAT (IS)

PREPARE OUTPUT FILE.

CALL OOPEN ('RKBO'.'DATL01')

ASK FOR NUMBER OF DATA JOINTS TO BE COLLECTED.

URITE (4.30) N
FORMAT (A2)

LOOP TO COLLECT N A/D C NVERSIONS AND STORE ON

DO 40 IIl.N

CLA

6521 ISELECT A/D CHANNEL 0.
NOP

NOP

6531 ICONVERT.

6532 ISKIP ON FLAG.

JMP JMP1

6534 /READ AID BUFFER.

DCA \IDATA

URITE DATA ON DISC.

WRITE (4.50) IDATA
FORMAT (A2)
CONTINUE

CLOSE DATA FILE.

CALL OCLOSE

PREPARE TO READ DATA FROM DISC FILE.

CALL IOPEN ('RKBO'.'DATA01')
READ (4.60) N
FORMAT (A2)

LOOP TO FILL ARRAY WITH DATA.

DO 70 J'l.N -
READ (4.80) IAD(J)
FORMAT (A2)
CONTINUE

DVD

DISC.

OOOOQOOO

n

.—
b

(11

(’04

Q00

410

16

OOOUO

O OF)
U ()0

”On”

C-N‘

nnnnnn—ro

134

PROGRAM LRGET.FT. PAGE I.

CALCULATIONAL ROUTINE FOR PHOTOMETRIC TITRATION DATA

DERIVED FROM THE PROGRAM LRT.FT. ALL DATA USED IN THIS PROGRAM
ARE STORED ON THE DISC. THE ONLY OPERATOR INPUT

IS THE NAMES OF THE DATA FILES TO BE ANALYZED.

DIMENSION S(SB).D(SB).FILE(SO)

URITE (1.14)

FORMAT ('HON MANY DATA FILES TO BE KTALYZED7‘)
READ (1.15 ) N

FORMAT (I2)

URITE (1.408)

FORMAT (' DO YOU DESIRE A LISTING FOR EACH DATA SET?'./.
1 'TYPE 1 FOR YES AND R FOR NO.')

READ (1.410) IYN

FORMAT (IS)

WRITE (1.1%)

FORMAT ('LIST THE NAMEE FOR THE FILES TO BE ANALYZED.‘)
DO 16 J3IJN

READ (1.18) FILE(J)

FORMAT (A6)

CONTINUE

WRITE (1.36)

FORMAT (' ARE ALL FILE NAMES CORRECT? 1=YES. O=NO.')
READ (1.31) INS

FORMAT (I1)

IF (INS-1) 35.32.35

WRITE (1.33)

FORMAT ('SPECIFY INCORRECT FILE HY NUMBER')
READ (1.34) NUM

FORMAT (I2)

WRITE (1.36)

FORMAT ('UHAT IS THE CORRECT NAM47’)

READ (1.37) FILE(NUM)

FORMAT (A6)

GOTO 29

DO 2I M=I1N

PREPARE TO READ IN DATA FILE FROM DISC.

CALL IOPEN ('RKBO'.FILE(M))

READ INITIAL PARAMETERS SPECIFIED IN DATA FILE FOR
TITRANT CONCENTRATION. FORMATION CONSTANT. MOLAF
ABSORPTIUITY 0F COMPLEX AN E SAM’LE.

READ (‘1. 23) A. 7K. AB. ABU
FORMAT (4A6)

LOOP TO READ DATA INTO ARRAY S(I: FROM DATA FILF.

DO 11 I=1.SO
READ (“.22) 5(1)
FORMAT (A6)
CONTINUE

('56

orafirw

f";

26

103
71

76

215

79
80

[CI

135

PROGRAM LRGET.FT. PHGE 2.

VTI!0.2
V0310.0

INITIAL ESTIMATE OF SAIPLE CONCBJTRATION.

B0'S(1)/ABU
IT80

WRITE (1.26) FILE(M)

FORMAT (/././.'FOR DATA FILE ‘.A6)
R80

U'0

CI0

DRITE (1.123) 80

FORMAT ('INITIAL ESTIMATE =‘.2X.E1a.8)

x-e

X1-0

T-O

VTa-VTI

DO 70 KaI.SO.I

D<x)ss<K)

VT:VT+VTI

V18VT/(UT+VO)

V2-VO/(VT+VO)

R-mt(VI*A+V2¥B0)+1.
UISORTcEtta-a.¢FK*t2*v1tU2EA*BG)
C=(R-C)/(2.6*FK)

D<K>=<E<x>-Aeu.<v2.BO-CII/AE

ABOVE CALCULATES ESTIMATED COMPEEX CONCENTRATION AT EACH
DATA POINT. BELOW CALCULATES SAMPLE CONC. FROM
ESTIMATES AND OBSERVED DATA.

X=X+2.#VI*V2*A*D(K)ttz-UlttztvztAtt2t0(K)~v2tD(K)t#3
XI'XI-VI‘VZtAtD(K)/FK+U2#D(K)¢‘2/FK
Y-Y+vI:*2.v2:*2.A..2+v2.¢2.D<K)..2-2..v2.“.—“2¢A:D(K)
CONTINUE

Ya-Y

BI<X+X1)/Y

UT=-VTI

IF (8—82-80/10960.60) 73.79.76

IF (B-BI+Efl/109@Z.E) 76.79.79

8038

IT-IT+1

WRITE (1.215) IT.RO
FORMAT ('ITEFATION NUMBER '.2X.I2. '3'. 2X. F14.8)
GOTO 71

WRITE (1.80) B

FORMAT ('THE SAMPLE CONCENTRATION IS'.2X.E14.8)
IF (1-IYIJ)21.101.21

VTI-VTI

PRINT READINGS FOR LISTING.

WRITE (1.217)
FORMAT (/./.'ABSORBANCE EATA'.6X.'MEASUFED [CI'.6X.
I 'CALCULATED (CJ')

DO 220 L=1.S0.1

VT=VT+VTI

VI-VT/(VT+VO)

V2IVO/(VT+VO)
R=FK*(V1*AOV2*E)+1.
U'SCRT(Rtt2-A.¥FK*¥2*U1*V2*A*E)
C.(R-U)/(2.0*FK)

WRITE (1.230) S(L).D(L).C

FORNAT (3E17.8)

CONTINUE

CONTINUE

WRITE (1.102)

FORMAT (/././.'TH-TH-TRATS ALL. FOLKS!!!!!1!')
BUD

136

.R FORT
tLRGET.FT/I/G

HOW MANY DATA FILES TO BE ANALYZED?
3
DO YOU DESIRE A LISTING FOR EACH DATA SET?

TYPE 1 FOR YES AND 0 FOR NO.

0

LIST THE NAMES FOR THE FILES TO BE ANALYZED.
LDDFA6

LRFILZ

LRDATA

ARE ALL FILE NAMES CORRECT? 1=YES. 0=NO.
Q

SPECIFY INCORRECT FILE BY NUMBER

2

WHAT IS THE CORRECT NAME?

LRFILE

ARE ALL FILE NAMES CORRECT? 1=YES. 0=NO.

1

FOR DATA FILE LEDFA6
INITIAL ESTIMATE =

ITEFATION NUMBER 13
THE SAMPLE CONCENTRATION IS

0.31976333E-02
0.31719894E-02
0.31721449E-02

FOR DATA FILE LRFILE

INITIAL ESTIMATE = 0.69478369E-01

ITERATION NUMBER 13 0.15675307E-01
ITEFATION NUMBER 2: 0.25946562E-01
ITERATION NUMBER 3' 0.23151435E-01
ITERATION NUMBER 4: 0.23897988E-01
ITERATION NUMBER 5: 0.23697424E-01
ITERATION NUMBER 2 0.23751224E-01
ITERATION NUMBER 72 0.23736787E-01
ITERATION NUMBER 83 0.2374066IE-01

THE SAMPLE CONCENTRATION

FOE

ITERATION
ITERATION
ITIRATION
ITIRATION
ITIRATION
ITIRATION
ITIRATION
ITLRATION

THE SAMPLE CONCENTRATION

DATA FILE LREATA
INITIAL ESTIMATE '

NUMBER
NUMBER
NUMBER
NUMBER
NLMBER
NUMBER
NUMBER
NUMBER

TR~TH-THATS ALL.

IS 0.23739623E-01

0.69478369E-01

I:
2:
3:
a:
5::
6:
7:
8:

0.15675307E-01
0.259A6562E-01
0.2315103SE-01
0.23897988E-01
0.23697424E-01
0.23751224E-01
0.2373(787E-01
0.23740661E-01

IS 0.23739623E~01

FOLKS!11!!!I

"‘WTMT‘TTTTTTTT