.. .IMIIIw . WI IR. 3.....sfi.mu1ur.....s.mu1v..m. Mwmnkfimnmwn. 5.. - .I 4.-.. NM...q .1... I..n...rI.I.....d.o JJAHIMQImINmI .IIHHI,.I....I;, III..u..II IIr . . I4“. .I.. :IIJH . .II. II. IIQIMJ.InIL I . I I " I. 1’1 'I. ‘II II, H ’ I I ‘ 'n‘1"""I ' “II 'I ’I ' 10‘1“,“ III ‘ I ‘1' c” III' 'vz‘I I I I‘IIM I ”III l l I I I'. t %I “nIIJIHquIIIIIII.IIII I I KII. II . {III I I IIIII III.II..IIH I I I . .vuuu cm .I .LI m...IIIhu_JMIIIII1. I ".II. ..II...I..I IIIIIII. II. II .II . I I . MINI... I {JIFHII n JIII“... I It..- I I I . \IIIII . mWIIHIIIIIIIIIw.II HIMIHWLIILIII IIIdUIWWHHII. . Q‘(. ‘I‘IIIInIl‘lII‘III‘I‘I A III! I‘I‘II- "J'll’ I \l I .IIII .II \IIHIInJIIHIIIIINIIIIII. IIII IJIIIWIIIIIIIMIVIIIWI. OIIIKIIIIIluIIr. R "u I II.vII\II>IIIIIIIIIIVIIIh~II..IIIW “I,“ IIMIIIIIIIII rII. “Inn“- I IIIIIIIIIIIIIIII II IIIIIEII 9 I III“- IIIIIIIIIIIIIIIRHI II.I.I.I I IIIII IEI I IIIII . III ”.3; II II ’II I I I; I II III III II I I I I I" . . III .I. I I IlflxfliIIIIuOII .- III I I I 6 . I IIIIIIII IIIIIIIIIII: III II . I Innull \IJIIIIIIII I III III I II II .II IIIIIIII.I III III III III II III I. I I . I I. I .1 n IIII .II I‘i‘IV I II‘ I III. I umIIII-I. IIIIIIL III,“ .II III IIIIILI III. III I 1 . I .II I III II .II I I .III I .IIII IIUWI. I IEI III I .4 IIIIINII .I. )II IIIIIJDfiIIII .I v I.‘ II’II- It Inllll III-II I .I . IIIIII b IR“ tel . I III-III IVI'IUfifiuIIIIWIPIIIIIII IIIIIITIIIII III II. IIIAHIII‘I‘It‘fiII‘HDYII “I quIIIIII. I I III-IIIIIIIIIIIIIIIIIII II III %h1 I I‘ n.1‘l A I.‘| . I I .IIII INCIlJ‘l-IJ'I H IIIIIIID IIIIIIIIIII I ll 'bl HI: .I .-I I I I II . I I ‘ II II I I II \.I I I ' 'I' 'III ' Mu". ,IIIlhI I .3 1"”: "I'.I §' -‘I" \l'. II III I I I4 I H I a I I l I .v I II I II I I II. . I I ,I W I I“ II II II I I I. I II . , ‘WII. II I I II I I I .. V I a I I .II I I I I I II I III I II I I VI. I III III III III II It. IIEII III III lIIIIIIIII I II I II.I II IIIIvIII II III: I I. III III IIIIIIIII IIIIIIIIIIIIIIIIIIIIIHIII I I III“ I II It’ll! II II II I \IIIIII .II. . III III! III Illfi IirlII III\I\II II II II I I... I I \IIIqIIIIII3 IIII wIIIII .IIIIIII I III I I I I IIIIIIII‘I’IIIIIIM‘ III .III III. IIIRUIIIFIIHII‘I II II “III III II III IIII‘II‘IIIII I III: II .II- I I II III. I I I. I I .II. IIIIIIIII I . I “IIII .IIIIIII III I I IIIJJI'I III I. I I “IIIHI IIIIJ. III II. III P1 I III - I .II. It“ II..I I 4. I III I I I II I I I I r I.II. HII A. In I I I III! II I I 2‘9 III III I ‘tII 'I‘II I\|‘ I I I nI.I III.IIIII III. III I 1. I I . I .I I I III-III IvIIIIIrIIII l I “IIII J: I I I I III! II I ‘1 IIIIII III II n I III I I 0 I «I III ‘II‘III I ‘ .lf Ill-\I I‘ PII IIWIII II IW II I-..“ II, I: III IIIIIMIIIIIII I II ”.II.“ IIIIHIIIIIIIMIWIMIIIIIII I II . III II IIIII, . .I "mVJIIIII. - -.I.IIIIMIWIM n...» I - . - .. HI... - II I.- .II- -HIIIIuII. n In II I .. . II- .II I III I IIIIIIIIIVIIIUIIII ILVIIT III. I n III. I I I . . IIIIIIIIIuIIIIflIIIIII" IlahflIIl. . IIAIWIIIINIIIIIIIIIIIIIIIIIII IIIIUIIIIvIIIHIEIIIIIIIIIII It...“ .vaI IIIIII I I I II I . . III INTER“: II IIIIIILNU. THESYS This is to certify that the dissertation entitled APPLICATIONS OF STOPPED-FLOW MIXING IN HETEROPOLYI‘DLYBDATE KINETICS STUDIES AND IN REACTION-RATE DETERMINATIONS presented by Carl C. Kircher has been accepted towards fulfillment of the requirements for Ph . D . degree in mm Mex/M béjor professor Date 4214.844352— MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES m V RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. ft APPLICATIONS OF STOPPED-FLOW MIXING IN HETEROPOLYMOLYBDATE KINETICS STUDIES AND IN REACTION-RATE DETERMINATIONS By Carl C. Kircher, Jr. A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1982 ABSTRACT APPLICATIONS OF STOPPED-FLOW MIXING IN HETEROPOLYMOLYBDATE KINETICS STUDIES AND IN REACTION-RATE DETERMINATIONS By Carl C. Kircher, Jr. The stopped-flow mixing technique was applied to chemical examples of fast-reaction kinetics studies and reaction rate analytical deter- minations. The pH—dependent heteropolymolybdate speciation in aqueous solutions was deduced, and the equilibrium constants for each species were found. With experimental data from an automated stopped-flow spectrophotometer, the kinetics of 12-molybdophosphate and 12-molybdo- silicate formation, decomposition, and conversion to other heteropoly- molybdates were monitored as a function of phosphate, silicate, molyb- date, hydrogen ion, and heteropolymolybdate concentrations and the solution ionic strength at 25.0°C. Rate law equations, rate constants, and chemical mechanisms for these reactions in nitric, perchloric, and sulfuric acid media were deduced with the help of computer simulations from various chemical models and mathematical equations. The implica- tions upon phosphate and silicate determinations are discussed. Reaction-rate determinations for phosphate and silicate in aqueous solutions based on 12-heteropolymolybdate formation were performed. Simultaneous determinations were possible in o.u-o.5 M acid solutions Carl C. Kircher, Jr. because the 12-molybdophosphate reaction reaches equilibrium quickly while the B-lZ-molybdosilicate formation proceeds at its initial rate. The measured reaction rate varied linearly with phosphate and silicate concentrations over 0.001-1.00 mMiand 0.050-1.00 mM ranges, respective- Ly. The analysis time was typically 20 sec. The measured reaction rates were reproducible to within a few percent relative standard deviation. The major sources of error were a synergistic effect of a ‘faster reaction rate for one analyte increasing the reaction rate for the other analyte (less than 8% error in.most cases) and the measurement of the B-lZ-molybdosilicate reaction rate when the sample has a high phosphate concentration and a low silicate concentration. To My Family with Much Love ACKNOWLEDGEMENTS I wish to thank Dr. Crouch for serving as my research advisor, for providing all the help when research was progressing only slowly and when research publications needed typing and revising, and for sharing many of the lighter, enjoyable aspects of chemistry. I also thank Dr. Dye for being my second reader and Dr. Enke and Dr. Chang for serving an my Guidance Committee as well. My deepest appreciation goes to Dr. Horne, Dr. Crouch, and Dr. Enke who helped me to receive a National Science Foundation Graduate Fellowship that aided me financially and expedited my research pro- gress. To secretaries Debbie Jahangardi and Debbie Wuethrich I give my gratitude for typing my manuscripts for publication and other official administrative forms. To Jo Kotarski goes my appreciation for doing my seminar slides and drawings for figures and my respect and appreciation for her artistic talent. I also thank graduate secretaries Lori Garn and Dorothy Byrne for being the best liaison people any graduate student could have with university administration. I also wish to thank the members of Dr. Crouch's research group, past and present: Charlie, Gene, Rob, Nelson, Rytis, Dave, Frank, Cor, King, Jim, Clay, Kim, John, Keith, Pat, Marguerite, Paul, Jim W., and iii Mark. Not only will I remember the wide variety of chemistry research you all undertook but also your friendship and fellowship, the good times we shared, and your ideas and experiences that you have brought from all over the nation and the world. Good luck in your careers, and I hope to hear from you often. My best wishes are extended to the other graduate students, faculty, and staff for the many fun times we had together and for the great parties I'll always remember--hazily at best. I also acknowledge all my "non-chemistry" friends who enriched my experiences here through Owen Graduate Center, Episcopal Ministry of MSU, Academic Governance, mid-Michigan Mensa, Repertory Concert Band, and the Lansing area party scene. Finally, I wish to thank my Mom, Dad, and other relatives whose love nurtured my brothers and me through our formative years and encouraged us to try our best in all we undertake and to be the beautiful people that we potentially can be. Without you, this thesiS'would not have been possible. iv TABLE OF CONTENTS Page LIST OF TABLES . . . . . . . . . . . . . . . . . vii LIST OF FIGURES . . . . . . . . . . . . . . .. . x Chapter I. Overview: The Stopped-Flow Technique in Analytical Chemistry . . . . . . . . . . . . . A. Description of the Technique . . . . . . . . B. General Principles of Stopped-Flow Mixing . . . . C. The Beckwith Automated Stepped-Flow Analyzer . . D. Some Recent Advances in the Use of Stopped-Flow . . 1 OU‘IUl-l-TA Chapter II. Chemistry of MO(VI) and Heteropolymolybdates . . 13 A. The Mblybdenum Blue Reaction . . . . . 13 B. Literature Background on Molybdate and Heteropolymolyb- date Chemistry . . . . . . . . . . . . . 17 C. Literature Background on Pertinent Kinetics Studies . 19 Chapter III. Reaction Models . . . . . . . . an A. Computer Programs to Calculate pH-Dependent MOlybdate Speciation . . . . . . . . . 2h B. Mbdeling for 12-M01ybd0phosphate and 12-Molybdosilicate Formation . . . .. . . . . . . . . . . . 29 Chapter IV. Equilibrium Studies . . . . . . #5 A. Equilibrium Studies for 12-Molybdophosphate Formation Us 1. Spectrophotometric Instrumentation . . . . . ”5 2. Reagents Used and Standardizations . . . . . #6 3. Stoichiometric Studies . . . . . . u7 4. Some Unsuccessful Studies with Methyl Violet Indicator and Strong Acid Potentiometry . . . . 55 5. Continuous Variation Experiments . . . 59 6. Computer Simulation: Molybdophosphate Equilibrium Constants . . . . . . . . . . . . . 68 7. MOdeling and Discussion . . . . . 74 B. Equilibrium Studies of 12-Molybdosilicate Formation . 85 1. Equilibrium Constants for’B-12-Molybdosilicate at pH 1.2 . . . . . . . . . . . . . 85 2. Discussion and Comparison with Molybdophosphate Equilibria . . . . . . . . . . . . . 89 Chapter V. A. D. Chapter VI A. B. C. D. E. F. G. REFERENCES APPENDIX - Selected Program Listings Kinetics Studies . Stopped-Flow Kinetics Studies of 12-Molybdophosphate Formation 1. 2. 3. 1. 2. 3. 1. 2. 3. Introduction and Literature Background Experimental Section Kinetics of B-12-Molybdosilicate Formation in Strong Some Preliminary Studies . Experimental Section Results and Discussion Stopped-Flow Kinetics Studies of B-12-Mblybdosilicate Formation and Decomposition Experimental Conditions and Preliminary Measure- ments Rate Equation and Chemical Mechanism in Acidic Solutions . . Simultaneous Reaction-Rate Determinations for Phosphate and Silicate . Acid Solution . Kinetics of 12-Molybdophosphate Formation in Dilute Acid Solution . Theoretical Considerations . Simultaneous Phosphate and Silicate Determinations Sources of Error . vi 3-12-Molybdosilicate Decomposition in Basic Solution 12-Molybdophosphate Deomposition Kinetics Studies Acid Decomposition Studies Decomposition with Excess Phosphate Decomposition in Basic Solution Heteropolymolybdate Formation at Different Ionic Strengths Page 91 91 9h 97 111 111 113 118 122 122 125 129 130 133 133 136 136 138 138 139 1“? 151 155 Table Table Table Table Table Table Table Table Table Table Table Table Table II. III. IV. VI. VII. VIII. IX. XI. XII. XIII. LIST OF TABLES Chemical Equations for Phospho- and Silicomolyb- denum Blue Reactions . . . . . . . . . pH—Dependent Speciation of Melybdate Complexes in Aqueous Solution . . . . . . . . . . Molybdenum (VI) Equilibria in Strong Acid Solutions . . . . . . . . . . . . Molybdenum (VI) Equilibria in Dilute Acid Solution . . . . . . . . . . . Proposed Rate Equations for 12-Molybdophosphate Formation . . . . . . . . . . . . A Mechanism for 12—Molybdophosphate Formation From HMOZO I O O O l O I O O O O 0 Possible Reaction Schemes for a-12-Molybdosilicate Formation (from Truesdale) . . . . . . . Concentration Data Used in 12-Molybdophosphate Stoichiometry Studies . . . . . . . . Stoichiometric Coefficients fer 12-Molybdophos- phate Formation in Different Ionic Strengths . Equilibria Between 12-Molybdophosphate and Predominant Molybdenum (VI) Complexes in Strong Acid Solution . . . . . . . . . . . Variation of (XP)maxAA with (CP + CM) . . . Mblybdophosphate Equilibrium Constants that Best Simulate 12-Molybdophosphate Experimental Data Comparison Between Experimental Data and Computer- Simulated 12-Molybdophosphate Concentrations . vii Page 15 18 25 28 39 HO “3 50 51 63 65 75 76 Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table XIV. “I XVI. XVII. XVIII. XIX. n. XXI. XXII. XXIII. XXIV. XXV. XXVI. XXVII. XXVIII. Linear Regressions of log K' vs. log [HI] for Different Acidic Media . . . . . . . . . . Models for 12-Molybdophosphate and Dimeric 9-Molyb- dophosphate Formation from Phosphoric Acid and WIdeate O O O O I O O O O O O I 0 Reaction Schemes Consistent with the Double Exponential Equation with 020” < O . . . . Theoretical Variation of the Reaction Rate with Reactant Concentration and the Rate Constants for the Consistent Reaction Schemes in Table .XVI . Linear Equations that Best Fit 12-Molybdophosphate Kinetics Data over Limited Ranges of Experimental conditions 0 O O O O O O O O O O O 0 Chemical Mechanisms for 12-Molybdophosphate Formation Corresponding to the Linear Equations 0f Table MII O O O O O O O O O O O O O 0 Rate Law Equation, Constants, and Chemical Mechanism for 12-Molybdophosphate Formation . Overall Chemical Scheme for Molydephosphate Reactions 0 O O O O O O O O O O 0 Rate Constants fer B-12-Molybdosilicate Formation in Various Acidic Media . . . . . . . . . . Rate Law Constants for 12-Molybdophosphate Conver- sion to Dimeric 9-Molybdophosphate . . . . . . 12-MOlybdophosphate and B-12-Molybdosilicate Formation Rates at Different Ionic Strengths . Ratios of Reaction Half-Lives which Impart Certain Errors to Differential Kinetics Measurements . . . Reaction Rates fer Heteropolymolybdate Formations at Different Phosphate and Silicate Concentrations . Reaction Rates fer Heteropolymolybdate Formations at Different Phosphate and Silicate Concentrations . Reaction Rates for Heteropolymolybdate Formations at Different Phosphate and Silicate Concentrations . viii Page 81 8h 95 99 . 103 105 . 109 112 116 128 . 132 1H0 1H3 1H” 1N5 Page Table XIX. Reaction Rates for Heteropolymolybdate Formation at Different Phosphate and Silicate Concentrations . 1H6 Table XXX. Simultaneous Determinations of Phosphate and Silicate o o o o o o o o o o o o o 0 1"9 ix Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 10. 11. 12. LIST OF FIGURES Block Diagram of the Beckwith Stopped-Flow Spectrophotometer . . . . . . . . Equilibria Among the Molybdophosphates . Distribution of Molybdate Complexes in Dilute Acid Solution . . . . . . . . Distributions of Melybdate Complexes in Strong Acid Solutions (HNO , HClOu) with Variable Total Mblybdate Con entration . . . . Distributions of Mblybdate Complexes in Strong Acid Solutions (pH < 0.9, H280 ) with Variable Total MOlybdate Concentration . . . . . . Distributions of Molybdate Complexes in Strong Acid Solutions (pH < 0.9) with Variable Acid Concentration . . . . . . . . . Variation of the Phosphoric Acid Coefficient (X) with Ionic Strength . . . . . . . Variation of the Molybdate Coefficient (Y) with Ionic Strength . . . . . . . Variation of the Hydrogen Ion Coefficient (2) with Ionic Strength . . . . . Visible Absorption Spectra of the Protonated and Deprotonated Methyl Violet Indicator Species . . . . . . . . . Variation of the Methyl Violet Color Transition pH with Solution Ionic Strength . . . . pH Glass Electrode Calibration Curve for Perchloric Acid Solutions at 25 0C Page 20 31 33 35 37 52 53 5A 57 58 60 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 13. m. 15. 16. 17. 18. 19. 20. 21. 22. 23. 2“. 25. UV-VIS Absorption Spectra of Three Phospho- molybdenum Solutions . . . . . . Comparisons Between Experimental Data and Computer Simulation with a 12-Molybdophosphate and Dimeric 9-Molybdophosphate Chemical Model Continuous Variations Plot for Molybdophosphate C + C = 0.01 M, [H ]= 0.50 M . . . P M Continuous Variations Plot for Molybdophos- phates, CP + CM = 0.01 M, [H ]= 0.20 M Continuous Variations Plot for Molybdophos- phates, CP + CM = 0.05 M, [H+]= 0.20 M Continuous Variations Plot for 3-12-Molybdo- silicate, CS 4- CM = 0.01 M, pH = 1.2 Semilogarithmic Plots of 12-Molybdophosphate Kinetics Data to Fit Exponential Equations Kinetics of 12-Molybdophosphate Formation as a Function of Initial Molybdate Concentration Kinetics of 12-Molybdophosphate Formation as a Function of Hydrogen Ion Concentration Semilogarithmic Plot of B-12-Molybdosilicate Kinetics Data . . . . . . . . . Plots of B-12-Molybdosilicate Formation Rate vs. Molybdenum (VI) Species Concentration in various Acidic Solutions . . Kinetics Plot of Heteropolymolybdate Formation ([H ]= O. u M) . . . . . Kinetics Plot of Heteropolymolybdate Formation (pH 1.8).. . . . . . . . xi Page 67 72 77 78 79 87 93 100 101 11" 119 131 1H2 Chapter I. Overview: The Stopped-Flow Technique in Analytical Chemistry A. Description of the Technique Ever since the stopped-flow mixing technique was developed in 19uol'3, researchers have discovered its enormous potential as a tool for studying many types of chemical reactions. The general description of stopped-flow mixing is simple. Two or more chemical reagents are mechanically sent into motion from their respective reservoirs, mixed so that the chemical reaction among the reactants may begin, and directed to an observation cell where the reaction progress may be monitored. A brief time period is required for reagent mixing and delivery to the observation cell; afterwards, this solution flow is mechanically stopped. The chemistry of the mixed solution can be studied over a predetermined, variable time period within the observa- tion cell by using common, convenient measurement methods such as absorption and fluorescence spectroscopy and conductimetry. With suit- able designs for the reagent delivery system, mixer, and observation cell, only small amounts of reagents are required for experiments, and measurements can be made within milliseconds after the mixing of the reagents. In addition, since measurements are made after the solution flow is stopped, the same mixed solution within the observation cell can be studied over a long length of time, minutes if so desired. Thus, the stopped-flow technique provides the capability of studying a wide variety of chemical processes occurring over time scales ranging from milliseconds to several minutes. The stopped-flow technique also has certain fundamental limita- tions. The detection system must have a fast response time in order to measure chemical changes that occur within millisecond time intervals. Also, because of the finite time of mixing required, chemical phenomena occurring within microseconds would not be discernable by stopped-flow methods, so small perturbation techniques (relaxation methods) such as temperature-Jump or electrical field variations would be required. However, in assuming the stopped-flow technique is applicable for a given chemical system to be studied, the stopped-flow results would provide information about the entire reaction scheme, from the initial stages to the attainment of equilibrium. Only the chemical reaction steps close to the equilibrium state can be studied with relaxation methods since the methods require only slight fluctuations from equi- librium. Large perturbation methods could enable the study of more chemical reaction steps; however, the species concentrations may not be calculable from a given perturbation or else the chemical system could relax to a different equilibrium state. In addition to studying the kinetics and elucidating the mecha- nisms of chemical reactions useful in analytical chemistry, stopped- flow mixing can also be employed in reaction-rate analytical methods. These methods are generally employed in chemical systems where prior studies of kinetics have verified first-order rate conditions with respect to the analyte involved or that pseudo first-order kinetics apply over a given set of experimental conditions. Consequently, the chemical reaction rate would yield the analytical information because this rate is linearly proportional to the analyte concentration over the course of the reaction. With a valid initial rate approximation, the initial, analytical species concentration is related to the initial reaction rate. Thus, no waiting time is required to reach equilibrium, so the analysis time is much shorter than that of a procedure based on attainment of equilibrium. Since the initial rate reflects only the initial steps of the reaction mechanism, any interferences or decreases in sensitivity caused by unforeseeable side reactions in subsequent mechanistic steps are reduced. If there are several components in the mixture that react similarly as the species of interest does, then, to the extent that these several components react at different initial rates, a multicomponent analysis is possible and the rates correspond- ing to the interferences may be subtracted from the total rate to give the initial rate corresponding to the analyte's reaction. Furthermore, because relative measurements of transmittance and changes in absor- bance with time are involved, constant interferences such as turbidity and background absorbance do not contribute to systematic experimental errors as would happen when equilibrium methods are employed. However, since the final equilibrium state is not reached during the analysis, measurements of the time-varying physical parameter inherently possess a lower signal-to-noise ratio and thus lower sensitivity and precision. Experimental conditions such as pH, ionic strength, and temperature must be carefully controlled over the measurement time interval so that the measured initial rates are reproducible. Stopped-flow instrumenta- tion may require a complicated design and fairly expensive accessories in order that mixing times are sufficiently small, total reagent volumes are small, cavitation can be eliminated, leakage of solutions does not occur, and the mixer and other components can withstand high solution flow rates and high pressure stresses occurring during the stoppage of flow. Since the chemical reaction rates may show an initial lag time and a subsequent decay as equilibrium is approached, the detection and data acquisition systems must be equipped to perform multipoint analy- ses and regression. B. General Principles of Stopped-Flow Mixing In reviewing the general principles of the stopped-flow technique, the ideal experiment would involve the instantaneous mixing of the chemical reagents and immediate transportation to the observation cell where the solution flow is abruptly stopped and the chemical reaction is monitored. The so-called "dead time", which is characteristic of a particular stopped-flow apparatus, reflects the finite time required between the initial contact between solutions in the mixer and the termination of solution flow in the observation cell. The dead time parameter is important in determining which chemical reactions can be studied. If this dead time is not significantly less than the half-life (the time required to convert half of the reactants into products) of a particular reaction, then the initial reaction rate measured will be significantly slower than the actual initial rate. In order to study these fast reactions, the reaction time must be evaluated as a function of position within the observation cell. Another important consideration for stopped-flow' involves the promotion of turbulent flow through the mixer and observation cell rather than laminar flow. Turbulent flow gives a more uniform flow velocity profile within a cross section of the flow channel as compared to the parabolic flow velocity profile resulting from laminar flow. This consideration facilitates a better reaction time resolution within the observation cell as well as more homogeneity within the mixed solution. In fact, the reagent mixing is accomplished better under turbulent conditions than with streamlined laminar flow. In addition, the flow velocity is less affected by the driving pressure required to initiate solution flow and by the viscosity of the solutions themselves. A more detailed presentation of these general principles may be found in an earlier dissertation.“ C. The Beckwith Automated Stopped-Flow Analyzer The stopped-flow spectrophotometer used for most of this research was initially designed by Beckwith.5 The block diagram of this instru- ment and the electronic computer interfaces is shown in Figure 1. The basic features of this stopped-flow system include a pneumatic syringe drive system to promote rapid mixing. The vertical flow system mini- mizes interferences from bubble formation. The valves that direct the solution flow and drain the waste solutions are actuated with air pistons mechanically controlled by solenoids. The stopped-flow operat- ing cycle and valve sequencing may be performed manually with push m>am> mmmoaoe mama: mcaaaonocoo Loccfiaho caumasosq ooumsuom.caocoaom gauge on onom owcasxm wcaqaoum noemoanmcasqm o>am> ommoaos some: maxim Haoo souam>eomno penance mcaxaz mnwo>eomos anommon o» unom amumhm mean no mLHo>somoe sud: nomnaamn ucommon msauoocsoo xooonoun hazam cannon unmeasam o>aen acowmom ‘m tn (3 c: a: a: c: a: +4 *3 a: o>am> >no>aaoe msaaaosucoo Loecaaho oaumanocq eoamsuom egosoaom soonfiaho o>ane oaumasocq ooumsuom vaccoaom nanosecosaosuooom soamivoaaoum spasxoom on» no amsmmaa xooam .F mesmHm Fiber Optic :: :E Quartz R°d Light Source .Chromotor Multiplier D . . F Mono- , ::\ —— Photo— . ‘G — Tube I/V Converter ADC with S 8 H Opto- Interruptor Trigger_: :1 I I II 1.] DAC MlNlCOMPUTER J J A Scope Printer . Terminal buttons or automatically with pulses originating from a PDP 8/e mini- computer (Digital Equipment Corporation). The measured light intensity is converted to electrical current with a photomultiplier tube (RCA/IP28A) and subsequently converted to voltage with a Keithley Mbdel U27 current amplifier. This voltage can be recorded on an oscilloscope or digitized automatically with a Datel DAS-16-M12B analog-to-digital converter and stored on a mass storage device (i.e. floppy disk). A trigger signal from a photointerruptor module beneath the stop syringe initiates the data acquisition sequence from the minicomputer. Up to four stopped-flow data pushes may be performed each time the drive syringes are filled. Although the basic block diagram of the Beckwith instrument has remained the same, several members of Dr. Crouch's research group have made various improvements and modifications. A water circulation system to thermostat the stopped-flow system, a new mixer, and an optoelectronic trigger module (G.E. H13B1) now augment the original stopped-flow system.ll The observation cell, quartz fiber Optic light guide, and quartz rod that interface the stopped-flow to.a.GCA/McPherson spectrophotometer system have been housed in a stainless steel and black Delrin assembly. The computer programs controlling the stopped-flow operation, data acquisition, and data analysis have been rewritten in higher level languages FORTRAN and SABR6’7 (Call and Balciunas). Some of the electrical computer interface components have been replaced to improve the integrity of the monostable pulses and other dialogue 7 between the stopped-flow instrument and minicomputer. The instrument has also been modified to perform conductance measurements within the observation cell.8 7 the operator may With the current stopped-flow computer program, set several options prior to performing stopped-flow pushes. The analog data-taking rate, for example, may vary from one per millisecond to one per second. The number of analog points averaged for each data point may vary from one to 1000. The time delay between the clock start and the stopped-flow trigger is variable; a 7 ms delay time works well for most stopped-flow measurements. Up to 100 data points may be obtained per stopped-flow push. A maximum of four data pushes is possible for each drive syringe filling, and one or two syringe fillings may be performed for each set of solutions in a stopped-flow experiment. The computer program contains automatic routines to rinse the stopped-flow channels with appropriate solutions, measure 100$ transmittance with blank solutions, and measure the dark current.u’6’7 For each stopped- flow experiment, the computer measures the solution transmittance at the specified data-taking rate and calculates the absorbance and its standard deviation at each time point. These data may be plotted as a function of time on an oscilloscope, stored on floppy disk under a specified filename, or listed on a line printer or terminal.7 Some important considerations for the operator include the volume of solu- tion required to perform the entire stopped-flow experiment for the - specified number of syringe fillings.Also, a slower data-taking rate and/or a larger number of analog points averaged per data point result in a larger time interval between data points and a longer time span covered in a stopped-flow run. Thus, over the last ten years, Dr. Stan Crouch's research group has made considerable improvements in stopped-flow equipment and interfaces 10 for computer automation, and the equipment currently available (5 1 1 ms dead time on the Beckwith system“) should be adequate for studying many types of chemical reactions. Consequently, this reseach investigates two applications of the stopped-flow technique in Analytical Chemistry. One such application is the kinetics study of two chemical reactions of analytical importance: the complexation of phosphate and silicate with molybdates in acidic solution and the acidic and basic decomposition of these heteropolymolybdate complexes. The other application develops reaction-rate determinations of phosphate and silicate in mixtures based on the rate of formation of their heteropolymolybdate complexes. D. Some Recent Advances in the Use of Stopped-Flow The recent literature is filled with new applications of stopped- flow mixing and improvements in existing systems. Most of these advances involve investigations of new and current designs for the essential stopped-flow components and their impact on instrumental performance, new detection methods to monitor chemical reactions within the observation cell, and new chemical reaction systems that can now be studied with stopped-flow techniques. To extend the analytical utility of the stopped-flow technique, several recent improvements have been reported in the mechanical design and evaluation of stopped-flow components. A complete stopped-flow system was designed and constructed that could withstand hydrostatic pressures up to 3 kbar yet would be able to follow chemical reactions with millisecond half-lives, thus enabling high pressure kinetics stud- ies of chemical reactions in solution.9 As an accessory to stopped-flow 11 instruments, a microprocessor-controlled reagent dilution system con- taining three-way proportioning valves and a homogenizing mixer on-line with the stopped-flow sample delivery loops was developed.10 For indi- vidual stopped-flow components, a chemically inert tangential Jet mixer was recently designed.11 Some considerations regarding the use of stainless steel drive syringes have been discussed.12 As an example of recent work in investigating stopped-flow instrument performance, the dead time for a stopped-flow that employs fluorescence detection in the observation cell has been reported.13 The increases in solution temperature during the mixing process were compared for several con- figurations of the drive syringes, mixer, and observation cell, and subsequent considerations have been presented about controlling these temperature variations.” A chemical reaction employing disulfide exchange was presented as another way to evaluate stopped-flow spectro- photometer performance15 16 in addition to the iron-thiocyanate reaction. The stopped-flow technique has been extended to chemical reactions that occur in media other than homogeneous solutions. For example, the gas phase reaction of ozone with organic sulfides was studied with stopped-flow instrumentation coupled to a beam sampling mass spectro- 17 meter. Also of note is a recent stopped-flow study of chemical reactions that occur in micelles.18 A significant portion of a recent advances encompasses different methods to monitor chemical processes in the observation cell. Calori- metry has been used to study reaction rates by stopped-flow methods. The thermistor used had a 15 ms response time and could detect temperature 12 19 By using Fourier methods to reduce the changes as small as 1 m°C. time scale of NMR measurements, a stopped-flow instrument was designed to monitor the kinetics and dynamics of a chemical reaction by line shape analysis of’NMR signals.20 Similarly, a stopped-flow EPR instru- ment has been used to monitor free radical reaction mechanisms with a time resolution of 4 ms.21 Furthermore, the use of circular dichroism as a stepped-flow detection method has enabled studies of millisecond conformational changes in biological macromolecules.22 Thus, recent advances in stopped-flow instrumentation and detection methods have greatly increased the number of chemical reactions than can be studied with stopped-flow techniques and have defined more clearly the limits within which useful information may be obtained. Chapter II. Equilibrium and Kinetics Studies of Heteropolymolybdates A. The Molybdenum Blue Reaction Chemical analyses for phosphate and silicate provide important information about the sample that contains these analytes. Two ofthe most important applications of phosphate determinations are performed on clinical and agricultural samples. Phosphate levels in biological fluids reveal metabolic information on how chemical energy is stored and utilized in the formation of teeth and bone (calcium phosphate minerals) and in the conversions involving adenosine triphosphate (ATP -9ADP 4- P1). Phosphate levels in soils and fertilizers may determine which crops can grow best in a given growing season. Determination of silicate in soils and rock samples can elucidate the geological history of a given land region or ocean bed, and its detection on mobile machine parts and air filters indicates contamination from dust particles. The use of Mo(VI) in acidic solutions has been employed extensively in analytical determinations of both phosphate and silicate. Many experimental procedures for these assays are based on the reaction of phosphate or silicate with acidified Mo(VI) to form the yellow 12- molybdophosphate anion (12-MPA) or the 12-molybdosilicate anion (12- MSA), respectively. The formations of these heteropolymolybdates are 13 14 23,2” most often monitored spectrophotometrically. Many other proce- dures use the so called "molybdenum blue" reaction in which the 12-MPA and 12-MSA complexes are reduced to intensely blue-colored species.25 In addition to equilibrium methods, the applicability of reaction-rate methods based upon the rate of formation of 12-MPA,26 12-MSA,6 and reduced 12-MPA blue27 has been demonstrated. As required in developing each experimental procedure, the phosphate or silicate concentration ranges in which the stable, colored complexes formed will obey Beer's Law and under which stoichiometric amounts of the complexes will be formed were evaluated for the equilibrium methods. For reaction-rate methods, those concentration ranges in which Beer's Law is followed and under which pseudo first-order kinetics with respect to phosphate or silicate prevail were evaluated. Despite the sensitivity and applicability of the heteropolymolyb- date system in analytical chemistry, many details of the chemical reactions involved have not yet been resolved beyond all doubt. Table I reflects what is known with certainty about these reactions. The specific Mo(VI) species which combine with phosphate or silicate were unknown prior to this thesis; however, 12-MPA and 12-MSA were known to form in amounts proportional to the total. phosphate: and silicate concentrations present in solution if the Mo(VI) present is in large, 27.28 constant excess. The uncertainties in the stoichiometric coeffi- cients Y Y 21, and Z 1’ 2' 2 Mo(VI) species. With different pH and total Mo(VI) conditions, reflect the uncertain nature of the complexing different isomers of 12-MPA and 12-MSA may form, each isomer having 29,30 different spectrophotometric properties. In addition, the compo- sitions of the reduced molybdenum blue species vary with different 15 Table I Chemical Equations for Phospho- and Silicomolybdenum Blue Reactions PI-DSPHATE DETERMINATION S pH 0.5 3- + H3P0u + YI MO(VI) >PM°120HO + 21H 3_ Sn2+ _ PMo(VI)120u0 >'PMo(V)4M'o(VI)80uo yellow 12-MPA Phosphomolybdenum Blue SILICATE DETERMINATIONS PH 1'5 li- + Si(OH)u + Y2 MO(VI) )'(0/B)SiMO120uO + 22H u- Sn2+ SiMo(V) Mo(VI) O H- SiMo(VI) O >r u 8 ”0 12 ’40 M- Ascorbic 4- SiMo(VI)120uo Acid 981M0(V)3M0(VI)9OHO yellow 12-MSA Silicomolybdenum Blue 16 experimental conditions. The reduction of 12-MPA and 12-MSA has been shown to proceed, at least initially, in two-electron steps.31’32 Many reagents can reduce 12-MPA and 12-MSA since these heteropoly compounds are good oxidizing agents.33 With the addition of two and four elec- trons, the molybdenum blue complexes would have Mo(VI) : Mo(V) ratios of 5:1 (i.e. PMo(V)2Mo(VI) 5‘) and 2:1 (PMo(V)uMo(VI)80uO7‘), respec- 3H- 1o°uo tively; the existence of such species and others have been reported. 36 The composition of the blue complex and its absorption spectrum depend in part upon which reducing agent is used and how long the reduction takes place. The blue color is most intense when ascorbic 37 acid is the reducing agent. Some reducing agents such as ascorbic acid, thiourea, and HI exhibit autocatalysis because the reaction products increase the reaction rate for unreacted species. By contrast, Sn(II), Fe(II), and hydroquinone are non-autocatalytic reducing agents. 37 According to Kriss and coworkers autocatalytic reducing agents form multiple oxidation products whereas non-autocatalytic reagents produce only one oxidation product. Strong reducing agents such as Zn-HCl decompose 12-MPA and 12-MSA.33 Reduction of the two different 12-MSA 33 isomers with Sn(II) results in different reduction products. The use of metallic reducing agents such as Sn(II), Ti(III), and Cr(II) can further complicate the molybdenum blue composition since one or two of the Mo atoms may be substituted with these other transition metal 38 atoms. The reversibility of certain redox reactions, though reported 39 by some workers, is still in question. 17 B. Literature Background on Mo(VI) and HeterOpolymolybdate Chemistry The speciation of Mo(VI) compounds as a function of pH is quite complicated. In aqueous solution, Mo(VI) forms either multiple bonds with oxygen as Mo == 0 or single bonds in the bridging systems Mo—O—Mo and Mo <3>Mo. At pH 7, the colorless, tetrahedral molybdate anion (MoOuz') is the predominant Mo(VI) Species. As the acid concentration increases, molybdenum becomes octahedrally coordinated and polymerizes rapidly to give various polyacid anions.“0 Predominant molybdenum species in the pH range of 1-6 include Mo(VI) monomers (HMoOu-, HZMOOu), 6- - u- u- heptamers (M20702n , HMo702u5 , H2Mo702u ), and octamers (M08026 ); their distribution as a function of pH has been plotted.“ Other clusters have been identified as having 2,3,“,6,10,12,16, and 2H Mo(VI) atoms.“ Mo(VI) precipitates as MoO at an isoelectric point that 3 occurs at pH 0.9. Under more acidic conditions the precipitate redis- solves as Mo(OH)6 (commonly written as M803) and accepts a proton to become the cationic monomer Mo(OH)5(I-120)+ (or HMoO +) These monomers 3 dimerize to form the cationic species M020(OH)9(H20)+ (or HM°206+)’ 2+ 2+ 3+ 3+ ”2 M020(OH)8(H20)2 (or H2M°206 ), and MoZO(OH)7(H20)3 (or H3M0206 ). These two monomers and three dimers are the predominant Mo(VI) species in strong acid solution (pH< 0.9).“ A sumary of the Mo(VI) speciation as a function of pH is given in Table II. Strickland first reported that two isomeric products are possible in 12-MSA formation.29 Other researchers”3 have argued that pH and total Mo(VI) concentration determine which of the o and B isomers form, rather than Strickland's acid-to-molybdenum ratio. The isomer a—12-MSA 18 Table II pH-Dependent Speciation of Molybdate Complexes in Aqueous Solution Mo(VI) pH >7 tetrahedral MoOua- — qr- '— — 2- o o °§§=H 1’0" H°\n!.r’o‘\\Jl/’OH pH 6-7 octahedral M0 ° / o/ \ HO/ | \ow HO I \ow (unsymmetrical) H2 ' I o” I OH OH OH i— - _ .— pH <6 polymeric Mo-Complexes 6- [ACIDItot , M0702" . when [Mo VI ] __1.1h u- ACID tot M08026 when W > 1.5 pH = 0.9 isoelectric point; MoO3 ppt. pH 7 MoOuz' + 8 HB+ depolymerization were studied at 25°C and 3.0 M ionic strength; the observed rate equation had the following form“8 6- The overall reaction kinetics were second-order with strong bases such as OH', 420-, and NHZCHZCOZ' and first-order with weaker bases. The dimerization kinetics of 2 HM003+—-9H2M02062+ were studied at 25°C 22 and 3.0 M ionic strength with temperature-Jump relaxation techniques.”9 5 1 A forward rate constant of (1.71 I 0.10) x 10 M. sec.1 and a reverse 3 rate constant of (3.20 :_0.02) x 10 sec-1 were observed. The kinetics of heteropolymolybdate formation and decomposition have been investigated by several researchers. Stopped-flow experi- ments at 25°C and 3.0 M ionic strength were used to study the OH- n-6 50 n-6 hydrolysis of the HnM°5P2023 and HnM°7024 species. The second- 1 order reaction rate constants observed were 1637 I 0.9 M' sec-1 and 1sec-1, respectively. Stopped-flow kinetics studies of 13000 :_1300 M- 12-MPA formation in nitric acid solution were first performed by Javier and coworkers.51 Since then, Beckwith has repeated these experiments in perchloric and sulfuric acids,5 and Beckwith, Scheeline and Crouch52 published a rate law and chemical mechanism for 12-MPA formation. Notzu and Call6 did subsequent rate measurements under the assumptions that some (2.5) analytical acid equivalents are consumed in the formation of the protonated Mo(VI) cations and that ionic strength must be held constant (at 2.0 M). The consequences of Beckwith not controlling the ionic strength affected only the magnitudes of the rate constants but not the reaction mechanism.6 Because 12-MPA formation proceeds more slowly than the dimerizations and protonations of Mo(VI), steady state is established rapidly among all the prominent Mo(VI) complexes prior to their coordination with phosphate. Gall also performed some prelimi- nary studies on 8-12-MSA formation kinetics.6 Despite all these past efforts, the rate equations were expressed in terms of total analytical Mo(VI) concentration without regard to Mo(VI) speciation. The first heteropolymolybdate formation rate equation expressed in terms of 23 individual Mo(VI) species concentrations was published recently.“1 In this paper, thecz- and B-iZ-MSA formation kinetics were studied in HCl media at 1.0 M ionic strength; the initial concentrations of prominent Mo(VI) complexes were calculated by using Aveston's equilibrium con- stant data53 and the HALTAFALL computer program.5n The repeat of 12-MSA and 12-MPA formation kinetics studies which consider individual Mo(VI) Species is important for two reasons. First, this research considers and appreciates the consequences of the rela- tive change in Mo(VI) speciation as the pH varies. Second, experimental rate equations in terms of individual species concentrations provide a better understanding of the actual chemistry of Mo(VI) polymerizations around phosphate or silicate and thus established a firmer foundation for the analytical methods used. ._E..A-n Chapter III. Reaction Models A. Computer Programs to Calculate pH-Dependent Mo(VI) Speciation In order to determine empirically the kinetic rate laws in terms of individual Mo(VI) complexes, equilibrium constant data must be used to calculate individual complex concentrations from given analytical acid and total Mo(VI) concentrations. In strong acid solutions (pH‘<0.9), the equilibrium constant data reported by Cruywagen, 22. al. from U.V. spectrophotometric measurementsuz were used. The acid and Mo(VI) mass balance equations were formulated under the assumption that only the five strong acid, monomeric and dimeric Mo(VI) cations)42 are present in significant concentrations. These equations are shown in Table III. Two FORTRAN II programs were written for the PDP-8/e minicomputer to solve the two simultaneous equations and calculate the concentrations of individual Mo(VI) complexes. The program MDLYB.FT was written for HNO and HClO 3 N programs first assume an initial estimate of [H+] = CH-3CM, corre- solutions while MOSOu.FT was used for H230“ media. The spending to three protons required to convert Moouz' to HMoO3+. Succes- sively, the lH+l is used to calculate [HMOO +1, then the [HMoO3+] is 3 used to calculate a new [H+], and so forth until the old and new values of [HMo03+l agree within a given tolerance. Usually, convergence of 2M 25 Table III Molybdenum (VI) Equilibria in Strong Acid Solutions + + 2+ 3+ , [HM003], [M003], [HMo206], [HZM’OZO6 l, [H3Mio206 ] 6 Unknowns: [H+] 6 Equations: Cruywagen's equilibrium constant data at 25°C and I = 3.0 ”(NZ) K [Hueo*] 1 + 3 = 11.36 HMoo+ -——-——=-Moo + H K = ——-———————- 3 l 3 1 [H*][Me03] 2+ K in Me 0 . i + (13 2+ 2 2 6 3 2+ K [H Mo 0 1 + + g, 2+ _ 2 2 6 _ [H llHMo o l 2 6 K [H Rd 03*] _ H2M020§+ + H+ 3 : H3M0202+ K3 = +3 2 6 2 - 0.2u‘: 0.07 Mass Balance equations _ + + 2+ 3 3+ CM - [M003] + [Hueo3] + 2[HM8206] + 2132Me206 ] + 2133Mo206 ] _ + + + 2+ 3+ CH - [H ] + 2[Mo03] + 3lHM003] + 5lHMo206] + 6lH2M0206 ] + 7[H3M020 l 6 Results: cM = [HMto;](1 +--——J—-—) + 21HM¢0§12( Kd + Kd + KdK3[H*]) K1[H*] K2[H*] c = [H+] + [HMoO+l( 2 + 3) + K [Hueo*]( 5 + 6 + 7K [H*]) H 3 K1lH+l d 3 K2[H*] 3 Thus, two quadratic equations with two unknowns must be solved simul- taneously. The positive roots from the quadratic formulas are used. 26 Table III (cont'd) Same situations as for the HNO3 and HClOu solutions except for one more unknown, [HSOE], and one more equilibrium, HSOE=F223PH+ + 80i- [H*1[sofi‘1 [HS _] C,[H*1 0 = ----- Sin°e K1 = [HSOu-] ' u Ka + [H*] Acid mass balance equation: 2 cA = [H*] + [H303] + 2lMoO3] + 3iHMb0§l + sinuezog] + 6[H2M020§+] + 3+ Result: 2K -+ 2[H*] + + _ +3 +2 4' a 2CH[H 1(Ka + [H l) - [H l + KalH ] + [HM0031( -‘ + 3Kalfl+l + K1 SKa + 5lH+l K2 2( + 6lH+](Ka + [H+l) + 7K3[H+]2(Ka + 3[H*12) . xdtnneogi [H+l)) + cH[H*] Thus, a quadratic equation and a cubic equation must be solved simulta- neously. The cubic equation may be solved analytically be setting + 2 A = 1 + 7K3Kd[HM003] + + 2 B = Ka - CA + 3[HMoO3] + Kd[HMoO3] (6 + 7K3Ka) 2[Huoo+ 3 + + 2 5 C = T— - ZCAKa + 3Ka[HMC)03] + Kd[HMOO3] (R2— 4- 6K8.) + + 2 0 .. 253.13%] . iidf‘al’yfqi K1 K2 and solving 26a Table III (cont'd) ,30-P2 3 C D 2P-P 2R P=%’Q=A’R=-A"a' 3 ' ' 90+? &b- 27 . 2 3 Because the discriminant E- + 37- < O for our chemical system, the cubic equation has three real roots. The solution which has physical meaning - - - X is X: cos 1(-gv-E%) and [H+] = 2 --3- cos? 27 consecutive approximations is achieved after three or four iterations. Both programs chain to MOOUT.FT, which displays calculated concentra- tions on the computer terminal. Based on the value of CA, which represents the H80; acid dissociation constant, MOOUT chains back to either MOLYB or M0803. A specified total Mo(VI) concentration (CM) of zero returns computer control to the 08/8 operating mode. Listings of MOLYB.FT, MOSOh.FT, and MOOUT.FT are found in the Appendix. For the dilute acid solutions, the'equilibrium constant equations and data from Aveston, 32, al.53 and mass balance equations are given in Table IV. The two resulting simultaneous equations can not be solved analytically, so a successive approximations algorithm must be used. The general-purpose HALTAFALL program mentioned earlier uses a succes- sive approximations routine and the secant method to calculate equilibe rium concentrations of complexes Aan from their overall (not stepwise) formation constants from n A and mlB.5‘4 This program, written in ALGOL- 60 and run on a high-speed CDC Cyber 760 computer system, calculates the analytical acid concentration and Mo(VI) apecies concentrations: (1) when CMo = 0.05 M and pH varies from 1.2 to 6.8, (2) when pH = 1.2 (where B-12-MSA forms exclusively) and CMo varies from 0.01-0.20 M, and (3) when pH = 3.6 (where c-12-MSA forms exclusively) and CMo varies from 0.01-0.20 M. The ALGOL (Algorithmic Language, a precursor of PASCAL) programming language works well for numerical computations and simple data struc- tures; the formal syntactic definitions express recursive procedures conveniently and give the operator the benefits of locally defined symbols. 28 Table IV Molybdenum (VI) Equilibria in Dilute Acid Solution . ‘ +. 2- - 6- - “- Unknowns. [H 1, [Moon ],[HMOOu].lH2MoOu]. [M0702u].[HMo702u].lH2Mo702u]. u- [“°8°261 Equations: Avestion's equilibrium constant data at 25°C and I = 1.0 M53 General form p H” + q hoof: HPWOOMS-ZQ -2 - [Hp(MoOu): q] eq - [3*]p[Moofi‘1q log K log K P. .9 .____£3 2. £1 _____29 1 1 3.53 (1 0.07) 9 7 57.16 (1 0.07) 2 1 7.26 10 7 60.8u a 7 52.80 12 8 71.56 Mass Balance equations CM = [noon 1 + [HMOOu] + [quoou] + 7[Mo702ul + 7laue7ozul + 7lH2Mo702u] 1.1.. + 8[M08026] OH = [H 1 + [HMoOu] + 2[H2MoOu] + 8lMo702ul + 9lHMo702ul + 10[H2Mo702u] + 12[MOBO:E] For the two simultaneous equations, one is an 8th degree polynomial in [MoOfi'] and the other is 12th degree in [8*]. 29 In the execution of the program, the concentrations converged to within the'given tolerance after about eight to twelve iterations through the successive approximations routine. In running the HALTAFALL program, an undefined array value for TOL(IA) was encountered when IA = 3 at the third line after label "SLINGOR:". The substitution of 'IF' IA=’NA 'THEN' 'BEGIN' IA: = IA-1; 'GOTO' SLINGOR 'END’; for the second line after label "NYA1:" corrected this problem because the program then compared the defined value in TOL(2) rather than the undefined TOL(3). In addition, when the calculations in subroutine TOTBER required a value for TOLY(1), the statement "IA: = IA-1" had to be inserted between "KARL[IA]: : 1" and "'END'" in the fourth line of "SLINGOR:" so that a value for TOLY(1) could be specified. Thus, from using HALTAFALL for the dilute acid solutions and MOLYBJ'I' and 110301131 for the strong acid solutions, the initial reactant concentrations were calculated. Rate equations were experi- mentally determined from stopped-flow measurements of the maximum reac- tion rate as a function of these initial concentrations. Some graphs of the distribution of Mo(VI) species with C CH’ CA’ and pH are plotted M! in Figures 3-6. B. Modeling for 12-Molybdophosphate and 12-Molybdosilicate Formation After the initial reaction rates were measured with various analy- tical acid, Mo(VI), and phosphate concentrations, these data were fit by a variety of rate laws with KINFITSS, which is a general purpose curve- fitting program executed on the CDC Cyber 760 computeru With input of a rate law equation, concentration data and their variances, rate data and 30 Figure 3. Distribution of Molybdate Complexes in Dilute Acid Solution With variable CM and constant pH (I = 1.0) 12:; 0.105 0.10 0.15 0.20 20- 15" (mM) '0 .. 0.05 0J0 0.15 0.20 pH 3.6 M671 Cm(M) M0, M0 M0l cmaw 31 32 Figure U. Distributions of Molybdate Complexes in Strong Acid Solu- tions (HNO H010”) with Variable Total Molybdate Con- centration 3 9 50 40 [ ] (mM) 30 20 10 4.. CH . 0.495 M [H2M02062] [HMozog ] [HMoog1 [Mo(OH)6] - : f :[H SMOZOBJ (M 0.02 0.04 0.06 0.08 0.10 Cm ) - [HZMozogfi cflaoeeom " ‘ [101020;] [HMoogj '/ I [H.Moeoé’i flow-1) 1 - - t 6 0.05 0.10 0.15 0.20 CmW') 33 3“ Figure 5. Distributions of Molybdate Complexes in Strong Acid Solu- tions (pH < 0.9, H250“) with Variable Total Molybdate Con- centration 20 15 1 1 (mM) '0 lu- Con- 50 40 [ ] (mM) 30 20 10 2+ [H2M0206 ] CA: 0.495 M [HMoOQ] [HMOzog] [M0(OH)6] ill-l3 M020: +1 A A l L 0.10 L C (M) m 0.08 002' 0.041 0.06 4- CA =0.990 M [1421010203 ] [HM6203 1 [H Moog] 1 '/ ' [H.Moeoé’i 0.05 0.10 0. 15 0.20 35 36 Figure 6. Distributions of Melybdate Complexes in Strong Acid Solu- tions (pH < 0.9) with Variable Acid Concentration ,5; H1103 Solutions 2+ - - [0211110206] [l 10- I A - - - 11134053.] (mM) q. 5 [H3Mo2021 - [HMOZOE] 1 J t 3 n c *0 ?[WOH)6] 0.5 1.0 1.5 2.0 CW) l- H2804 Solutions 15 2. ‘ ‘ [HgMozoel ll 10- - (HMSO‘ST (mM) A [H3M02021 5r- [HM02031 l 'L A 34 ‘ : f[M0(OH)§1 CA(M) 0.5 1.0 1.5 2.0 37 38 their variances, and initial estimates of the rate law constants, KINFIT uses a Runga-Kutta procedure to adjust the rate law constants to give the best fit of the given rate equation within the variances of the rate and concentration data. Initial estimates for the adjustable rate law constants should be as close to the actual values as possible, or else the Runga-Kutta procedure might converge to a false residual minimum. The criteria for the best fit rate equations include a low sum of the residuals squared from KINFIT, low average relative standard deviations of the adjusted rate law constants, a low number of experimental data points not within 10% of the values calculated from the rate equation, and a random distribution of residuals as a function of the measured rates. To reduce the number of possible rate equations to be fit with KINFIT, the rate equations from past research and from chemical modeling were considered. Javier, gt. al. discovered that in the low acid limit (0.1“ M ‘< CH < 0.20 M) 12-MPA formation follows second-order kinetics independent of the [H+], but that the kinetics shows a complicated 51 The various dependence upon [H+] in more strongly acidic solutions. rate equations for 12-MPA formation, along with their rate constants, are tabulated in Table V. Because the [H+] range that can be varied spans less than two orders of magnitude, there is nota [H33 range where only one of the denominator terms in [H+] predominates significantly over the others. It is thus not surprising that several different rate equations have been reported. The chemical modeling was based upon the stepwise coordination of a Mo(VI) species about phosphate. Table VI shows one such consecutive step mechanism with HM0206+ as the ligand; a 39 Table V Proposed Rate Equation for 12-MPA Formation (51) Javier, 33.12;. k1[H3 POu][MO(VI)] RATE : (.113 lll k2[Mo(VI)] Kinetics carried out in HNO3 solutions only. Notzu 1 RATE cc 9 K1[H+] + K2[H+] Constants K1 and K2 are dependent upon [H3P0u] and [Mo(VI)]. Beckwith5 K1[ Mo(VI)] [33mg RATE = 3 ¥+n K2[H+] k3[H 1 + 2 ————5 + ———fi + Ku[H 1 [Mo(VI)] [Mo(VI)] for HNO HClOu Sol'ns 3’ K [ Mo(VI)][H PO RATE = 1 3 1‘] + 2 +K3[H+]u Ku[H+]6 1 + K2[H ] +-————-—-2 + -——-——- 4 [Mo(VI)] [Mo(VIH for H280“ Sol'ns Although Mo(VI) was specified as HMOZO6’ no conclusive evidence has been given as to the specific Mo(VI) complexes involved. Scheeline52 K1lM°(VIN [H3P0u] RATE = + 7 + u 1‘2“” + 1‘3“” + Kuih‘lz +1 [Mo(VI)] " [MO(VI)]; 7 (Same RATE as Beckwith reported for HNOB) Call In this study, ionic strength was controlled, measurements were made under automated computer control, and [H ]= CH'3CM’ rather than CH’ were used for [H+ ] dependence. no m.:.m.m;.o m.=.m.m._.o mimdio m.=.m.m._.o mite m.m...o mp #933 ale: 0:5 m + xi + .7 mm WI- Vm . m Illluv . +m +N. +PmN I Im mmump anaum All... No a + N mm mm 02a: a m Acévozm m f. + E m ml. P x m+mmm+mmm+ mmozmcuaumm llv mozmmm + mmmtmmoxacueumm :A:+av mx + . 3| mmm u ~+~a ~+J~ozmn..a..m= Ilmv. moszm + $592.“??? + 13.15 Ala“! + lasts Ml zJ + ~+J~ozmcénmm Ilyx mosz + mozmcuaumm + 121.5 filmlx- + Act—5 mu: m cuaum llv m sum +2 : + IAG+EvoZm 34M“ +33 3 N Elm |||.VN 3 m +IAP+EV + IEOZm 24.7023 + Cam 3 mcwsflagouoanoumm on on oceanoom : 3.3m op noon on» cam momozm eon.“ nofiumanom 52:3 com smacmnooz < H> 0.3m“. an nAc+av Howl"; E"? ..----..._a_ ~-x -x mux -x -x x x x x oxmxzxmxmx - m exexmxzxmxmx - m we - a F - P m- z- m- m- P- - c on m- =. m- m- P- - a flu: - a x - a x x x x x x x x x x x x moon: Ac": on assoc Acu_m on unsov iouamumm an assoc N m e m o m m FOE: + P+a_+m_m~+ozm_ M + F.I.~+s_..._:wn7.m"+0.25H x + 9+:+S+Pmfl+mamm+zm+mm F052 M + Acupmummumm ca assoc moupmum um "am an ego» man» pnaov H N _ m: z m .mozm.zx + P m m _ m. m _mo=m_mx + m _ m.mm H+c+e+ m+ m.+ m+ m + F+c+a+ m+ m+ m + m + mnc+a+c + u ma agony In similar model and rate equation can be diagramed and derived for the other Mo(VI) complexes. The model considers the possibility that one or more of the proton dissociations may not be rapid compared with the Mo(VI) complexation steps (thus, nonnegative integers m and n were defined such that m + n 5 3) and that one or more of the Mo(VI) polymerization steps may not proceed rapidly (thus, nonnegative inte- gers P1, P2, P3, Pl!’ and P5 were defined such that P1 4- P2 4- P3 + Pu 4- P5 = 5). This model assumes that the last step to form H3_m_nPMo:2+")' is rate-determining; other models with different forms of the rate law may be derived if any other step is presumed to be rate-determining. The rate law and mechanism reported by Javier and coworkers51 presumed the PMou(m+n)-) to be rate-determining, with the 52 second step (to form H3-m—n subsequent steps taking place more rapidly. Beckwith's mechanism fits the model presented in Table VI, with m=1, n=2, P1=P2=P3=O, Pu=4, and P5=1. The mechanisms for HZSO,4 solutions involve the H nPMo1o(m+n)' formation (the Pu step) as the rate-determining step with 3-m- m=1, n=2, P :2, and Pil=2 for Beckwith's rate law and with m=1, n=2, 3 P3=3, and P14“ for Scheeline's rate law. All of these mechanisms assume that phosphate combines with a Mo(VI) species in the first step, as shown by Javier, gt _a_l_._.51, and that the final product 12-MPA is completely deprotonated in the acidic solution, as reported by Murata and Kiba.56 Based upon preliminary measurements, Gall found that 12-MSA forms in a similar mechanism as in Table VI.6 However, the preliminary measurements from this research (see Section B, Chapter V) indicate that 12-MSA is not formed appreciably in strongly acidic solutions where H2 Mo(VI) complexes are positively charged; this observation has been reported and utilized in analytical techniques in which phosphate and silicate are determined simultaneously}!7 Similarly, 12-MPA was not observed to form in more dilutely acidic solutions, although phosphate can react with Mo(VI) anions to form a colorless heteropoly intermedi- ate.58 Thus, the chemical system under which 12-MPA formation kinetics will be studied is noticeably different than the system to be used for the 12-MSA kinetics study. In dilutely acidic solution, the kinetics of .a- and B-12-MSA formation in term of individual Mo(VI) anions and polyanions was studied spectrophotometrically."1 The B-12-MSA formation followed pseudo first-order kinetics with reSpect to silicon; the overall rate K1[Mo1][Moa][Si] d[B-12-MSA]_ with K = 8.71! x was given by measurements dt - K2 + [M08] 1 107 1 g-ion-1min-1 and K2 = 6.17 g-ion 1'1 at 25.o°c., 1.0 M ionic strength, and pH 1.2 (acidified with HCl). Mo1 refers to monomeric Mo(VI) species, andMo8 refers to octameric species. Since the protona- tion of the various complexes proceeds rapidly, specific species of 2- Moon , HMoOu', and H Moo).I were not considered in the rate equation. 2 The M08 species in the equation was postulated because the pH where the pseudo first-order rate constant is maximum ( pH 1.8) corresponds to the pH region where M08026u- is the predominant Mo(VI) species. c-12-MSA formation followed more complicated kinetics, requiring an exponential model with two exponential terms. The reaction schemes consistent with the model and observed chemical phenomena are shown in Table VII. X and Y refer to reactants or intermediates that convert chemically tocx-12- NBA. The postulation of these schemes requires a third silicon species “3 Table VII Possible Reaction Schemes for<1-12-MSA Formation Competitive- Consecutive Parallel Consecutive Reactions with Reactions Reactions Reversible Step K __EE;5 x——>2 Y fV—J Xk k ’ k3 . 1 2 k3 k1 \/ a-12-MSA a-12-MSA a-12-MSA uu along with the silicon reactant (X) and a-12-MSA (2). At present, it is unknown whether the third species is another condensed or depolymerized silicate or another molybdosilicate; the possibility of B—12-MSA as the third species has been ruled out, however.u1 Chapter IV. Equilibrium Studies A. Equilibrium Studies of 12-Molybdophosphate Formation 1. Spectrophotometric Instrumentation In hopes of eliminating some possible mechanisms and rate laws from further consideration, the chemical equilibria among 12-MPA, H3P0u, and Mo(VI) were re-examined. Theoretically, the sum of all the steps in a valid chemical mechanism should yield an overall reaction equation consistent with the stoichiometry of the process considered. The equilibrium studies were carried out on a spectrophotometer con- sisting of several individual modules (GOA/McPherson EU 700 Series). The individual components are listed below. Heath Tungsten light source EU-701-50 GCA/McPherson monochromator EU-700-56 with programmable filter attachment Heath sample cell module EU-701-11 Heath photomultiplier module EU-701-30 Heath photometric readout module EU-703-31 Constant temperature bath The slew-driven monochromator contained its own programmable filter attachment which was kept on an automatic adjustment mode- throughout the equilibrium experiments. The slit width was 1000 um, N5 M6 which corresponded to a spectral bandpass of 2.0 nm. The accuracy of the monochromator was checked by a spectrophotometric scan of a holmium oxide filter. The monochromator exhibited a positive error of +1.7 nm with an average deviation of 10.1 nm about this positive error. The sample cell module was modified to house a brass, thermostated cuvette block constructed by Ingles9 and a water-driven magnetic stirrer (G.F. Smith Co.). Because of the large monochromator light throughput with the wide slit width, a photomultiplier tube voltage of around -400v was sufficient to read solution transmittances on the 10.6 amp scale of the readout module. A potentiometer on the readout module was used to adjust the 01 T (shutter closed), and the 1002 T was adjusted by varying the photomultiplier voltage to give full-scale meter deflection for the distilled-deionized (DDI) water blank. Except for spectrophotometric scans conducted from 350 nm to #50 nm with a Cary 17 instrument, all the spectrophotometric experiments for the equilibrium studies were measured with the modular Heath system at 400, 410, or “30 nm. Beckman quartz cuvettes were used, even though ordinary glass cuvettes would have sufficed for the wavelengths used. 2. Reagents Used and Standardizations The reagents and solutions used in the 12-MPA and 12-MSA studies were prepared as follows. Stock solutions approximately 5 M in HN03, H280”, and HClOu were prepared by dilution with DDI water of the corresponding concentrated acid solutions available commercially, from Fisher Scientific Co. (HNO and H280”) and Mallinckrodt, Inc. (70% 3 HClOu). The acid stock solutions were standardized by titrations with NaOH solution which had been previously standardized with KHP. H7 Contamination from H P0,l and silicate in these concentrated acids was 3 determined to be negligible from blank measurements. The 0.5 M stock Mo(VI) solution was prepared, without further purification, from the NaZMoonoZHZO reagent (Baker Chemical Co.). A stock 10.2 M phosphate solution was similarly prepared from reagent grade KHZPOA 2 (Mallinckrodt, Inc.). A 10' M silicate stock solution was prepared from Na SiO ~9H20 (Allied Chemical and Dye Corp.) and standardized by 2 3 titrations with a diluted HNO3 solution immediately after preparation in order to avoid interference form airborne C02. The titration with acid was carried to the bromphenol red endpoint as in a carbonate-HCl titration. A 5 M NaClOu (G. Frederick Smith Chemical Co.) reagent was determined to be free of silicate contamination. The solutions were stored in polyethylene bottles to avoid additional silicate contamina- tion from the glassware. The second H280“ acid dissociation constant pK , required for computer calculations with the MOSOH.FT program, was 2 measured at ionic strengths 1.0 M and 3.0 M by titrations of 0.1 M NaHSOu with 1 M NaOH. The pH was measured with a glass electrode as various increments of NaOH titrant were added. The electrodes and pH meter were calibrated with 0.05 m tetraoxalate buffer (pH = 1.679 at 25°C).60 The value of pKz, calculated from pH data nearly halfway to the equivalence point, was found to be 1.51 I 0.0a at I = 1.0 M, and 1.12 3.0.0” at I = 3.0 M. 3. Stoichiometric Studies The studies of equilibria among 12-MPA, H3P0’4’ were conducted with two different experimental designs. One such design and Mo(VI) was a continuous variations study carried out at constant ionic strength l18 and temperature, but at different [H+] for the different acidic media. The stoichiometric studies were implemented with the experimental 31 design used by Crouch, 22. El' These experiments were repeated here with ionic strength control and more nearly monochromatic incident radiation. In the previous research31 the ionic strength was not controlled, and a filter photometer was used. The basis for the experimental procedure begins with the logarith- mic expression for the 12-MPA equilibrium constant. log Kf = log[12-MPA] + z log[H+]- x log[H3P0u]- Y log[Mo(VI)] from x H3130” + Y Mo(VI):(12-MpA)3' + 2 H“ (1) With the assumptions that only 12-MPA absorbs light of a particular wavelength and that 12-MPA obeys Beer's Law, AAA = €Ab[12-MPA], at that wavelength, a rearrangement of terms in the above equation gives log AAA = log Kt. eb + x log[H3P0u] + Y log[Mo(VI)] - z log[H+] . The use ofHAA to denote an increase in absorbance corrects for a negligibly changing background absorbance at wavelength.l. If’12-MPA were the only molybdophosphate present, the use of mass balance equations in phos- phate (total concentration 01,), molybdate (CM), and hydrogen ions ([H+] ) would transform the equation to log AAA = log Kfexb + x log(CP-X[12-MPA]) + Y log(CM-Y[12-MPA]) -2 log( [11"] + Z[12-MPA]) The use of [H+] instead of CH considers that a portion of the acid protons were consumed in protonating the Mo(VI) complexes from M00" #9 If experimental conditions were established such that a small but measurable absorbance increase is measured and that 12-MPA has a high absorptivity EA’ then the second terms in the log expressions may be ignored. Thus, by varying CP while holding CM and [H+] constant and measuring AA at each C followed by subsequent linear regression of A P’ log AAA versus log CP’ one obtains the coefficient X as the slope of the regression. Similarly, the linear regressions of log AAA with log CM or log[H+] should give values for stoichiometric coefficients Y and Z, respectively. Cary 17 spectrophotometric scans from 350-fl50 nm were made on a 0.05 M MO(VI) and 1.386 M H280" solution and a 0.025 M Mo(VI), 0.693 M H230“, and 0.0025 M phOSphate solution. The latter solution exhibited considerably more absorption in the u00-h40 nm wavelength range rela- tive to the absorption by the first solution, presumably due to the polymerization of twelve Mo(VI) atoms in 12-MPA.56 Thus, the increase in absorbance at #30 nm was measured as C C P’ M’ and [H+] were successively varied in HNO and H280" media at different 3 ionic strengths. The temperature bath and brass cuvette block main- tained the solution temperature within the cuvette at 25.0 1 0.1°C. The actual concentrations and ionic strengths are tabulated in Table VIII. The slopes of the linear regressions are tabulated by ionic strength in Table IX. Plots of the stoichiometric coefficients as a function of ionic strength are graphed in Figures 7, 8, and 9. The ionic strengths for the solutions used by previous researchers31 were calculated, and the values for X, Y, and Z in HNO3 and HZSOH solutions coincide, within experimental error, with those values predicted from the graphs (Fi- gures 7-9) at those ionic strengths. Varied KHZPOA NazMoou HNO3 KHZPOH NazMoOu H280“ 50 Table VIII Concentration Data Used in 12-MPA Stoichiometry Studies Concentration Data for HNO3 Solutions (25°C) Eu 3:; 3:1. g 0.050 M 0.1-0.“ mM 0.995 M 0.75, 3.0 M “.95-1u.6 mM 2.50 mM 0.395 ’4 0.75, 1.75, 3.0 M 0.025 M 2.50 mM 0.495-1.014 M 1.2,1.5,2.0,2.5,3.0 M Concentration Data for H230“ Solutions (25.000) 0.090 M 0.1-0.9 mM 0.792 M o.75,2.0,3.0 M u.95-1I+.6 mM 2.50 mM 0.195 M 0.5,1.12,1.75,3.0 M 0.0125 M 2.50 mM 0.69-1.09 M 1.0,1.5,2.5,3.0 M Linear Regression log log log log log log log log log log log log log log log log log log log log log 51 El 51 E: 51 El E: E: E: 51 AA AA AA AA AA AA AA AA AA AA AA V8. V8. V8. V8. V8. V8. V8. V8. V8. V8. V8. V8. V8. V8. V8. V8. V8. V8. V8. V8. V3. log C log C log C log C log C log [H log [H+] log [3*] log [H+] log [H log log log log log log sflsf’zf’sf’nf’nf’nf’ log log [H+] log [H+] log [H+] log [H+] Ionic Strength(M) 51 Table IX Stoichiometric Coefficient X, Y, Z for 12-MPA Formation HNO3 Solutions :1: N U) 0 WN—a—Jw—AAOWNC43 0. 3. .75 .75 .0 .20 O UJNN—b—AWA OU'IOU'I 75 0 Solutions .75 NNNNNNNNNN NNNNNMHNNNN as a Function of Ionic Strength Slope 0.99 z 0.05 0.96 3 0.05 6.25 I 0.110 n.53 I 0.20 2.66 I 0.15 3.3“ 1.0'35 2.90 I 0.25 1.29 i 0.20 1.03 0.10 0.81 0.10 H- H- 0.95 3 0.05 0.95 :_0.04 0.94 I 0.05 ".59.: 0.55 2.36 I 0.20 1.77 :_0.15 0.89 1 0.10 5.u3.1 0.45 ”.22 I 0.30 3.12 I 0.20 3.02 i 0.10 Figure 7. Variation of the Phosphoric Acid Coefficient (X) with Ionic Strength o.u95 M; cP = (1.0-9.1) x 10'" M CM : 0.050 M (KNOB); CH = 0.090 M (32301-1) CA = 0.792 M 2.0 "' Error bars are within the circles drawn. l.£5" HNO 3 2304 ().ES" 1 L 1 LC) 22(31 31C) Ionic Strengthw) 52 Figure 8. Variation of the Molybdate Coefficient (Y) with Ionic Strength cM = n.95-1u.6 mM; cP = 2.50 mM; CH = cA = 0.u95 M; T = 25.o°c IO' .9- 8" 7__ I 61- 5.. 4 - Previous 0010 3.. 2.. |.. 1 l l 1.0 2.0 3.0 Ionic Strength (M) 53 Figure 9. Variation of the Hydrogen Ion Coefficient (2) with Ionic Strength CM = 25.0 mM (HNO3); CH = 0.495-1.014 M; CP = 2.50 mM 12.5 mM (HZSOH); C = 0.690-1.09 M 5 I A NOJ-bmemCO l " ' HNO LO 2.0 3.0 Ionic Strength (M) 54 55 The values for X seem to be independent of ionic strength. This result is expected since most of the phosphate exists as neutral H3P0u in the strong acid solutions and the neutral species would not be as subject to ionic strength effects or variations in the activity coeffi- cient as charged species would. Furthermore, the value X = 1.0 is consistent with the known crystallographic structure and empirical formula of the 12-MPA complex.61 The values for Y and Z vary signifi- cantly with ionic strength, in contrast. The Y values observed are all less than twelve; even extrapolation of the graphs in Figure 8 to zero ionic strength produces Y-intercepts less than twelve. The proposed chemical model of 12-MPA formation from H3P0u and Mo(VI) monomers and dimers cannot explain the large variation of Y and Z coefficients with ionic strength. In addition, much difficulty exists in determining the activity coefficients of triply-charged 12-MPA and other charged species in aqueous solution as the ionic strength is varied from 0.5 M to 3.0 M. A. Some Unsuccessful Studies with Methyl Violet Indicator and Strong Acid Potentiometry Two other experimental procedures were tried in order to obtain a value for the H+ coefficient 2. Both procedures attempted to measure the increase in [H+] after a specified amount of phosphate is added to an acidified molybdate solution. The spectrophotometric method was supposed to measure the absorbance change of a pH indicator as hydrogen ions are released during 12-MPA formation. The electrochem~ ical method measured the increase in pH glass electrode potential as additional H+ ions are released into solution. 56 The pH dye indicator methyl violet was selected for its availabili- ty and its color transition pH (0.1-3.0) being within the [H+] range in the strong acid solutions. Cary 17 spectrophotometric scans of acidic (pH 0.0) and basic (pH 7.0) solutions with equal methyl violet concen- trations revealed 580 nm as a suitable working wavelength where the methyl violet deprotonated form absorbs substantially better than the corresponding yellow, protonated form (see Figure 10). The structure of methyl violet and its acidic sites is shown below. Methyl violet--violet, deprotonated form The experiment was not conducted further for two reasons. First, since the colorimetric dye species are all cationic, the transition pH range between the yellow and violet colors theoretically should shift to greater pH values as ionic strength increases, if the Debye-Huckel Equation is valid. Methyl violet solutions were prepared in a series of HNO3 solutions of different pH values and at two different ionic strengths. Absorbance measurements of these solutions and the subse- quent plot shown in Figure 11 support the theoretical predictions. As a result, the 580 nm absorbance change is not large enough in the pH 0.3 Figure 10. Visible Absorption Spectra of the Protonated and Deprotonated Methyl Violet Indicator Species (vs. DDI-H20 Blank) o Ionic Strength Controlled with NaClOu, T = 25.0 C Violet A + =— N O O I .539 9‘?” Absorbonce .0 0 O 04 4: l l Yellow H 3* ,0 N I Q I 500 600 700 Mnm) 400 57 Figure‘11. Variation of the Methyl Violet Color Transition pH with Solution Ionic Strength 0 A 580 = Absorbance of Same Methyl Violet Solution at pH n.0, A : H80 nm HNO3 Solutions 0.9 0.8 0.7 0.5 A580 5 0580 0.5 0.4 0.3 0.2 0.l 58 59 0.3 to 0.7 range to detect millimolar increases in [H+] at high ionic strengths. Second, and more important, methyl violet was observed to reduce 12-MPA to the molybdenum blue species. Thus, methyl violet indicator was unsuitable to monitor the release of H+ as 12-MPA forms. After having been conditioned for a day in 1 M acid solution, the Orion pH glass electrode gave reproducible, stable potential measure- ments in strong acid solution. Calibration curves of electrode poten- tial versus [H+] were prepared for both HNO3 and H280” solutions (see Figure 12). However, the [H+] increase when phosphate was added to an acidified molybdate solution was detectable only for HMO3 solutions with initial [H+] values between 0.20 M and 0.55 M. For four such solutions the [H+] increase was measured, and the 12-MPA concentration was estimated from C data under the assumption that essentially all P phosphate is complexed to form 12-MPA when Mo(VI) is in a 100:1 excess.31 Because of the instrumental limitations of measuring small changes in [H+] when the initial [H+] is large, a substantial propagated standard deviation in 2 resulted. The 2 value, as calculated from Z = .. + A2214“? was 16.6 2 14.8. 5. Continuous Variations Experiments The Job method of continuous variations62 was also used to determine the Z coefficient and to calculate the 12-MPA formation constant K12. A series of solutions was prepared in which the combined analytical concentrations (CP + CM) and the ionic strength were held constant at 0.01 M and 3.0 M, respectively. The mole fraction of C phosphate (XP = 5—5120-9 was varied within each series. The computer P M programs MDLYB.FT and MOSOH.FT were used to calculate the analytical Figure 12. pH Glass Electrgde Calibration Curve for Perchloric Acid Solutions at 25 C Ionic Strength Controlled with NaClOu 4'0" Electrode contains its own internal reference electrode 400’ 390* E (mv) 380 ' 370* 360- ] l l (18 10 l l J 340 ofe [ H‘] (M) l l 1 02 (M1 60 61 HN03, HClOu, and H280” concentrations necessary to maintain the initial solution [if] at 0.2, 0.3, 0.11, and 0.5 M for each acid (thus, there were twelve series of solutions prepared). The background absorbance at "30 nm was measured for the XP = 0.0 solution (0.01 M Mo(VI)), and the increase in #30 nm absorbance was measured for each solution in the twelve series. In addition, as a slight experimental modification, the increases in absorbance at 1130 nm were measured for an acidified molybdate solution in which successive amounts of a phosphate solution with the same [H+], acid type, and ionic strength were added. The brass thermostated cuvette block and temperature bath maintained the cuvette solution temperature at 25.0 3'0.1°C. When solutions were added to the ' cuvette, a micro stir bar (Nalgene) and water-driven magnetic stirrer facilitated rapid mixing and ensured solution homogeneity. The quanti- ties Z and K12 may be calculated from these experimental absorbance data if the following assumptions are valid. (1) With a large excess of MO(VI) (i.e. X :_0.010), essentially P all the phosphate is complexed as 12%MPA. Thus, the proportionality cb between AA“ and [12-MPA] is established. 30 (2) The complexation stoichiometry between phosphate and molyb- date in 12-MPA is 1:12. (3) H3P011 + 12 Mo(VI);-é12-MPA is the only molybdophosphate equilibrium present in solution. Free, uncomplexed [H3POA] and [Mo(VI)] may then be calculated from mass balance equations Cp-[12-MPA] and CM-12 [12-MPA ] , respectively. (4) The release of H+ during 12-MPA formation does not alter the solution [H+] significantly. 62 (5) The relative concentrations of HM003+, H2M02062+, and the other prominent Mo(VI) species do not vary in the pH range 0.3 to 0.7; otherwise, the coefficient 2 will vary with pH (see the mathematical modeling, Table X). A conditional 12-MPA equilibrium constant K;2 can be calculated for each [H+1 according to: [12-MPA ] K. [H P0 11M (111)]‘2' 3 11 ° 12= and these K' values are related to K 12 constant in I = 3.0 M solutions, by 12, the overall equilibrium log K' - log K - Z log [H+]. 12 ' 12 Thus, a plot of log K32 versus log[H+] should give a linear graph with slope -Z and Y-intercept log K12. To check the validity of some assumptions used, the experiments were repeated for series of solutions with [H+] = 0.20 M (HN03) in which the combined CP + CM concentrations were varied from 0.008 M to 0.05 M. In addition, the spectrophotometric measurements were repeated at 410 nm for the [H+] = 0.20 M (HN03) solution series with CP + CM = 0.010 M. No experimental data were obtained for 0.“ M and 0.5 M [H+] in H280“; the absorbance increases were too small to be detected with the instru- mentation available. Several experimental observations invalidate most of the assump- tions used. The presence of an inflection point at XP < 0.10 in the plot ofAAu3o versus XP (which has the same shape as the 12-MPA curve in Figure 16) contradicts a result predicted from assumption (3). One- equilibrium chemical systems studied with Job's method would be 63 Table X Equilibria Between 12-MPA and Prominent MO(VI) Complexes in Strong Acid Solution (a) H3P0u (b) H3P0u + 12 10100;: (12-MPA)3" + 15 H“ (0) “39°11 + 6 HMozo‘g e——-> (12-MPA)3' + 9 H” (d) H390" + 6 HZMo20§2——=‘= (12-MPA)3' + 15 H+ (e) 3390“ + 6 H3M0202+€-——"- (12-MPA)3' + 21 H“ + 12M003= (12-MPA)3' + 3 H+ H390” + Y Mo(VI) \_—.—_-"‘.(12-MPA)3' + 2 3” Because of the steady state re-equilibration among all Mo(VI) complexes, the Y and Z coefficients observed will be a weighted average of those coefficients depicted above, weighted with respect to the relative contribution of each Mo(VI) species concentration to the total CM' These relative contributions vary with [H‘] (see Figure 6). Equilibria Between the Dimeric 9-MPA and 12-MPA in Strong Acid Solution (a) (12-MPA)3' + n 9°11 + 6 Moo3 = (9-10115' + 3 11* 3 (b) (12-MPA)3' + H3P0u + 6 10100;: (9-MPA) (c) (12-MPA)3- + H3P0ll + 3 PMOZOE:(9-MPA) 3- 2+ ((1) (12-MPA) + 113190,, + 3 H2M0206 ;_—\(9-MPA) (e) (12-MPA)3‘ + H.330,4 + 3 H 6-+9H+ 6-~1-6H+ 6-+9H+ 3M0202+:—(9-MPA)6- + 12 H+ 611 expected to have only upward or downward concavity in the mole fraction plot; the appearance of both concavities indicates that two or more molybdophosphate equilibria are present. In addition, the X? where AA has its maximum value is less than the = 0.0769 value 1 1130 12 + 1 expected for a 1:12 stoichiometry in 12-MPA. The position of the maximum AA varies with C + CM (see Table XI). Within experimental 1130 P error, the [H+] does not affect the XP where A11,430 has a maximum, though. However, if assumption (1) held and CP = [12-MPA] for XP : 0.01, then an increase in [H+] would decrease the 12-MPA absorptivity. Because the increase in absorbance at #30 nm results only from increase in 12-MPA , according to Beer's Law, the experimental data shown that [12-MPA] is proportional to CP when XP :_0.010 but that the relative amounts of [12-MPA] for a given C decrease with increased 1H”. No P method or experimental observation to check assumption (’1) was con- ceived. The graphs plotted in Figure 6 show a significant variation in the relative distribution of Mo(VI) species with [3*] ranging from 0.2 to 0.5 M; thus, assumption (5) is not valid. The implication of two or more molybdophosphate equilibria is that independent measurements for each of the heter0poly species are required or else conjectures and approximations must be introduced. Three aqueous solutions with identical C + C [11+]" and ionic strength, P M’ but different XP were prepared. Rapid cyclic voltametric scans with a stationary Pt electrode between -0.5 v and +0.5 v (vs. SCE) produced one or two distinct, reversible waves and what appears to be several unresolved waves due to reductions and oxidations of the 12-MPA, other molybdophosphates, and the Mo(VI) species present in the solutions. 65 Table x1 Variation of (XP)max AA with (CP + CM) [3*] = 0.200 M (inHNOB) I = 3.0, T = 25.0% CF * CM (M) (XP)max AA 0.008 0.0119 1 0.0011 0.010 0.050 t. 0.005 0. 020 0.055 1 0.0011 0.050 0.063 3. 0.0011 66 Cary 17 spectrophotometric scans between 350 and 1150 nm of these solutions, shown in Figure 13, reveal similar absorption spectra for the XP = 0.00 and 0.50 solutions. The XP to contain more 12-MPA, shows considerably more absorption for H10-u40 = 0.05 solution, which is expected nm radiation. Thus, because of large, overlapping Mo(VI) background absorption and interfering 12-MPA absorption, independent measurements of other molybdophosphates would be nearly impossible spectrophotome- trically. To substantiate this conclusion further, the experimental + . . data of AA versus XP with cP + cM = 0.010 M and [H I = 0.20 M (HNOB) shows proportionality between the absorbance increases at 1110 nm and those increases at 430 nm. Deviation from proportionality at the larger XP values probably arises through the systematic error of larger Mo(VI) absorbance changes relative to 12-MPA absorbance changes encountered when XP > 0.20. Another implication for two or more molybdophosphates present is that [H3P0u] a c - [12-MPA] and [Mo(VI)] as GM - 12 [12-MPA 1. In the P stoichiometry studies the concentrations of the other molybdophosphates may be significant, even though the [12-MPA] may be negligible, compared to CP and CM' Thus, in the Y-coefficient determinations, the log [H3P0u] term may not be constant over the range of CM studied (A.95-1H.7 mM), and log [Mo(VI)] may not be proportional to log CM' In yet another implication, the results for 2 from the glass electrode measurements are invalid. Those calculations for Z assume that all phosphates exists as [12-MPA] and that no additional protons are released when other molybdophosphates are formed. Figure 13 . UV-VIS Absorption Spectra of Three Phosphomolybdenum Solutions . 0 With (cP + CM) = 0.010 M, [H*] = 0.2 M (HN03), I = 3.0, T : 25.0 c 1.0 0.9 0.8 0.7 ’ 3 0.6 C: 3 0.5 .9: < 0.4 0.3 0.2 01 350 400 460 500 550 Wovelength,nm 67 68 6. Computer Simulation: Molydephosphate Equilibrium Constants Because of the concentrations of other heteropoly compounds present in solution are difficult to measure, the H+ coefficient and 12- MPA equilibrium constant had to be obtained by approximation. The HALTAFALL program was used to provide computer simulation of the continuous variations data. Under the assumption that only 12-MPA is responsible for the increase in “30 nm light absorption, the stoichiome- tries and overall formation constants for two or more molybdophosphates were adjusted until the computer-calculated [12-MPA] values match the experimental AAA30 values proportionally for each solution. The best fit of the experimental data for a given series of solutions satisfied the following criteria: (1) Computer-calculated [12-MPA] values (C1 in HALTAFALL).have their maximum value at CP and CM conditions matching the experimental conditions that give the maximum AA value measured. “30 (2) A ratio of [12-MPA] values from the fifth data point to the c (5) second data point (51(57) matches the ratio of absorbance increases at 1 AA(5)) 3M9) ' criterion considers the different proportionalities between [12-MPA] A30 nm for the fifth to the second experimental points ( This and C for the different [H+] in solution. P (3) A ratio of [12-MPA] values from the fifth data point to the C (5) ninth data point 601(5)) matches the ratio of absorbance increases at 1 1130 nm for the fifth and ninth experimental points ($3). This criterion attempts to approximate the overall, general shape of the < <‘ AA430 vs. XP graph, especially in the region 0.05 __XP _ 0.20. 69 (u) The K12 conditional equilibrium constants vary with pH in such a.manner as to give reasonable Z values through linear regression. This criterion was formulated because several choices of molybdophosphate equilibrium constants could satisfy the first three criteria for a given experimental data set, but subsequent linear regression could give either poor linear fit or aH+ coefficient that is physically impossible (see Table X). The HALTAFALL computer simulation data was run in the titration mode to duplicate exactly the experimental conditions in which 50 ul successive volumes of’an acidified phosphate solution were added to 2.50 ml initial volume of acidified molybdate. In all cases except for the 0.5 M H+(HN03) solutions, the maximumlSAu was observed after the fifth 30 addition of phosphate (XP,= 0.0A8), thus rationalizing the choice of the fifth data point in criteria (2) and (3). The correct choice of equilibrium constants could be verified through precise computer fitting of the continuous variations data and other experimental (i.e. stoichiometric) data where the same conditional equilibrium constants should apply. A two molydephosphate chemical model with 12-MPA and 9-MPA was assumed, and various estimates of their conditional, overall formation constants, K;2 and Ké, respectively, were supplied as HALTAFALL input in the attempts to approximate the experimental observations. The species 9-MPA was chosen because of its reported formation when ph03phate is 33 present in excess over molybdate. The species 9-MPA actually exists as a dimeric complex with a 2:18 phosphate to molybdate stoichiometry. 70 Several interesting features are observed in the computer- simulated data for CP + CM = 0.01 M. First, unless the K12 value at which C1 ([12-MPA]) is a maximum is value is sufficiently large, the XP less than the value 0.0769 expected for a one-to-twelve complex, even if 30 " ° 2 ' ! [9 MFA] is negligibly small For example if K = 10 the value of 12 X at maximum.C1((XP) ) is always 0.067 or less, and if K' -1020, P max Cl 12 this limiting XP valueloigs KO" 038. Second, the proper (XP) max C1 is obtained whenever the m ratio is 1.62 _-o_- 0.02. However, this 12 condition no longer holds true when K' is so small that (X 12 P)max C1 never reaches the value 0.098. Third, whenever K;2 >'1027, essentially all the phosphate exists as [12-MPA] when X i 0.020. A paradox P develops because if the K' values are too high, then one has difficulty 12 explaining the apparent change in 12-MPA absorptivity with [H*] changes since [12-MPA] is constant for corresponding data points (i.e. the same XP conditions). Alternatively, if K;2 values are too low, difficulties arise in reconciling the larger experimental (XP)max C values which are 1 constant with [H+] variations, within experimental error. Finally, and most importantly, another paradox arises fromlattempts to choose K12 and K5 values that satisfy criteria (1), (2), and (3) simultaneously. Even if criterion (1) has been satisfied, choosing K12 and K6 to satisfy (2) C (5) always results in a —T7 ratio which isC tzlso) small and choosing K12 and K5 to satisfy (3) always results in a.——T—7 ratio which is too large. These observations are illustrated in Figure 19. The chemical model must be modified to include at least one additonal molybdophosphate in equilibrium with H P09 and Mo(VI). 3 71 Figure 111. Comparisons Between Experimental Data and Computer Simulation with a 12-Molybdophosphate and Dimeric 9- Molybdophosphate Chemical Model computer simulated results; eb adjusted to match the experimental AA“30 data. ..... eiqoerimental.AA.”3O data at I = 3.0 M, T = 25 0°C, cP + 0M : 0.010 M, [H*] = 0.30 M (HN03) (A K' d K' c0(5) AA(5) ) 9 an 12 are chosen to fit -—-5C1(2 = A—m 2 . As a result, C16) < AA(5) C1(9) AAZ9)’ K' ' . C1(5) AA(5) (B) 9 and K12 are chosen to fit my =m. As a result, (31(5) > AA(5) C1(2) AA(2)' I v C (5) C (5) . 1 AMS) 1 K - - (C) K9 and 12 are chosen to fit C1(‘,£,7.AA(27and E:(9) - AA(5) AA(9)' AS a result, (XP)max C1 7‘ (XP)maxAA. XP XD 1 2 1 O m 1 m 2 1 O A a ( o a. 1 l. 11.°1\e\ 0 19110.1 L _ . O 5 O 5 2 m o o O . O Onv< (C) 0.20 ' 5 O. 0 XP 72 73 A chemical model with 12-MPA, dimeric 9-MPA, and monomeric 11-MPA was postulated. The 11-MPA species was selected because of its stoi- chiometric position between 9-MPA and 12-MPA and because 11-MPA is reported as the first product in the basic hydrolysis of 12-MPA (see Figure 2).”u From the attempted computer simulations with this model, the relative magnitudes of the conditional, overall 12-MPA and 11-MPA formation constants, K' and K' , respectively, primarily determined 12 11 c (5) 01(5) 7 the value of 0172')” Only fairly large K9 values affected the 672-) 1 1 ratio. Therefore, the general procedure of fitting each set of experi- c (5) 1 t I mental data involved choosing K12 and K11 such that the-E:(§7 ratio matched the experimental %%%g% ratio and then adjusting the K' value 9 C (5) AA(5) 1 1 until.E:7§7-matched the experimental AKT§T° While a continuum of K12, K11, and K5 values satisfies these two conditions, only certain values of these constants produced maximum C concentration and the C and C 1 P M conditions that match the experimental conditions. Yet, most sets of K K and 10 had to be eliminated because linear regressions of I V 12’ 11’ 9 their logarithms with log[H+] gave physically improbable Z values or poor linearity. For instance, the set log K12 = 30.3, log K51 = 28.6, and log K5 = 119.1 which fit the experimental data for 0.200 M [11*] in H280” had to be discarded because the closest precise computer fit for the data at [H+] = 0.300 M in H280“ would have given a.Z value of'nearly twenty-four from the log K' versus log [H+] linear regression. Not 12 3+ even if all the CM existed as H3M0206 into solution for each 12-MPA complex produced (see Table X). would so many H+ be released Nevertheless, some values for K32, K11, and K6 satisfy all four criteria listed earlier; these molybdophosphate equilibrium constants 7A are tabulated in Table XII. As a check on these values, the ratio between the experimental AAABO and the computer-simulated [12-MPA] was calculated for each data point. These proportionalities are tabulated for two sets of experimental data in Table XIII. The relative standard deviation of these proportionality constants are approximately 10% within each experimental data set as well as among all the data sets. As a grand average, 12-MPA's molar absorptivity at A30 nm is 1061 I 78 l 1cm'1. The variances within each data set appear to be randomly mole- distributed. Another chemical model with monomeric 12-MPA and 10-MPA and dimeric 9-MPA was tried in an attempt to simulate the experimental data C (5) 1 7 better. However, the K9 value could not be adjusted so that 57-97 matched %%%3% under given experimental conditions. 7. Medeling and Discussion The molybdophosphate chemical model with 12-MPA, 11-MPA, and dimeric 9-MPA provides the best fit of the experimental measurements. Other molybdophosphates may be present, but their equilibrium concen- trations are not expected to be significantly large. The 12-MPA, 11- MPA, and 9-MPA conditional equilibrium constants were observed to fit the continuous variations data as well as the data in which phosphate was successively added to molybdate. The computer calculated the concentrations of each molybdophosphate species for the various mole fractions; some of these results are graphed as Job plots in Figures 15, 16, and 17} The XP values where [12-MPA] and H1-MPA] are maximum increase with increased CP + CM and decreased [H+] while the XP values 75 mzo.o omo.o Nao.o 0:0.o mao.o mzo.o Nao.o 0:0.0 m:o.o omo.o u was m xv Anemonucocma ea co>aw one nowumm <4 Hmucoaasoqxo mcwocoonoenoov Aamm.Pv Ammm.Pv Ammm.Fv Amom.pv Amao.9v Aum>.mv Ammo.Fv AF~>.PV Aauo.Pv Apmm.mv mmm.P mmm.F 9mm.P mom.F amo.. mom.N awe." :Nm.F >>0.F mmm.N soon Haemoaasooxm one one P Axons oo. mNu Acms.av ems.. z com.o “mom.Pv Pem.P z ccm.o necessaom scam m Amms.Pv mms.e z com.o Ammm..v .mm.F z ooa.o flame.Pv cmc.a : com.o Amee.ev cmc.P z oom.o nsoaosHom soaom Aeem..v oc=.P z oom.o Asmm.ev msm.F : cos.o Amse.Pv mee.. z com.o Apee.PV see.a z cc~.o nsoaoaaom mozm as. a E: me o a o. m1 Hv soaumassam Lousosoo nwsoana onom page neoconcoo asascaaasom one HHx manna x . mNoP o m x . NNOP P N mNoP x N.P omop x N F Pmop Mo m x . mmop o o mNoF x N.P x . omop o P x . Fmo m m mmop x c a .me 22 h:op cmcP ch ems mmc. op or op om Nm am mm x o.> x >.N N.P 0.0 m.P 0.: NKNK o.: N.F m.P >.N XNNX as x m.P x N.F o.m o.F o.m o.F NKNN o.m o.F o.m o.P ”XXX .M 75a No0 mp -. a: aH>voz mmo maozau Asaz1ppv11uv .1 scam: mo me o:_o= a 1wsom av 1mo was aH>voz mo Voz ASH—215741» : m loo 10mm a m as m as m "assoc: cacaonoa Ac.ucoov HHx manna 76 Table XIII Comparison Between Experimental Data and Computer Simulated 12-MPA Concentrations (C1) [H*l = 0.500 M (HN03) CP + CM = 0.01 M log K12 = 30.7, log K5 log K11 = 29.1 .33 C1(mM) ‘32322 0.0100 0.026 0.0299 0.0198 0.095 0.0556 0.0299 0.058 0.0731 0.0388 0.063 0.0765 0.0981 0.061 0.0759 0.0572 0.052 0.0609 0.0798 0.038 0.0959 0.0883 0.035 0.0927 I : 3.0, T = 25°C = 50.6, cb(M-1) 1131 1236 1260 1213 1235 1162 1199 1220 .1]. [H ] = 0.200 M (H2809) CP + CM = 0.02 M log K12 = 29.1, log K6 = 97.6, log K;1 = 27.3 XE C1(mM) 35329. €b(M-1) 0.0100 0.097 0.0868 1118 0.0198 0.175 0.160 1099 0.0299 0.232 0.209 1110 0.0388 0.267 0.291 1108 0.0981 0.275 0.250 1100 0.0572 0.250 0.293 1029 0.0660 0.200 0.219 935 0.0798 0.167 0.172 971 0.0883 0.198 0.135 1096 0.0917 0.136 0.113 1209 Figure 15. Continuous Variations Plot for Molybdophosphate c + c = 0.01M, [n+1 = 0.50 M P M ' 3° ' - 5° ' = . 1029 I = .0, T = 25.0°c K12-MPA = 5.0 X 10 , K9-MPA - 9.0 x 10 , K11-MPA 1 2 X , 3 0.5" 04 '- 2 €03— c .9. 4— o b 1.- 5 ll-MPA 8 0.2 " o C) SB-NMQA 01 -~ pawn l l I I i i k Cl 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9 |.0 XP 77 Figure 16. Continuous Variations Plot for Molybdophosphates, cP + on = 0.01 M, [11+] = 0.20 M I 58 K12-MPA = 1.0 x 1036 = 2.7 x 10 , K = u.0 x 1033, I = 3.0, T = 25.0°c I 1 1-MPA ' ' K9 -MPA 0.5 ‘ .0 A I 8 O N 1 j Concentration, mM or AA430 0.l L 1 L 1 L ' ° '-- . . . O.l 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 Xp 78 Figure 17. Continuous Variations Plot for Molybdophosphates, cP + cM = 0.05 M, [3*] = 0.20 M 7 K : 1, 36 ' _ 58 ' 12-MPA o X10 , Kg‘MPA - 2.7 X 10 ’ K11_MPA = [4.0 x 1033, I = 3.0, T = 25.00C 3.0 ‘- 2 E c" l2-MPA .2 8 ll-MPA : 20 r 9-MPA t: m o c 0 U 1.0 r A I l l l l 0.1 02 0.3 0.4 0.5 0.6 0.7 0.0.9 1.0 XP I y 79 80 where [9-MPA] is maximum decrease under the same Cp 4- CM and [PH] variations. Linear regressions of the logarithms of these conditional equilibrium constants with log [H+] were performed, and the resulting Z and log K values are tabulated in Table XIV. In fitting the experimen- f tal data, a slight variation in log K12, log K;1, or log Ké by 0.1 or 0.2 c (5) 01(5) logarithmic units caused a significant change in the and 01(2) 01(9) ratios. Experimentally, there were three main sources of error. One source, in the [H+]va1ue, resulted from the propagation of errors in Cruywagen's equilibrium constants”2 used to calculate [H+](possib1y 10% RSD). Another source of error occurred in the AA93O measurements, which resulted from errors in measuring spectrophotometric transmittance (1 0.1% T) and from a possible systematic error in assuming that the background Mo(VI) absorbance at 930 nm changed negligibly as H P0“ was 3 added. The other error source occurred in determining the X where P AA“30 is maximum. Even though the extrapolation error between data points was reduced with more data points and a smaller XP interval between the two data points encompassing (XP) the» error in maxAA’ [H9 resulted in an (X ) variance of + 0.003. If the conditional P maxAA - equilibrium constants could be calculated directly from the experimen- tal data, propagation of the above experimental errors would impart a': 0.5 variance to log K12. Because of a large relative error propagated in [MO(VI)] = C -12[12-MPA] and of’propagating the [Mo(VI)] error to the M twelfth power, the equilibrium constants may be determined only to within one order of magnitude or power of ten. Thus, the errors in the choices of constants used for the computer simulation were smaller than the pr0pagated experimental errors. 81 11111 ~.m mm.om 11111 s.om Fm.mm 11111 5mm.o m.FF 2:.mN me.o m.FN om.m= amm.o mmm.o m.FP mm.mN mmm.o >.mp mm.:: mam.o n M mod xx» N am on mxu .M mod scam muasnom cofinmonwom Leona; >Hx manna m.oP mm..m m.ma ca.e~ m.me ce.e~ .m was moH «az1ma cm m can: 02m QHU< 82 The equilibrium constants reported were in terms of the total concentration of all Mo(VI) species rather than the concentration of any particular species. Because the stoichiometric and the continuous variations experiments both involved equilibrium methods, no informa- tion was available on which Mo(VI) species actually coordinate(s) with the phosphate anion. Instead, stopped-flow studies of 12-MPA kinetics must be performed to provide this information. Linear regression of log[12-MPA]- X log[H3P0u] versus log[Mo(VI)] (see Eqn. (1) on P. 98) was performed on the computer simulation results to determine the stoichiometric coefficient Y. From the regression slope the Mo(VI) coefficient was 12.0 (rxy = 0.999+) for both HNO3 and H280“ media. Thus, the proposed chemical model appeared to fit the experimental data from the stoichiometry studies, though the assump- tions used to calculate Y from log AA930 versus log CM regressions were not valid (i.e. [H3P09] not constant; more than one molybdophosphate present). The Z coefficient associated with 11-MPA formation is less than the Z value associated with 12-MPA formation. If the 11-MPA were present in solution as a result of basic 12-MPA hydrolysis, then three or four additional equivalents of H+ would have been released to the solution according to the reaction 3_ OH- 7_ + +111 12090 + 2 H20'—->PM011039 + HMoO3 + 3 H (12-MPA) (11-MPA)l (MO(VI)) PMo Instead, the postulated third molybdophosphate seems to be a colorless intermediate between uncomplexed H P09 and Mo(VI) and the yellow 12-MPA 3 complex. According to this model, the complexation of'H3P0u with.Mo(VI) 83 produces the third molydephosphate prior to complexation with more Mo(VI) to form 12-MPA. The constitution of this third molybdophosphate is uncertain, but the relative magnitudes of its Z, log K11, and standard error of the estimate values are consistent with a 1:11 phosphate to molybdate stoichiometry. K and K are a few orders of 12’ 11’ 9 magnitude lower in H280" than in the corresponding HNO3 and HClOu solutions. These lower values result from probable sulfate complexa- 63 The equilibrium constants K tion with Mo(VI) species or else incorporation of sulfate into the heteropoly cage.6” Either case would lead to a lower 12-MPA concentra- tion for a given CM' The equilibrium reactions for 12-MPA and 9-MPA with and without Mo(VI)-sulfate complexation are modeled in Table XV. Here, sulfate complexation with Mo(VI) is assumed. From the results for the Z coefficient in HNO and HClOu, the m values are calculated to be 3 0.858 (12-MPA equilibrium), 0.761 (9-MPA in HNO3), and 0.867 (9-MPA in HClOu). These 01 values reflect the occurrence of HM003+, H2M02062+, and HM0206+ as the three most predominant Mo(VI) species in strong acid solution. The 1': value reflects smaller, observed Z coefficients in H280“ solutions; the sulfate displaced from Mo(VI) species accepts a proton equivalent to become H30“: the prominent sulfate-containing species in strong acid solution. However, since the decreased free [Mo(VI)] alters the relative distribution of Mo(VI) species in solu- tion, the m values calculated above may not be used in the calculation of n. With an assumed value in = 0.858, n = 0.208 for the 12-MPA model, and n = 0.067 for the 9-MPA model. 89 Table XV An Overall Model for 12-MPA and 9-MPA Formations From H3P0u and Mo(VI) Without sulfate complexation 3- + H3P0u + 12 HmMo(VI)m+~:-‘PM01ZOHO + (3 + 1211:) H (12-MPA) m = 0 for MO(VI): 3(Mo(OH) 6)’ m = 0. 5 for } HM0206 111-.- 1.0 forMo(VI) =HMoO§+§H22M006+ ,m: 1. 5foriH3M ozog" 2 H390, + 18 mom)“ a-—>(9-MPA)6' + (6 1. 18m) u+ With sulfate complexation “39°11 1 12 HmMo(VI)(SOu)n(m-2n)+$§(12-MPA)3- + (3 1. 12111-13011" 1 12n 1130; 2 113m, 1 1e HmMo(VI)(SOn)ém-2n)+$ (9-MPA)5‘ + (6 + 18m-18n) 11* + 18n H80; 85 Certain analytical implications were observed from the experimen- tal data of the relationship between [12-MPA] (or AA930) and C In P' most cases the [12-MPA] was pr0portional to CP for XP : 0.01, but the proportionality varied with the different acid media used (HN03, H230“, and H010”) and the solution [H+]. The ratio-91259291 ranged in values P from 0.25 in the 0.5 M [H+] HClOu medium to 0.80 in the 0.2 M [H+] HNO3 medium. 'This information emphasizes the need for reproducible CH(or CA) and CM concentrations between the samples and the phosphate standards. Nevertheless, a 100-to-1 excess of molybdate over the phosphate present in the samples guarantees a proportionality between UZ-MPA] and CP‘ These considerations are important for reaction-rate methods based upon the initial reduced molybdenum blue formation rate as well as analytical procedures based on the equilibrium 12-MPA concentration. B. Equilibrium Studies of 12-MSA Formation 1. Equilibrium Constants of B-12-MSA at pH 1.2 To measure the equilibrium constants fer 12-MSA formation, continuous variations studies were carried out at constant pH in three acidic media (HNO HClOu, and H280 ). The total molar silicate and 3’ molybdate concentration (CS + CM) was varied between 0.01 M and 0.05 M but has held constant within each series of solutions as the mole 0 fraction of silicate (X = -———§———) was varied. These experiments 8 CS + CM were carried out at pH 1.2 where only the B isomer of 12-MSA forms and at pH 3.6 where c-12-MSA forms exclusively at 1.0 M ionic strength.u1 The required amounts of acid necessary to bring the solution pH to the specified values were calculated with the HALTAFALL computer program 86 described in Section A, Chapter III. Also, once the conditional formation constant was determined from one set of data, HALTAFALL was used to predict B—12-MSA concentrations for other data sets. Absorbance measurements were made with the modular GCA/McPherson UV-visible spectrophotometer with the modified sample cell module. The temperature of the cuvette block (see Section A and ref. 59), cuvette, and solution was maintained at 25°0.I.0°1OC by circulating water from a constant temperature bath through the cuvette block. Absorbance measurements were made at 910 nm and 930 nm (monochromator spectral bandpass of 9 nm) where 12-MSA absorbs strongly and the Mo(VI) back- ground absorption is small. A plot of one set of continuous variations data for the formation of 8-12-MSA is shown in Figure 18. The absorbance change at 930 nm S corresponds to a 12:1 complexation ratio of molybdate to silicate. (AA930) is a maximum at a silicate mole fraction X of 0.077, which Continuous variations studies were also carried out with different total concentrations of silicate and molybdate, C + C and with S M’ absorbances measured at 910 nm. All continuous variations plots have a similar shape to that shown in Figure 18. Within experimental error all plots have a maximum absorbance change at a 12:1 mole ratio of molybdate to silicate. Because of the identical results at two different wave- lengths and several different total CS and CM values, it is concluded that B-12-MSA is the only molybdosilicate species that forms at pH 1.2. The formation constant was calculated in the following manner. From the linear increase in “930 with CS at low mole fractions of silicate (see Figure 18), it was concluded that essentially all the Figure 18. 0.8 430nm) AA430 ( A 0.7 Continuous Variations Plot for B -12-Molybdosilicate, CS + CM = 0.01 M, pH = 1.2 Ionic Strength 1.0 M, T = 25°C l l I. l l 1 1 OJ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 LC 87 88 silicate present was converted to B-12-MSA for XS‘: 0.02. The slope of the AA.versus CS plot under these conditions yielded the molar absorpti- vity e of s-12-MSA (rxy values typically were 0.9998). Equilibrium B- 12-MSA concentrations were then calculated from.AA values measured near the maximum in the continuous variations plot (0.07 < X ‘< 0.10). The S conditional formation constant K} was then calculated from K' _ [B-12-MSA] _ [8-12-M3A] f - 112 ~ 112 [Si(OH)u][Mo(VI) (cS - [8-12-MSA1)(CM - 12[B-12-MSA i values in HNO3, and 31.6, respectively. These values are conditional constants at pH Results for the log K HClOu, and H280“ were 32.2, 31.9, 1.2 and 1.0 M ionic strength. From the variances in therAA.measurements and the value of‘e obtained, a propagation of error analysis showed that the experimental error in log K; values is about i 0.5. To test how well the calculated K}. continuous variations curves, B-12-MSA concentrations were calculated values describe the entire at each XS value and compared to the measured values of AA. A proportional relationship (constant cb value) over the entire range of XS values would indicate excellent agreement. For one data set in HNO3 at 930 nm, the average and standard deviation of the 6b value were 790;: 37 M’1(RSD = 5%). Similar correspondence was obtained for all three acidic media and for the different total CS + CM values. The grand 1 at A30 nm, average for the B-12-MSA molar absorptivity was 803 M-1cm' 25°C., and 1.0 M ionic strength. Thus, a single equilibrium with the given conditional fermation constants adequately describes the forma- tion of B-12-MSA. 89 2. Discussion and Comparison with the 12-MPA Equilibria The pH dependence of the K? values was observed by obtaining continuous variations data at pH 1.6 and pH 2.0. However, significant amounts of the o isomer were found at pH values higher than 1.8 as noted by an apparent decrease in the molar absorptivity of B-12-MSA with increasing pH. The determination for the formation constant of the a isomer at pH 3.6 was attempted. However, the lower molar absorptivity of the o isomer in comparison to that of the B isomer led to uncertain- ties in the K} values that Spanned several orders of magnitude. Within experimental error, the B—12-MSA formation constants are the same at pH 1.2 and pH 1.6 for a given acid solution. Some interesting similarities and differences exist between these results and those for the corresponding molybdophosphate system. The conditional formation constants of B-12-MSA at pH 1.2 in all the acidic media are of the same order of'magnitude as those for 12-MPA at pH 0.6 in HNO3 and HClOu (one should note that the ionic strength was 3.0 M in the 12-MPA study). In addition, the molar absorptivity for 8-12-MSA at 930 nm is within 25% of the molar absorptivity for 12-MPA at the same wavelength (again, ionic strengths were different in the two studies). On the other hand, the optimum pH for forming 12-MSA is higher than that for forming 12-MPA. The B-12-MSA formation constant is nearly indepen- dent of the type of acid used to adjust the pH, whereas the formation and HNO . 9 3 Apparently, the degree of sulfate complexation with Mo(VI) or with the constant of 12-MPA is much lower in H280” than in HClO heteropoly anion is much less in the 12-MSA case because of the smaller total sulfate concentration. 90 The (3-12-MSA species appears to be the only' molybdosilicate species formed at pH 1.2. As a consequence.stoichiometric amounts of 8- 12-MSA can be formed from silicate and molybdate when the mole fraction of silicate is less than 0.02. Because the molybdophosphate system is influenced by equilibria among three species, only 20-80% of the available phosphate is converted to 12-MPA at mole fractions of phos- phate less than 0.02. Chapter V. Kinetic Studies A. Kinetics of 12-MPA Formation 1. Some Preliminary Stopped-Flow Studies To verify the reproducibility of the 12-MPA rate data of 5 and Gall6, some preliminary stopped-flow measurements of 12- Beckwith MPA and 12-MSA were performed at 900 nm on solutions with various analytical concentrations of acid, Mo(VI), and phosphate. The ionic strength was not controlled in these preliminary experiments, but the temperature was maintained at 25.0 :_0.1°C. The stopped-flow components were allowed to equilibrate with the 25°C temperature bath for five minutes before rate measurements were made. The stopped-flow system*was operated in the manual mode, and photographs were taken of the time- dependent spectrophotometric readout on an oscilloscope. The rates measured here agreed with Notz's previous data within experimental error)4 but were twice those measured by Gall6 for 0.1 mM phOSphate solutions. This discrepancy resulted from my not controlling the solution ionic strength (see Section D for studies of ionic strength effects). As long as the 0% T and 100% T voltage levels were accurately known, the rate data obtained here showed good day-to-day reproducibil- ity. The ten minute equilibration of acidified molybdate solutions in 91 92 the 25°C temperature bath reduced the standard deviation in the trans- mittance data and initial reaction rates, as earlier observations had indicated.31 In addition, rates obtained when acidified phosphate solu- tions were mixed with molybdate agreed with rates obtained when neutral, aqueous phosphate solutions were mixed with molybdate solutions having twice the acidity (same Cfifor the mixed solutions) within experimental error. Hence, the heat of mixing between neutral and acidic solutions did not seem to alter the initial reaction rates significantly. In another preliminary study, the reaction time for 12-MPA forma- tion was extended to 10 8 (100 analog points per run on the Beckwith stopped-flow system) in order to observe a greater range of kinetics as equilibrium is attained. The absorbance at 930 nm was plotted versus time for each kinetics run; afterwards, the data were fit to a simple exponential equation of the ferm Y = 0 + 02 exp 03t or to a double 1 exponential equation of the form Y = 01 + 02 exp 03t + 0" exp 0 t. The 5 absorbance value at equilibrium A,» was obtained from an asymptotic extrapolation of the data points between 6 s and 10 3. Then ln(Am-At) was plotted against time. A linear plot over the entire time range would indicate that the simple exponential equation fits the experimen- tal kinetics data. Experimental plots, however, (see Figure 19 for an example) showed negative concavity in the first few seconds of the chemical reaction and linearity for all times thereafter. This observa- tion suggested that a double exponential equation would better describe the formation of 12-MPA and that the parameters 0 and 0“ would be 2 opposite in sign (020” < 0). All the experimental kinetics runs fit this equation well, as evidenced by the low value of the sum of squares of the residuals and by the satisfaction of the necessary constraint Figure 19. Semilogarithmic Plots of 12-Molybdophosphate Kinetics Data to Fit Exponential Equations 0.29- ' cM = 0.300 M + a [H ] = 0.965 M both HClOu b [11*] = 0.200 M 0' = 25.0°c, I = 3.0 M, A = 930 nm 0.06 0.06 0.04 ' . 0.021- 0.0l 0.008 I g 0.006 0.004 A,ot X=430nm Trlrril 0.002 - 0.00|llllllllll 0|23456789l0 Time (sec) 93 99 that 01 = 02 + 99 when the background absorbance at 930 nm was negligible. Though constant within a given kinetics run, the 0 para- meters varied in the different experimental runs as the molybdate and H+ concentrations in solution were varied. Some chemical reaction schemes consistent with this mathematical function are shown in Table XVI. 2. Experimental Section ‘ The stopped-flow observation cell path length and the solu- tion volumes delivered from the drive syringes were determined with potassium dichromate solutions in 1 M H280". A dichromate solution was mixed with water in which the dichromate was introduced first from the sample syringe and then from the reagent syringe. The variance in absorbance was within 1%; both drive syringes thus delivered the same volume of solutions to the mixer. When compared with the absorbance measured in a 1.00 cm quartz cuvette on the Cary 17 Spectrophotometer, the stopped-flow cell path length was found to be 1.87 i 0.02 cm. The solutions used in the kinetics studies were prepared with the consideration that both the sample and the reagent solutions are diluted 1:1 during stopped-flow mixing. The reagent solution contained spec- ified amounts of the stock acid solutions, molybdate, and NaClOu as calculated with the FORTRAN program MOLYB. If the acid, molybdate, and perchlorate concentrations required for a given pH or isopolymolybdate concentration at 3.0 M ionic strength were CH’ CM’ and [NaClOu], 20 respectively, then the reagent solution of composition 2C and H, M! 2[NaClOu]-3 gave the desired concentrations when mixed with the phos- phate sample solution. A Latin square experimental design was employed, with the final acid concentrations varying from 0.150-1.00 M and CM 95 o u o u m .o u s ca m a x s 01ml». m—9¢ "nooum 332960 03... :33 ncofiuomom 333002.80 “NV s s Aps1ms + «spams + use ms + oms 1 P 1 a s1mx+ms 1 ms + New - uAmxnmxnvoxo o N P o :m+ A» x v we NM 0 o o 1 o a x s %s can A 0+ m+ as we P 1m m m: ;m m m a 1m P m m uAmanxuvoxo h,x 3+ NVANx+ P 0V 10M+ M N + Aupxnvoxo om 3+ x + o ” +om x u m s s a as + m x oafi s1 so A 0+ m+ «cm a H o S .398 a w a ms omWWmAPIIa. "noun 33228 m :33 ncofiuomom o>3=oomcoo CV x x o 0 so we con: omo awe so + omo owo mo + Po 1 » coaumsom amusoswnoaxm cannon can saw: acoumwmcoo noaonom cofluomom H>x canoe 95a A mA AA1AAA A1 1A AA: - NAAAA1AA1AA1 AA>AHAAA 1A1 NA1 AAV A- ASAA wnwnz m 1A m N 1A A m A AuNAuvqu AIKIIKNIK I AAAAIVme A A Av A + A A u o m A o m A o m A 0A A A A A A A A A m 1A m 1A A m A AuNAnvnxmo A MA I“: + AuAAIqum 0A M M» M + o m H u m AA AA AA AA A1 AA A A A A A 1A A 1A A m A AumA1Aaxoo A A AVN A 1 AAAA1V980 A A AA A + A A u A N m m N A A o A m oAA A1 A1 NAA AA AA A1 A1 Av A A A A AUAucooV H>N wflnmh 96 varying from 0.01-0.05 M at each [H+] used. The sample solutions employed mostly 0.1 mM phosphate in 3 M NaClO,l after the 1:1 dilution, but 0.2 mM and 0.5 mM phosphate solutions were used occasionally to verify the first-order dependence of the reaction rate upon phosphate concentration. All these solutions were prepared by dilution of the apprOpriate stock solutions with distilled, deionized water. The concentrations chosen provided two related advantages. First, since the molybdate concentrations were present in a 100-fold excess over the phosphate concentrations, the total molybdate concentration (and thus the Mo(VI) species concentrations) did not change appreciably as 12-MPA formed. Second, the formation of 1.2-MFA did not increase the 3+ concentration of the mixed solutions by more than 3%. These two advantages were important because the double exponential kinetics equa- tion requires that some of the phosphate and molybdate be reacted before the maximum 12-MPA formation rate is attained. In addition, since the molybdate was present in such a large excess, a negligible amount of the dimeric 9-MPA was complexed from 12-MPA and 11-MPA. The stopped-flow experiments were performed on the automated Beck- with system thermostated at 25.0 1 O.1°C with water circulating from a constant temperature bath. The solutions, stored in polyethylene bottles to prevent silicate contamination from prolonged contact with the volumetric glassware, were placed in the constant-temperature bath for ten minutes to increase their temperatures to 25° prior to analysis. The monochromator was set at 1430 nm (2 nm bandpass). Generally, only the first ten seconds of the 12-MPA reaction were monitored. The maximum rate of 12-MPA formation was taken as the largest slope from 97 linear regression of the absorbance versus time data over any given 0.5 s interval; there were-six data points within such an interval, with 100 analog points averaged for each data point. The dependence of the maximum 12-MPA formation rate upon free molybdate and acid concentrations were examined with log-log plots of the maximum rate with concentration, with linear regressions on a programmable pocket calculator, and with KINFIT. The logarithmic plots reveal the range of reaction orders in H+ or any Mo(VI) isopoly species. ' Use of the inexpensive pocket calculator to do linear regressions on the maximum rate data showed which linear equations fit the data the best and provided values for the rate law constants to be used as initial estimates for the adjustable parameters in the KINFIT program. The description of KINFIT's utility and operation was described earlier in Section B, Chapter III. 3. Results and Discussion The modeling with a double exponential equation implies that the chemical mechanism for 12-molybdophosphate formation has two rate- determining steps. The initial rate of 12-MPA formation is no longer the maximum rate since g—E = Wwwld be zero at the start of the reaction (see Table XVI; the intermediate B must form before any 12-MPA forms). Furthermore, if reversible steps are involved in the mechanism, 9.2.2 dt2 0) no longer coincides with the time at which the intermediate B reaches the time at which 12-MPA formation proceeds at its maximum rate ( steady state, i.e. at its maximum concentration 0%? = 0). To determine the relationship between %% and the rate law constants for each mecha- nism, the integrated rate expressions in Table XVI were differentiated 98 twice with respect to time with the second derivative set equal to zero. The expressions for the maximum»%% value are derived for both.mechanisms in Table XVII. In both cases, the maximum rate is directly proportional to A0, the initial phosphate concentration. However, the expressions for the proportionality constants are quite complex, and the rate constants k1, k2, RB, and kn may vary with both H+ and Mo(VI) species concentrations. Thus, as was the case for analytical methods based upon the equilibrium 12-MPA concentration in solution, the need for reproducible acid and molybdate concentrations between samples and standards reacted with acidified molybdate reagent is emphasized. Different acid and molybdate concentrations can alter the proportionality constant re- lating the reaction rate with phosphate content and lead to inaccurate analytical results. Log-log plots of the time rate of change in absorbance versus the acid and molybdate concentrations reveal effective reaction orders ranging from -0.5 to -6 for 8+, 1.0 to 6.0 for HM003+ and.MoO3(MO(OH)6), and 0.5 to 3.5 for HM0206+ and H2M°062+' Most of the graphs in Figures 20 and 21 show some degree of curvature, possibly from the convolution of two or more terms which have different orders in H+ and.Mo(VI). Each curve in Figure 21 represents one particular total molybdate concentra- tion. Though the total Mo(VI) dimer and monomer concentrations are fairly constant with [H+], the degree of protonation required to form each particular Mo(VI) complex is pH-dependent (see Figure 6). In the low acid solutions where [H+]< 0.30 M, the measured maximum rates were significantly lower than the rates observed in the 0.3-0.5 M 99 Table XVII Theoretical Variation of the Maximum Reaction Rate C%%) with reactant concentration (A0) and rate constants (k1,k2,etc.) for the reaction schemes in Table XVI Att=0,B =0 :0 O O (1) k k A A k k dC_ 1 2 o 1 2 o —— exp(-k t) - -—— exp(-k -k )t dt=k2+k3-k11 k2+k3-k1 2 3 (Fa £11210 1:111 11221111 11103” -- = o - -———-——- exp(-k t) + exp(-k -k )t dt2 M2+k3 -k 1 k1+k3-k1 1 . 2 3 => exp(-k1t) - 112::— => t —_1—_ In 2.1:; exp(-k2-k3)t k1 k2+k3-k1 k1 -k1 -k2-k3 k +k-k1k +k -k 9.9) _ k1k2Ao (1‘2 R3) 2 3 _‘k2*k3 2 3 1 dt c,” k2+k3-k k1 R1 (2) k1k A k1k2Ao dc: 20 exp(-A t) +— oexp(- t) 1'"2 1 1 ”‘2 A2 2 A k k A A k k A 1% = 0 = 317172;) exp(-A1t)- '—:=_1_)12-£ exp(-)12t) dt 2 1 2 exp(-A t) A => 2 = *1 1 1 expz-A1t) 3; => 1: = m; ln T; -A 2 AA (92) k1k2A° (h) A1"A2 A1 )1 -1 dt C": )‘1-A2 X2 - (T) 1 2 Rate (590") (A= 430nm) Maximum Rxn Figure 20; Kinetics of 12-Molybdophosphate Formation as a Function of Initial Molybdate Concentration 0"0 l 1 T l l l I I [ I 0.08” .. 006" In HC1011 .1 a [H*] = 0.u00 M 004— b [11*] = 0.150 M c [3*] = 0.500 M + - 0.02_ d [11+] - 0.750 M e [H ] = 1.00 M 0.0101- 0.008- 0.006- 0004- 0.002— 000! l 1 1 1 1 1 g 1 1 0.1 0.2 0.4 0.60.8 1.0 l‘zo 4.0 6.0 8.0l0.0 [MO(VI) SPECIES] (mM) 100 Figure 21. Kinetics of 12-Molybdophosphate Formation as a Function of Hydrogen Ion Concentration 0J0 I i I I I , 0.08- 0.06 "' \CMO 0.05M 004— \\ 0.03M 0.02 *- ‘OQIO '- 0.02 M 0.006 ‘- ,0 8 an I 0.004 +- 00. M 0.002 _. Maximum Ranote (secf') (>\=430) 0.00, l 1 l I 1 0.! 0.2 0.4 0.6 0.8 l.0 2.0 + M [H ]O( ) 10] 102 acidity range. Because the pH of these lower acidity solutions is close to the Mo(VI) isoelectric pH of 0.9, the amount of Mo(VI) species available to complex with phosphate was reduced through Moo precipita- 3 tion, Though no white precipitate was visible, any suspended particles in the solution could scatter the incident 1430 nm radiation and cause spectrophotometric background interferences. With the linear regression program on a HP-ZSC pocket calculator, various linear equations were found that could fit the variation in the 12-MPA maximum formation rate with molybdate concentration over limited acidity ranges or the variation in the maximum rate with hydrogen ion concentration over certain ranges of molybdate concentration. The equa- tions, slopes, and Y-intercepts that fit the selected experimental condi- tions in HNO and HClOu solutions are tabulated in Table XVIII. For the 3 molybdate dependence in the low acid limit between 0.3-0.5 M 3+, the _l_ - m RATE [H2 M0 2062 1 linear equation -+ b gave the best fit of the [H 2M020 6 2] = experimental data. Other equations that were tried included RATE m [HZMOZO6 m m log[HzMo2 062 1+ b, and“ 1 -———-———-+ b. Since the equation RTTE (mo; + b held over a fairly wide acidity range, the variation of the + b, RATE = m[H2Mo o 2 62*]+ b, RATE = m[HMoO3+]+ b, log(RATE) 2+] __1_._ RATE m m2M°2°6 constants m and b with [8+] was tested. The constant b was found to be 2+ independent of [3*] while m varied with the square of [8+ L This equation no longer described the 12-MPA formation data well at [H+] = 0.750 M; m [HZMOZOE instead, the equation“ 1 RATE: 2+12+ b gave an excellent fit for the 103 Table XVIII Linear Equations Which Best Fit 12-MPA Kinetics Data Over Limited Ranges of Experimental Conditions [H*jo = 0.30-0.50 M 1 mm+12 -———- = 2+ RATE [HZMOZOG ] 4. [H 10 - 0.750 M 1 m .___. = b RATE 2+ * 4. [H 10 - 1.00 M ._J_. - m + b RATE ‘ 2+ 3 [HzMo206 1 cM = 0.01-0.02 M 1 _ m[H+]8 RATE ' 2+ 3 [HZMoZOG 1 + b 2+ [82M0206 1 CM > o.ou M 1 _ m[H+] RATE ‘ [H O 02*]2 * blfizM° 2M 2 6 HNO3 b = 10.n8 sec m = 0.580 sec/M ny = 0.9996 b = 16.97 sec m = 1.12x10‘3MZsec ny = 0.9988 b = 22.53 sec m = 1.56x10‘5M3sec = 0.999+ b = 0.1515 M sec m = 1.53x10' 1“ = 009997 5sec/M5 2 2 206+] HClO 4 b = 9.96 sec m = 0.396 sec/M r = 0.998" xy b = 17.187 sec = 5.21x10-nM23ec ny 3 009998 b = 16.3" sec 6 m = 6.26x10' Mssec I‘xy = 0 09999 b = 0.1105 M sec m = 6.00x10‘6sec/M5 rxy = 0.9991 This is the best fit linear equation for the experimental data in this 0 concentration range, but is still a relatively poor fit. ‘M (rxy < 0.98) 10" data obtained from 0.75 M H+ solutions. In 1.00 M H+ solutions the equation -—l—»= m + b fits the data slightly better than the RATE [H MO 0 2+]3 2 2 6 1 m equation EITE = [H2 Mo 02+]h + b, but both gave correlation coefficients 02 O6 greater than 0.9990. To analyze the variation of the mmmimum rate with acidity, the equation RTTE = m[H+]8 solutions low in molybdate concentration (CM< 0.02 M). In this equation, 2+ + b provided the best fit of the kinetics data for b varied inversely with [H2 Mo ], and m varied inversely with 02+]3. 206 [HZM At higher molybdate concentrations, the equation 1 °2 06 RATE = m[H+ ] [H2MoO 02+]2 + b[H2M02062 12 provided the best fit, but this fit was poor 2 6 compared to the other equations described above. There are at least two possible explanations for the observation of different linear equations to describe the experimental observations. First, the linear equations reflect the predominant terms in the complex rate expressions shown in Table XVII. The relative magnitude and signifi- cance of each term in the equation vary with changes in either acid or molybdate concentrations. The second explanation suggests that the same chemical mechanism for 12-molybdophosphate formation does not occur over the entire range of‘H+ and Mo(VI) concentrations studied. The mechanisms consistent with the linear equations for the 12-MPA formation are tabu- lated in Table XIX. The molybdate linear equation in the low acid limit (0.3-0.5 M 1*?) suggests that an acidic dissociation occurs prior to combination with H 2M0 #06 (or HM0206+ or H Mo2 0 63+) in the latter rate- 3 determining step. The other mechanisms displayed suggest either a change 105 Table XIX Chemical Mechanisms for 12-MPA Formation Corresponding to the Linear Equations in Table XVIII 2+ K [H P0 ][H MoO 1 + RATE: 1 3 " 2 5 (0301410111050 M) + 2 2+ K2[H ] + [H2M0206 ] INTERMEDIATE __>. INTERMEDIATE + A v——- B + 2 H B + HzMozogf———————€> Products 2 3 K1 [331304] [ HzMozoé‘j 4. RATE = + 2 2+ 3 ([H] = 1.00 M) K2[H ] + [H2M0206 ] A = B + Z H+ B + 3 H2M020§+----€> Products 1 g mm" + ——--‘?--5— (CM 10.02 M) RATE [H2M020§+] 3 [H2Mo206‘] 11:3 +2H+ B + H2M020§+F2 c + 23" 2+ 3 + C + H2M°206 ‘——D + 2H 2+ + D + H2M0206 —-912-MPA + 2H No reasonable mechanism could be postulated for the acid dependence at high molybdate concentrations. 106 in the number of molybdate species bound to phosphate in the intermediate involved in the rate-determining step or else a switch in rate-determining steps to form different molybdophosphates. Various attempts to integrate the observed rate equations and fit the experimental data with this integrated equation were performed with KINFIT. The rate equation which gave the best fit had the form 2+ 3 d-flZ-MPA]_ K1[H3P0u]l'H2MoZO6 ] dt " + 5 _ + 2 2+ 2 2+ 3 K2 [H 1 .. K3[H ] [H2Mo206 ] + [H2M0206 ] and the values for the rate constants were 10103 H010, K1 = 0.1430 1 0.028 sec“; 0.1106 1 0.019 sec‘1 K2 = (6.7 11.8) x 10'7M'2; (2.7 31.0) x 10"7M'2 K3 = 0.03911 1 0.00113 M“; 0.0222 : 0.0028 M"1 This was the best fit possible with consecutive step modeling. However, the chemical mechanism consistent with the above rate equation suggested that HBPOH dissociates two proton equivalents before being coordinated by molybdate. Because of'the negligible formation rate of’12-MPA.measured in solutions more alkaline than pH 2,66 this proposed mechanism cannot be correct. 0n the other hand, a consecutive step mechanism in which HBPOM initially complexes with a Mo( VI) species (as illustrated in Table_VI with related forms of the rate equation in Table V) cannot be valid either. The sum of the squares of the residuals from KINFIT were significantly large, and the standard deviations associated with the final values of the adjustable parameters were ten times larger than the parameter values 107 themselves. Though the consecutive step mechanism with only one rate- determining step postulated by previous workersu’s’7’51’52 may have been valid since their solution ionic strengths were 2 M and below, a consecu- tive reactions mechanism with two rate-determining steps had to be postulated for the 12-MPA formation kinetics as observed in 3.0 M ionic strength solutions. For a consecutive reactions mechanism the equation relating the maximum reaction rate with AO(CPO ) and rate constants k and k is k1 k2 (99-) - -———-—k1k2A° HEB)" kz'k1 - (1(3)‘ k2’1‘1] (1) d C”=o ‘ k -k k k 2 1 1 1 The constants k1 and k2 each depend on the acid and molybdate concentra- tions present. Though these dependencies may seem difficult to deduce at first, the 12-MPA formation rate is first-order with respect to the molybdate concentration and follows a simple exponential equation more closely when 0.3 M : [H+] 5 0.50 M and CM‘: 0.02 M. The reaction rate decreases with increased [11+], and 12-MPA formation releases 11" into solution. Therefore, k2 must increase faster than k so that the kinetics 1 are controlled by the first rate-determining step in the low acid limit. Mathematically, as k increases relative to k the above equation becomes 2 1’ k 1 d0 kz'E’ k2 -1 (2.111901=0 = k1Ao[cE:) 2 - (E?) ],which approaches k1Ao(1.0-0.0) = k1Ao as k2 approaches + w. When 0.3 M1: [n+1 : 0.50 M and CM': 0.02 M, the maximum reaction [HMo03+] rate was proportional to [ +1 H This relationship was consistent with 108 previous publication of bimolecular 12-MPA kinetics in the low acid limit51 if Mo(OH)6 is the species that reacts with HBPOH. If one sets k1 [HMoo3*1 equal to K1-——j:———T K constant, the range of the measured reaction rates [H1 1 from 0.8 A0 to 650 A0 confined k to values ranging from 10 to 650 and set 1 1.0 as the lower bound for k2. The k2 1 values, reproduced the measured maximum rates were observed to vary values which, along with the k + 9 directly with [HMOOB ] . [n+110 [HMo03+] [EMo03‘1 9 With k1 = K1 -—--- and k2 = K2 --——--1-o— , the consecutive [11+] [3"] reactions rate equation (equation 1) converged during KINFIT execution after only nine iterations. 'The final values for the adjusted parameters K1 and K2 are tabulated in Table XX. The residual errors were reduced by as much as 60% by using this rate equation, and the relative standard deviations associated with the adjusted parameter values were the smallest observed from all the KINFIT executions. The chemical mechanism tabulated in Table XX reproduced the rate equation when equilibrium approximations were applied to MoO3 (same as Mo(0H)6), (HPOu)(Mo0 )2-, 3- 3’9 ' The proposed mechanism has the following support. First, with 3 and (Pou)(M00 equilibrium approximations on the molybdophoSphates following the second rate-determining step, the overall 12-MPA reaction stoichiometry +___\_, 3' + H3P011 + 12 HMOOB <7"PM°12040 + 15 H is reproduced. Secondly, the second rate-determining step (k2) is the same as the rate-determining step postulated from the 12-MPA acid 109 Table XX Rate Law Equation, Constants, and Mechanism for 12-MPA Formation (I = 3.0 M, T = 25.0°c, A = A30 mm)! -k 1 _k2 MAx. _ (d[12-MPA]) _ 1‘11‘2CP0,1 (53 k2"“1 1E3) 2—k11 RATE 'at ' Ax = 0 ' [ k ‘ k kz-k1 1 1 + + 9 where k1 = K1 [HMOO3] and k2 = K2 [HMOOB] [H+] [11"]1o HN03 H010l1 K1 10.3 i 0.2 sec-1; 12.5 :13 sec-1 K2 (6.2 i 0.7) x 1017 M sec-1; (A.3 :11.u) x 1018 M sec”1 P111003: Moo3 + H". fast H3P01l + M003__k_1_)(HP0u)(M003)2- + 2H+ slow 2- 3- + (HP011)(Mo03) + 8 Mo03;.3(1’011)(M003)9 + H fast (P04)(M003)g- + M003 k2 E Products slow 3- (Pou)(M°°3)10,11,12 'At ”30 nm, €12-MPA = 1061 +178 1 mole.1 cm-1 b : 1.87 i 0.02 cm in the Beckwith stopped-flow observation cell. 110 decomposition rate equation (see Section C). Thirdly, the k step 2 reflects the competing mechanisms of monomeric 9-MPA coordinating another phosphate to form dimeric 9-MPA or more molybdate to become 12- MPA. Finally, in the higher acid solutions ([H+] > 0.5 M), with equilibrium approximations on all intermediates prior to the k step and 2 consideration of only the first term in the acid decomposition rate equation (see Section C), the 12-MPA equilibrium constant is KeQ. = E: -_- ELF—Egg. = W [H+]13 . kr kd k1r [IMoO3+]2 [I-Irio03"']10 [H3P011] All of the stoichiometric coefficients are reproduced in the equilib- rium constant exponents. The H+ exponent represents the weighted average of the number of H." released from each Mo(VI) species during complexation rather than from HMo03+ or M003 only (see Table X). As pointed out by Truesdale, gt. glut” knowledge of molybdate and heteropolymolybdate speciation in strong acid solution has revealed more definitive information about the chemistry and kinetics of 12-MPA formation. Analytically, the experimental considerations for reaction- 51,52 rate phosphate determinations already published have amplified significance. Not only does performing the experiment at 0.3 M _f [Hflj 0.50 M and C _<_ 0.02 M give the best sensitivity with faster reaction Mo rates, but the 12-MPA kinetics are simpler and more well-behaved. Under these conditions, the MOD precipitate does not interfere with absor- 3 bance measurements. Also, the sensitivity is enhanced in perchloric acid solutions since the rate constants are larger in HClO,1 media. The reproducibility of the solution pH, molybdate content, and ionic 111 strength between reagent solutions mixed with samples and with phos- phate standards is again emphasized as each factor affects the 12-MPA reaction rate measured (see also Section D). An overall chemical scheme that displays the pathways for the molybdophosphate reactions studied in this section (and in Section C) is shown in Table XXI. The intermediate 9-mo1ybdophosphate monomer repre- sents a branching point in the pathways. Depending upon which chemical reagent is present in excess, the 9-molybdophosphate can associate more molybdate to form 10,11, and 12-molybdophosphate products, associate another phosphate and more molybdate to become dimeric 9-MPA, or decompose in the presence of excess H+ back to phosphoric acid and molybdate constituents. B. Stopped-Flow Kinetics Studies of B-12-MSA Formation and Decomposi- tion 1. Experimental Conditions and Preliminary Measurements As mentioned in the previous section, the reproducibility of the B-12-MSA formation rate data was tested with preliminary stopped- flow measurements at 1100 nm on solutions with various analytical concentrations of acid, Mo(VI), and silicate. As for the 12-MPA system, the rates measured here agreed with previous data6 within experimental error, except for solutions with total acid concentrations greater than 0.2 M. Under these experimental conditions, the reaction rates measured here were significantly slower than those rates measured by Gall.6 The discrepancy was related to the formation of cationic Mo(VI) species rather than the heptameric and octameric polyanions. When C > 0.2 M, H the solution pH was below the Mo(VI) isoelectric pH of 0.9. 112 Table XXI Overall Chemical Scheme for Molybdophosphate Reactions Mo(OH) 8 Mo(OH) Mo(OH) 1132011 _—_ri_s2' —‘—H+=——$(P0u)(Mo03)g- __ (90,1)(11003135 2 H + 3 H ”113m,1 1LMO(OH)6 6- 3- (p0,)2(Mo03>9 (POu)(M003)11 11-MPA 6 Mo(OH)6 1LMO6 6- 3- (P011)2(Mo03)15 (P011)(M'oO3)12 12-MPA 3 Mo(OH)6 5- (201)2(Mo03)18 dimeric 9-MPA 113 The overall kinetics profile for the. formation of 8-12-MSA was observed in the pH range 1.2-1.8. A reaction time of 100-200 s was required for equilibrium to be approached. The B—12-MSA absorbance time profile followed a simple exponential equation of the form A = 01 + 02 exp 03t. A sample ln(Aa7At) vs. t plot for 8-12-MSA formation is shown in Figure 22 and can be compared with a similar plot for 12-MPA in Figure 19. The monochromator wavelength was #30 nm (2 nm bandpass). The solutions used for the B-12-MSA kinetics study were prepared in a similar manner as for the 12-MPA study, except that silicate was used instead of phosphate, the solution ionic strengths were 1.0 M, and the CH and CM concentrations required to give the correct pH or molybdate content were calculated with the HALTAFALL computer program instead of MOLYB.FT. A typical stopped-flow kinetics run lasted 100 3 rather than 10 s as in the phosphate case. The Mo(VI) concentrations varied from 0.01 M to 0.10 M, and the solution pH's employed were 1.2, 1.5, and 1.8. All the solutions were allowed to equilibrate thermally in the 25.000 temperature bath for ten minutes prior to the stOpped-flow run. 2. Rate Equation and Chemical Mechanism in HNO and H01011 Solu- tions 3 Because a simple exponential equation was observed to fit the overall B-12-MSA kinetics profile, the mathematical modeling and the postulation of a chemical mechanism for B-12-MSA formation were easier to perform than for the 12-MPA formation. This absorbance-time behavior indicated that there is only one rate-determining step in the mechanism. The maximum absorbance increase with time corresponded with the initial reaction rate. However, the pH 1.2 solutions exhibited suspiciously low Figure 22. Semilogarithmic Plot of 5-12-Molybdosilicate Kinetics Data I = 1.0M, T = 25.0°c, pH 1.5 in H0101, csi = 0.1 mM, cM = 0.05 M 0.2 - Am- Ar of 430nm ,0 8 I l l I I 00' 20 40 so so 100 TIME(sec.) 114 115 initial rates which were independent of the total molybdate concentra- tion at values exceeding 0.0“ M. This behavior suggested precipitation of’MoOB, since pH 1.2 is close to the Mo(VI) isoelectric pH'of'O.9. For the solutions at pH 1.5 and 1.8, which did not exhibit any evidence of Mo(VI) precipitation, several linear equations were used in an attempt to fit the experimental data. The equation that best fit the data for [H Moon] _ m molybdate dependence had the form W ——11—_- + b. An [Mo 0 ] 8 26 identical equation has been found to fit similar data in HCl solu- tions.“1 l The rate constants of Truesdale in HCl solutions at pH 1.2 and the rate constants found here in HNO H280”, and HC1011 solutions at pH 1.5 3’ and 1.8 are shown in Table XXII. The values differ slightly because of the different pH values and different acidic media used for the experi- ments. In HCl solutions the 01- could have complexed some Mo(VI) species to reduce the free molybdate concentration; under such condi- tions Truesdale and coworkers would have studied 8-12-MSA kinetics without significant MoO3 precipitation. Truesdale's chemical mechanism consistent with the rate law consists of a.rapid equilibration step between silicate and H MoOl1 followed by a rate-determining step between 2 the molybdosilicate intermediate and M08026u-.u1 Of course, other Mo(VI) species may be represented in the empirical rate equation, such - A- 4- as HMoO,1 in place of H2Mo011 and H2Mo70211 in place of M08026 . However, in the pH range 1.2-1.8, H Mo011 is the predominant molybdate 2 monomer, and MO8026u- is more prevalent than any of the heptameric species. 116 Table XXII Rate Constant for B-12-MSA Formation in Various Acidic Media K I! K is Acidic b' m’ r _11 _1 2H Medium (M sec) (M2 sec) _gy (M sec ) 'LM) 1.2 301 ---- ---- 0.987 0.122 0.000791 1.5 HN03 0.952 3.25x10'" 0.988 0.332 0.000719 1.5 H0101 0.508 3.13x10‘” 0.999 0.296 0.000616 1.5 ”2309 0.199 3.08x10’" 0.996 1.01 0.00210 1.8 HN03 0.219 3.30x10-u 0.986 0.69 0.00150 1.8 00101 0.226 3.23x10‘” 0.972 0.67 0.00190 1.8 11230,1 0.119 2.79x10'“ 0.981 1.31 0.0029 [H MoO 1 m 2 9 = -—-—-———T + b *Constants in -—-——-—- 4- RATE [Mo8026] n. K [Si(OH) ][H MoO ][Mo 0 1 *‘Constants in d[B-12-MSA] = 1 9 2 9 8 26 dt K + [Mo 0“” ] 2 8 26 d[8-12-MSA] RATE [31(0H)9] m where dt ' ebo ' K1 ‘ Eb0 ‘17—" and K2 =3 e = 803.3 M'1om'1 for 8-12-MSA at 930 nm, T = 25.0°c, I = 1.0 M 00 = 1.87 cm in the Beckwith stopped-flow observation cell. 117 Because of the propagated errors in calculating the solution pH from HALTAFALL, the experimental errors in measuring the initial reac- tion rate ( 51 RSD), and the narrow pH range in which only the 8 isomer of 12-MSA is fermed, the acid dependence for the B-12-MSA reaction was difficult to determine. Inapection of K1 and K2 values tabulated in Table XXII suggested inverse relationships between both K1 and K2 and the solution [H+] in HNO H280”, and H01011 solutions. To explain these 3’ experimental observations, Truesdale's'mechanism was modified to in- clude an acidic deprotonation of HzMoOl1 (or 1128103) and an acid- catalyzed rate-determining step with McBOZGu': k1 H MoO =HMo0" + H” 2 A k A -1 k2 31(011),1 + HMo0§;=EA k -2 9 k3 A + H“ + M08023—-—> Products (slow) 9. d[B-12-MSA] _ K1[Si(°H)9lm2M°°9HM°8°261 dt ' + H- - K2 + K3 [H ][M08026] + [HZMoOu][Mo8026] k k k -1 -2 -1 where K = k , K = , K = ——— 1 1 2 k2k3 3 k2 This modified mechanism accounted for all the observed dependences in H+, HZMOON’ and Moaogg. The rate law equation converged during KINFIT execution with the experimental data; however, the final values for the adjusted parameters had large standard deviations associated with them. Much of the problem originated from the nonlinearities of the graphs of 118 [HZMOOE] vs 1 shown in Figure 23, though the linearities RATE ° [Mo 0 ‘1 8 26 obtained from other equations were considerably worse. Because of both acid and molybdate dependence in the 8-12-MSA rate equation, reproducible acid concentrations between samples and silicon standards mixed with molybdate reagent are essential to insure the integrity and accuracy of silicate determinations. Not only are both the solution pH and isopolymolybdate concentrations affected, but CH and CM.also determine which 12-MSA isomer is complexed. Each isomer has a different molar absorptivity at a given wavelength. Two other comparisons of the constants in Table XXII should be 1 and K2 with [H+] in sulfuric acid solutions was not as large as in the other acidic media. Perhaps the noted. First, the variation of K buffering capacity of the HSOE-Soi- system regulated the hydrogen ion concentration as the molybdosilicate reaction progressed; the second pKa for H280111 is 1.51 in 1.0 M ionic strength solutions (see Chapter IV, Section A). Secondly, for any given molybdate concentration, the K1 rate constant for H23011 solutions was significantly greater than for any other acid solution over the pH range 1.2-1.8. Thus, because of its inherently faster initial reaction rate, a reagent molybdate solution acidified with H280,1 would provide the best sensitivity in a reaction- rate determination for silicate. 3. B-12-MSA Decomposition in Basic Solution To complete the characterization of the molybdosilicate chem- ical system, the decomposition kinetics of B-12-MSA in basic solution were studied. Acidic solutions containing B-12-MSA were mixed with excess hydroxide in the stopped-flow system, and. the decrease in Figure 23 . [H2M004] RATE (M sec.) Plots of B-12-Molybdosilicate Formation Rate vs. Molybdenum (VI) Species Concentration in Various Acidic Solutions 0.9 - 0.8 - 0.7 ‘- 0.6 - 0.5 1- H so .pH 1.5 0.3— 2 4 H2304, pH 18 0.2 '- Error bars are the some 01 on each curve I l l l J l 200 400 600 800 1000 l200 ' -1 [MOBOZ'G] (M ) 119 120 absorbance was measured at 930 nm. These molybdosilicate solutions were prepared by adding silicate to the acidified molybdate solution. Enough silicate was added to produce a substantial yellow color, yet leave the solution pH and .free molybdate concentration relatively unchanged. When molybdate was present in a hundred-fold excess, HALTAFALL showed that essentially all silicate was complexed as 12-MSA. The ranges of molybdosilicate, molybdate, and hydroxide concentrations employed were 0.1-0.5 mM, 0.01-0.05 M, and 0.1-0.5 M, respectivelyu Because the 8-12- MSA decomposition occurred rapidly, the fastest data acquisition rates available on the stopped-flow system had to be used. One data point was measured each millisecond, and the reaction time spanned only 100 ms. Refractive index changes caused by temperature changes in the stopped- flow system were negligible, as evidenced by an absorbance change of less than 0.002 when 0.1 M HNO was mixed with 1.0 M NaOH. 3 The B-12—MSA decomposition kinetics follows the rate law below. 1d[B-12-M3A] _ K1[E-12-MSA][OH'] dt - K2 + [OHTP (2) with K1 = 20.7 1 0.5 sec-1(HN03) and 20.1 1 0.5 see"1 (H0101) and K = 0.192 1 0.015 M (HN03) and 0.176 i 0.01" M (HClOu) 2 Here, [0H7 represents the excess hydroxide after all acid has been neutralized ([OH'] = COH- - CH+). A possible mechanism for the basic hydrolysis, without hydrogen or oxygen atoms listed, is: k 1 B-SiMO -<————-I 12111 _ k I + 0H AProducts 121 If intermediate I is assumed to be a steady-state intermediate, the k above rate law is reproduced, with K1 = k1 and K2 = Ell. Furthermore, 2 within experimental error, the rate law constants do not vary signifi- cantly with different acidic media. The experimental errors associated with measuring the spectro- photometric transmittances and calculating the rate constants are small because of the large decreases in absorbance observed as B-12-MSA decomposes. However, several model errors arise since the reaction rate is observed to increase slightly for a few milliseconds before decaying exponentially thereafter. This observation could not have resulted from an initial lag period for the decomposition to begin, because the 7 ms delay time between the stopped-flow push and the start of the data- taking sequence should have covered such a lag period. Thus, the reaction speed that can be measured accurately is limited. On the other hand, if the intermediate I also absorbs 930 nm radiation, the apparent increase in the reaction rate results as the steady state in I is established. The error in the measured initial rate is expected to be less than 51 since steady state is attained quickly. The 01 isomer of 12-MSA is an attractive possibility for the intermediate since o-12-MSA absorbs 930 nm radiation, though consider- ably less than the B isomer. Also, the 01 isomer predominates in solutions slightly more basic (pH 3.8-9.8) than those that favor the B isomer (pH 1.2-1.8). In addition, the equilibration between B and a isomers does not consume any hydroxide equivalents or change the stoichiometry of Mo and Si atoms in the polyanions. As experimental support, the decomposition rate is independent of the free molybdate 122 concentration and appears to be nonlinearly dependent upon the excess hydroxide concentration. If the B-12-MSA equilibrium had involved a deprotonation, then a linear, first-order hydroxide dependence would have been indicated. The reversibility of the B to a isomeric conver- sion has not been reported, however, and the constants (equation (2)) give no clue as to what the "equilibrium constant"«-:-l1 might be. C. 12-MPA Decomposition Kinetics Studies 1. Acidic Decomposition Studies Since hydrogen ions were released into solution as 12-MPA formed, a kinetics study of 12-MPA decomposition when mixed with a concentrated acid solution provided information about the molybdophos- phate reaction in the reverse direction and a comparison with the results in the forward direction. In this study, the ranges of 12-MPA, molybdate, and H+ concentrations were 0.02-0.20 mM, 0.01-0.05 M, and 0.75-2.00 M, respectively, after stopped—flow mixing. For molybdophos- phate solutions in which phosphate was added to acidified molybdate solutions ([H+] = 0.50 M prior to mixing with the acidic reagent), the general shape of the 12-MPA decomposition profile at 930 nm showed an exponential decay in the absorbance to the final value, as expected. In fact, a semilogarithmic plot of 19(At-A.) versus time was linear over the entire time range (10 s) with no initial curvature as the reaction began. In contrast to the 12-MPA formation reaction, its decomposition followed a simple exponential function. Preliminary reaction rate measurements revealed first order dependence of the decomposition rate upon the initial 12-MPA concentration, also as expected. Surprisingly, no dependence upon the H+ concentration was observed, which meant that 123 the rate-determining step involved the dissociation of molybdate from a molybdophosphate and that the molybdate species accepted hydrogen ions in subsequent, rapid equilibrium steps not related to the rate-deter- mining step. The general shape of the 12-MPA decomposition profile showed anomalous behavior for solutions in which 12-MPA was prepared from the solid reagent. In a typical reaction the 930 nm absorbance increased significantly over a time period that decreased with decreased [12-MPA] and increased [H+], before the absorbance decreased with time. This initial absorbance increase interfered with decomposition rate measure- ments as a proportionately higher amount of 12-MPA was formed at the same time that the excess H+ decomposed the complex. No molybdate or acid was added to these 12-MPA solutions, only enough Na01011 to bring the ionic strength to 3.0 PL. Subsequent experiments showed that the initial absorbance increase is indeed due to additional 12-MPA being complexed in the strong acid solution; in the kinetics run where [H+] = 1.00 M, the final )130 nm absorbance was greater than the initial absorbance (the pH of the unmixed 12-MPA solutions was about 3). Because the absorbance increased and then decreased instead of decreas- ing over all time, the formation of 12-MPA occurred at a faster rate than the decomposition under these experimental conditions. In addi- tion, as explained in the next paragraph, the 12-MPA decomposition was inhibited during the course of the reaction as the concentration of unbound molybdate in solution increased. In solutions where pho3phate was added to acidified molybdate to form 12-MPA, the 12-MPA decomposition rate varied inversely with 129 molybdate concentration. However, none of the linear equations such as m m 2 [H2Mo206 *‘J [EMo03‘1 very well. Several nonlinear equations were tried. One unsuccessful RATE = + b or RATE : + b fit the experimental data candidate was RATE = k1[12-MPA]-k_1[HMoO3+][11-MPA], which was derived with the 12-MPAé11-MPA equilibrium as the rate-determining step and 11-MPA present in significant concentration at the start of the reac- tion. One nonlinear equation reproduced all the measured decomposition rates to within 5% error; this equation is: d[12-MPA] K1 K2 - ( + —--—-) [12-MPA] dt - 2+ + [H2M0206 ] [HMoO3 ] K1 = (2.21 1 0.06) x 10-uM/sec; K2 = (9.69 1 0.20) x 10'"M/sec for 111103 solutions 1; K1 = (2.15 1 0.08) x 10' M/sec; K2 = (9.99 1 0.17) x 10"”M/sec for 11010,1 solutions Because the decomposition rate equation involves two terms, a branched chemical mechanism is indicated. The mechanism below repro- duces the above rate equation with K1 = k1K10K11KiKd and K2 = k2K11Ka. K11 3'.......:; 3- POu(M003) POu(Mo03)11 + MoO 12‘<———' 3 K10 3-.......:; 3- POu(M003)11 ‘r—————. P09(M°03)10 + MoO3 K a MoO3 + 11+: HMoO'; K 2HMo0; i HsMo20§+ 125 k P011(Mo03)?6 ———1——) Products POu(MoOB)?1i—-—E§—->»Products According to this mechanism, the acid decomposition occurs through both 11-MPA and 10-molybdophosphate (10-MPA) intermediates; equilibrium approximations on these intermediates reproduces the observed 12-MPA and molybdate dependences. Since [Mo03] is a constant linear multiple of [HMOOE], there is still no dependence upon [H+]. One implication from this mechanism is that 10—MPA coexists in equilibrium with 11-MPA and 12-MPA in higher acidity solutions, even when molybdate is in a hundred-fold excess over phoSphate. Even so, the molybdophosphate equilibrium constants determined over the lower acidity range 0.2 M‘: [H+] f 0.5 M in Section A, Chapter IV should not be affected signifi- cantly since 10-MPA would not predominate in, these lower* acidity solutions. 2. Decomposition with Excess Phosphate In this set of kinetics studies, the conversion of 12-MPA to the dimeric 9-MPA when 12-MPA.was mixed with a large amount of’phosphate was observed. Preliminary measurements revealed a first-order depen- dence upon the 12-MPA concentration. A second-order dependence would have been observed if the rate-determining step had involved a dimeriza- tion such as between two 9-molybdophosphate monomers. Also, plots of ln(At-Am) versus time were linear over the entire time range for most of these kinetics runs, with no curvature indicative of an initial lag period. Thus, as for the acid decomposition kinetics, the 12-MPA to 9- MPA conversion kinetics followed a simple exponential function. The 126 linear equation that best fit the experimental data for ph03phate 1 m dependence was RAT-E- - 1131,0141 + b, which indicated that the rate determining step is not the initial dissociation of molybdate from 12- MPA. The molybdate dependence was more complex. The equation.fii%§ = m[HMoO'§]3 + b described the rate data in lower acidity solutions (0.90- 1 m + 0.50 M 11*) while = _— + b fit the data for 0.75 M H RATE [HMo0*]3 solutions. As was the case in éhe acid decomposition studies, Mo(VI) dimers could be substituted for HMoO§ in the equations with the expo- 1 _ 2+ 3/2 nents halved (i.e. HATE - m[H2M0206 1 + b) and still give the same linearity. The observation of these third-order molybdate dependences and the appearance of trimeric molybdates arranged tetrahedrally about the central phosphate in the 12-MPA structure61 implies that molybdate dissociation from molybdophosphates could occur with three Mo atoms at a time. Experiments to test the acid dependence of the 12-MPA conversion exhibited zero-order kinetics with respect to [H+], as was observed in the acid decomposition studies. Either the free molybdate species accept or donate hydrogen ions in rapid equilibrium steps not related to the rate-determining step, or the errors associated with the initial rate measurements are so large that the acid dependence could not have been detected. From all these experimental observations and linear functions, a consecutive step mechanism was formulated in which there is dissocia- tion of molybdate, combination with the second phosphate, and then molybdate complexation to form the dimeric 9-MPA. With steady state approximations on certain intermediates and equilibrium approximations on the others, various forms of the rate law equation were derived for 127 use during KINFIT execution, and the constants m and b from the linear equations were used as the initial estimates for the KINFIT adjustable parameters. The best equation is: + 6 11112411,“ - K1 [12-MPA] [1132011] [HMoO3] dt - + 3 + 9 +16 KZIHMoO3] + K3[HMo03] + [H3PO11HHMo03 (3) and the rate law constants K1, K2, and K3 are tabulated in Table XXIII for HNO3 and H010,1 solutions. The residuals (differences between the theoretical and experimental initial reaction. rates) appear' to be randomly distributed over the entire range of initial reaction rates measured. Thus, in spite of the complexities of this molybdophosphate system, the given rate equation converged satisfactorily during KINFIT execution. The chemical mechanism consistent with the rate equation, with hydrogen and oxygen atoms omitted from the chemical species, is shown below. Mo1 + Mo1;;:::fi2 Mo2 __>. 1 + M02?— M03 k1 1 012 ?— PMo9 + Mo3 k-1 Mo PM k2 P + 13mg: P2Mo9 -2 k 6 Mo + P2Mo ——3%P2Mo 9 15 128 Table XXIII Rate Law Constants for 12-MPA Conversion to Dimeric 9-MPA Obtained From KINFIT Execution T = 25°C, I = 3.0 M + 6 K1 [12-MPA ] [HBPOI1 ] [511003] -d[12-MPA] _ dt ' + 3 + 9 + 6 K2[HMoO3] + K3[HMo03] + [HBPOu][HMoO3] HNO3 Solutions HC109 Solutions K1 = 0.215 11.029 sec-1 K1 = 0.272 1 .012 sec"1 K2 = (2.8 1 2.9) x 10"9 M" K2 = (3. 1 3.) x 10'10 M" K = (1.9 1 1.1) x 10" M‘2 x = (9.3 1_0.6) x 10" M‘2 129 RAPID A P Mo M°15 \— 2 18 3 Mo + P2 From this mechanism steady state approximations on intermediates PMo9 and P Mo and equilibrium approximations on M02, Mo3, P2Mo15, and P2Mo12 2 9 are made. The rate equation (equation (3 )) is reproduced with K1 = k1, k k k K = _;_1__:_§’ and K = -'-1-. This mechanism does not elucidate whether six 2 k2k3 3 k2 moles of Mo(VI) monomers, three moles of dimers, or any combination thereof combines with P2Mo9 in the k3 step. The results from the 12-MPA decomposition studies are incorporated into the overall chemical scheme in Table XXI (Section A). These decompositions proceed via the same pathway up to formation of the monomeric 9-MPA intermediate. The 12-MPA acidic decomposition follows a simple exponential equation because the step preceding 9-MPA forma- tion is rate-determining at 25°C and [H+] greater than 1.00 M. When 0.50 M < [H+]<1.00 M, the association reaction between H3PO,1 and Mo(OH) 6 occurs as slowly as the other rate-determining step so that the 12-MPA formation kinetics is observed to follow a double exponential equation. With higher phosphate concentrations in the 12-MPA to dimeric 9-MPA conversion studies, the steps to associate the second phosphate or additional molybdate species become rate-determining at 25°C, and simple exponential kinetics is observed. 3. 12-MPA Decomposition in Basic Solutions A kinetics study of 12-MPA decomposition in concentrated hydroxide solutions was attempted. In spite of the fastest stopped-flow data acquisition rates where one data point was measured per millisecond over a 0.100 s reaction time, the 12-MPA decomposition appeared to be complete by the time the solutions were mixed and delivered to the 130 observation cell. Even though the initial 12-MPA concentrations varied from 0.02 to 0.20 mM, the 430 nm absorbances decreased by nearly the same, unexpectedly small amounts (less than 0.1 absorbance units in all cases). The shape of the decomposition kinetics profile was peculiar; an initial increase in absorbance followed by a decrease with time was observed. A blank kinetics run in which 1 M H+ solution was mixed with 3 M NaOH (for all solutions I = 3 M, but no 12-MPA was present) showed the same behavior. Thus, what was observed in the experiments resulted from the heat of mixing the strong acid and strong base solutions. This heating effect changed the solution refractive index and caused an apparent absorbance change with time. D. Heteropolymolybdate Formation at Different Ionic Strengths The previous experiments of’ heteropolymolybdate .kinetics ‘were conducted at specific solution ionic strengths so that the relationship between measured reaction rates and Mo(VI) species concentrations could be deduced. Because the formation constants for the equilibria among cationic Mo(VI) monomers and dimers were determined in 3.0 M ionic strength, the 12-MPA kinetics studies reported earlier were carried out at I = 3.0 M. Similarly, the B-12-MSA kinetics were studied at I = 1.0 M because the concentration equilibrium constants for the anionic iso- polymolybdates were determined in solutions of 1.0 M ionic strength. Although no extrathermodynamic assumptions such as the Debye-Huckel equation are generally applicable to high ionic strength solutions, the variation of the heteropolymolybdate reaction rates with ionic strength was studied for two purposes. The increase or decrease in reaction rate with increased ionic strength was noted, and the data were examined for 131 changes in the overall kinetics profile as the ionic strength was varied. The measured reaction rates were tabulated with ionic strength in Table XXIV. The initial B-12-MSA formation rate decreased while the maximum 12-MPA formation rate increased with increased ionic strength. With a larger concentration of ions, the increased 12—MPA reaction rate at higher ionic strength indicated that the rate-determining step involves a reaction between oppositely charged ions (Mo(VI) cation and a molybdophosphate anion), and the decreased B-12-MSA formation rate indicates a reaction between ions of the same charge (M08026n- and a molybdosilicate anion) in the rate-determining step. The plots of ln(Am-At) versus time revealed no changes in the overall kinetics profiles; the B-12-MSA formation followed a simple exponential function while most 12-MPA formation reactions followed a double exponential function, regardless of the solution ionic strength. These complexa- tion reactions do not appear to be as sensitive to ionic strength variations as the conversion from B-12-MSA to a-12-MSA.6n Neverthe- less, the variations are significant enough to emphasize control of ionic strengths among samples and standards for phosphate and silicate determinations that use reaction-rate measurements. 132 Table XXIV 12-MPA and B-12-MSA Formation Rates at Different Solution Ionic Strengths + . CP or CSi - 0.1 mM, CMo - 0.02 M, [H ] - 0.50 M for 12-MPA experiments, pH = 1.5 for 3-12-MSA experiments, H0101 acid medium, T : 25.0°c. 12-MPA B-12-MSA Ionic Maximum Ionic Strength Rate Strength In1tial Rate (M) (sec )(1=430nm) (M) (sec )(A=930nm) 1.50 0.0168 0.50 0.00257 2.00 0.0195 1.00 0.00219 2.50 0.0212 1.50 0.00180 3.00 0.0267 2.00 0.00147 Chapter VI. Simultaneous Reaction-Rate Determination for Phosphate and Silicate A. Introduction and Literature Background As previously mentioned in Chapter I, analysis methods based upon measuring chemical reaction rates offer several advantages such as shorter analysis times and elimination of time-invariant interferences. The difficulties associated with reaction-rate methods can be allevi- ated through good instrumental designs, computer automation of instru- mental tasks, judicious choice of the chemical systan used in the analysis, and proper preparation of the samples, standards, and rea- gents. All these considerations help insure that the measured reaction rate will be proportional to the analyte concentration. The methods used in these determinations differ with the experi- mental conditions such as the analyte concentrations relative to the concentration of the reacting species.67 The broad categories of kinetics methods are the pr0portional equations method, the graphical extrapolation methods, masking methods, and methods involving changes in the kinetics of the system. Masking methods generally involve shifting the equilibria of interfering species or the equilibria among several analytes that react with the same species.68 Other methods that 133 1311 alter the kinetics of the system help the chemical analysis either by accelerating or decelerating the reaction rates to a speed that can be measured more precisely or by differentiating one reaction rate rela- tive to another rate so that other selective differential kinetics methods can be applied.69’7° In most cases, these two methods cannot differentiate rates of reaction of mixtures. The proportional equations method and the graphical extrapolation methods are used more generally, and each 71 Because the method has its own advantages over the other method. proportional equations method is based upon a constant fraction of the analyte being reacted at any given time, regardless of‘ the initial concentration, this method is less limited with respect to relative initial concentrations of the analytes or to their relative reaction rates than the graphical extrapolation methods. The proportional equations method involves less work and time, which is more suitable for analysis of a large number of samples. The total concentration of all analytes reacting with the same chemical reagent does not need to be determined. In addition, chemical systems with complex kinetics (such as pseudo first-order) may be employed, and mixtures with several components may be analyzed. On the other hand, graphical extrapolation methods offer certain advantages because the rate constants do not need to be known prior to the analysis. Thus, experimental conditions do not need to be controlled so carefully for all the analyses, and samples containing varying amounts of catalysts can be analyzed more easily. 135 Some mixtures contain components that react at significantly dif- ferent rates with the same chemical reagent. In such a case, the analysis of the mixture is simple because only one species will be reacting at a significant rate over certain time intervals. In order to perform differential kinetics procedures to determine two analytes in a mixture, we need to be able to measure the slower chemical reaction proceeding at its maximum rate while the faster reaction has already reached equilibrium. If the kinetics is pseudo first-order with respect to both analyte concentrations, the reaction rate measured at the start of the reaction corresponds predominantly to the faster reaction. Since both phosphate and silicate react with molybdate in acid solution, the presence of silicate would interfere with determinations of’phosphate, and visa-versa. Several approaches have been presented to avoid these interferences. The heteropolymolybdates can be extracted selectively with varying compositions of n-butanol, acetate esters, and chloroform in the organic phase,72-75 but the multiple extraction procedures are cumbersome and introduce considerable error. The use of oxalate or tartrate masks the presence of either 12-molybdophosphate or 12-molybdosilicate, depending upon whether the tartrate is added before or after the heteropolymolybdates are complexed.76 However, the analy- sis sensitivity is drastically reduced. Acetone may be used to enhance the 12-MPA molar absorptivity, but the absorptivity does vary with the acetone concentration present.76 Reduction.methods to heteropolymolyb- denum blues suffer problems of slower reaction times, deviations in Beer's Law (too much analytical sensitivity), unstable reagents and products, and high blank absorbances. Another procedure measures the 136 12-MPA formation and then the combined 12-MPA and 12-MSA concentrations in.a less acidic solution.57 As pointed out in the previous chapter,13- 12-MSA, 12-MPA, and other heteropolymolybdate concentrations vary with acidity, so this procedure does not produce valid results. Neverthe- less, in all these experimental procedures, at least two separate experiments have to be performed to distinguish 12-MPA from 12-MSA formation. B. Experimental Section All the kinetics experiments were performed automatically with the Beckwith stopped-flow spectrophotometer (see Chapter I, Section C). The computer programs described in Chapter III, Section A were used to calculate the analytical acid and molybdate concentrations required to give the prescribed H+ and Mo(VI) species concentrations. Thus, 0.05 M molybdate reagent solutions were prepared with 0.11 M and 0.5 M H+ present from HNO and HClO,1 stock solutions and with pH 1.5 and 1.8 from 3 HClOl1 and H SO media. In the preliminary kinetics studies, the 2 9 molybdate concentration was varied from 0.01 M to 0.05 M. The sample solutions for phosphate and silicate determinations were prepared with both species present. In a Latin square experimental design, the phosphate concentrations varied from 0.01-1.00 mM with the silicate concentration held constant. The silicate concentration was varied successively from 0.01-1.00 mM for each series of phosphate solutions. C. Kinetics of B-12-MSA Formation in Strong Acid Solution The kinetics rate law equation, constants, and chemical mechanism for 12-MPA formation in strong acid solutions (pH < 0.9) is reported in Chapter ‘V. In the low acid limit where 0.3 M ‘ 20.23.8 C=HC GO TO 15 SHLT=(6.0-HC*CH*((HC+4.8*CH)/(CR+HC))-2.0$CH-FC~CD*FC**2*(1.8/ i(C2*HC)+4.0+9.0*C3*HC))X2.8 CHLL CHHIM (’HOOUT’) END 156 MOOUT.FT C THIS PROGRHH NEITES OUT THE EQUILIBRIUM CONCENTEHTIONS OF RLL C HO SPECIES HND H+ IN STRONG HCID SOLUTION. COHHON C1.C2.C3.CD.CH.CH.HC.FC.SHLT.CH HRITE(1.1?) CH 1? FORHHT(’TOTHL IRCIDJ = ’.E15.4.’ H’) NRITE(1.18) CH 15 FORHHT(’TOTHL IfiOLVBDENUHJ = ’.E15.4.’ H’) NR1TE(1.EO) FC 3? FORHHT(’[HHOOE+J = ’.E15.4.’ H’) HRITE(1.35) HC 55 FORMRT(’IH+J = ’.El$.4.’ fi’) FC22=CD*FC**2 NRITE(1.36) FC22 36 FORHHT(’IH2HO2OSJ = ’.E15.4.’ H’) FC12=CD*FC**2*1.0/(C2*HC) NEITE(1.3?) F012 3? FORHHT(’IHHO2OG+J = ’.E15.4.’ H’) FC32=CD*FC**2*C3*HC NRITE(1.33) F032 33 FONHHT(’IH3HOZOSJ FCOl=FC/(C1*HC) NRITE(1.39) FCOi ’.E15.4.’ H’) 39 FORHHT(’IMO(OH)SJ = ’.Ei$.4.’ fi’) NEITE(1.40) SHLT 43 FORHHT(’INHCLO43 REQUIRED FOR 3.8 IONIC STRENGTH = ’.E15. IF (CH) 41.41.42. 41 CRLL CHHIN (’HOLVB’) 42 CHLL CHRIN (’HOSO4’) END 157 (“'21 1‘; 1":- 0 ("It 1‘? w~ Lam m a: [0‘1 ,J 1'.“ 0'1 NW! N1 0‘ 0131-? In. [-0 US 13' EXPO.FT THIS PROGRHM FITS OHERHLL CHEM. KINETICS DHTH TO H SIMPLE EXPONENTIHL CURUE Y= H + B EXP(CT) OR H SUM-OF-THE-EXFONENTIHLS CURVE V= H + B EXP(CT) + D EXP(ET). IN THE LHTTER CHSE THE USER MUST SPECIFY H TIME INTERNHL IN NHICH H SIMPLE EXPONENTIHL MODEL 15 FOLLONED HS HELL HS HN INTERVHL NHEN HLL TERMS HRE SIGNIFICHNT. THE F NHLUE HT INFINITE TIME MUST HLSO BE SPECIFIED. DIMENSION TIME£188).H8(188).SDEV(188) REHD(i.iO) FNHME FORMHT(’HHICH FILE NOULD YOU LIKE TO NORM NITH2’H6) REHD(1.15) ICHECK FORMHT(’HRE YOU SURE? 1=YES. 8=NO 2’12) IF (ICHECK’i) 5.16.5 REHD(1.25) MODEL FORMHT(’NHICH TVPE OF MODEL NOULD VOU LIKE TO FIT?’/’O = SIMPLE i EXPONENTIHL’X’i = SUM-OF-THE-EXFONENTIHLS’f’2 = STOP’K’F’I2) IF (MODELri) 2?.2?.3i CHLL IOPEN (’FLP2’.FNHME) REHD(4.20) IRHTE.NFHVG.ITIBD.NDP.IDLP.NSFPR.NDPF FORHHT(?I4) REHD(4.21) (TIME(I).HB(I).SDEV(I).I=1.NDP) FORMHT<3E13.6) REHD(1.26) HMHX FORMHT(’MHX.(MIN.) P-VHLUE FOR THIS RUN?’Fi0.0) IF (MODEL-i) 38.33.31 REHD(1.35) TMIN FORMHT(’ENTER THE TIME INTERVHL NITH ONLV ONE SIGNIFICHNT EXPONENTIHL 1 TERM’/’BEGINNING TIME (SEC.) ?’FI0.0) REHD(1.36) TMHX FORMHT(’ENDING TIME (SEC.) 2’F18.0) REHD(1.3?) ICHECK FORMHTC’HRE VOU SURE? i=PES. O=NO 2’12) IF (ICHECK~1)30.39.3O PHGE 158 39 0 131 13". ‘xi N=8 SUMX=8.8 SUMX2=8.8 SUM?=8.8 SUM?2=8.8 SUMX?=8.8 DO 48 I=1.NDP IF (TIME(I)~TMIN) 48.41.41 IF (TIHE(I)-TMHK) 42.42.48 SUMX=SUHX+TIME(I) SUMX2=FUMX2+TIME(I)*m2 SUM?=SUM?+HLOG(HBS(HMHX-H8(I))) SUM?2=SUM?2+HLOG(HBS(HMHX-HB SUMX2=5UNX2+TINE(I)**E SUHV=5UNT+HLOG(HBS(HHHX+THETH2*EXP(T3*TIHE(I))-HB(I))) 5UN?2=5UH?2+HLUG(HBS(RNHX+THETH2*EXP(T3*TIHE(I))-H3(I)))**2 SUWXT=SUNX?+TINE