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L meszstormwegtéémh a MICHIGAN STATE fineveasm V- “‘ :3; , LAWRENCE CHUNG LEE su ' . r5: , ,v‘ Q: a. fi’ This is to certify that the v, 4"!- thesis entitled THEORY OF THE COMPENSATION LAW AND ITS APPLICATION IN BIOLOGY presented by Lawrence Chung Lee Su has been accepted towards fulfillment of the requirements for PhD degree in BiODhVSiCS Dew.“ Major professor Date M 0—7 639 THEORY OF THE The study 1 hampered by dig and evaluation more important been found to t Cation of the c Sation temper31 interval and ti A Statist equations prov elistence of C Strated With t A“Other 11 applications f to demmstrat. confidence in less depen den ABSTRACT THEORY OF THE COMPENSATION LAW AND ITS APPLICATION IN BIOLOGY By Lawrence Chung Lee Su The study of reaction kinetics using the compensation law has been hampered by disputes over a reliable criterion for the determination and evaluation of the existence and temperature of compensation. The nmre important criteria have been studied. Exner's method (1970) has been found to be the most appropriate. The method involves the appli- cation of the concept of least squares in the evaluation of the compen- sation temperature. Explicit rules for the evaluation of a confidence interval and the existence of compensation were inadequate. A statistical method, called the F-test, is proposed and explicit equations provided for the evaluation of a confidence interval and the existence of compensation. Its application and effectiveness is demon- strated with two practical examples from.published data. Another method, called the computer method, has been developed for applications in experiments which a researcher is planning to carry out to demonstrate the existence of compensation in a process and to find a confidence interval for the compensation temperature. This method is less dependent on statistics but requires data with high precision. It is also usable f tures. This me! which will be di The conduc The comductivi t‘ of Hemoglobin w is accomplished plots are obtair Applicatic shows that the compensation t: 5 ) Mo K 10' It is con involved in pr ately, many bi compensation a criterion Whip Finally . 05 the confid tion of exper (”Stance of t POimt. Lawrence C.L. Su is also usable for processes which have negative compensation tempera- tures. This method is applied to a set of data collected by the author which will be discussed next. The conductivity of a hemoglobin-ammonia system has been studied. The conductivity is measured in terms of the current across a pellet of Hemoglobin with an applied voltage of ten volts. Current variation is accomplished by changing the amount of ammonia adsorbed. Arrhenius plots are obtained with a temperature range of from 20°C to 50°C. . Application of Exner's equations (1970) and the computer method shows that the conductivity data satisfies the compensation law with a compensation temperature of -9380K and a confidence x-interval of (-139 x 10's, -74 x 10’5). It is concluded that some biological entities very probably are involved in processes which satisfy compensation even though, unfortun- ately, many biological processes which were reported to have satisfied compensation are in fact the result of the use of the entropy-enthalpy criterion which has been shown to be unreliable. Finally an equation has been found which establishes the magnitude of the confidence interval for the compensation temperature as a func- tion of experimental errors, the experimental temperature range and the distance of the compensation temperature from a predetermined reference point. in THEORY OF THE COMPENSATION LAW AND ITS APPLICATION IN BIOLOGY By Lawrence Chung Lee Su A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Biophysics 1973 To Ruth, Billy and John 11 lwish to e Heldmam for his this thesis is l Duggan without v sihle and to th my Thesis Commi Pant, T. Hamilt ACKNOWLEDGMENTS I wish to express my sincere gratitude to Professor Dennis Heldman for his continual encouragement during the past year when this thesis is being prepared. My gratitude also goes to Dean Duggan without whose guidance this thesis would not have been pos- sible and to the members of the Biophysics Department who are on my Thesis Conmittee. I also wish to thank Drs. J. Stapleton, H. Pant, T. Hamilton and Mrs. L. Su for their valuable discussions. iii AMOWLEDGEMENT LIST OF TABLES LIST OF FIGURE: CHAPTER I. INTRODUC 1) comes ii) soups ) cones iv) NEED ) HOPE v1) OBJE( 11. THEORY . 1) some} 1) TI 2 C: moomopg oouo vvv TABLE g§_CONTENTs ACKNOWLEDGEMENTS LIST OF TABLES . LIST OF FIGURES. CHAPTER I. INTRODUCTION . i) COMPENSATION IN HETEROGENEOUS CATALYSIS . ii) COMPENSATION IN SMALL MOLECULES . iii) COMPENSATION IN BIOLOGICAL ENTITIES . . iv) NEED FOR IMPROVED METHODS IN COMPENSATION STUDY . v) HOPE IN COMPENSATION STUDY OF BIOLOGICAL ENTITIES . vi) OBJECTIVES OF THE THESIS. . . . . . . . . . II. THEORY . 1) COMPENSATION LAW. 1) The Entropy- Enthalpy Criterion for Compensation. 2) Criticism.of theAAHT and AS* Method. 3) Exner' s Criticism of the Entropy- -Enthalpy Method . 4) Other Sources of Possible Error. . . . 5) Exner' 3 log k1 vs log k2 Method. . . . 6) Criticism.of Exner' 3 log k1 vs log k2 Method . 7) Other Methods for the Study of Compensation. 8) Exner's Statistical Method . . . . . ii) EXNER' S STATISTICAL ANALYSIS. . . . . . . . . 1) Derivation of Exner' 3 Equations. . . 2) A Discussion of Exner' 3 Statistical Method . iii) ANALYSIS OF DATA USING THE F-TEST . iv) COMPUTER STUDY OF COMPENSATION. III. EXPERIMENTAL............... 1) EXNER' s EQUATIONS . . . . . . . . . . . . 1) Flow Chart for Exner' 3 Equations . . . 2) Types of Analysis to be Carried Out. iv Page iii vi vii CDVU'ILAJNt—e H 10 10 ll 12 20 22 24 25 26 27 28 29 33 36 39 44 44 45 45 Ii) THE 00‘ 1) F101 2) We iii) MEASUR IV. RESULTS 1) ANALYs 1) Dat 2) Dat ii) COMPU'] 1) Re] 2) An: 3) De iii) ANALY: 1) Ex 2) C01 V. CUNCLUS 1V. RECOMME LIST OF REFER APPENDICES . APPENDIX A APPENDIX B APPENDIX 0 APPENDIX D Page ii) THE COMPUTER METHOD. . . . . . . . . . . . . 49 1) Flow Chart for the Computer Method. . . . . . . . . . 50 2) Types of Analysis to be Carried Out . . . . . . . . . 53 iii) MEASUREMENT OF CONDUCTIVITY OF PROTEINS. . . . . . . . . 55 IV. RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . 62 1) ANALYSIS OF EXISTING DATA. . . . . . . . . . . . . . . . 62 1) Data of Barnes et a1. . . . . . . . . . . . . . . . . 63 2) Data of Luedecke. . . . . . . . . . . . . . . . . . . 67 ii) COMPUTER ANALYSIS OF GENERATED DATA. . . . . . . 68 1) Relationships Concerning the Confidence Interval. . . 70 2) Analysis of the Graphs Involving Su and su. . . . . . 82 3) Determination of Whether Compensation is Satisfied. . 90 iii) ANALYSIS OF CONDUCTIVITY DATA. . . . . . . . . . . . . . 103 1) Experimental Results and Discussion . . . . . . . . 103 2) Compensation Study with the Computer Method . . . . . 104 V. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . 110 IV. RECOMMENDATIONS. . . . . . . . . . . . . . . . . . . . . . 113 LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . 116 APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 APPENDIX A . . . . . . . . . . . . . . . . . . . . . . . . . . 118 APPENDIX B . . . . . . . . . . . . . . . . . . . . . . . . . . 119 APPENDIX C . . . . . . . . . . . . . . . . . . . . . . . . . . 123 APPENDIX D . . . . . . . . . . . . . . . . . . . . . . . . . . 129 V Table (4-1) Estimat by the (4-2) The val (4-3) Values (4-4) Confidi il°/. wi' (4-5) x-inte variou Table (4-1) (4-2) (4-3) (4-4) (4-5) (4-6) (4-7) (4-8) (4-9) (4-10) (4-11) LIST OF TABLES Estimated compensation temperatures of protection by the HPoa ions. A The values of H , T , S and S . o c o 00 H) fl Values of x , o c Confidence intervals for D-values of i5%, i2% and i1% with various values of TC. x-intervals for D-values of i5%, i2% and il% with various values of Tc' . . . . f(Tc, D) for various values of TC and D. R-values for various values of Tc and D . R/D values for various values of Tc and D. The largest So-values for various values of D and TC. Experimental points used for analysis. R , T , S and 8 obtained from conductivity data. . o c o 00 vi , S , and S from the data of Luedecke. o 00 Page 64 65 68 73 73 74 74 79 97 104 108 Plot < A p10 of P. amoun Plot Plot Two c withi regi< whicl are' Flow in E A pl reg: ment line the mini F101 the Figure (2-1) (2-2) (2-3) (2-4) (2-5) (3-1) (3-2) (3-3) (3-4) (3-5) (3-6) (3-7) (4-1) LIST OF FIGURES Plot of AH* vs AS*. A plot of l/T vs kd, where kd is the rate of killing of P. fragi, in various types of milk with various- amounts of fat content. (Luedecke, 1962). Plot of AE vs 00. Plot of l/T vs 0. Two of the ideal lines (solid) with the regions within which the experimental lines must lie (dotted regions). xk is the x-component of the point about which the changes in slope of the experimental lines are rotated. Flow chart for the calculation of the parameters in Exner's paper (1970). A plot of log k vs l/T showing the least-square regression lines (solid lines) of the actual experi- mental points (*) and the least-square regression lines moved in such a way as to make them all pass the point (x0, yo) while the So-value is minimized (dotted lines). . . . . . Flow chart for the generation of points used in the computer method. Flow chart to join parts 1 and 2 of the program. A plot of k vs 1/T showing the relative positions of the ideal Tc's used. . . . . . . . . Schematic diagram of vacuum micro-balance apparatus. An illustration of the transformation from a plot of current vs amount adsorbed to a plot of current vs 1/T x 1000. . A plot of Sn vs T, using the HPOZ data of Barnes et al. (1969). . . . . . . . . . . . . . . . . . vii Page 14 16 18 19 40 47 52 52 54 58 61 66 figure (4-2) A plot (4-3) Frequen temper (4-4) Freque x-vslu (4-5) Superp ill an (4-6) A plot values (4-7) A plot values (4-8) A plo (4-9) Aplo (4'10) A plo (4-11) Aplo comprv (4'12) Aplo for a (113) Am i1 perim havir reprt thel (1'14) ipll for 4'4) (pl Figure <4-2) (4-3) (4-4) (4-5) (4-6) (4-7) (4-8) (4-9) (4-10) (4-11) (4-12) (4-13) (4-14) (4-15) (4-16) (4-17) (4-18) (4-19) (4-20) A plot of Su vs T, using data from Luedecke (1962). Frequency histogram of the estimated compensation temperatures for Tc = 4000K and D = i5%. Frequency histogram of the estimated compensation x-values for Tc = 400°K and D = i5%. Superposition of histograms with D = i5%, i2% and i1% and Tc = 4000K. . . A plot of Su vs T when T = 4000K with various values of D for combination 11,001. Su vs T when Tc = 4000K with various D for combination 111. of of A plot values of S vs u for the data of Barnes et al. A plot u of S vs u for the data of Luedecke. A plot u of S vs u for the conductivity data. A plot u A plot of Su vs u for the conductivity data with u-axis compressed 8 times relative to that of Figure (4-10) . A A plot of So vs Tc’ with TC = 400°K and D = i2% for all the combinations. . . . . . . . . . . . An illustration of the relationship between the ex- perimental points * and points on the boundary lines having the same x-coordinates °. The solid line represents an ideal line and the dotted lines are the boundary lines. . . . . . . . . . . . . . . . . . A plot of S vs T , with Tc = 4000K and D = 0 or 2% for all the combinations. . . . . . . . . . . . . . _ o D, Tc — 400 K. A plot of largest So-values vs A plot of largest So-values vs d(xo, xk), D== i5%. A plot of largest So-values vs d(xo, xk), D== i5% on semi-log paper. . . . . . . . A plot of largest So-values vs d(xo, xk), I): 15% on log-log paper. . . . . . . . A plot of I vs % adsorbed for conductivity data. A plot of I vs 1/T for conductivity data. . . . . . viii Page 69 71 76 77 83 84 86 87 88 89 92 94 96 98 99 100 101 105 106 to exist in man Cvmstable repor persons catalys action is enhan (Ashore, 1963) imdmstrial app rents and its tion of sulphur proposed to exp The slim sites has been 1111 the fact 14111311 4111111 I. INTRODUCTION 1) COMPENSATION IN, TEROGENEOUS CATALYSIS _’ The compensation law or the isokinetic relation has been reported to exist in many chemical and physical processes. As early as 1925, Constable reported the existence of a compensation effect in a hetero- geneous catalysis which "occurs whenever the rate of a chemical re- action is enhanced by the presence of an interface between two phases." (Ashmore, 1963) Much research has been done in this area since it has industrial applications such as the synthesis of ammonia from its ele- ments and its oxidation to nitric oxide and nitric acid and the oxida- tion of sulphur dioxide to trioxide. A number of mechanisms has been preposed to explain this compensation effect. The distribution of sites theory using a Boltzmann distribution of sites has been suggested by Cremer (1955). This theory is compatible with the fact that many multicomponent catalysts exhibit compensation even though when very pure substances are used the compensation effect observed disappears. This, of course, suggests the possibility that the impurities present in the catalysts are responsible for the active sites. Cremer partially supported this theory with data which show that adsorption on catalysts depend on the temperature of catalyst- Pretreatment in accordance with a Boltzmann distribution of their surface centers. Eley et a1. (1957) favor the entrOpy of activation theory which leads to the camel surface structure how the entropy va also suggest that might be needed t experimentally . Quantum mech anism to account reaction rate clot in too slow using Iatively large ma tunnelling yield reactions which transition as an tunnelling calcu ii) CTII’ENSATIO Since 1930 for reactions (2‘ 41m equation 2 leads to the conclusion that the entrapy term must be sensitive to the surface structure of the catalyst although they cannot even determine how the entrOpy varies with the geometry of a given catalyst. They also suggest that an activated complex consisting of a number of atoms might be needed to accountiknrthe‘large range of entrepies encountered experimentally. Quantum mechanical tunnelling has also been suggested as a mech- anism to account for compensation (Born; 1930, 1931). However, the reaction rate obtained, for an atom to pass through the energy barrier, is too slow using the mechanism of tunnelling since the atom has a re- .latively large mass. The suggestion that it is the electron that is tunnelling yields a more reasonable reaction rate, although many of the reactions which exhibit compensation do not seem to require an electron transition as an essential step. Cremer (1967) has also published some tunnelling calculations recently. ii) COMPENSATION L11 SMALL MOLECULES Since 1930 many cases of compensation behavior have been reported for reactions (involving small molecules) whose rates satisfy the well- known equation kBT e-(AHI/RT - As*/R). (1-1) “KT These cases include processes of solvation of ions and non-electrolytes, hYdrolysis, oxidation-reduction, ionization of weak electrolytes, etc. (Lumry, 1970). In fact Leffler et a1. (1963) had already collected a list of almost one hundred cases which were considered to be of interest Ind whose correlatio which will he discus Leffler et al. relationships" base they claimed to hav relation. The theo dependent on the en The method states t between 1111* and AS* be obtained from t 111* vs AS*. The v into by using Equa often used assunpt Involved in the re lid) ClliTEllShTION More recently bed by certain to 1914) and protein 111 been reports 4111 11 3 and whose correlation coefficients, using the entrOpy-enthalpy method, which will be discussed in the THEORY, are 0.95 or better. Leffler et a1. (1963) prOposed a "theory of extrathermodynamic relationships" based on the above mentioned experimental results, which they claimed to have satisfied the compensation law or the isokinetic relation. The theory of extrathermodynamic relationships is strongly dependent on the entrOpy-enthalpy method for the study of compensation. The method states that compensation exists if and only if the relation betweenPAHT andPAST is linear and that the compensation temperature can be obtained from the slope of the straight line obtained from a plot of AHTk vs AS*. The values of AH:t and AS* are obtained from experimental data by using Equation (l-l). One important result of the theory is the often used assumption that when compensation exists the substituents involved in the reactions interact by a single mechanism only. iii) COMPENSATION IN_BIOLOGICAL ENTITIES More recently the isokinetic relation has been claimed to be satis- fied by certain reactions involving water solutions of proteins (Lumry, 1970) and proteins in general (Likhtenshtein, 1966). In fact it has even been reported to exist in biological processes such as the thermal killing of bacteria and phages by Likhtenshtein et a1. (1963, 1965), Rosenberg et a1. (1971) and Barnes et al. (1969). Lumry et a1. (1970) seem to think that the existence of the iso- kinetic relation in reactions involving water solutions of proteins is "real, very common and a consequence of the properties of liquid water as a solvent regardless of the solutes and the solute processes studied." It has been suggested that the phenomenon is a "consequence of the properties of water solutes effect than (limry, 1970) Stud csrboxypeptidase-A (Lipscomb, 1968). I protein can occu may affect the rat solving lysozyme a between these smal compensation may 5 physiological fun Likhtenshtei: in the area of c tempted to unify shoring that all pensstion tempera that compensatior recently, Likhte: notation temper not take on on) Rosenberg 4 prOperties of water which require that eXpansions and contractions of solutes effect changes in the free volume of the nearby liquid water." (Lumry, 1970) Studies using x-ray methods have shown in the case of carboxypeptidase-A that atomic readjustments do occur in the molecule (Lipscomb, 1968). It seems that small displacements in some regions of a protein can occur without causing it to unfold and these displacements may affect the rate of unfolding of the protein. Certain reactiOns in- volving lysozyme and hemoglobin are examples. Even though the relation between these small changes and the compensation law is not yet known, compensation may still play a role in the area of protein stability and physiological function. Likhtenshtein (1963, 1965, 1966) has contributed a number of papers in the area of compensation involving biological entities. He has at- tempted to unify catalytic and denaturation reactions of proteins by showing that all these reactions follow compensation with a common com- pensation temperature using the entrOpy-enthalpy method. He suggested that compensation plays an essential role in enzymic mechanisms. More recently, Likhtenshtein (1970) also proposed a theory in which the com- pensation temperature of all reactions satisfying the compensation law must take on only one or one of several given values. Rosenberg et al. (1971) proposed the hypothesis that protein dena— turation is the cause of thermal death in unicellular organisms. They showed as support a ARI vs AS* plot for data obtained from thermal killing of yeasts, viruses and bacteria which seem to follow "a compen- sation law with constants which are in very good agreement with the constants for thermal denaturation of proteins." Barnes et a1. (1969) found that the protection of Sindbis virus against thermal ina sation law with a c cases, depending on of the compensation entities . n) N_m_s_n ms, IMPR From what has existence of comps pensation temperat series of reaction beaver, unless t the existence of the compensation tervnl, it is use sation law. blhen the exi of heterogeneous that it was a tru experimental tern] hundred degrees 1 experimental tern (1955). In most ovate to determi compensation 15‘ range attainablr most processes 5 against thermal inactivation, using HPO4 and 804, satisfies the compen- sation law with a compensation temperature of 52.700 to 64.80C, for both cases, depending on the method of analysis used. They suggested the use of the compensation temperature as a physical constant for biological entities. iv) NEED FOR IMPROVEQyMETHODS l§_COMPENSATION STUDY From what has been said so far, it should be clear that the existence of compensation and the knowledge of the value of the com- pensation temperature, Tc, are useful in the study of a reaction or a series of reactions and the substances involved in the reactions. However, unless the method used for the study of compensation can show the existence of compensation with reasonable confidence and determine the compensation temperature within a relatively small confidence in- terval, it is useless to discuss about the application of the compen- sation law. When the existence of compensation was first reported in the study of heterogeneous catalysis, there was no doubt in the minds of chemists that it was a true phenomenon, since Arrhenius plots were used and the experimental temperature range covered was usually as large as several hundred degrees Kelvin. Sometimes the value of Tc even falls within the experimental temperature range such as shown on the paper of Cremer (1955). In most cases, a glance at the Arrhenius plots would be ade- quate to determine if compensation exists. With the application of the compensation law in other areas of study, the experimental temperature range attainable decreases drastically. This is due to the fact that most processes studied involve substances which cannot withstand such high temperatures reliability of vis mental temperature as a more reliable Unfortunately entropy-enthalpy n mtical transform: the All* vs ASt pl loped between the and those who fav shouted, with a ri temperature obta' This leaves no d for a precise st In the mean compensation in l and useful tool. change in temper Kelvin are very killing of bacte thirty minutes, determination 0: temperature is perature is not Thus there compensation, ‘ biological ent'. high temperatures as those encountered in heterogeneous catalysis. The reliability of visual judgement decreases with the width of the experi- mental temperature range and the entrOpy-enthalpy method was proposed as a more reliable substitute for the study of compensation. Unfortunately the researchers who advocated the use of the entropy-enthalpy method failed to realize the consequence of the mathe- matical transformation implicit in a change from the Arrhenius plot to the AH* vs AS* plot until Exner (1964) pointed it out. A dispute deve- loped between those who are committed to the entrOpy-enthalpy method and those who favor Exner's criticism of the method. Recently Exner showed, with a rigorous mathematical proof, that even the compensation temperature obtained with this method is "erroneous". (Exner, 1972) This leaves no doubt that the method should be rejected as a criterion for a precise study of compensation. In the mean time, researchers are beginning to use the concept of compensation in biological studies since it seems to be an interesting and useful tool. Most biological substances cannot withstand a large change in temperature. Temperature ranges of less than ten degrees Kelvin are very common in papers such as those dealing with thermal. killing of bacteria. Moreover, many bacteria replicate in less than thirty minutes, leaving the rate constant range rather limited. Visual determination of the existence of compensation and the compensation temperature is totally unreliable, especially if the compensation tem- perature is not within the eXperimental temperature range. Thus there is an urgent need for a better method in the study of Compensation, especially if the compensation law is to be applied to biological entities. A number of other methods have been proposed, but none of them are us Some of these metho 1:) none g courses Fortunately th usable in the study involves the appli obtained from the analysis should in improved reliabili to biological enti The use of Ar such as Equation ( exact description the Arrhenius plo might be introduce It is also free 01 tortions Exner (15 Actually Exm statistical analy to estimate the c related to it. '1 Ihich uses the st plots. lihen thi: need. 7 none of them are very satisfactory except Exner's prOposal in 1970. Some of these methods will be discussed in the THEORY. v) HOPE 111 COMPENSATION STUDY on; BIOLOGICAL ENTITIES Fortunately the statistical method prOposed by Exner (1970) may be usable in the study of the compensation law in biology. The method involves the application of statistical analysis to the Arrhenius plots obtained from the experimental data. This application of the statistical analysis should improve the reliability of the results obtained. The improved reliability may make the application of the compensation law to biological entities meaningful. The use of Arrhenius plots does not require an extraneous equation such as Equation (1-1) which, although very well-known, may not be an exact description of the rate constants obtained experimentally. Thus the Arrhenius plots are inherently free of any physical error that might be introduced by the use of an equation such as Equation (l-l). It is also free of the type of mathematical, transformational dis- tortions Exner (1964) described. Actually Exner's major contribution was not his suggestion to use statistical analysis , but in the derivation of the equations required to estimate the compensation temperature and to calculate the errors related to it. The equations, however, are applicable to the experiments which uses the same experimental temperatures for each of the Arrhenius plots. ‘When this condition is not satisfied, the equations can not be used. vi) OBJECTIVES g lhe following (i) To illustra method avai cially when logical ent N V To evaluat w v To deve10p existence the compen equations A k V To illustr situations u v To present of the he of adsorbs A a v To show by thesis the compensat: perature . A N v To derive design of vi) OBJECTIVES 92 THE THESIS The following is a list of the objectives of the thesis: (1) To illustrate that the statistical method is the best method available for the study of compensation, espe- cially when dealing with existing data involving bio- logical entities. (2) To evaluate the methods of analysis prOposed by Exner. (3) To deve10p new methods for the determination of the existence of compensation and for the evaluation of the compensation temperature based on some of the equations derived by Exner. (4) To illustrate the use of the new methods in practical situations. (5) To present a set of data dealing with the conductivity of the hemoglobin-ammonia system with various amounts of adsorbate and at various temperatures. (6) To show by the use of the new methods deve10ped in this thesis that the hemoglobin-ammonia data satisfies the compensation law with a negative compensation tem- perature. (7) To derive approximate equations which are useful in the design of experiments in the study of compensation. Before closing no doubt that compel small molecules stur compensation does e: this thesis is, the but to suggest new the compensation la the smallest confid Perature; since the given set of data : sation temperature 9 Before closing the INTRODUCTION, it must be mentioned that there is no doubt that compensation exists in heterogeneous catalysis. Even in small molecules studied in chemistry, the consensus of Opinion is that compensation does exist in some reactions at least. The purpose of this thesis is, therefore, not to prove the existence of compensation, 'but to suggest new methods to determine if a given set of data satisfies the compensation law with a fair amount of confidence and to calculate the smallest confidence interval possible for the compensation tem- perature; since the disagreement among researchers involves whether a given set of data satisfies the compensation law and what its compen- sation temperature is. 1) commnsnnon l_u The concept 0 can be represented may as to "compens Ifk is indeed con exact. In the cas which compensation abbreviated as Tc' More specific vhich vary with or example, many ch91 where k is the m absolute tempera t thalhl’, A31: is th II. THEORY i) COMPENSATION LAW The coucept of compensation can be applied to any variable k which can be represented as a product of two exponents which vary in such a way as to "compensate" each other so that k remains relatively constant. If k is indeed constant at a given point, the compensation is said to be exact. In the case of Arrhenius type equations, the temperature at which compensation occurs will be called compensation temperature and abbreviated as Tc' More specifically, the two exponents must depend on parameters which vary with certain experimentally variable conditions. For example, many chemical reactions follow the well-known rate equation kBT e-(Afli/RT - As*/R). (2_1) k=KT where k is the rate of the reaction, K is the transmission coefficient which is usually taken to be one, kB is Boltzmann's constant, T is the # . absolute temperature, h is Planck's constant, AH 15 the activated en thalpy, AS* is the activated entropy and R is the gas constant. 10 1) Iii Wig in the above c expad/a). If hot 3 temperature Tc’ t vhich is MTC) age lav is satisfied. The above Slt where t is a was experimental Var 1 or where W is agai for the “ii-stem is satisfied. I many authOrs s 1‘ 0f the medium 11 1) lgg_Entr021-Enth§lpy Criterion for Compensation In the above case, the two exponents involved are exp(-Afl}/RT) and expCAS*/R). If both.AH§/RT and AS*/R vary by the same amount, say @, at a temperature TC, then Equation (2-1) becomes k T 1: a: k(Tc) = K_§_.e'[KAH-/RTC + @) - (A8 /R + @X]. _ KFBT -(AHi/RT ) -@ As*/R @ '— h e C e e e .. If}: "(AHiz/RT -As*/R) - K h e c which is k(Tc) again. Thus k is unchanged at TC and the compensation law is satisfied. The above situation implies, mathematically, that -(AH*/RT - AS*/R) _ -t e C -e where t is a constant. This condition holds even though one of the experimental variables, say pH, is varied. The linear relation a AH*/RTC - as /R = t or A31;- (l/TC)AH* + tR (2-2) Where tR is again constant, is used by many researchers as a criterion for the existence of compensation or to claim that the compensation law is satisfied. Equation (2-2) is also called the isokinetic relation by many authors since the rate constant k does not change even though one Of the experimental variables is varied. The temperature Tc is the compensation tempe The above dis that a reaction or if a plot of A31: \ provides a method from the 310pe of 2) triti_ci.s_a 2f. Unfortunatel the existence of failed to appreci iectly reasonable solution of the 1 mentally determii Mi. If the tem' (ten to twenty d values are only With most availal by the solution the AHiASi‘lilane t0 yield a relat “'1“ due to a 1 The abOVe Consider the eq 12 compensation temperature or isokinetic temperature. The above discussion is the justification given for the assumption that a reaction or a reaction series satisfies compensation if and only if a plot of AH* vs As" yields a reasonably good straight line. It also provides a method for the determination of the compensation temperature from the slope of the straight line obtained. Unfortunately, the use of the.AH* vs.AS* plot as a criterion for the existence of compensation creates problems which some researchers failed to appreciate, especially since the above discussion seems per- fectly reasonable. First, the plotting of.AH* vs.AS* involves the solution of the rate equation by substituting into the equation experi- mentally determined values of k and T to obtain the values of.AH§ and ZSS*. If the temperature range of a series of measurements is small (ten to twenty degrees Kelvin) and the measurable range of the rate values are only four or five orders of magnitude, which is the case with most available biological techniques, then all the points obtainable by the solution of the rate equation are confined to a narrow region on theaAHiAS*-plane. Even a set of random points in the region will appear to Yield a relatively straight line. Compensation may be claimed to exist due to a mathematical artifact. The above argument can be demonstrated more clearly as follows: Consider the equation - * -s*R __I.3.. k=ce(AH/RT A/), c—Kh. Taking the natural vhich is the equati a , . b8 -1ntercept of (. (Ti, ki) determine mental values dete is the set of all possible points in the smallest regic 133* = (l/T. This region is bo' AS* and A 8* Where T and max T, melt and k max Figure (2.1: region b°unded b‘ Points °bt31uab1 ‘1 (1 sec ’ 105 13 Taking the natural logarithm of the equation yields In it = In C + AS*/R - Allah/RT or As* = (1m and + (R In k - R ln c) (2-3) which is the equation of a straight line with a slope of UT and a [Bit-intercept of (R In k - R In c). Each pair of experimental values (Ti’ ki) determines a line in the AH*AS*-plane. Any two pairs of experi- mental values determine a point in the plane. If ”1’ ki)’ i = 1,...,m, is the set of all possible experimental values, then the set of all possible points in the corresponding AH*AS*-plane must be contained in the smallest region containing the lines 1: # . As = (l/Ti)AH + (R In ki - R 1n (1), 1 = 1,...,m. This region is bounded by the lines 4: 1A8 (l/TminflfiH + (R In kmax - R In C) and A S 1: ; (l/Tmax)AH + (R In kmin - R In c) where T and T . are the maximum and minimum temperatures respec- max min tively and k and k . have similar definitions. max min t 1: Figure (2-1) shows a graph of three lines in aAHAS ~plane. The region bounded between lines 2 and 3 is the region in which all the Points obtainable from experiments with a rate-constant interval of (1 sec-1, 105 sec-'1) and a negligible temperature change from 2830K must WGHUIHOE\HNU "on 500 0 w 300 l00 100 '100 50 Figure (2.1 l4 AS¢ cal/mol-deg 500 400 300 200 100 0 50 100 150 200 250 300 AH” kcalories/mol Figure (2-1) . Plot of AH* vs AS *. lie. When the temp nah), the region the same rate inter tween lines 1 and I perature range, si' measurements the l the AbiASt-plane m grees Kelvin, it 3 region bounded by Ali-range of the p The above at; values (Ti’ ki) aw however, most res. culateAHi, so th set of points whi Alibi-plane coul from which the Ar region defined at As mentiOnec‘ method f0r the SI sation exists am The entrOPY‘enth‘ OfAHi Vs Asi yi bsi plot Which y existence of mm are Arrhenius ty (a . fragi), in V 15 lie. When the temperature can be varied by fifty degrees Kelvin (2830K, 3330K), the region in which all points obtainable from experiments with the same rate interval must lie has now increased to that bounded be- tween lines 1 and 3. These regions show the importance of a wide tem- perature range, since the larger the temperature range covered in the measurementsthe larger is the region in which the possible points in the AHtAS*-plane must lie. Even with a temperature range of fifty de- grees Kelvin, it should be clear that a random set of points in the region bounded by lines 1 and 3 will form a straight line when the Adi-range of the points is relatively large. The above argument illustrates that any two pairs of experimental_ values (Ti’ ki) and (Tj’ kj) can determine a point in the AH*AS*-plane. However, most researchers use the slepe of each Arrhenius plot to cal- culate A114: so that one point is obtained for the AHZS 1:--plane from each set of points which form an Arrhenius plot. Since this point in the AflfiAs*-plane could be considered as a type of mean for the set of points from which the Arrhenius plot is obtained, it is also contained in the region defined above. The argument there still holds. As mentioned in the INTRODUCTION, the most important questions a method for the study of compensation must answer are whether compen- sation exists and what the compensation temperature is if it exists. The entrepy-enthalpy method states that compensation exists if a plot of AH}: vs A81: yields a straight line. We have shown above that aAH¢ vs AS* plot which yields a straight line does not necessarily guarantee the existence of compensation. As an example, consider Figure (2-2) which are Arrhenius type plots of the data of the thermal killing of bacteria (P. fragi), in various types of milk, obtained by Luedecke (1962). It Figure (2-2' 0f P. fragi ( content . l6 -1) kd (sec 10 1. 3.08 3.09 3.10 3.11 l/T x 1000 (T in OK) Figure (2-2), A Plot of l/T vs kd, where kd is the rate of killing of P. fragi, in various types of milk with various amounts of fat content. (Luedecke, 1962), taken from a plot by Hamilton (1971). should be clear can hardly be 53 presented later) pensation exists straight line. The experimenta' rate range poss tions of the mi sented above, t must fall (1.9,. to cover these Recently I the intervals and obtained a' yielded a Stra A Simple Plot usually (1. and for variet (equivalent tt “timing th. k3 is Boltzma COHStant . Tl‘ 17 should be clear by just looking at the graph that the compensation law can hardly be satisfied (further analysis of this set of data will be presented later). However, Rosenberg et al. (1971) claimed that com- pensation exists since an entropy-enthalpy plot yields a near-perfect straight line. The reason these points fall on a line is quite simple. The experimental temperature range is only four degrees Kelvin and the rate range possible is two or three orders of magnitude due to limita- tions of the microbial technique used. Using the type of analysis pre- sented above, the region in which all the points obtained from the data must fall (i.e. the region between lines 1 and 3) is barely large enough to cover these points. Recently Banks et al. (1972) used arbitrary values of k and T in -l 6sec-1) and (2950K, 3450K) respectively the intervals (IO-Baec , 10- and obtained an entropy-enthalpy plot containing fifty points which yielded a straight line with a slope correSponding to a TC of 3220K. A simple example will show that the slope of the entropy-enthalpy plot usually'doesrmot give the correct TC. For simplicity in plotting and for variety, consider Figures (2-3) and (2-4) which are the gave AE (equivalent to AH; vs As*) and the Arrhenius plots of a set of data satisfying the semiconduction equation 0 = 00 enAE/ZkBT whereCI is the conductivity, AB is the semiconduction activation energy, kB is Boltzmann's constant, T is the absolute temperature and!%, is a constant. The slope of Figure (2-3) is the inverse ofTéwhich turns out H I AEU IS Figure (2 18 - 1 0’0 (11- Cm) 10 10' 10“ 1.4 1-6 0.8 1.0 V 102 as (eV) Figure (2‘3). P101: 0f AE V8 00. 1H I AceOIS ,0 Figure (2. H -l O (fl-cm) 10 10' 10 19 Figure (2-4). 0.25 0.50 (l/T) x 1000 (T in OK) Plot of l/T vs 0. O .75 1.0 to be -5ou°1<. Usi compensation tempe temperature is abr 1.31 which yields just about the win proposed later. A number of concerning the ab Some of the impo] 3) Exner's 9E5 Exner has to ticized the use Study of compens analysis and On be looked for by the “‘3le of not exist." (E: to the classica' according to Eq Consmeril (r1, k1) an a (. 20 to be -5040K. Using lines 2 through 7 in Figure (2-4), the estimated compensation temperature is -546.970K. The difference in compensation temperature is about 430K. What is really bad is the fact that S0 is lu31 which yields a D-value of about t§OZ and a confidence interval of just about the whole x-axis using the computer method which will be prOposed later. A nwmber of other criticisms have been raised by various papers concerning the above method for the study of the compensation law. Some of the important ones will be presented in the next few sections. 3) Exner's Critici§__gj_the EntropyrEnthalpy Method Exner has written a number of papers (1964, 1970, 1972) which cri- ticized the use of the entroPy-enthalpy method as a criterion for the study of compensation. He claimed that he had proven by "mathematical analysis and on practical examples that the isokinetic relation cannot be looked for by a direct correlation of activation parameters and that the majority of these relations described in liturature in fact does not exist." (Exner, 1964) He also noted that his argument apply both to the classical Arrhenius theory according to Equation (2-5) which is in the same form as Equation (2-4) and to the activated complex theory according to Equation (2-1) k = Ae‘AE/RT. (2-5) Considering only Equation (2-5) and assuming two pairs of values (T1, k1) and (T2, k2), he arrived at the following equations: log This pair of equa- coordinates from ' illustrated with the whole second asmall region in example, which or Exner also r since they are m: 1964) A graph 0 indePendent poin lotion coefficie shoved with actu lines of the em: Dentures. Many do not exist who he Presented ex; the “’EChanisms . Eight Year satiOn temper-at ferent item the 21 T1 T2 log A = Ef-ffjf— (log k1 - Er-log k2), 1 2 l 2.3RT1T AE = —_—T1 _ T2 (log k1 - log k2). This pair of equations can be considered as a transformation of the coordinates from log k2 to log A and from log k1 toaAE. Exner (1964) illustrated with a graphical representation of the transformation that the whole second quadrant of the log kzlog kl-plane is taken into only a small region in the log AAE-plane. The ratio of T2/T1 is 0.9 for this example, which corresponds to a temperature range of about 300K. Exner also claimed that log k1 and log k2 are "mutually independent since they are measured in independent separate experiments." (Exner, 1964) A graph of log k vs log k2 which consists of randomly chosen and 1 independent points is transformed into a set of points with a corre- lation coefficient of 0.9986 for the case T2/T1 = 0.9. Exner (1964) showed with actual experimental data that the slapes of the regression lines of the entrOpy-enthalpy method yield incorrect compensation tem- peratures. Many cases of compensation reported in literature in fact do not exist when tested with the log k1 vs log k2 method. Moreover, he presented examples where incorrect conclusions have been drawn about the mechanisms of reactions when the 2311* and AS": method is used. Eight years later, Exner (1972) was able to show that the compen- sation temperature obtained with the entrOpy-enthalpy method is dif- ferent from the'right" value. The method yields the correct result only in a special (unrealistic) case. 4) upper. £99.22 Even before published, a mm of possible errl tical error in where 1: nd r r 1 a 2 respectively, '1 gas constant. cal/mole for a: o and 308 K I'BSpe maximum error ( error in k inc: While the erro: Crease Of the I Patersen . used Afil is C01 and (T2, k2) , nth a max 1mm 22 4) Other Sources g£_Possible Error Even before Exner's criticism of the Afl} and AS* method was published, a number of authors have cautioned against certain sources of possible error. Purlee et a1. (1956) has shown that the statis- tical error in the Arrhenius activation energy AE is of the form RTlT % 2 2 2 sz T1 (rZ/rl) +(r1/r2) wherepr1 and r2 are the absolute values of the errors of k1 and k2 respectively, T1 and T2 are the experimental temperatures and R is the gas constant. From this expression, they obtained an error of't256 cal/mole for an error of 1% in both k1 and k2 when T1 and T2 are 2980K and 3080K reSpectively. With appropriately distributed errors of 1%, a maximum error of £365 cal/mole is possible. As the percentage of the error in k increases, the error in AE increases by about the same ratio while the error in.AE decreases by about the same ratio with the in- crease of the experimental temperature interval. Petersen et a1. (1961) has shown that regardless of the technique used AH1r is computed in principle from two pairs of values (T1, kl) and (T2, k2), as was mentioned in an earlier section, and is given by AH*=R-—-;-—--1n with a maximum possible error of approximately where n is the m Equation (2-1) , is given by the Thus if "measu 1! a plot of All The slope of t This slope is As an ex: of Luedecke (i fled compensa together with calculation 0 perature of 5 by Rosenberg Rosenberg et Even Le book which Thus even t 23 where a is the maximum possible fractional error in RI and k2. Using Equation (2-1), it was shown that the maximum possible error A in AS* is given by the equation A = (>[1/1‘1 + (T2 - T1)/2T2T1j, (1 << 1. Thus if "measurements of rate contain a sufficient range of errors..., a plot of.AH* vs As* will be a straight line...." (Petersen, 1961) The slope of this line is given by m = 2T2T't/(3T2 - T1). (2-6) This $10pe is sometimes referred to as the "error slope". As an example of this type of error, consider the bacterial data of Luedecke (1962) which Rosenberg et a1. (1971) claimed to have satis- fied compensation with a compensation temperature of 3310K.when lumped together with the data of Walker (1964) and Beamer et a1. (1939). A calculation of the "error slope", using Equation (2-6) gives a tem- perature of 327°K.which is almost the compensation temperature obtained by Rosenberg et al. (1971). Thus the compensation reported by Rosenberg et al. (1971) is probably an artifact. Even Leffler et a1. (1963) discussed this type of error in their book which contains the theory of extrathermodynamic relationships. Thus even those who are committed to the.AH* and As* method for the study of compen beenberg et a1 an observed ent line of slope T on the line, th error." the study of c the use of log method involve constant at a temperature T2 parameter suc compensation e straight lines where b is tht (Ta/T1). T1 The average 0 mental temper temperature 24 study of compensation must reject the claim of compensation made by Rosenberg et a1. (1971). According to Petersen et al. (1961): "When an observed enthalpy-entrOpy of activation plot presents a straight line of 310pe T Lm in my notation] with each point situated perfectly on the line, this plot is very likely a demonstration of experimental error." 5) Exner's log k1 Kg log k2 Method After criticizing the entrOpy-enthalpy method as a criterion for the study of compensation, Exner (1964) proposed, in the same paper, the use of log k vs log k2 plots as a more correct method. This 1 method involves the plotting of log k vs log k where k1 is the rate 1 constant at a temperature T1 and k2 is the rate constant at another 2 temperature T The values of k and k are varied by changing a given 2' l 2 parameter such as the pH of the solution. It can be shown that if compensation exists the log k vs log k2 relationships will yield 1 straight lines with the compensation temperature given by Tc = T2(1 - 6)/(1 - $6) <2-7) where b is the lepe of a given log k1 vs log k2 plot, $ is the ratio (TZ/Tl)' T1 and T2 are the temperatures involved in the given plot. The average of the Tc's for all the possible combinations of experi- mental temperatures in pairs is sometimes taken to be the compensation temperature of the whole series of measurements. An alternate method used to determine a mean value for TC is to map the log k1 vs log k 2 plot into the.AH£AStplane and determine the slope of this new line. Exner (1964) gav utures obtained the entropy-ant the correct val h) m 0 Since the pointed out tha useful", is not context of what nodynanic relat Grunvald (1963) substituent in mechanism, the interaction me except in a fee straight lines of the log k1 ‘ the points. A the log k1 vs -uo°x to 450° Lumry et as "so strict false cases." data must be enraged site cedure _ " 25 Exner (1964) gave a number of examples where the compensation temper- atures obtained by this method are different from those obtained with the entropy-enthalpy method and claimed that the proposed method gave the correct values. 6) Criticism QEDExner's log k1 y§_log k2 Method Since the publication of Exner's paper (1964), Leffler (1965) has method, although ”potentially pointed out that the log k vs log k l 2 useful", is not without limitations. He evaluated the technique in the context of what was formally referred to as the ”theory of extrather- modynamic relationship" contained in a book written by Leffler and Grunwald (1963). According to Leffler, this theory shows that if a substituent interacts with a reaction zone by a single interaction mechanism, the isokinetic relation is often obeyed. When two or more interaction mechanisms are operative, the relation does not hold except in a few special cases. When one of these cases applies and straight lines are obtained from the plots of both methods, the slope of the log k1 vs log k plot becomes very dependent on any scatter of 2 the points. An increase of 10% in the estimated value of the slope of the log k vs log k2 plot can change the calculated value of Tc from 1 -140°K to 4500K. Lumry et al. (1970) also criticized the log k1 vs log k method 2 as "so strict that...true compensation cases are thrown out with the false cases." In addition, if more than two temperatures are used "the data must be fitted for each independent pair of (Ti’ Th) and then averaged after application of a correctly weighted least-squares pro- cedure." Finally, it criticisms abov method is equiv only two experi also abandoned experimental te cause it is "im the most probab 7) mpg Meth A number pensation sinc According to E based on wrong haremde (1965) Emory at based on the e where A0201) homologous se corresponding “Pt of the e 26 Finally, it should be pointed out that even though not all the criticisms above may be valid since Exner (1972) has proved that this method is equivalent to the new method he prOposed (Exner, 1970) when only two experimental temperatures are available. However, Exner has also abandoned the use of this method except in the case when only two experimental temperatures are available. The method is abandoned be- cause it is ”impractical...and gives no correct possibility to choose the most probable value [TC] among a lot of results" (Exner, 1972). 7) Other Methods for the Study Q£_Compensation A number of other methods have been proposed for the study of com- pensation since Exner's (1964) criticism of the entropy-enthalpy method. According to Exner (1970), most of the methods proposed "are again based on wrong statistics". He cited the papers by Palm (1966), Maremae (1965) and Pihl (1965) as examples. Lumry et al. (1970) proposed a AG;(T) vs AH;(T) method which is based on the equation Asia) = a(T/Tc) + AH;(T)[1 - (T/Tc):) (2-8) where AG;(T) is the standard free energy change of the ith member of a homologous series in the reaction under consideration,.AH;(T) is the corresponding standard enthalpy change and o is the constant or inter- cept of the equation AH; = or + TCAS° (2-9) which is the co Note that Equat According compromise and entropy‘enthalp of an illustrat method is equi terion is stil the method pro Lumry's method liability of t been gained. is still requi use of Equatio may lead to Equation (2-1) pecially when consideration. statistical e1 entropy-entha‘ 8) unease Due to t cussed above, analysis to results obta tica1 analys 27 which is the condition used for the derivation of the above equation. Note that Equation (2-9) is equivalent to Equation (2-2). According to Lumry et al. (1970), the above method is "a valid compromise and is certainly preferable to the use of either" the entropy-enthalpy method or the log k1 vs log k2 method. An examination of an illustration presented by Lumry et al. (1970) reveals that his method is equivalent to the original Arrhenius-plot method. A cri- terion is still needed to decide if the straight lines obtained with the method proposed by Lumry et al. (1970), which will be called Lumry's method, intersect at a point. Thus no improvement in the re- liability of the determination of the existence of compensation has been gained. A statistical method such as that proposed by Exner (1970) is still required to improve the reliability of Lumry's method. The use of Equation (2-9) which is derived from a form of Equation (2-1) may lead to two possible sources of error. First, the fact that Equation (2-1) may not be applicable in a process being studied, es- pecially when dealing with biological entities, must be taken into consideration. Second, the use of Equation (2-9) may again introduce statistical errors similar to those encountered in the use of the entropy-enthalpy method. 8) WWW Due to the inadequacies or shortcomings of all the methods dis- cussed above, Exner (1970) suggested the application of statistical analysis to the Arrhenius type plots to improve the reliability of the results obtained. As mentioned in the INTRODUCTION, the use of statis- tical analysis requires certain equations which were unavailable until the publication c devoted to the d! applications . m) EXNER'S _s_T_A; This sectic vation 0f Exner derivation prov tical method, to statistical met tained from it clarified. It must b. only the equat temperature an points from th conditions. ( the data or t) Paper contain. be Performed the analysis. Some of the s Finer (1972) Program and 1 data. 28 the publication of Exner's (1970) paper. The next section will be devoted to the derivation of the equations and a discussion of their applications. ii) EXNER'S STATISTICAL ANALYSIS This section will be divided into two subdivisions: 1) the deri- vation of Exner's equations, which will be given briefly (with detailed derivation provided in the Appendix); 2) the discussion of the statis- tical method, which will be given in detail. The discussion of the statistical method includes the type of conclusions that can be ob- tained from it and the areas where the method might be refined or clarified. It must be mentioned here that Exner's first paper (1970) contains only the equations necessary for the estimation of the compensation temperature and the sum of squares of the errors of the experimental points from the least-square regression lines under two different conditions. Only limited discussion is devoted to actual analysis of the data or the applicatiOn of the equations presented. Exner's (1972) paper contains several actual examples of the type of analysis that can be performed although very little is said about the actual procedure of the analysis. Hopefully, the procedures can be clarified by providing some of the steps missing in the application of the statistical method. Exner (1972) referred to another paper which will contain a computer program and can be used for the analysis of any appropriate set of data. 1) Derivation 9 Exner's sta aset of experim li’ i= 1,...,1 Then statistical is satisfied am perature togeth: To decide it is satisfied at a common poi the best estima using the math: tions of the p( lated and Comp; Points from th. Pensation exis amount of erro 1“89-1” than S c dered is said Before 81 Calculated. . and is Simile deviations of 29 1) Derivation g£_Exner's Equations Exner's statistical method can be described as follows: Suppose a set of experimental data is found to form a family of straight lines 1i’ i = 1,..., L, such that each line satisfies the Arrhenius equation. Then statistical analysis can be used to decide if the compensation law is satisfied and to estimate the best value of the compensation tem- perature together with a confidence interval for this temperature. To decide if compensation is satisfied, adopt the hypothesis that it is satisfied (exact compensation), i.e., that the L lines intersect at a common point (x0, yo). To test the validity of this hypothesis, the best estimate of the point of intersection (x0, 90) is calculated using the method of least squares. The sum of squares of the devia- tions of the points from their corresponding lines, So’ can be calcu- lated and compared to the sum of squares of the deviations of the points from their corresponding lines without the assumption that com- pensation exists, 300' This comparison gives an estimate of the amount of error introduced by the hypothesis. If So is significantly larger than 300’ the hypothesis is rejected and the set of data consi- dered is said to have failed to satisfy the compensation law. Before any comparison of SO and S00 can be made, they have to be calculated. The sum S00 is by definition 3 =ZZ(y.-b.x.-a)2 00 1,3 i] 1 i] i and is similar to the standard definition for the sum of squares of the deviations of the points from a single regression line. In the present case, when it is assumed that xij = Xj and M1 = M for all values of i, 0( Note that the do which are ob tait Similarly 1 since the devie ”him are obta However , of B ‘ 1’ X0 and sation temper from Equéltion only the cone A °°mP1ete ar algEbraiC 11181 30 z p, 300 = z yfj - M219: - M2$(: f ;)2 (2—10) id 1 j" J' y. = b.x. + 5., i = 1,... L (2-11) which are obtained without the hypothesis of compensation. Similarly So must be calculated from the equation .. .x.. . (2~12) 13 1 13 o 1 o ., L (2-13) which are obtained with the hypothesis of compensation. However, Equation (2-12) cannot be used without knowing the values 8 e e a . . . of 1’ x0 and y . Moreover, x0 18 the inverse of the estimated compen- o sation temperature and must be calculated. These values can be solved from Equations (2-14), (2-15) and (2-16) which can be derived using only the concept of least squares and lengthy algebraic calculations. A complete and detailed derivation of all the equations used in this thesis are given in the Appendix since some of them involve considerable algebraic manipulation. H: ll II k<33 where ii stands . . E ‘ ihj/ :Mi q 1,} 1 where )1: x 51 1 that the paramw Unfortuna available in a the above equg equations in 1 M L The so: 2 . ‘Eu (pi ‘ P) equal to 3g 0 31 == -0 A + A... - y y0 xo'iZM b./%Mi .iiMibixi/iXMi (214) i 1 ‘ a A A2 ‘ - ‘2 = Z .. .. + ' - o yO lMibi x0 iZMib].L f Mlbiyl EMibixi (2 15) p - -(§ - x )(y - y ) + b (Zixz /M - 2% i + £2) (2-16) 1 o 0 j ij i o i o where xi stands for zix j/Mi’ yi stands for Zyij/Mi and y stands for i J j Z yij/ 2M1. Equation (2-16) is obtained with the general definition 1,1 i . = Z x. ../M - x-. p1 j inj 1 Y1 where i = ii since xij = xj. Note also the a hat * always indicates that the parameter is a statistical estimate of the true value. Unfortunately, a general solution of the above equations is not available in any statistical literature, probably due to the fact that the above equations are non-linear. Exner (1970) has solved the equations in the special case when x = xj and Mi = M for all values ii of i. The solution is given in the form of a quadratic equation. 2 - 2 - - = - __1. _-2 - _=2 EU(pi-p)(yi-y)+u[‘ii(pi-p) MEAX X) $071 501+ 1%)?“- - 5'02 ‘21.“(131 - 13) (91 - '3?) = 0 <2-17) J J + where u = x - i, x = fiij/M. One of the roots of this equation, ho, is .1 equal to xo - x where x0 is the estimated value of the inverse of the compensation temperature, to be referred to as the compensation x-value. To find b1 and Y which are ob tail After calc obtained from t 2 s=Z o uh Unfortunat Used to test it derived with t! statistical te Speaking, be u linear. Exner amAles because manner" and ex sufficient in sation. More for the 00me e(lustrous men compensation 32 To find bi and yo, the following equations can be used: 3" = ; + :1 Min/L HMj - )2) {2-18) 1 + lzwx, - $02] (2-19) Mj J _ which are obtained by simplifying Equations (2-14) and (2-15). After calculating the values for x0, yo and bi’ SO can be obtained from the following equation: 2 . - - = {pi - uoguai ~ p)(yi ~ y) s =.Z.y.2. -MLy -M (2-20) 0 1"] 13 Z(x. ~x)2 j J Unfortunately, the most common statistical test which might be used to test for the eXistence of compensation is the F-test which is derived with the hypothesis of "linear models" as are many of the statistical tests. According to Exner (1972), it cannot, strictly speaking, be used since Equations (2-14), (2-15) and (2-16) are non- linear. Exner (1972), however, did use the F-test in some of his ex- amples because he stated that it was only used in a "qualitative manner" and even ”a qualitative comparison of So’ 300’ and 6 may be sufficient in most instances" to establish the existence of compen- sation. Moreover, he claimed that a confidence interval can be found for the compensation temperatureixispite of the non-linearity of the equations mentioned above. To find this confidence interval for the compensation temperature, let x be the inverse of the compensation 1 temperature which is assumed to be known. Then, since x1 = i + u, by definition, t similar to that 2 .. I -2uZ( .‘F zipi 1P1 M—_——— 2) ADiscussi As mentio at the analysi satisfied and V917 important discuss in des Four 3831 a) The valid terval of EXP confide-Nd as °°mPensation menta1 errors (For the Sp e‘ taken to be . Plete. Before sat 0f expel 33 by definition, the residual sum Su can be derived with a technique similar to that used to obtain SO and is given below: = 2 = Z - _. 2 - - = 2 - = 2 M ' 2 " 2 2 §p1_2u§(pi-p)(yi-y)+u [gm-y) +1:82:31) /3§(xj -x) :I M - 2 2 (2‘21) Z(x.-X)‘+MU j J 2) A_Discussion of Exner's Statistical Method As mentioned before, Exner's method is the first correct attempt at the analysis of Arrhenius plots to determine if compensation is satisfied and to estimate the compensation temperature. It is thus very important to examine the effectiveness of this analysis and to discuss in detail the conditions under which it can be applied. Four assumptions have been presented by Exner (1972). They are: a) The validity of the Arrhenius equation-~at least within the in- terval of experimental temperatures. b) Experimental temperatures are considered as an exact quantity rather than a random variable. c) If compensation exists, the quantities, eij’ are identified with experi- mental errors in log k and assumed to have a normal distribution. (For the special case studied by Exner, the absolute error in log k is taken to be constant.) d) The set of experimental data must be com- plete. Before the application of this new criterion for the study of a set of experimental data, the researcher must make sure that these assumptions are the calculated V viations of all from the standai line only by th large, no furth to satisfy sinc amuch greater especially true ditions. In bf l . can be quite d; Assumption c) required to co try his best t cisiom. The c since most par measurements . ments are plat Many existing The abov tain a set of “be“ small mg are it“’Olved. with the 8130' are much 1a,. weidhts whic ordinary inc 34 assumptions are satisfied. The validity of a) is easily verified using the calculated value of Soo‘which is the sum of squares of the de- viations of all the points from.their corresponding lines. It differs from the standard sum of squares of the deviations for a regression line only by the fact that 800 is for L lines instead. If S00 is too large, no further analysis is necessary. Assumption b) should be easy to satisfy since experimental temperatures can usually be controlled to a much greater precision than the other measured quantities. This is especially true with experimental data obtained under equilibrium con- ditions. In biological experiments involving kinetic data, conditions can be quite different. More will be said about this point later. Assumption c) can be difficult to satisfy if more than one technique is required to cover the kinetic rate range desired. The researcher must try his best to bring all techniques used to the same level of pre- cision. The use of existing data for analysis can present a problem since most papers do not include an analysis of the precision of the measurements. Assumption d) is not too difficult to satisfy if experi- ments are planned with the application of this new criterion in mind. Many existing data do not satisfy this assumption. The above discussion indicates that it is not too difficult to ob- tain a set of data which will satisfy the above mentioned assumptions when small molecules, such as those commonly encountered in chemistry, are involved. This is especially true when experiments are carried out with the above assumptions in mind. In biology, the entities involved are much larger and more complex. A protein can have molecular weights which are thousands of times larger than those encountered in ordinary inorganic or organic chemistry. A single cell can contain an uncountable numb has a can W31] in temperature. case. Great ca: peratures are m the biological temperatures . the applicatior this is the ty; The next : can determine judgement. Ex than 300, the reiected." (E Corresponding detenuine whet one Place when Since Exner ( As to th temPérature, thing an an: be °btained a 8u as an admi [°°mPensaum Nest is no think it pro I Ethel“ S meth 35 uncountable number of protein molecules and other types of molecules. It has a cell wall to protect the inside of the cell from external changes in temperature. Thus assumption b) is not as easy to satisfy in this case. Great care must be taken to make sure that the experimental tem- peratures are maintained as precisely as possible and where feasible the biological entities considered must be maintained at the desired temperatures. For the time being, the discussion will be limited to the application of Exner's statistical method to small molecules since this is the type of molecules he has in mind. The next important question to be answered is whether the method can determine the existence of compensation more reliably than visual judgement. Exner only stated that: "If so is not significantly larger than see, the hypothesis of a common point of intersection cannot be rejected." (Exner, 1972) Here 30 and 800 are the standard deviations corresponding to S0 and S00 respectively. No definite rule is given to determine when so is actually significantly larger than 300' This is one place where more definite rules will be desirable, especially since Exner (1972) does not think the F-test can be used. As to the question of a confidence interval for the compensation temperature, Exner (1972) claimed that it can be constructed by de- riving an expreSSion for Su from which the standard deviation, su, can be obtained as a function of u. ”If one now chooses a fixed value of su as an admissible limit, one obtains the confidence interval of E [compensation temperature] from the graph." (Exner, 1972) Again the F-test is not explicitly mentioned. It is possible that Exner does not think it proper to use the F-test for this purpose. Nevertheless, Exner's method can yield a confidence interval if the magnitude of the experimental er limit of su can is the lack of allow other res compared. It can be statistical me the old method terion which c iidemce interu meaningfully s fidence inten Similarly, a ( ciding if the Exner (1972) '. may contain 3 Papers. iii) ANALYSI \- AS mentj of the F‘tes1 examPles in y 1eBitimate t c°mpensation The use of t pemlisSible 36 experimental errors are known so that a reasonable value for the upper limit of su can be estimated. The only drawback in Exner's proposal is the lack of a clearly defined value for the upper limit of su to allow other researchers to obtain confidence intervals which can be compared. It can be seen from the above discussion that even though the statistical method prOposed by Exner (1970) can be an improvement over the old method of visual judgement, it still lacks a clear-cut cri- terion which can be used by all researchers. This will allow the con- fidence intervals, obtained by different researchers, to be compared meaningfully since it should be obvious that the magnitude of any con- fidence interval depends on the choice of the upper limit for an. Similarly, a clear-cut criterion should have been established for de- ciding if the compensation law is satisfied. As mentioned before, Exner (1972) refers to a third paper which is not available yet. It may contain some of the things which have been lacking in the first two papers. iii) ANALYSIS 9; DATA USING THE F-TEST As mentioned before, even though Exner (1972) claimed that the use of the F-test is, strictly speaking, not legitimate, it was used in the examples in his paper. Stapleton (1972) is of the opinion that it is legitimate to use the F-test to estimate a confidence interval for the compensation temperature if this temperature is assumed to be known. The use of the F-test to determine if compensation exists is also permissible even though it may reject some data which should not have been rejected a satisfy compens usually larger To test fc that compensatl (S0 - SOO)/f i: S and f is tl no consideration, pectively, whe the number of level for the less of not 52 holds, where f having (L _ 2 T0 calcu fies cOIIlpensa Petature is 1‘ on the X'axig by E(dilation c equation Can 37 been rejected at the significance level considered. Those found to satisfy compensation will satisfy the criterion since the sum 80 is usually larger when the vectors used are not linear. To test for the existence of compensation, make the hypothesis that compensation does exist, i.e., the difference between Soc/f0 and 0 (So - Soo)/f is insignificant. Here f00 is the degrees of freedom of S00 and f is the degrees of freedom of (So - SOC). For the case under consideration, f00 is equal to (ML - 2L) and f is equal to (L - 2) res- pectively, where L is the number of lines in the set of data and M is .the number of points in each line. If 0.05 is used as the significance level for the F-test, then the data have a probability, P, of 5% or less of not satisfying the compensation law if the inequality (80 - Soo)/(L - 2) 800/ (ML - 2L) £170.05 (2'22) holds, where F is a value obtained from an appropriate F-table 0.05 having (L - 2) and (ML - 2L) degrees of freedom. To calculate a confidence interval for a set of data which satis- fiescompensationeitzis necessary to assume that the compensation tem- perature is known. By assuming a value which corresponds to u = x - i on the x-axis, the sum Su can be derived and is the same as that given by Equation (2-21). Using a similar argument as before, a probability equation can be obtained (8 - S )/(L - 1) Sn /(M([).o- 2L) 3 Fe = 3 (2-23) 00 P ' a V: where F6 13 of G and having can be rewritte Thus, at a sigr values smaller give the conf It should 3U, can usuall 0f EXperimenta torous calcula Crame'r (1946) Single rfigres: Vol"ES the Us. theorems . 38 where Fe is a value obtained from the F-table for a significance level of G and having degrees of freedom (L — l) and (ML - 2L). The equation can be rewritten in the form L - l PESu S 800(1 + Fe m)] = 6 (2-24) Thus, at a significance level of G, the values of u which yield Su values smaller than give the confidence interval. It should be noted that the degrees of freedom for a sum, such as Su, can usually be obtained by taking the difference between the number of experimental points and the number of parameters involved. A ri- gorous calculation of what it should be is quite difficult and involved. Cramer (1946) has given a derivation of the degrees of freedom for a single regression line with a given number of data points which in- volves the use of linear algebra, statistical theory and some specific theorems. iv) COMPUTER ' The compu (1970) paper. terval and dec with the assun data are knowr assumption may existing papei lyzes his own For simp‘ the errors as mius plot res maximum absol given set of Which the cha that yi(xk) i Since D shou] the data, the dotted regior dotted regiOI should be oh the x-coordi. lines no that Even for mor regl‘tssion 1 0f inter-Set“: 39 iv) COMPUTER STUDY Q£_OOMPENSATION The computer method will make use of the equations in Exner's (1970) paper. Instead of using the F-test to obtain a confidence in- terval and decide if compensation is satisfied, computer calculations with the assumption that the errors associated with the experimental data are known or can be estimated is employed for the purpose. This assumption may not be realized when a set of data is taken from an existing paper, but should be easy to satisfy when a researcher ana- lyzes his own data obtained from.well-planned experiments. For simplicity in the present discussion, it will be assumed that the errors associated with the experimental points for a given Arrhe- nius plot result in an error in the slope of the plot. Let D be the maximum absolute value of this error in $10pe for all the L lines in a given set of data. Let (Xk’ yi(xk)) be the point of rotation about which the changes in slope occur. (Note that it is also assumed here that yi(xk) is different for i = 1,..., L while xk remains the same.) Since D should be the largest change in slope caused by the errors in the data, the experimental regression lines should lie inside the dotted regions like those illustrated in Figure (2-5), where two such dotted regions are shown with their corresponding ideal lines. It should be obvious in this case that the interval (x1, x2) must contain the x-coordinate of the point of intersection of the actual regression lines no matter where they actually lie in their corresponding regions. Even for more than two lines the actual points of intersection of the regression lines must lie in an interval which contains all the points of intersection of the family of lines. For a family of L lines, the is —ax Y Figure Wl thin is S OPE y-axis 40 I I (o '22. I o I 00, O O O O O I .y. I O ‘I I O O O O I V '1‘ Iooooooo’ I .0000’ O o I I I I I I s s s a a I I a a a r s a a r I I I I ¥¥¥¥ >l ,e. x-axis Figure (2-5). Two of the ideal lines (solid) with the regions within which the experimental lines must lie (dotted regions). xIf is the x-component of the point about which the changes in s ope of the experimental lines are rotated. boundaries of where bi is at line and ai is There art which correspl there exists lines and the For example, binary number tion of lines would come. Which conta lx ' l X m“ max 41 boundaries of the regions are defined by Equations (2-25). _ + ' = ., yi bixi ai, 1 1,..., L (2 25) where bi is equal to Mi + D or Mi - D, M1 is the slope of the ith ideal line and a1 is the y-coordinate of the point of rotation (Xk’ yi(xk)). There are a total of 2L possible combinations of L lines, each of which corresponding to a different value of i. It can be shown that there exists an isomorphism between the 2L different combinations of lines and the set of all binary numbers with L digits including zero. For example, if bk = Mk'+ D corresponds to a 1 in the kth digit of the binary number while bk = Mk - D corresponds to a 0, then the combina- tion of lines D) + + (M1 D)x1 ‘4 N I _ + + (M2 D)x2 a - - '1' Y3 (M3 D>X3 a3 = + + ya (“4 D>X4 34 = + y5 (M5 + D)x5 35 Y6 = (M6 ' D)X6 + as would correspond to the binary number 110,110. The smallest interval which contains all the points of intersection of the 2L families is (X , x ) where x ax min m min is the x-coordinate of the point of intersection w: nition. This t temperature wh nitude of the pensation temp The above interval for t linement on tl the estimated the estimate, lines are fix. compensation temperatures since the lin Possible fami error in slot Still be smal fidence lute] When Ca: °°mpensation intEI‘Val can plot is illipo the We of as the ideal left, the un creases, Tl 42 intersection with the smallest value in x and xmax has a similar defi- nition. This interval gives the range of the estimated compensation temperature when the ideal compensation temperature is known. The mag— nitude of the interval is an estimate of the reliability of the com- pensation temperature obtained experimentally. The above interval, however, is not the best possible confidence interval for the compensation temperature under investigation. A re- finement on the interval would be the smallest interval containing only the estimated compensation temperatures for the 2L combinations using . the estimate, Tc’ according to Exner (1970). Since each combination of lines are fixed, its estimated compensation temperature is also its true compensation temperature. The interval containing the compensation temperatures for all the combinations should be a confidence interval since the lines defined by Equations (2-25) should include almost any possible family of regression lines obtained experimentally with an error in slope of at most D in absolute value. The interval should still be smaller than the one given above and is chosen to be the con- fidence interval for the computer method. When calculations are carried out for a number of different ideal compensation temperatures, a plot of the magnitude of this confidence interval can be made against the ideal compensation temperatures. This plot is important since it helps a researcher to plan in advance for the type of precision he desires in his experiments. It is known that as the ideal compensation temperature moves along the x-axis to the left, the uncertainty in estimating the compensation temperature in- creases. The exact relationship between the magnitude of the uncer- tainty and the position of the ideal compensation temperature along the x-axis has not ‘ some calculatio that the uncert pensation tempe iicult ever to dence." ThiSl has not been t benefit from t be valuable to liability of 2 Equation: regression li' fectiveness o (1970,1972). lead to the i The type Cilhtd in th, 43 x-axis has not been reported in literature. Hammett (1970) has made some calculations using the methods available at the time and claimed that the uncertainty can increase by forty-eight times when the com- pensation temperature is negative and concluded that "it will be dif- ficult ever to assign such a quantity any considerable level of confi- dence.” This may bee why report of negative compensation temperatures has not been taken seriously. Hammett's calculations, however, did not benefit from the statistical method proposed by Exner (1970). It would be valuable to see if this new method gives any improvement in the re- liability of a negative compensation temperature finding. Equations (2-25) can also be considered as actual, experimental regression lines and analyzed. This will give an idea about the ef— fectiveness of some of the methods of analysis suggested by Exner (1970,1972). It may also be helpful in providing the insight which may lead to the formulation of new methods of analysis. The types of analysis actually conducted in this thesis are des- cribed in the EXPERIMENTAL. Since thi search, this c subdivision wi (1970) equatic second subdivi proposed in tl The third subc set-up that w; adsorption of ii EXNER'S E. x- In this were Used sin conSidei‘ed mc Some of the l here 3'1th tl 0f compensat: Chart Centai. at Several v III. EXPERIMENTAL Since this thesis includes both theoretical and experimental re- search, this chapter will be divided into three subdivisions. The first subdivision will contain a flow chart for theicalculation of Exner's (1970) equations and the types of analysis that will be done. The second subdivision will contain a flow chart of the computer analysis prOposed in this thesis and the type of analysis that will be performed. The third subdivision will contain a description of the experimental set-up that was used to measure the conductivity of proteins with the adsorption of various amounts of a given adsorbate. i) EXNER' S EQUATIONS In this theSis, only those equations in Exner's first paper (1970) were used since they contained all the parameters that were needed and considered most useful in the analysis of the compensation phenomenon. Some of the parameters in Exner's second paper (1972) were not included here since they did not deal directly with the existence and temperature of compensation which are the main topics in this thesis. The flow chart contained the equations needed to obtain x0, T , S , S and Su C 0 00 at several values of u. 44 1) i191 Elli-EP- The flow . the program. for the symb01 was written in in Exner's equ which calculai predetermined 2) Tress 2i . The prog actual, exist examples he p method is mos that knowledg necessary, '. the exoerimes Exner's anal: since only 0 Sidered, (S The val ti"hill error make them a] Figure (H: 8or) by a Si; caused by m 45 1) Elgw_§ha£t_for Exner's Equations The flow chart was written with symbols similar to those used in the program. The program itself, together with a list of definitions for the symbols used in it, is provided in the Appendix. The program was written in ALGOL since it is most adapted to the many matrices used in Exner's equations. Figure (3-1) Shows the flow chart for the program which calculates the values of £0, T , S , S00 and Su at a number of C O predetermined values of u. 2) Iy2g§_gf_Analysis tg_be_Carried 9g; The program in this section was intended for the analysis of actual, existing data such as those done by Exner (1972) in the examples he presented. The F-test method was used, however, since this method is most adapted to the analysis of existing data due to the fact that knowledge of the precision of experimental measurements is not necessary. The only assumption required was that the distribution of the experimental errors should be normal, which was also required by Exner's analysis (1972). This assumption was assumed to be satisfied since only one technique was employed in each of the two examples con- sidered. (See also the discussion in Section II-ii-Z.) The values of So and S00 obtained were used to study the addi- tional errors caused by moving the experimental regression lines to make them all meet at a common point as required by exact compensation. Figure (342) shows the situation graphically. If So was larger than S by a significant amount, it was,assumed that the added errors 00 caused by moving the lines to make them meet cannot be attributed to Figure (3'1) . in Exner' s pa] # M Read KT(I,J) Calculate YIJ Read T(l), T( Calculate Yl( Calculate XJC Calculate PSl Calculate X = Calculate PC Calculate PA Calculate Y = Calculate SP Calculate SY Calculate SP Calculate S): Calculate B Calculate U Calculate X( Calculate T4 Calculate 3‘ Calculate 8 Calculate 8 Calculate S Output U,“ calchlate 5 \ Figure (3-1). in Exner's 46 Flow chart for the calculation of the parameters paper (1970). Read KT(I,J) for I=1 to L, J=l to M Calculate YIJ(I,J) = LN(KT(I,J)) for I=1 to L, J=l to M Read T(l), Calculate Calculate Calculate Calculate Calculate Calculate Calculate Calculate Calculate Calculate Calculate Calculate Calculate Calculate Calculate Calculate Calculate Calculate Calculate T(2),..., T(M) YI(I) = ZYIJ(1,J)/M for I=1 to L XJ(J) = l/(T(J)+273) for J=l to M PS(I) = EXJ(J)*YIJ(I,J) for I=1 to L X = XXJ(J)/M for J=1 to M P(I) = PS(I)/MrX*YI(I) for I=1 to L PA = ZP(I)/L for I=1 to L Y = ZYI(I)/L for I=1 to L SPY = 2(P(I)-PA)*(YI(I)-Y) for I=1 to L SY = Z(YI(I)-Y)2 for I=1 to L SP = Z(P(I)-PA)2 for I=1 to L sx = 2(XJ(J)-X)2 for J=l to M B SP- (1 /M*SX*SY) u = -(-B+ J'FFESPY’M/Msesm/(2espv) x0 = U+x TC = 1/xo SYY = 22(Y1J(1,J))2 for I=1 to L and J=l to M SPP = 23(P(I))2 for I=1 to L $0 = SYY-M*L*Y2-M2*(SPP-U*SPY)/SX 300 = SYY-M*(Y(I))2-(M2*SPP/SX) Output U,SO,SOO,XO,TC Calculate SU = SYY-M*L*Y Output U,SU for U = .2, 2-M2*(sPP-2*U*SPY+U2*(SY+M/L*(13"?) 2 /SX) ) / /(SX+M*U2) for u= .2, .1, o, -.1, -.2,... .1, O, -.1, -.2,... log k I '1 Figure reBrass and the make tl mized ( 47 s‘ \ ~‘\ “ ( ) \‘s X a y \‘ ‘s O O \‘ ‘ s “ x x \ ‘s ‘x \ ‘s \ ‘~ \ \ \‘ ‘s x 3" ~‘~ \ s \ ‘x \ “ ' “ .k *‘S A‘s \ “ \\ x‘ \ \ ‘\ x ‘ \ \ \\ s‘ \ ‘\ “\ \ \ ‘\ 7“ \ \\ \ x \ ‘\ 9;. \\ w ‘ ‘ “ o ‘ ‘\ ‘ H \\ \ ‘ \ \ .‘k \ \ \ \ \ ’F \ \ \ \ \ ‘\ \\ \ \ \ \ \ \ \\ ‘\ \ \ \ \ \ x \ s \ \ Figure (3-2). A plot of log k vs l/T showing the least square regression lines (solid lines) of the actual experimental points(*) and the least square regression lines moved in such a way as to make them all pass the point (x , y ) while the S -value is mini- mized (dotted lines). 0 o o pure errors 1' sation for U dw assumptit compensation not meet at away from wh also obtaine was 0.05 unl Exner': of compensa' the example (1972) does was especia since it we limit" (Ex: The second explicit p] especially The v.- °riginal p The v Were Used EffeCCTVen Exner (197 a diagram 0f the is derivedecn that the ‘ 48 pure errors in experimental measurement and the existence of compen- sation for the process under investigation was rejected. (Note that the assumption used here was that if the process actually satisfies compensation, then the fact that the experimental regression lines did not meet at a point was caused by experimental errors which moved them away from where they should have been.) A confidence interval was also obtained using the F-test method. The significance level used was 0.05 unless otherwise stated to Show a point. Exner's method for determining the temperature and the existence of compensation was not used for two reasons. First, the precision of the examples used was not mentioned in the papers and Exner's method (1972) does require the knowledge of the experimental precision. This was especially true in the determination of a confidence interval since it was done by choosing "a fixed value of su as an admissible limit" (Exner, 1972) which was then applied to the graph of su vs u. The second reason was the fact that Exner really has not given an explicit procedure for the application of his criterion. This was especially true in the determination of the existence of compensation. The values of £0 and TE were compared with those given in the original papers. The values of Su obtained at various predetermined values of u were used to plot Su vs T or Su vs u curves in order to determine the effectiveness of these curves as an aid in the study of compensation. Exner (1970) claimed that the "most valuable result is, in our Opinion, a diagram like Figure 2[ Su vs T] which shows objectively the validity of the isokinetic relationship and the meaning of the quantities derived." An examination of the figure mentioned by Exner revealed that the minimum which existed was very broad. In fact, in some cases, aminimum exis cannot be give supposition 0f (Exner, 1970) vs u curve has jetted. Howe‘ cance of Su 01 with one asym' point u = no. this criterio cannot be re j must not be is Prompted him for the stud} lich ~~ As disc assumPtion t 1“ Slope whi exPerimentg tive t0 the the Special sum“ t0 co: Peinte 0n e fl the assu equations . 49 a minimum exists which is so broad that the compensation temperature cannot be given any distinct value. "This may happen even when the supposition of a common point of intersection cannot be rejected." (Exner, 1970) Thus Exner seemed to suggest that as long as the Su vs u curve has a minimum the existence of compensation cannot be re- jected. However, he stated elsewhere in the paper that: "The depen- cence of Su on u can be represented by a curve of the third order... with one asymptote parallel to the x-axis and with a minimum in the point u = no." This seemed to suggest that every analysis made with this criterion would indicate that the process under investigation cannot be rejected as having failed the test for compensation, which must not be what Exner had in mind. This may be the reason that prompted him to publish a second paper which contained other methods for the study of compensation and will be discussed in the next chapter. ii) THE COMPUTER METHOD AS discussed in the THEORY, Equations (2-25) were derived from the assumption that each line in the data took on the maximum possible error in 310pe which could be estimated from knowledge of the precision of the experimental measurements. These errors were, of course, taken rela- tive to the ideal Arrhenius lines which were assumed to be known. In the special case considered in this thesis, the set of data was as- sumed to consist of six lines each of which containing six points. The points on each line were given the same x—coordinates in order to satis- fy the assumption, xij = xj, used in the derivation of Exner's equations. To simplil to intersect a‘ may consider t the more appro The progr temperature cc binations obta they were actt used above. binations wer tion was used Again a data required tioned before 1) 13m Q13; Since t the calculat thud, the £1 The first pa the analysis points into were fed tn he third p hithou thesise the 2L 50 To simplify the program, the lines in Equations (2-25) were made to intersect at the lowest experimental temperature. Some researchers may consider the mid-point of the experimental temperature range to be the more appropriate temperature to use. The program generated a set of points, for each ideal compensation . . L . temperature con31dered, corresponding to one of the 2 p0331ble com- binations obtained from Equations (2-25) and analyzed the points as if they were actual experimental points in the same manner as the program used above. The procedure was repeated until all the possible com- binations were exhausted. Thus the program presented in the last sec- tion was used as part of the new program. / Again a complete program for the generation and analysis of the data required can be found in the Appendix. The list of symbols, men- tioned before, also includestfimeones used in this section. 1) Flow Chart for the Computer Method Since the computer method involved the generation of points and the calculation of the same parameters as those in the statistical me- thod, the flow chart for the whole prOgram was divided into three parts: The first part of the flow chart generated all the points required in the analysis from Equations (2-25). The second part converted these points into the proper form corresponding to the 2L combinations which were fed into the program for the statistical method which was used for the third part of this program. Although only six points in each of six lines were used in this thesis, the program was written for the general case where each of the L 2 sets of simulated experimental points contained M points for each of its L lines. It generated ing an ideal other the min given ideal ] The sec: binations of have two alt: could be est system of bi lines. The numbers. Tb the first pa were part 01 Possible co: third part . IYSiSo Eac sible lines corresPonde For “ample the larger the shame] The it although a combinatio As me 51 its L lines. The first part of the flow chart is shown as Figure (3-3). It generated the two lines which formed the boundary of a region contain- ing an ideal line. One of these lines had the maximum lepe and the other the minimum slope which could be obtained experimentally for the given ideal line. The second part of the flow chart selected the 2L possible com- binations of L lines which were different. Since each ideal line could have two alternatives and there were L ideal lines, an isomorphism could be established between the possible combinations of lines and the system of binary numbers with the same number of digits as there were lines. The combinations were named after their corresponding binary numbers. This part of the flow chart arranged the lines generated in the first part in such a form that the set of 2L binary numbers which were part of the input to the program could be used to pick out all the possible combinations of the lines, one at a time, andfeed.it to the third part of the program which calculated the values needed for ana- lysis. Each digit of a binary number picked out one of the two pos- sible lines corresponding to an ideal line. The digit to the far left corresponded to the ideal line with the smallest y-value and so on. For example, a 1 for any digit would be used to pick out the line with the larger SlOpe while a 0 for any digit would pick out the line with the smaller slape as described in more detail in the THEORY. The flow chart for the second part of the program is rather short although a total of 2L input cards were required to generate the 2L combinations of lines. It is shown as Figure (3-4). As mentioned before, only the special case where the set of eXperimental points contained six lines with six points on each line Figure (3‘3) computer metl Read TK Calculate K Read CYY(1), Calculate Y3 Read T(l) , .. Read TC, CY Calculate Y! Calculate ll Calculate D Calculate M Calculate l4 Calculate )i. Calculate 14 Calculate 1 Output m: \ Figure (3. ‘2: Read (1(1) : c"illfilllate Calculate \ 52 Figure (3-3). Flow chart for the generation of points used in the computer‘method. Read TK Calculate K = l/(TK+273) Read CYY(l),..., CYY(L) Calculate YY(I) = LN(CYY(I)) for I=1 to L Read T(l),..., T(M) Read TC, CY Calculate YC = LN(CY) Calculate MM(I) = (TC-YY(I))/(l/TC-K) for I=1 to L Calculate Calculate Calculate Calculate Calculate Calculate D(I) = E*MM(I) for I=1 to L and E = 0.01, 0.02,... MD1(I) = MM(IT+D(I) for I=1 to L MD2(I) = MM(I)-D(I) for I=1 to L XJ(J) = l/(T(J)+273) for J=1 to'M KY(2*I-l,J)=MD1(I)*(XJ(J)-K)+YY(I) for I=1 to L and J=l to M KY(2*I,J)=MD2(I)*(XJ(J)-K)+YY(I) for I=1 to L and J=l to M Output KY(2*I-1,J), KY(2*I,J) for I=1 to L and J=1 to M Figure (3-4). Flow Chart to Join Parts 1 and 2 of the program. Read C(l),. ..,C(L) Calculate V(I) = 2*1-1 for I=1 to L Calculate S'+ C(I)+V(I) for I=1 to L Calculate YIJ(I,J) = KY(S,J) for I=1 to L and J=l to M would be stud: 10°C to 60°C, constants was experimental rate constant translation : The only thi terval which in the study The sir the existent chosen for moo°r whic 2000°K, as represent 1 become hare 2) Mei An at the COmPer SatiOn ter The . were Chos umber of Values of Ca“ be dt Change ll 53 would be studied. The temperature range chosen for this study was from 100C to 600C, which covered a range of 50°C. The interval of the rate 1 5 -l constants was arbitrarily chosen to be from 1 sec- to 10 sec at the experimental temperature of 10°C. The choice of the position of the rate constants along the ordinate was arbitrary since only a simple translation is required to move the interval to a different position. The only thing that mattered was the magnitude of the rate constant in- terval which should represent the result of the best possible technique in the study of biological entities. The six ideal lines were made to intersect at a point to simulate the existence of exact compensation. The compensation temperatures chosen for the study were 4000K, 8000K, 2000°K,(106)°K, -8oo°K and -4000K which were evenly spread over the x-axis, except for the one at ZOOOOK, as shown in Figure (3-5). It should be clear that these values represent increasing distances from the experimental temperatures and become harder and harder to estimate precisely from experimental data. 2) Metastaseteeesmleeeur An attempt was made to find a correlation between the position of the compensation temperature and the precision of the estimated compen- sation temperature. The values of the change in slope, D, due to experimental errors were chosen to be‘ilZ,'t2%,'t5%,‘t10%,'t20%,':§0% and :l00%. The large number of D-values were used in order that any relation between the values of D and the various other parameters obtained from the program can be determined. The program was also run for the case where the change in slope, D, corresponding to each ideal line was 0 or 2%. lllll l.’"'l. ’d’luhdlvu,” """"" Il"""'"”u’"”’ """""""""""""""" "’ """’ MU 0 kn M0009...“ .H a. owl." H U bun. OHnpo M00 A 00 Uva MOOOON“ .H. M00 Va 0 h 0 0 n H MOOOsV _ _ - H I V _ O — H\ AM as so coca xoema om.o- o O . A on o . on H 8 . and 8 N om.m co m 54 I II I ’ ’ f ’ l I I I I I II I’ ’I I, ’I I I, I ’ I I I ’ ’ I ’ II I , I, I’ ’ I, I’ I ’ I, I” [I ’ I I ’ I I I I, z I: I ’ I I 'I I, I z I a I i ' I I I, II ’ I, ’I I ’ I ’ I I ’ I I I, I, I I I, ’ .I I I I, I, , I I ” I ’ ” I III I, I, I II ..II II I, III ” ’ " ’ I ’ I I I ” ’ ’ ’ ’ II I, ’ ’ I, ’I I ’ ” ’I I, ’I i I I I III II I, I II II I I ’ ’ I ’ '10 I'll ” I, I, ’1', loll ” I, I ’ I, I, ’ ” ' 'I I, I, " II I, ’ III! III II II 'I"l' I,” ”’ ’I'I' I"l I.” ’1'”, ’I” I, III, ” ’ I 'l'l' II." ’I ' ’I' I. "l.’ I, '1-"" "" I, ""”"'l’ ""'l' "I'lvd' ""' """"" " ""' 'I”.""""""' ll. M oosm H 0 III uumwm mo uon < .Amemv m m> M com. o 10H M .oMm H M M oowm H Mooooml om O Frequency vestigated. Tl give an indica perature. The value since they we: into the line tained were t check to make The vah the rotation at a point. correspondin Plots c vestigate tt 55 Frequency histograms of i0 and TE were drawn for all the cases in- vestigated. The width of these frequency histograms might be used to give an indication of the precision of the estimated compensation tem- perature. The values of S00 for the generated sets of data should be zero since they were obtained by substituting the appropriate values of x into the lines represented by Equations (2-25). The small values ob- tained were the result of computational errors. THuurwere used as a check to make sure that the points were generated correctly. The values of So, therefore, reflected only the errors caused by the rotation or the translation of the generated lines to make them meet at a point. A plot was also made of the values of 80 against their corresponding estimated compensation temperatures. Plots of Su vs u and Su vs T were obtained for some cases to in- vestigate the dependence of the shape of these curves on the values of D. iii) MEASUREMENT Q§_CONDUCTIVITY QE_PROTEINS In a search for a set of experimental data, involving a biological entity, for the study of compensation, the conductivity of various pro- teins with various amounts of different adsorbate were measured. Pro- tein conductivity was chosen for investigation since there were a number 0f papers available which pointed to a large change in conductivity if an appropriate adsorbate was used. Moreover, both Rosenberg et al. (1968) and Eley (1967) reported the existence of a linear relationship of the form l__; where 0 and B equation is e (H). Theu fact that it due to the fa ductivity wi sulphide, et because it e magnitude wh method. Act fortunately Parts of th. above room sorbed alwa was reached as that “F acGunnercia hemoglobin Seravac (B. Pellet Use about One measuremer. thick. Prel: 56 log 00 = GAE + B where 0 and B are constants. (See Rosenberg et al., 1968) This equation is equivalent to the isokinetic relation given by Equation (2-2). The protein finally chosen was hemOglobin, partly due to the fact that it could be obtained in a pure form rather easily and partly due to the fact that it did exhibit a relatively large change in con- ductivity with a number of adsorbates such as water, ammonia, hydrogen sulphide, etc. Ammonia was used as the adsorbate in the present study because it exhibited a change in conductivity of about five orders of magnitude which is the range chosen for the ordinate of the computer method. Actually, the conductivity of water was also measured. Un— fortunately considerable condensation must have occurred at the coldest parts of the vacuum chamber when the temperature of the samples was above room temperature. This is true since the amount of water ad- sorbed always dr0pped by a considerable amount before quasi-equilibrium was reached and the range of the change in conductivity was not as wide as that reported by Rosenberg and Postow (1970). The ammonia used was a commercial-grade anhydrous ammonia available in small bottles. The hemoglobin used was twice crystallized bovine hemoglobin obtained from Seravac (Batch UKll). The hemoglobin was pressed into a pellet. The pellet used for conductivity measurements was one inch in diameter and about one hundredth of an inch thick. The pellet used for adsorption measurements was one centimeter in diameter and ab0ut one millimeter thick. ‘ Preliminary experiments show that a pellet similar to the conduc- tivity pellet described above would change its conductivity by about {our to five sorbed at w desorb ammoni range of 100 vacuum for 3 weight as be use the‘ same drying and a influenced 1 compensatim A sket consisted o 0f ammonia hundred gra weighing me Which had 1 located. 7 Changing. ductivity and insert the hang d on their a Pressed be heads fr< seals and amount of COuld be 57 four to five orders of magnitude with increasing amounts of ammonia ad- sorbed at room temperature. The hemoglobin also seemed to adsorb and desorb ammonia without apparent denaturation throughout a temperature range of 100C to 600C. When heated to a temperature of about 60°C in vacuum for several days, it would usually go back to the exact same dry weight as before it had adsorbed an adsorbate. Thus it was pOssible to use the‘same pellet to do all the measurements by adsorbing, measuring, drying and adsorbing again for a given temperature. These factors also influenced the choice of a hemoglobin-ammonia system for the study of compensation in a biological entity. A sketch of the experimental set-up is shown in Figure (3-6). It consisted of a Cahn microbalance which was used to measure the amount of ammonia adsorbed. This amount was recorded in units of one gram per hundred grams and was abbreviated as % throughout the thesis. The weighing mechanism of the balance was enclosed in a vacuum tight bottle which had two hang down tubes where the sample and the weight pans were located. The hang down tubes were detachable for weight and sample changing. The adsorption pellet was placed on the sample pan. The con- ductivity pellet was sandwiched between two stainless steel electrodes and inserted into a hole in a teflon block which was placed in one of the hang down tubes. The stainless steel electrodes had threaded holes on their side for electrical connections. The hemoglobin pellet was pressed between the electrodes to insure good electrical contact. Leads from the electrodes were fed through vacuum tight glass to metal seals and connected to an electrometer outside the vacuum chamber. The amount of ammonia adsorbed and the conductivity of the hemOglobin pellet could be determined simultaneously. Even though two separate pellets eeeurE ml _ hlLL MUZAV‘HE IOMHMVHE 58 mesh Ado Eva: .wsumumaam moamamnuouowfi 8552; m0 Emuwmwv owumfiosom .Aoumv onowam mg 948 mHo>mmmmm aHaoHa HmHzH m<0 l8 sumo a E; u HHUDQZB 9 g _ a as \_\ Mme 55242 g MMDUMMZ IIID\III mozaaem-OMUHs were used the) baths around 1 The desi‘ through an 0p ammonia and a curacy of one amount of arm partial pres: A secon- connected to for dielectr the hang don the chamber were not us: A more in the thes the equipme The as through the desired pa: chamber an over a da PIESsure C tablished signing; taken at . 59 were used they were both maintained at the same temperature with heating baths around the hang down tubes. The desired amount of ammonia was fed into the vacuum chamber through an Opening which was connected to a manifold. A small bottle of ammonia and a mercury manometer which could measure pressures to an ac- curacy of one millimeter were both connected to the manifold. The amount of ammonia adsorbed by the pellets was roughly controlled by the partial pressure of the gas allowed to enter the vacuum chamber. A second chamber which contained another hemoglobin pellet was connected to the manifold. This chamber was made of brass and was used for dielectric measurements under the same conditions as the pellets in the hang down tubes. When dielectric measurements were not required, the chamber can be isolated from the manifold. Since the data obtained were not used in this thesis, it would not be described in detail here. A more detailed description of the experimental set-up can be found in the thesis by Postow (1968). Although the set-up was not identical, the equipment and instrumentation used were the same. The experimental procedure involved the measurement of the current through the sample together with the amount of ammonia adsorbed after a desired partial pressure of the gas was allowed to enter the vacuum chamber and equilibrium had been established. It took several hours to over a day for equilibrium to be reached, depending on the partial pressure of the ammonia in the chamber. Equilibrium was considered es- tablished if the current and the weight measurements did not change significantly for over a period of one-half hOur. Measurements were taken at a constant temperature for a number of weights of ammonia ad- sorbed before changing to another constant temperature. Before starting meets for at least had desorbed dry weight. Since a occasionally due to the d changing to The dal rent vs amm y the Arrheni‘ as shown in 60 starting measurements at each temperature the vacuum chamber was pumped for at least two day at 60°C to make sure that all the ammonia adsorbed had desorbed and the weight of the sample had returned to its original dry weight. Since a complete set of points took over a month to measure, it was occasionally necessary to correct the dry weight reading of the sample due to the drift of the balance. This was done, when necessary, while changing to a new temperature. The data obtained from the above measurements gave a plot of cur- rent vs amount of ammonia adsorbed for each temperature used. To obtain the Arrhenius type plots, a transformation was necessary. This was done as shown in Figure (3-7). 1“igur cum 61 Amou t n of Ammonia Adsorbed (7) 10’8 0 1 2 . 3 4 z : 5 6 ' I ' l ' I : ‘ I I I / ' I l I : 7’: 10'9 - : ///// I I : 3': i a I=3r3°K I I I 1 10'10- : l I I I A I a. I E I m I v I I H l ' . ' '.\ 7' /l I I . I ' I | l I l I : I I ' ' I : : 1c --------- ' ' ------------ I | ---------------- i / | - I I ' ° : : 24 ' I ' I . I : : 10-13r : : ' I ‘ I | I I 2.5 2.6 2 1 .7 2.8 2.9 3.0 3.1 3 2 /T x 1000 (T in OK) . 3.3 3.4 3.5 . Some of separate wit criticism oi into one ma; will contair t) YSI E In the know with c especially “Ch p0int researcher always rep inappropri the (:0thC w“Ch rec Witted. ACtu. Precision sation in which may IV. RESULTS AND DISCUSSION Some of the results and discussion in this thesis are hard to separate without lesing the clarity of the presentation or risking the criticism of too much repetition. Thus the two tOpics will be combined into one major division. Those sections which required a discussion will contain one throughout the section or at the end of a section. i) ANALYSIS 9:5; EXISTING DATA In the analysis of existing data, it is usually not possible to know with confidence how precise the experimental values are. This is especially true with experiments involving microbial techniques where each point is generally the mean of several readings. Even though the researcher himself may know the error limits of the points, he may not always report these limits in his paper. It is, therefore, considered inappropriate to make a guess of what these errors might be. Except for the conductivity data which this author collected, all the analysis which require the knowledge of the experimental errors will be omitted. Actually most biological data really do not possess the type of Precision required in the study of compensation. The study of compen- sation involves the determination of whether a set of straight lines, which may be almost parallel, intersect at a point. This determination 62 is not as ea: their points ments may m0 the whole pr ciding if a in later ane often too hf analysis. 1 her of expe Two sets of illustratin et al. (196 1) Dates Accorl best set 0 et a1, (19 1‘18 that c Strated tp lected us: the time, log k2 me The activatic Only the Since the tempera” 63 is not as easy as it may appear. For lines which form small angles at their points of intersection, an error of a few percent in the measure- ments may move the points of intersection by large distances. (Actually the whole problem of compensation study lies in the difficulty in de- ciding if a given set of lines intersect at a point.) As will be clear in later analysis, experimental errors of ten or twenty percent are often too high for any meaningful analysis with available methods of analysis. Moreover, most of the papers do not contain sufficient num- ber of experimental points to make any statistical analysis meaningful. Two sets of existing data were analyzed, as examples, for the purpose of illustrating the use of the F-test method. They are the data of Barnes et al. (1969) and Luedecke (1962), the latter from Hamilton (1971). 1) Eggs 2; Barnes g; 1L; According to a comprehensive review by Lumry et a1. (1970), the best set of microbial data available was contained in a paper by Barnes et al. (1969) who collected the data expressly for the purpose of show- ing that compensation exists in bi010gica1 entities. The authors demon- strated that the compensation law was satisfied by the data they col- lected using some of the well-known methods of analysis available at the time, which included the entrOpy-enthalpy method and the log k1 vs 10g k method. 2 The reactions studied involved the protection against thermal in- activation of Sindbis virus by HPO4 or 80; ions in a suitable solution. Only the case of protection by the HPO; ions will be considered here since the analysis would be identical for both cases. The c0mpensation temperature estimated by the various methods used by Barnes et al.(l969) isreproduce Table (4'1) h%imu. ..,fl # Method ” Arrhen EntrOp Exner Exner Exner Exner K Source: B thenmm Exner me The 5 fun: liner than adeqr adequate (1972) se for a mea least f1, Squares, at a Sin' than tWe 64 is reproduced in Table (4-1). Table (4-1) Estimated compensation temperatures of protection by the 'HPOA ions. Method of Estimation Compensation Temperature Arrhenius 56.3fi'; (CC) + . Entrapy-Enthalpy 60.4_§ ; ‘k + Exner 44, 48 52.7 1'0 -O.6 Exner 44, 52* 56.7f3'2 + . Exner 48, 52* 60.9 4 9 , -204 Exner mean+ 56.3:3.3 Source: Barnes et al. (1969). o i The numbers after Exner represent temperatures in C. Exner mean is the mean of the three compensation temperatures aboveiJ; The set of data contained twelve points, which were made up of four lines of three points each. Even though twelve points is more than adequate when one straight line is involved, it is not really adequate for the determination of the existence of compensation. Exner (1972) seemed to think that a minimum of twenty points are necessary for a meaningful application of his statistical analysis. It takes at least five or six points just to determine if they form a single straight line with any degree of certainty using the method of least squares. It takes at least three lines to determine if they will meet at a single point. However, most available data do not contain more than twenty points and the data of Barnes et al. (1969) will be treated as if there the values C 303.85 x 1C Source: Be It is favorably ' (2-22) is which is 1 handbook cannot b The BiVes a in Fig,“ 65 as if there are an adequate number of points. Table (4-2) gives some of the values obtained from the program. A Table (4-2). The values of i , T , S and S . o c o oo . o -1 * 0 X0 ( K) TC ( K) S 800 2 2 303.85 x 10'5 329.11 2.16465 x 10' 2.02004 x 10' Source: Barnes et al. (1969). It is evident that the estimated compensation temperature compares favorably with those obtained by Barnes et al. (1969). Inequality (2-22) is easily satisfied since the ratio (SO - Soo)/(L - 2) = 0.143 goo/(ML - 2L) which is much smaller than the FO OS(2,4) value of 6.94 taken from the handbook by Burington et al. (1970). Thus the existence of compensation cannot be rejected. The expression l L - l M = Soo(1 F0.05 Rfl.- 2L ) 0°120 gives a confidence interval of (3200K, 4050K) using the Su vs T curve in Figure (4-1). The width of the above interval is 85°C. Figure (4-1) is a plot of Su vs T where Su is defined by Equation (2‘21). A clear minimum can be observed at about 3300K which is a 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.5 0.4 0.3. 0.2 0.1 Fi et 66 Figure (4-1). A plot of Su vs T, et al. (1969). - it i ..W - ' ’ ' J i . I . 1* ' ' 1 -500 -400 -300 -200 ~10:OK 0 100 200 300 400 500 600 using the RFC; data of Barnes necessary C not be a so 2) Date .1 The dz part of the sation witl was report- perature o compensati 3270K obta sation is the F-test The l sixteen 1: temperatu: This is m' of the ex We of s of the va Pensatiop ROSenberE since Which is 67 necessary condition for the existence of compensation although it may not be a sufficient condition. 2) Data gf Luedecke The data of Luedecke (1962) was used by Rosenberg et al. (1971) as part of the points in an entrOpy-enthalpy plot which satisfied compen- sation with a c0mpensation temperature of 3250K. Luedecke's data alone was reported to have satisfied compensation with a compensation tem- perature of 3310K. However, it has been shown in the THEORY that the compensation temperature of 3310K corresponds rather closely to the 3270K obtained from the error SIOpe and that the existence of compen- sation is probably an artifact. The same data will be analyzed with the F-test method in this section. The Luedecke data contains forty-eight points and was made up of sixteen lines each of which contains three points. The experimental - temperatures were 3210K, 3230K and 325°K.which have a width of only 40C. This is much too narrow a temperature range for a useful determination of the existence of compensation, althOugh it is quite suitable for the type of studies Luedecke (1962) was conducting. Table (4-3) lists some of the values obtained from the computer program. The estimated com- pensation temperature compares favorably with those obtained by Rosenberg et al. (1971). However, Inquality (2-22) is not satisfied since (SO - Soo)/(L - 2) 5 /(ML - 2L) 00 = 2.98 which is larger than the E (14,16) value of 2.37. Thus the chances 0.05 of the data hundred and Table (4'3) 300.80 x 10 Source: Lt It is the confid. COmpensatil Final This minim et a1, (19 0f the cla alone can This obser ii) comm M This 68 of the data satisfying the compensation law is less than five in one hundred and the assumption of compensation must be rejected. Table (4-3). Values of £0, Tc, 80’ and 800 from the data of Luedecke. o -1 t o o ( K) Tc ( K) So Soo 5 300.80 x 10' 332.44 8.0066 x 10'1 2.2368 x 10"1 Source: Luedecke (1962). It is obvious from Figure (4-2), which is a plot of Su vs T, that the confidence interval, even though it is not useful for a case where compensation is not satisfied, is the entire T-axis. Finally it should be clear from Figure (4-2) that a minimum exists. This minimum appears to be sharper than the Su vs T curve of Barnes et al. (1969). Figure (4-1) and (4-2) also serve as practical examples of the claim that neither the existence nor the sharpness of a minimum alone can be used as a criterion for the existence of compensation. This observation will be discussed again in the next section. ii) COMPUTER.ANALYSIS QE GENERATED DATA This subdivision contains new results which should be helpful in the collection and analysis of experimental data. 1.5 ' 1.0 - 1.3 ' 1.2 1.1- 1.0 - 0.9 - 0.6 - 0.5 . 0.4 . 0.3 . 0.2. 0.1. Not 69 1.7 1.5 - 1:: l / 0.4 - 0.3 ~ 0.2 r 0.1 e 0 1 L c J J ..L l _L J_ I I -500 -400 -300 -200 -100 0 100 200 300 400 500 600 T K Figure (4-2). A plot of Su vs T, using data from Luedecke (1962). Note: based on some calculated results of Hamilton (1971). w eaten Since 2 answer the 1 tence and tl of experime'. There must precision. relationshi experiment: Figurr correspond perature, to any kno lc with va not appear coUrsa, or will not I The can be us was discu f°r a set ideal com the fI‘Eqr teIVal f o 400 K an Which ha c0ml’ehsa 70 1) Relationships Concerning the Confidence Interval Since any study of the compensation phenomenon must be able to answer the two questions asked throughout the entire thesis--the exis- tence and the temperature of compensation--it is imperative that any set of experimental data obtained should be adequate for this purpose. There must be a sufficient number of points which are of the required precision. This is very time consuming. In this section, some useful relationships have been derived which will be helpful in the design of experiments which can accomplish the purpose without excessive work. Figure (4-3) shows the frequency histogram of all the values of TC corresponding to the 2L binary numbers for an ideal compensation tem- perature, Tc’ of 4000K and a D-value of :5%. They do not seem to belong to any known type of frequency distribution. Frequency histOgrams of Tc with various other ideal compensation temperatures and D-values do not appear to be similar to any known distribution. This is, of c0urse, no proof that if sets of random data are used the distribution will not have a known distribution. The interval covered by the frequency distribution or histogram can be used as a confidence interval for an apprOpriate set of data. As was discussed in the THEORY, it can be used as a confidence interval for a set of data which is known to satisfy compensation with a known ideal compensation temperature. For example, the interval covered by the frequency histogram of Figure (4-3) can be used as a confidence in- terval for a set of data having an ideal compensation temperature of 4000K and a D-value of i5%. Conversely, a set of experimental data which has errors corresponding to a D-value of i5% and an estimated compensation temperature of 4000K should have the same confidence 7l .NnH u a was Moooq u lofiflemmusumuanou dowummaomsoo coumfiflumo osu mo EmnwOumwn >0uo3voum AMOV mousumuomaoa aowummcomsoo woumafiumm ¢N¢ NNq omq de cad «Au NH¢ oaq woo 00¢ dos Now ooq mom 0mm «mm Nam 0mm wwm Amid 0.33m own «mm Nwm interval for the confiden Figure (4'33 Table 1 ill. and i170 20000K, 1,01 tained from clear from is not very interval ir temperaturw Intact, a magnitude same magni more apprc Table Which wilf given in ihtervah Petature is Possit componsm be denot. 72 interval for its true compensation temperature. In this particular case the confidence interval is (3830K, 4230K), which is obtained from Figure (4-3). Table (4-4) shows such confidence intervals for D-values of i5%, i2% and il% when the ideal compensation temperatures are 4000K , 8000K, 2000°1<, 1,ooo,ooo°1<, -800°1< and -4oo°r<. These intervals are all ob- tained from frequency histograms such as Figure (4-3). It should be clear from the table that a confidence interval in terms of temperature is not very satisfactory. It is biased in such a way that a confidence interval in a region near x = O (T =(D) appears very large in terms of temperature even though it actually is quite small in terms of x-values. In fact, an x-interval with a given magnitude near x = 1 has a larger magnitude in T than that which corresponds to the x-interval with the same magnitude but located near x = 2. This is the reason that it is more apprOpriate to cOmpare confidence intervals in terms of x-values. Table (4-5) gives the confidence intervals in terms of x-values, which will be called x-intervals for convenience, corresponding to those given in Table (4-4). Thymust be noted that the magnitude of the x- interval does increase with the distance of the ideal compensation tem- perature from the experimental temperatures in a predictable manner. It is pOSsible to find a relationship between the position of the ideal compensation x~value, x0, and the magnitude of the x-interval which will be denoted by f(Tc, D), where TC and D have their usual meaning. The p0sition of x0 is defined in terms of its distance from a reference point xk which is the point of rotation of the lines defined by Equations (2-25). This distance is denoted by d(xo, x k)' Table (4-6) gives the magnitude of the x-intervals for various Table (it-[W Jgl‘l. with war 2000 6 10 -800 400 *These int Actually, larger. Table (4- various v 2000 10 800 '400 Table (4-4). 73 jj%.with various values of Tc' Confidence intervals for D-values of i5%,'i2% and O TCK D = :57. (°1<) D = :27. (OK) D = :17. (OK) 400 383 423 393 -- 408 396 --- 404 800 673 1053 740 -- 878 768 --- 836 2000 1229 9866 1577 -- 2837 1759 --- 2336 106 -2160 2730* -5859 -- 6366* —12085 ---12368* -800 -1325 -533 -963 -- -675 -876 --- -734 -400 -533 -304 ~488 -- -358 -423 --- -378 *These intervals correspond larger. Table (4-5) . various values of Tc' to the largest intervals along the x-axis. Actually, the values of all the Tc's have absolute values that are x-intervals for d-values of‘i5%,‘t2% and il% with TZK Dé:5% (T'1x10'5) D=:2%(T‘1x1o'5) o=iaw (T’1x10‘5) 400 236 261 245.0-- 254.6 247.6 -- 252.3 800 94.95 148.6 113.9-- 135.1 119.6 -- 130.2 2000 10.14 81.36 35.25-- 63.40 42.81 -- 56.86 106 -46.29 36.63 -17.07-- 15.71 -8.274 -- 8.085 -800 -187.9 -75.5 -148.2-- -1o3.9 -136.3 -- -114.2 -400 -329 -188 -297.3-- -223.3 -246.3 -- -236.4 Table (4-63 2000 2000 ~800 ~400 74 Table (4-6). f(TC, D) for various values of TC and D. (TC)OK D = :57. D = :27. D = :17. 400 25.0 x 10‘5 9.6 x 10~5 4.78 x 10'5 800 53.6 21.2 10.58 2000 71.2 28.2 14.05 106 82.9 32.8 16.36 -800 112.3 44.3 22.16 ~4oo 141.0 56.0 27.94 Table (4-7). R-values for various values of Tc and D. (TC)°K D = :57. D = :27. o = :17. 400 24 9.2 4.6 800 23 9.5 4.6 2000 23 9.4 4.6 106 24 9.5 4.8 -800 23 9.3 4.6 -400 23 9.3 4.6 values 0f T Tc and D no If r(1 remains rel Note that i value corn mm specia? rotation 0 always be pr0per poi Table tude of tb from the 1 Equation 1 where A 1 later. Befc noted her 0f the 2I 3“ examp] hum di: queney ht sation t illustra 75 values of TC and D. It should demonstrate the dependence of f(TC, D) on TC and D more clearly than Table (4-5). If f(Tc’ D) is divided by d(x0, Xk)’ then the ratio, denoted by R, remains relatively constant for a given D-value as shown in Table (4-7). Note that in the special case under consideration, xk is also the x- value corresponding to the smallest experimental temperature. Thus, in our special case, x and xM are the same. In fact it is the point of k rotation of the error lines represented by Equations (2-25) that should always be used as the reference point since it is theoretically the prOper point to use. Table (4-5) demonstrates that the relationship between the magni- tude of the x-interval, f(Tc’ D), and the distance of the x-value, x0, from the reference point , d(xo, xk), is given approximately by Equation (4-1). f(TC, D) = A x f(D) x d(x0, Xk) (4-1) Where A is a constant and f(D) is a function of D which will be obtained later. Before deriving an expression for the function, f(D), it should be noted here that the x-intervals are obtained from frequency histograms of the 2L pOSSible values of £0 for a given D and Tc. Figure (4-4) is an example of such a frequency histogram, which does not seem to have a known distribution again. However, the superposition of several fre- quency histOgrams with D-values of i§%,‘i2% and il% for an ideal compen- sation temperature of 4000K does slcerw a little to the left as illustrated by Figure (4-5). This is reasonable since the nearer two .NnH n O was Moooq u 0H pow mopam>ax coauMmGoQEOU woumeumo can we Emumoumwn >oao=voum .Aqudv unawam H AMOV mosam>ux cowummcomaoo moumfifiumm oo.m mm.N om.m um.N mm.N om.~ m¢.N wq.N ¢¢.N N¢.N o¢.N wm.~ om.N . . . . a . . _ _ _ n _ _ 76 o sowummomuomom .Amsqv shaman .Moooq m 09 can NfiH was NN+ pxnfl n a coma mamuwoumfic m AMOV monam>ux aowummcomeoo woumeumm . . . m.N om.~ -eaxoo N mm.N em.N am.N Nm.N om.N we.N ea.N ae.N Na N 04 N m 77 h _ d 1 . rd 0 I 0 D I I I I I n ‘ I d 1 I I I — ‘7 i In .1 q 1 . d d d d 1 I d :m -oH .rmH ideal Arrh their poin ltis beyo histogram distributi with randc be the to] The l ation con for all t be magni of 0. Eq v01ved in Where B 3 Not values 1 within f to be a faCt. ft to the c shOuld p precisir cotrfide1 D‘Value 78 ideal Arrhenius lines are to the experimental temperatures the smaller their point of intersection will shift for a given experimental error. It is beyond the sc0pe of this thesis to investigate if the frequency histogram of the estimated compensation temperature belongs to any known distribution. Such investigations involve the analysis of sets of data with randomly distribtuted errors within given limits which can easily be the tOpic for a thesis in applied statistics. The expression for the functiOn, f(D), can be derived from inform- ation contained in Table (4-8) which shows the values of the ratio R/D for all the various values of D and Tc investigated. From Table (4-8), the magnitude of the x-interval is directly proportional to the value of D. Equation (4-2) gives a relationship for all the variables in- volved in the analysis. ) (4-2) f(Tc’ D) = B x D x d(xo, xk where B is again a constant. Note that the above equation may not be an exact equation. The values in Table (4-8) would indicate that the equation is accurate to within five percent. In addition, the above arguments are not intended to be a proof of the validity of Equation (4-2) for all values of D. In fact, for much higher values of D, the equation does not hold according to the data obtained for the particular case considered here. It should be adequate for use as a guide in the estimation of the type of precision required in experimental measurements to achieve the desired confidence in the results obtained. This is true for experiments with D-values of up to l0%. Since no special restrictions are used in the choice 0f sis, exceP and points to data wl in fact so the condo Table (4- Th Uncerta Peratu: Posed j dence ; X‘valu geSted Pehsat of T C Arrhe, 79 choice of six points for each of six lines as the case for our analy- sis, except the fact that there should be a sufficient number of lines and points for a reliable analysis, the equation should be applicable to data which do not have exactly the same number of lines and points. In fact some of the results obtained in this section will be applied to the conductivity data later. Table (4-8). R/D values for various values of TC and D. TC(K) D=:57. D=:27. D=:17. 400 4.8 4.6 4.6 800 4.6 4.8 4.6 2000 4.6 4.7 4.6 106 4.8 4.8 4.8 -800 4.6 4.7 4.6 -400 4.6 4.7 4.6 The above study also shows that Hammett's (1970) claim that the uncertainty can increase forty-eight times when the compensation tem- perature is negative is not true in the case of the new methods pro- poSed in this thesis, especially if the x-interval is used as a confi- dence interval for the compensation temperature. Indeed the use Of the x-value corresponding to the compensation temperature is strongly sug- gested and will be call the compensation x~value. The use of the com- Pensation x-value is also mathematically more appr0priate since the use of TC involves temperature which is infinity at the origin of the Arrhenius plot. Finally, according to Equation (4-2), the uncertainty ofa proc only abou if everyt terms of [actor a: Ano whether tioned b periment sation t cesses a derivati contain x-inter sibilit eSpecia interse tures. tUTES ; also d. dePend depend botwee tore, t“4111181 affec- 80 of a process with a compensation temperature near minus infinity is only about 3.5 times larger than that of a process with a TC of 4000K if everything else is the same. If a confidence interval is given in tenns of temperature, the uncertainty would increase by a much larger factor and in an unpredictable manner. Another application of the x-interval is the determination of whether two processes have the same compensation temperature. As men- tioned before, Equations (2-25) define the regions within which the ex- perimental regression lines must lie. Suppose the estimated compen- sation temperatures obtained from the experimental data of two pro- cesses are used in place of the ideal compensation temperatures in the derivation of Equations (2-25). Then the x-intervals obtained must contain the true compensation temperatures of the processes. If these x-intervals have a relatively large intersection, there is a good pos- sibility that the processes may have the same compensation temperature, especially if the x~intervals are small. If the two x-intervals do not intersect, then the processes must have distinct compensation tempera- tures. In the language of set theory, the two compensation tempera- tures are distinct since the intervals containing them are disjoint. Finally it must be mentioned that the magnitude of the x-interval also depends on the width of the experimental range, d(xl, xM). This dependence is hidden in the estimation of the error in slepe, D, which depends on d(x , xM). More specifically, d(x , xM) is the distance 1 1 between the x-value corresponding to the largest experimental tempera- ture, x1, and the x-value corresponding to the smallest experimental temperature, xM. If d(xl, xM) is decreased by a given factor without affecting the error limits of the experimental measurements, then D is . ._ . Al,__.._ increased tion for D-value f where t n (1970), and Thus th Equatic 81 increased by a known factor. For example, using the standard equa- tion for the confidence limits of the SIOpe of a regression line, the D-value for the line is given by Equation (4-3). D = h (4-3) tn-2;Y/2 Sey 2 where tn-2'Y/2 can be obtained from Table XIII of Burington et al. (1970), 2 2 q — 2 S = (1 - r ) L(y. - y) /(n - 1) (4-4) ey J'J 2 ~ 2 and h = l/Z(x. - x) . (4-5) 2 j 3 Thus the only terms which involve x are h2 and r where r is defined by Equation (4-6). 2 _§L(xi - §)(y-L- 37)]2 26.. - $022.30. - £02 J J J J (4-6) r It should be clear that D can be writteneusa polynomial of the variable (Xj - i) which is dependent on the width of the temperature range of the data. In fact the polynomial will contain only negative powers of (Xj - i), so that a decrease in the x-value range will cause an in- crease in D. Since a decrease in the x-value range corresponds to a decrease in the temperature range, D also increasesausthe experimental temperature range is reduced. The increase in D as a function 0f (Xj - i) can be calculated from the above equations in principle. 2) deal It which we seem to to have 3 vs u u stead o the rel dard de evident curve 1 The ex. Barnes of a n1 compen mum a1 or exa ments. the 3 numbe- the 3 yield this Value COmbj 82 2) Analysis gj_the Graphs Involving EE.§EQ'SU It has been mentioned in the EXPERIMENTAL that the Su vs T plot which was considered by Exner (1970) to be "the most valuable" does not seem to live up to this high expectation. Exner (l972) himself seemed to have changed his mind somewhat on this subject. First, su vs l/T or su vs u curves have been used to determine the confidence interval in- stead of the Su vs T curve in his second paper (Exner, 1972) although the relation between su and Su is known since the former is the stan— dard deviation corresponding to the latter. In addition, there is no evidence in the second paper that either the Su vs T or the su vs 1/T curve has been used to determine the existence of compensation. The examples in Sections IV-i-l and IV-i-2 dealing with the data of Barnes et al. (1969) and Luedecke (1962) have shown that the existence of a minimum in the Su vs T curve is not a sufficient proof that the compensation law is satisfied. In addition, the sharpness of the mini- mum alone does not always give a reliable indication of the existence or exactness of compensation. The results of computer analysis seem to support these state- ments. For example, for an ideal compensation temperature of 4000K, the Su vs T curve for the combination which corresponds to a binary number of 11,001 shows a minimum even for a D-value of $501. Some of the sixty-four possible combinations for thisspecialcase are already yielding estimated compensation temperatures which are negative with this D~value. Figure (4-6) shows a partial plot Of Su vs T for D- values up to i§0% with an ideal compensation temperature of 4000K for combination 11,001. Figure (4-7) shows the minimum of the Su vs T curve for various 120- 110- 100- 90 - 70 - 60 - 50 ~ 40 - 30 - 20 10 83 7'c :17. :107. :207. :507. COCO llllll 120- 110- 100- 90 L 60 - 50 - 40 e y 20 “flit—7' wIV 4"! 0 +/ 1 \t/ 1 L L L 4 1 1 1 1 200 300 400 500 600 700 800 900 1000 1100 1200 1300 T (OK) Figure (4-6). A plot of Su vs T when TC = 4000K.with various values of D for combination 11,001. 120 - 110 - 84 120 110 100 90 80 70 60 50 40 30 20 10 *D=:m +n=:m% n=:2o7. . D = :507. /° /./ +______________... / //*M L J. l l _L J Figure (4-7). 200 300 400 500 600 700 800 .900 1000 1100 1200 T <°K> A plot of S vs T when TC = 4000K.with various values of D for combinatioh 111. l values 0 At least 111, the trary to cannot a is satis Fir Barnes sharper data, in Figu compute -9oo°1< should times 1 sing H graph plotti still satior Toanti eXistr towar than done graph 85 values of D up to j§0%.with an ideal compensation temperature of 4000K. At least for this combination, which corresponds to the binary number 111, the minimum is sharper as the value of D increases. This is con- trary to what Figure (4-6) shows. Thus the sharpness of the minimum cannot always be used as an indication of how W811 the compensation law is satisfied. Figures (4-8) and (4-9) are the graphs of Su vs u for the data of Barnes et al. (1969) and Luedecke (1962). These figures do show a sharper minimum for the better set of data. However, the conductivity data, which will be presented later, has a very broad minimum as shown in Figure (4-10) even though it satisfies compensation according to the computer method. It is true that this set of data has a To of about ~9000K as compared to about 3300K for the last two sets of data. This should require the width of the conductivity plot to be about nine times larger than if it had a TC of about 3300K. Even after compres- sing the abscissa by a factor of eight, as shown in Figure (4-11), the graph is still not sharper than those of Luedecke. Thus even though plotting with u (or l/T) on the abscissa improves the situation, it is still not satisfactory yet as a criterion for the existence of compen- sation. Moreover, the concept of sharpness is rather abstract and a quantitative criterion is required for a reliable judgement of the existence of compensation. The above fact is probably what prompted Exner (1972) to lean more toward statistical analysis of the calculated data in his second paper than to limit himself to just a plot of Su vs T as he seemed to have done in his first paper (Exner, 1970). In fact, Exner (1972) uses graphs of sU vs l/T in his second paper most of the time. Even then 1.3 - 1.2 - 1.1 . 1.0 - 0.9 - 0.8 r- 0.7 . 0.6 . 0.5 . 0.4 . 0'3 F- 0.2 . 01 I 86 * \ P 0 °\° \o \\\\b -3 72' 3 1 fl DJ 11 x 10 (OK) Figure (4—8). A plot of Su vs u for the data of Barnes et a1. 87 1.5~ 1'40 1.2r l 1.1- \ 1.0r \\ 009L 0 \. 008 '- 0,74 0.6n 0.50 0.4-1 003" 002" 0 _3- _§ -1 of -1 u x 103 (OK) ' f r the data of Luedecke. Flgure (4-9). A plot of Su vs u o 88 Figure (4-10). A plot of Su vs 11 for the conductivity data. 89 0.4 - 003 - 0 0.2 - 0 ‘\N 001. " O -24 -22 -20 -18 -16 -14 -12 -1o -87 -6 -4 -2‘ 0' 2' 42 u x 103 (°1<)"1 Figure (4-11). A plot of Su vs u for the conductivity data, with u-axis compressed 8 times relative to that of Figure (4-10). 90 they are mostly used as a qualitative illustration of general cate- gories such as the isoentropic and the isoenthalpic cases. His cri- terion for the existence of compensation does not seem to require any graphs at all (Exner, 1972). The only time the graphs are used is in the determination of the confidence interval which is defined as those values of T which yield values of su which a1:e smaller than the experimental error 0. 3) Determination 9£.Whether Compensation i§DSatisfied Exner was not very explicit in his papers (1970, 1972) about the criterion for deciding if compensation is satisfied. He only stated: "If 30 is not significantly larger than 800’ the hypothesis of a common point of intersection cannot be rejected." It seems that a more ex- plicit criterion would be of value. An explicit method has been sug- gested earlier in the EXPERIMENTAL based on the F-test. Another method will be suggested here which does not relycnistatistics except the con- cept of least squares which has been used to derive Exner's equations. An examination of the Su vs u curves illustrated by Figures (4-8), ..., (4-11) will show that even though the sharpness of the minimum does not guarantee the existence of compensation, those sets of data which are shown to satisfy compensation do have lower values of So for the three examples considered. This suggests the use of So in some manner as a criterion for the existence of compensation. Theoretic- ally, So should be about as good a term to use as any that is avail- able. It is usually the sum of squares of the deviations of all the points after the Arrhenius lines have been moved to make them intersect at a point. In the case of the computer method, since the points used are ran are 1m 91 are generated from straight lines, there should be no deviations due to random deviations of points from.the Arrhenius lines. This is sup- ported by the fact that the Soo values obtained by the computer method are of the order of the errors of computation, which is about 10"10 or larger. Thus the value of 30 obtained here represents only errors caused by the rotation or translation of the lines. Figure (4-12) is a plot of the values of 80 corresponding to all the possible combinations of the lines against their corresponding TC. Except for being somewhat symmetrical with respect to a vertical line passing through To = 4000K, the points are randomly scattered. One thing to notice here is the fact that the points tend to approach a maximum.in S0 at the ideal compensation temperature which acts as an axis of symmetry. I suggest the use of this maximum as a criterion for the existence of compensation. Even though this suggestion does not have any rigorous mathema- tical support, there are several reasons for choosing the maximum.va1ue of SO as the cut-off point above which a set of data with cmmparable conditions should be rejected as satisfying compensation. First, if the process under investigation satisfies compensation, then an ideal experiment would yield a set of points which form a family of lines which intersect at the x-value corresponding to the compensation tem- perature of the process. Since no measurement is free of errors, the experimental points obtained will not lie in perfect straight lines. This, of course, is the reason for the use of the method of least squares to obtain the best estimates of the ideal lines. The sum of squares of the deviations from.the lines obtained gives an indication of the goodness of fit of the experimental points and is called So. 92 5832350 2: Sm tom fin... u a was Moooe u as fig moan Axes 0H mom .oH m> om mo uoaa 4 .ANH;QV ouswwm j]— mo.» mos Hos «an mam O mam The ' the the thei same Sitl ate: pet is err Sup the sel pe1 0U mu 93 The lines defined by Equations (2-25) were derived in such a way that all the experimental points lie in the regions they contain as discussed in the THEORY. Any point on any of these lines must be farther away from their corresponding ideal lines than any experimental points having the same x-coordinate. Figure (4-13) is an illustration of the above situation. An examination of Figure (4-12) reveals that the largest SO-value obtained from computer analysis is approached when the estim- ated compensation temperature approaches the ideal compensation tem- perature selected for analysis. It should be clear that this So-value is larger than any So-value obtained from any experimental data having errors which are not larger than that used for the computer method. SuppoSe a set of experimental data has an So-value which is larger than the largest So-value obtained from the computer method for a generated set of data with a given D-value of t§%. Then the errors of the ex- perimental data must correspond to a D-value which is larger than.t§%. Otherwise the assumption of compensation for the process being studied must be rejected. The second reason for the above proposal is the fact that the largest So-value can be obtained for any set of data whose errors are known and whose D-values can be estimated. The D-values, of course, should not exceed 1107.. Equation (4-3) is a method for estimating the D~values for any set of data which has an adequate number of points per line. Finally, this method does not make use of the F-distribution and should not be affected by the non-linearity of Equations (2-14), (2-15) and (2-16). This is a significant advantage, especially for those re- searchers who prefer not to use the F—test in the analysis of theirdata. Fde 94 log k l/T Figure (4-13). An illustration of the relationship between the ex- perimental points * and points on the boundary lines having the same X-coordinates ° The solid line represents an ideal line and the dotted lines are the boundary lines. COIIG neare ones reli ment are WOU the WW 95 An examination of Figure (4-12) also reveals that the values of So corresponding to the estimated compensation temperatures which are nearest to the ideal compensation temperature are not smaller than the ones which are farther away. Thus the magnitude of So itself is not a reliable indication of the accuracy of the estimate even for experi- ments with the same error limits. So far the cases considered assume that the errors in measurement are distributed symmetrically about the ideal lines. Figure (4-12) is an example of one of these cases. It is for a D-value of t2%. It would be interesting to find out what happens to the So-values when the errors are not symmetrical. Figure (4—14) illustrates the case when D is O or 2% and the ideal compensation temperature is 4000K. The plot of So against their corresponding estimated compensation tempera- tures is still symmetrical about a vertical axis. The vertical axis is displaced by about 20 Kelvin toward the experimental temperatures which corresponds to a displacement of about 30% of the width of the con— fidence interval of the case considered. Hence it may be concluded that the proposal presented above is not too sensitive to a skewness of the errors to one side of the ideal lines. Table (4-9) is a list of the largest SO-values for all the various values of D and TC considered in the thesis except for some of the larger D-values. Those with large D-values are not included since they are The values in Table (4-9) are used to not considered very reliable. Plot the curves shown in Figures (4-15), (4-16) and (4-17). These ' ' - ues which figures are used to obtain values of D or the largest SO val are between those listed.. These values are needed in the analysis of data such as the conductivity data which are presented in the next L no-0 m 0 .maoaumannsoo we“ saw you em go o u a use Moooe u as spas .o@ as m «0 soda < .Aea-ev «Human U AMOV a , Noe Hoe ooe son mam ham mam mmm as ca 0 i . a [a M Hoo.o 6 o o 9 O o O o . . . . - Ho.o o o o O o o o o oo o o oo o o oo o o o o o o o o 0 00 MO 0000 o 0 ”00 m ”0 mo.o sectior their < a para] square the sm corres that t tance d(xo, the la peratx fall 1 In fa obvio a p10 yield Figur Table 97 section. Figure (4-15) is the graph of the largest So-values against their corresponding values of D. The graphturns out to be a branch of a parabola and shows that the maximum value of SO increases with the square of D to within a few percent. This is reasonable since So is the sum of square of deviations of the experimental points from their corresponding regression lines. What is really surprising is the fact that the maximum value of So increases with the decrease of the dis- tance d(xo, xk) at such a fast rate. This is especially true when d(xo, Xk) is small as can be seen in Figure (4-16) which is a graph of the largest SO-values against d(xo, xk) at various compensation tem- peratures. The value of D for this case is 15%. It appears that the fall in the largest So-value with d(xo, xk) is more than exponential. In fact, Figure (4-17) shows the same plot on semi-log paper. It is obvious that the graph is still not a straight line. As it turns out a plot of the largest SO-value against d(xo, xk) on log-log paper does yield a reasonably smooth straight line. This is illustrated in Figure (4-18). Table (4-9). The largest So-values for various values of D and TC. 1.— Tc(°K) so for D i5% 30 for D i2% 30 for n i1% 400 59900 x 10‘5 9549.6 x 10"5 2385.1 x 10'5 800 8903.7 1422.8 355.53 2000 4733.7 756.63 189.09 106 3397.2 543.08 135.73 -800 1774.9 283.80 70.934 ~400 1088.2 174.03 43.499 98 60 - 50 P p~ O T w 0 O I Largest S -values N O 10 - D (%) _ 0 Figure (4-15). A plot of largest So-values vs D, Tc - 400 K. -values 0 b I O Largest S O U) l O. 2 .1r 99 7C *.___ ‘4’: Ht Figure (4-16). 6 wt»!- 4> mt- - =+° A plot of largest So values vs d(xo, xk), D _54. 100 1.0L- * U) (D :3 H (U > I O (I) u 0) OJ DO :30 1 A ' * ‘k\ 7': * . r 0'01 i i 3 4 5 6 3 -1 o d(xo, xk) x 10 ( K) =+°° Figure (4-17). A plot of largest So-values vs d(xo, xk), D _§/, on semi-log paper. o . .umama moH-on so .snw u a .Axx .oxve m> amass»- m smmwums mo soda < .Awa-ev magmas flees mes x Axe .oxos w o M w H- . tl4 . ‘r - I 101 N nth H Ho.o H.o o.H 8181 SBDIBA- s 389 w 102 Finally, it should be mentioned that plots similar to Figure (4-18) for D-values of i2% and il% also yield a straight line. Thus even though the experimental errors of a process under study do not corres- pond to one of the D-values considered in the study, it can still be analyzed as to whether the compensation law is satisfied. This also applies to processes whose compensation temperatures turn out to be different from.anticipated. For the convenience of researchers who may be interested in using the largest So-value as a criterion for the existence of compensation, an equation is derived as follows: From the graphs presented above, we have two relations represented by Equations (4-7) and (4-8). o 1 (4-7) = + - log S0 C2 log d(xo, xk) C3 (4 8) where Ci, 02 and C3 are constants in the sense that Ci is independent of SO and D while C and C 2 3 are independent of SO and d(xo, xk). Equation (4-8) can be rewritten as Equation (4-9). So== C4 log [d(xo, Xk)]C2 (4‘9) where C3 is equal to log C 4. Combining Equations (4-7) and (4-9) gives C11)2 80 = C4 log [d(xo, xk)] (4‘10) where C1 and C4 must be determined from the graphs or Table (4-9). 103 iii) ANALYSIS 9}; CONDUCTIVITY DATA This is the only set of data which is collected by the author. It will be analyzed in more detail since more is known about the precision of the data. It is also the only set of data which contains sufficient number of lines and points to make the statistical analysis meaningful. There are, however, weaknesses in the data which must be mentioned. If compensation turns out to be satisfied it should only be taken as a preliminary proof of the definite existence of compensation. 1) Experimental Results and Discussion Since this set of data is used only as an example in the thesis which is mainly interested in the study of the compensation phenomenon, some of the experimental details have been omitted. However, some of the experimental steps which can be improved will be described here. First, the weighing mechanism of the microbalance used was inside the vacuum chamber and was exposed to any gas which was chosen as the ad- sorbate. For example, some of the adsorbates,enufi1as hydrogen sulphide and methanol, caused the weighing mechanism to become inoperative and give incorrect weight readings even though they both caused an increase in conductivity when adsorbed by the sample. A balance with a weighing chamber which is completely isolated from all the weighing mechanism is recommended. For example, the Sartorius Magnetic Suspension Balance, Model 4201, would be appropriate. Another improvement would be the enclosing of both protein pellets in the same chamber which can be accomplished easily with the use of a balance such as the Sartorius balance. There is also the possibility 104 of using the same sample for both adsorption and current measurements. Figure (4-19) is a plot of the experimental data. The plot is in terms of current against amount adsorbed since these are the experi- mentally measured values. Figure (4-20) is a graph of current against l/T which is the graph from which the set of points for analysis are obtained. It should be mentioned that these figures are only illus- trations of the actual graphs which were drawn on a 17” x 22" sheet of ten cycle semi-log paper. Table (4-10) shows the values used for ana- lysis. Note that the current values have been multiplied by 1013. Table (4-10). Experimental points used for analysis. Temp. (0C) Current Readings (amp) 1% 2% 3% 4% 5% 6% 10 2.8 15.0 93.0 470.0 2800.0 15200.0 20 7.0 37.0 210.0 1080.0 5900.0 31500.0 30 16.3 84.0 . 445.0 2230.0 11900.0 60000.0 40 37.5 185.0 940.0 4600.0 22000.0 112000.0 50 82.5 390.0 1780.0 9000.0 41000.0 205000.0 60 172.0 770.0 3500.0 16200.0 77000.0 348000.0 Note: the columns under Current Readings represent the various Arrhen- ius lines with the amounts of adsorbate given in %. 2) Compensation Study with the Computer Method Since the points on the Arrhenius lines are not the original ex- perimental points, the application of statistical analysis must be carried out with caution. There are several alternative approaches that can be adopted in the analysis. First, the regression lines which pass through the original current vs % adsorbed curves can be 105 t- 1 I I l O l 2 3 4 5 6 % Adsorbed Figure (4-19). A plot of I vs % adsorbed for conductivity data. 106 10- 10% 10- 1. (2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 1/T x 103 (0K)”1 Figure (4-20). A plot of I vs l/T for conductivity data. 107 transformed mathematically into the current vs l/T curves. An equiva- lent number of points can be generated from the current vs l/T curves at predetermined values of 1/T and analyzed. The SOC-values obtained are, of course, just computer errors again and are not very useful. All the other values obtained from the program (x0, TC, 80 and S“) are still useful, especially if they are analyzed in the light of the com- puter method which will be discussed later. The analysis which in- volves the F-test is impossible for this particular alternative. The second alternative approach would be to calculate 800 from the original curves which should have about the same magnitude as the Soo- values obtained from the Arrhenius-type plots. S00 is by definition the sum of the squares of the deviations of all the y values of the experimental points from their corresponding regression lines. Whether the two SOC-values are equal depends on how an SOC-value is affected by a change in scale of the abscissa. Assuming that the answer is that S00 is invariant under a change in scale of the absciSsa, then the Soo- value obtained from the current vs amount adsorbed plot can be used for analysis with 80, using Equation (2-24) and Inequality (2-22). Finally an equal number of points can be obtained for the current vs l/T curves by the method shown in Figure (3-4). The method involves taking the current values on the same vertical lines, that is current values with the same amount of adsorbate, and replotting them in the I(1/T)-plane where I represents current. The method can be used to 0b- tain as many lines as desired, but the number of points on each line are limited by the number of experimental temperatures investigated. Again there is doubt that the Soc-value obtained will reflect the true experimental errors since the points come from regression lines in the 108 I(% adsorbed)-plane which is essentially a mean of several experimental points. The regression lines obtained fnmnpoints on the I(l/T)-plane should still yield reliable values of x0 and Tc. The value of So ob- tained can be compared with that obtained from the computer method if the errorsdue to the mapping of the points and the inability to read the points off the map precisely can be eliminated. Since all these errors are contained in the value of 800 it is considered appropriate to use the difference So - S00 when a comparison is made with the So- value obtained from the computer method. The method actually adopted in the analysis of the conductivity data is a modification of the above. Instead of using the actual points for the analysis, six points are taken from.each of the six lines (ob- tained as described in the EXPERIMENTAL). The plotting is done on a 17” x 22" sheet of graph paper. Table (4-11) gives the values obtained A for x , T , S and S . From this table the value of S - S is o c o oo o oo 7.91 x 10-3. From Table (4-9), it can be seen that this corresponds to Table (4-11). i0, Tc, So and 800 obtained from conductivity data. —’:— . o -l ‘ o , xo( K) Tc( K) So Soo 5 3 3 -106.6 x 10' -938.05 1.519 x 10‘ 7.28 x 10' a D-value of less than't5%. Thus the conductivity data is consideredtx> have satisfied the compensation law if the experimental errors involved correspond to a D-value of t5% or larger. Since experimental errors 109 corresponding to a D-value of i5%enxeabout the best that can be ex- pected from the kind of experimental set-up and the type of points obtained, the existence of compensation cannot be rejected. Using Equation (4-10) or Figures (4-15) and (4-18), the So - 800 value of 7.91 x 10.3 corresponds to a D-value of less than i3%.which is very good. Using Table (4-6) and Equation (4-2), an x-interval of about 5 65 x 10- (OK).-1 in magnitude is obtained. This corresponds, approxi- 5, 74 x 10’5> in (°I<)'1 which in mately, to the x-interval (-139 x 10- turn corresponds to the temperature interval (-1,350°K, -720°K). The temperature interval may seem large; but it is due to the fact that the compensation temperature has a large absolute value. If the com- pensation temperature were at about -300°K an x-interval of about 65 x 10.5 (0K)”1 would yield a temperature interval of (-333°K, ~2730K) which is much smaller. This is one of the reasons the x-interval is strongly recommended in place of the temperature interval as discussed earlier. Although a negative compensation temperature is uncommon, it is possible and has been reported in several instances (Exner, 1964; Ritchie et al., 1964). Exner (1972) used the existence of a negative compensation temperature as an argument for his claim that the compen- sation temperature is only the result of "a linear extrapolation of a non-linear dependence of log k on T“1 and that it lacks any immediate physical meaning." (See also Exner, 1968.) Finally, a minimum exists for the plot of Su vs T as illustrated by Figure (4-11). As discussed in Section IV-ii-2, the minimum is very broad. V. CONCLUSION The most important purpose of this thesis is to propose some cri- teria which will aid in the study of the compensation phenomenon, es- pecially in the area of biology. These criteria should assist in answering the two most important questions in any compensation study: the existence and the temperature of compensation. As to the objectives of the thesis mentioned in the INTRODUCTION, they can be briefly concluded as follow: (1) The statistical method is found to be the only method which has the correct approach to the problem encountered in the study of compensation. It contains a set of equations which are mathematically correct and are derived in detail in the Appendix. However, the statistical analysis prOposed to determine the existence of compensation needs to be improved. The method prOposed for the evaluation of the confidence interval can be further refined. It should also be more eXplicit. (2) Exner's method of analysis is the only method prOposed which uses the equations mentioned above. Thus the comments above apply to Exner‘s method. The method of analysis proposed also suffers from being too vaguely defined. (3) Two different explicit methods have been proposed in this the F-test method, which is most useful in dealing thesis: 110 111 with existing data although there may some theoretical question as to whether it is rigorously correct to use the F-test here; the computer method, which is mostly useful in dealing with data whose errors are known or with experiments planned with this method of analysis in mind. It should be pointed out here that the limita- tion to data with known errors is not unreasonable since Exner's analysis also made such an assumption (1972). (4) The F-test method has been used in two examples and found to be adequate. It gives the correct conclusions to the questions con~ cerning the existence and the temperature of compensation. (5) A set of conductivity data have been presented in the RESULTS AND DISCUSSION which satisfy the Arrhenius-type equation. A brief description of the experimental procedure used to obtain the data and the graphical method employed to obtain the Arrhenius-type plots desired has also been presented in the EXPERIMENTAL. (6) The conductivity data has been shown to satisfy compensation with the computer method with a maximum deviation of the slape of the experimental lines of about i3%. The compensation temperature is found to be ~938OK. (7) Equation (4-2) gives a relation between the magnitude of the x- interval and the distance of the compensation temperature from a 4 point in the experimental temperature range. It also gives a re- lation between the magnitude of the x-interval and the errors of a set of measurements. This equation will prove valuable in the design and evaluation of an experimental procedure for the study of compensation. 112 A number of valuable conclusions have been arrived at that were not included in the objectives. For example, we have found that the claim made by Hammett (1970) concerning the poor level of confidence in the analysis of processes which have a negative compensation tem- perature has been eliminated with the new methods proposed in this thesis. Using a confidence interval in terms of x-values, the mag- nitude of the x-interval is increased only by about four times when the compensation temperature changes from about 4000K to a large ne- gative value. Due to the increased precision possible with the methods proposed here, it is also concluded that the identification of a biological en- tity by its compensation temperature in a process which satisfies com- pensation is now feasible. This possibility was also suggested by Barnes et al. (1969). Finally it is also concluded that a strong possibility exists for some biological processes to satisfy the compensation law. However, the use of the entrapy-enthalpy method as a criterion for the existence of compensation in the study of biological processes is completely inadequate and should be discouraged. VI. RECOMMENDATIONS As mentioned in the RESULTS AND DISCUSSION, the analysis carried out in this thesis using the computer method is not complete. IA number of important calculations remain to be done. First, for simplicity in the computation and for more flexibility in the application, the method has been developed with experimental errors in terms of D-values which have been defined simply as the maximum change in slope of the Arrhenius lines corresponding to the experimental errors involved. This defini— tion appears to lack the kind of explicitness which the author seems to stress throughout the thesis. However, it was done for the sake of the flexibility of the method which is only in its infancy. Equation (4-3) has been mentioned as a possible method for the determination of the D-value for a given experiment when the errors involved cannot be ob- tained from an independent evaluation of the precision of the experi- mental set-up. Other methods may prove to be more appropriate if the results of further study of the computer method should necessitate a better method for the evaluation of the D-value. If the D-value can be obtained with an independent evaluation such as the statistical analysis of a large sample of measurements or knowledge of the pre- cision of all the instruments used, then the present formulation of the computer method is perfectly usable. A second area where more work is needed is the determination of the distribution of the compensation x-values using points with a 113 114 random distribution of errors with only two constraints: they should have a normal distribution and should have absolute values which are smaller than a predetermined limit. Even though the confidence in- terval proposed can be used to determine a reasonably small interval within which the compensation x-value must lie, it is not the smallest confidence interval possible. For example, an interval could be chosen which will contain the true compensation x-value 95% of the time if the distribution of the compensation x-values is known. The Fetest method, for example, is a method for determining a confidence interval and the existence of compensation using a known distribution. However, as claimed by Exner (1972), the F~test may not be the appropriate method for the study of compensation since the theory is deve10ped with the assumption of "linear models" which is not satisfied by the equations involved in the study. Therefore the knowledge of the distribution of the x-values would be valuable. Finally it seems that any application of the sums So and S00 to determine the existence of compensation cannot be valid unless the rate constant range is wide enough. Consider the extreme case where the rate constant range is zero. Then, it should be obvious that Soand S00 should be equal if they can be obtained. As the rate constant range increases 80 is greater than or equal to 300' But when the range is small 30 cannot be greater than S00 by a large factor and a test such as the F-test will always yield the conclusion that So is not significantly larger than S00 which in turn will be interpreted as the existence of compensation. Thus statistical analysis fails to be reliable when the rate constant range is not sufficiently large. The knowledge of the minimum range necessary for a reliable statistical 115 analysis is, therefore, Very important since it will rule out any study of processes which do not satisfy the minimum rate constant range and save a lot of wasted effort. The computer method may be useful in the determination of the minimum range since programs can be run with various rate ranges. The values of SO obtained will provide a depen- dence of SO on the magnitude of the rate constant range. LIST F REFERENCES ma—_ Ashmore, P.G., Catalysis and Inhibition of Chemical Reactions (Butterworths, London, 1963). Banks, B.E.C., Damjanovic, V., and Vernon,C.A., Nature, 240, 147 (1972). Barnes, R., Vogel, H., and Gordon, 1., Proc. N,A,§,, 62, 263 (1969). Beamer, P.R., and Tanner, F.W., Zentralblatt Bakt., 100, 81 (1939). Born, M., and Frank, J., Nachr. Ges. Wiss. Gottingen, ll, 77 (1930). Born, M., and WeisskOpf, V., Physik. Chem., B12, 206 (1931). Burington, R., and May, C., Handbook g£_Probability_and Statistics with Tables, 2nd ed. (McGraw-Hill, New York, 1970). Constable, F.H., Proc. R, Soc., A108, 355 (1925). Cramer, H., Mathematical Methods pf Statistics (Princeton Univ. Press, Princeton, 1946). Cremer, E., Advances ig Catalysis and Related Subjects, vol. VII, edited W.G. Frankenberg et a1. (Acedemic Press, New York, 1955). Cremer, E., Allg. Prakt. Chem., 18, 173 (1967). Eley, D.D. and Rossignton, D.R., Chemisorption, edited by W.E. Garner (Butterworths, London, 1957). Eley, D.D., g, _£_Polxmer §g,, Part C, No. 17, 73 (1967). Exner, 0., Nature, 488 (1964); _(_3_p_l_l_. _(_3_zLech. Chem. Commun., g_9_, 1094 (1964). Exner, 0., Ind. Chima Belgg,, 3;, 343 (1968). Exner, 0., Ngture, 277, 366 (1970). Exner, 0., ggll, Qgggh, thm, ngmgg,, 31a 1425 (1972). Hamilton, T., personal communications, (1971 ~ 1973). Hammett, L.P., Physical Organic Chem. (Reaction Rgtes, Equilibrigband Mechanisms), 2nd ed. (McGraw-Hill, New York, 1970). 116 117 Leffler, J., and Grunwald, E., Rates and Eguilibria Q£.Organic Reactions (John Wiley & Sons, New York, 1963). Leffler, J., Nature, 205, 1101 (1965). Likhtenshtein, G.I., and Sukhorukov, B., gg, Fig, Rhym,, 38, 747 (1963). Likhtenshtein, G.I., and Sukhorukov, B., Biofizika, 19, 925 (1965). Likhtenshtein, G.I., Biofizika, 11, 23 (1966). Likhtenshtein, G.I., Eh, Fig, Chim,, 44, 1908 (1970). Liscomb, W., Hartsuck, J., Reeke, G. Jr., Quiocho, F., Bethge, P., Ludwig, M., Steitz, T., Muirhead, H., and Coppola, J., Brookhaven §ymp, Quant. Biol., 21, 24 (1968). Luedecke, L.O., thesis, Michigan State Univ. (1962). Lumry, R., and Rajender, S., Biopolymers, 2, 1125 (1970). Maremae, V.M., and Palm, V.A., Reaktsionnaya Sposobnost Organ. Soedin. (Tartu), g, No. 3, 209 (1965). _ Palm, V.A., Principles gf_the Quantitative Theory g£.0rgan. Reactions (Khimiya, Leningrad, 1966). Petersen, R.C., Markgraf, J.H., Ross, S.D., 1- Am_. Chem. _S__c_>_c_.. _§.. 3819 (1961). Pihl, A.E., Pihl, V.O., and Talvik, A.I., Reaktsionnaya_Sposobnost Organ. Soedin. (Tartu), 2, No. 3, 173 (1965). Postow, E., thesis, Michigan State Univ. (1968). Postow, E., and Rosenberg, B., Bioenergetics, l, 467 (1970). Purlee, E.L., Taft, RJW., De Fazio, C.A., ;, Am, Chem. Soc., _1, 837 (1956). Ritchie, C.D., and Sager, W.F., Prog. Phys. Org. Chem., 2, 323 (1964). Rosenberg, B., Bhowmik, B.B., Harder, H.C., and Postow, E., J, Chem. Phys., 32, 4108 (1968). Rosenberg, B., Kemeny, C., Switzer, R., and Hamilton, T., Nature, 232, 471 (1971). Stapleton, J., Personal Communications. Walker, G.C., thesis, Michigan State Univ. (1964). APPENDICES APPENDIX A Terms Used ig_the Flow Chart KT(1,J) = k.. 1] YIJ(I,J) - yij * 1n kij T(J) = TJ, YI(I) = y1 XJ(J) = xj x = i P(I) = El Y = §/L U = u X0 = x 0 TC = T c SO = S o 800 = S 00 SU = S u -1 TK = (xk) CYY(I) = yi(xk) CY = y(xo) MM(I) = M1 D(I) = D C(I) = digits of a binary number 118 L . _ APPENDIX 13 Progr§m_for the Calculation pf T , x , S , S and S at Various Values of u c o o oo -- u‘- '—- 119 .OO.J.J~PZD.«.QMFw.~fl¢o~.aou. . ..2\x\1\.nx+Xu..x .OO.E.JHFZD.uaawkw.muooo.a0u. ..Ooouooaooocouoo>ooOoOnooX ...ozm...i\o.a\.DH>*A\D\VDX+A\~\.manoo.\H\.mn .OO.Z.JLFZD.~.QMFW.“no.1.aou..2~0mm. .OQ.J.JHFZD.~.nwkm.«uooH.aou. ...ozm...a\x\3.a\.5a>+x\a\ca>u...\a\aH> .OQ.Z.JHFZD.«.kam.“No.1.aou..Zuowm. .OO.J.JHFZD.H.QMPm.~noo~.&OL. ...sz. ..u.ou..“\chmn..o.ou..x\i\cH>.zaomm. .OU.J.JHHZD.~.awkm.MuooH.aou. ..Ax\o.a\coa>.«\m.~\cou>.x\e.u\.1~> .x\m.i\coa>.x\m.a\cni>.“\H.H\.n~>....\.nmmom.om+co.c..oo.s3n2~ .OO.J.JHFZD.~.QMkm.auoon.aou. ..x«\o\.nx..\m\cox.x\¢\cnx 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..6m**3*z+xmc\.xxm\m**an*4.*4 \z+>w9*m**3+>nm*3*mlnnmV*AN**ZVIN**>*J*EI>>mu..Dm.zuowm. .OD.OooolomooolohoooI.GOO-I.moool.maooolodooolomoooluooD.dou. ..aoow..a.\Om+.om«.o+mN....wH 00m....c..u0ch3akDO ..Aoe.ox..x.\on+.oma.o+mm ..A.mH Us.c..moaom+.omaoo+mN..A.m~ OX.V..V..~®kauFDO APPENDIX 9_ Program for the Computer Method 123 124 .OO.J.JHFZD.~.awkm. noou.aom. ..x“\o\.>>o..\m\c>>o..\¢\.>>o.“\n\.>>u .A\N\.>>o.x\a\c>>o..A.\.oo+.mom+.moe+.mon+.moo+.mo+.c.9oock3nza ..,mhm+¥k.\~u..y .oaxh..a.\.oo+....oovk3azn ...\J..«\.>>u.>y.“\z..«.4..a\.a>¥..\J..~\.>>.>U.J.J.>.>..\2..a.2..~\.>¥.>.>.oow.0k.ox.J>m.m.>m.xw.>nm.n.3.>.x.J4me. ..z.m.awowhz~. ..E.J.D.H.amowkzm..zmowm. .OO.E.JHFZD.~.kam.«uooo.aou. .oa\~\c>+.\H\VUuo.w.Z~0wm. .OD.J.JHFZD.«.awkm.auo¢«.aou. ..«Iasmu..a\~\.> .OO.J.JHPZD.~.ame.auoo~.aou. ..xx\0\vo..\m\cuox\e\.u..\m\cu.x\m\cuoa\d\vo..n.\..mov0.c..oo.k3n2~ ooooa ...ozm. ..x\H\.>>+.¥I.\1\.1xc*.\~\.N02uoo.\1.~*N\.>y .OO.Z.J~FZD.~.&wbm.~nooo.aou..zuomm. .00.J.JHPZD.~.nwkm.«uooa.aou. ...ozw. 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.OO.Z.J~FZD.H.anm.«no.7.aon..Zawwm. .OO.J.JNFZD.~.nmhm.nuoou.a0m. .o.mnm+a\fi\vk.\«uoon\o\vox .OD.2.J~PZD.~.kam.«uoofi.a0&. ...ozm...z\.\o.a\.na>+x\axc~>u...\a\ca> .OO.E.JHFZD.«.ame.«No.3.aou..Zuwwm. .00.J.JHPZD.~.umem.~uoo~.aou. ...ozw. ..o. n...\~\ama.oo.ouooa\~\v~>.Z~0wm. .OQ.J.JHPZD.H.awkm.aucon.d0u. ...Ozw. ..A\o.m\v>¥uooa\1.~\.7u> 127 ...m**3*2+xmc\..xm\N*#an*J.*J Innmc*nN**E.IN**>*J*ZI>>muoebm.2~wmm. .00.Noclo«ooiomooolomNOOol .0NOO.I.¢NOO.I.NNOO.I.Nooolom~oooto«OOoI.mOOOoI.mOOoo«OeuooD.aOm. ..AOOW..n.\Om+.OmaoO+mN..n.m~ 00m...9v..«O~PDQPDO ..xm\dnm*.m**zc1m>*ze>>muo.oom .oaom.3....\om+.omaoo+mN ..A.mH Om.c..moaom+.omuo0+mm..a.wa D.......«OVFDQPDO ..xm\A>nm*Dlnnm.*”N#*z.IN**>*J*ZI>>Wu.-0m ..N**n\~\vHQ+Q&muooaam .OO.J.J~PZD.~.uwkm.nuoom.&0u. ...ozm...N**A\o.H\cna>+>>mu..>>m .oo.:.uas23.a.amhm.an..n.aou..zauwm. .OO.J.JHFZD.~.nwkw.«uoou.aou. ..Ooouooaamooooouoo>>w ...UF.OX..n.\on+.Ofl«oO+mN ..0.W~ UP.a..mod0m+.0fl«00+mmo.a.m~ OX....V..MOVPDQPDO ..OX\HuooUP ..x+3u..ox ..A>nm*mv\1Axm*z\a*1m**>nm.*¢+m**mceaom+m1c1uo.3 ..A>m*xm*2\ac1nwu..m ..N*#0XIn\o\aox.+xmuooxm .OO.Z.JLFZD.~.QMFm.«uooo.aOm. ...ozw...N**.nlx\a\canc+nmu..nm ..N**.>IA\H\.~>.+>muo.>m ..m**x\~\.a>+m>u..m> ..A>IA\H\VH>.*xnux\a\cHac+>nmu..>nm.z~owm. \z+>m.*m**a+>nw*3*m 8 2 l .uom. .Ozm. ...Ozw. ..a.OFOO..ZwIF.m0.mmMJ.fiH.um. oon+fizuflooalm ...Ozm...63m.kk.3....\Om+.omaoo+mm..a.mm Dm.v. .Omfioc+mm..6.mH Pk.”..mmom+.omfioo+mmo.a.m~ D.v..v..~0ka&kDO .oOX\«uooFF ..x+Dn..0x .mm0m+ APPENDIX D Derivation Q£_Exner's Equations Let E (xij’ yij) i = 1,...,L, j = 1,...,M 1 be an array of ex- perimental points through which we can draw a family of L regression lines A A where biand ai are the estimates of b1 and a1 which are the constants of the ith line. Note that bi and 51 are the estimates of the cons- tants of the true lines y1 = bixi + ai, i = 1,...,L which the regression lines approximate. To obtain the estimates b1 and 5i for the regression line i we minimize the sum over j of the squares of the errors eij where, by definition, eij = yij - bixij - ai. In most cases, when one regression line is involved, this sum 18 _ ‘ 2 - ixij 31] for line i. I... H L: l L... r==1 s4 H 1.. I 0") 129 130 However, we want all the lines to pass through a common point and must minimize the errors of all the points (xij’ yij) in one sum. Thus we need to minimize the sum 2 e?. = [:y.. - b.x.. - 3.12 1.1 13 i.j 13 1 13 1‘ in general. Moreover, if the estimated common point (i.e., point of intersection of the regression lines) is (i0, yo), it must satisfy all the regression equations. That is A ”A 0 yo - bixo + 31 which implies ai = (y0 - bixo) for all 1. Thus we have 2 e2 = X [:y - b.X.. - (9 - 6.8 >12- i,j ij l,j lj 1 1] O 10 Since we need to find bi’ £0 and 90, minimize with respect to them. Thus we have . 2 ( 2 e .) . . . a .iJ =ZZ(yi‘boX--9+b.x>=0 3y laj O == . -* ‘ + MA...M ZN. Y iijo/gMi XogMibi/gmi §( ibl xlj/ i)/i 1 . = = e _ * ‘ +- A '. 2P1 (A‘l) " y yo xo EMibi/ZMi iZMIil/i 1 where '3': = iijij/ §Mi and xi = ‘Exij/Mi by definition. 2 3(12jeii) , . . *4 ’ 2 Z - - A + 9 = 9R0 I,jb1(yij bixlj yo b1 0) 0 Thus 2 (b y - bzx - b y + 32% ) = 0 i,j i ij i ij i O i 0 i (A-2) To minimize with respecttx>bk, we should use only the sum of te over j since the slope of each Arrhenius line is independent of all the other lines. Thus we have, for the kth line, 2 . — 2 (xkj xo)(ykj bkxkj - yo bkxo> - 0 abk - A = - A +25 - G -26.. . - A Or j3y1j(xij X0) EIBiXij(xij X0) j yo(xij X0) j 1xo(x13 x6 A A 2 A A A _, A A A A2 +2. - Eb. .. L ,-Zx -x Zh.x..+x Eb. §Ixijyij Xogyij jbixij xoj 1X1J+y0jxlj j oyo Oj 1 1] Oj 1 A A A A A 2 A A2 = + - Z + b. 2 ., - 2x,Zx., + M.x ) XO§yij yo§3xij jxoyo 1(jxij oj I] 1 o B definition M. = Z x..y.. - 8.9.M.. Y pi 1 i,j 11 13 1 1 1 - - - - A A 2 A " ‘2 ° = . ~M +b. Zx.,-2x M.x:+M.x ) ° piMi Mixiyi+xoMiyi+yoMixi ixoyo 1(3 ij 0 1 1 1 o o _ A - A - A 2 - A - + A (A‘3) ' pi ” '(Xo ' Xi(yo ' yi) +'bi( §xij/M' 2X x' x0) 132 Equations (A—l), (A-2) and (A-3) have been derived for the general case. We now apply the restrictions xij = x1 and Mi = M for all 1. Then the equations simplify to the following equations: y = yo - (xO - X)? bi/L (A-4) A2 A - a“ A - = — - + 2. - A“ 0 Ibib‘o X) ibi(yo yi) ( 5) = -(g _ §)(§ - § ) + B (23x2/M - 2; i + £2) (A-6) pi o o i i .j j o 0 To simplify calculations, let u = x - x, 60 = x0 - i. If we 0 eliminate bi from Equations (A-4) and (A-5), we would have two equations left with two unknowns £0 and yo. We can then solve for these unknowns. From Equation (A-6), A - A - 2 A " A2 = A - - Z , M ‘ 2 X + X ) I)i [pi + (Xo x)(y0 yi)]/(-ij/ X0 0 = A A - - A [pi + uox + Z- A ipB( o 0 After cancelling identical terms in C and D which are indicated with arrows and identical numbers, C = D becomes 2 2. 2 ' + Z - =2 — +2 — + + - * $913 no ZipiB (y yim0 i piB (y yi)u0 iPiB ()7 yi) p B Luo §B (y y)uO Note that the sum 2(; - y.) = 23y../LM — EX y../LM = 0. This fact . 1 . . ij .. ij 1 1,3 1J has been used several times. Simplifying, we obtain .. + .. _ .. .. . .. . B = 0 11014139107 171)] 110115131 13 L B >13(yi y) 1 §pl(y yl) After adding the term. Z<§i - ;)5 which is equal to zero to the i above equation and rearranging, we obtain 52 _ _ = A _ 2 - = 2 - - = = - _§u0|:(Pi“P)(yi'Y)]+uO[_4:J(Pi'P) -B§(yi-y) ]+Biz(pi-p)(yi-y) 0 (A 8) A - + :1 (S, - 5;.) Also b. = p: 0 0 12 (A-9) 1 $1 + 2(X. - x) /M So far we have derived Equations (2-14),..., (2-19) given in the 135 THEORY. They correspond to Equations (A-l), (A-2), (A-3), (A-8), (A-7) and (A-9) respectively. The derivation of S00 for our case is similar to that for a single regression line and will not be included here. We present the derivation of So next. 80 is by definition the sum of squares of the.deviations of the experimental points from the family of regression lines which are re- quired to pass through the point (£0, §o). Thus =2 -2 _ _" So i,j[ yij bixij (yo bixo)] which, in principle, can be simplified to the form of Equation (2-20). A simpler approach would be to note that S0 is really the minimum of Su. The So can be obtained by substituting do, which can be obtained from Equation (2-17), into Su. It is necessary to solve for a: My. - $2 + LfiZ/Bl only from Equation (2-17) since this corresponds to one of the terms in the expression for Su. Thus we have - 1 - = 2 02 " ' = - - _= so [21?(pi-p)2-BZ{ (yi-y) ] 513%(131-9)(yi-y)-B§J (Pi'P)(Yi y) A .2 2 -2 2 = _ 2: Y uO [:pri -Lp -B§Yi__l u ZP.Y. uoBiPi i where Y1 = (y1 - y) and Pi = (pi - p). + {i3 ZRY. - a BZP.Y. o i 1 1 ‘ o i 1 1 .2 2 -2 2 2 3 ZIY + L B-l= * Z: - . z + . L1 , 0E1 1 p / 1/B{:uini uO iPiYi uOB§?PiY£] 2 .. . 2p].L - Z uOZ PiYi + 1/B[u:z: p: - mimiyi‘l . 2 =2 Thus 8 = .Z. .. - ML - o 1,3le y A/M A2 ., A A - 2 (B + u )Zsp? - (B + u2)u z p.y, = 2: y - REL—3; _ O 1 0 0 1 1 i,j ij BA/M 2 . a 2 _2 Zpi - uO LPiYi = Z .. - ML’ - ~ - which is equivalent to Equation (2-20). To find the expression for Su’ assume that the family of regression lines intersect at a point (u, yu) and calculate the sum of squares of the deviations as was done in the case for SO. u is used in place of A no, bu instead of bi and yu instead of yo since by definition 3 = E [y - b x - (y - b X)]2 u i j ij u ij u u This sum gives an indication of the effect of an error in the estimation of the compensation temperature. Expanding the above equation, we have 2 2 2 2 = + " ‘ + -b '2 .. -bx Su i?j [yij+buxij (yu bux) Zyijbuxij 2buxij(yu Ux)~ y1J(yu u f] 2 2 2 2 2 2 + . 2 = +1, - x+ — 2b ,, -2b xx .-2 .. +2b X ..) iEj-piMu(Y+U)-uMYi(uyi-pi)+u (y+U)Yi+uMU(uYi-pi)- -u2MU<'§r'+U) where we have used the fact that P1 ~< >< ll .. . EX... -le. .. = Mx-. - Zx. .. = M”. x-i) + M(xy.42x. ../M) j 1J J j yl] j Jle y1 j inj y1( 1 j Jle '. - .. + - myi(x x) < Mp1) -E§‘éf)'-§ (-2pi'+ Zpiuyi - 2piuy 2ppiu /B Zpiuy 2piuyi =_ __ 2- - 3.1 4-2 2 f 2u2yyi - Zuzyi + 2u3pyi/B - 2u ppi/B - 2pu y/B-2u p /B ) - 2 2 (C) = ‘{[Pi + u(y - yi>]/A} xj 2 2 2 - 2 - 2 A (C) = [pi + u (y - yi) + 2PiU(y - yi)]Xj = _ 2 = - §A2(C) = AMEp: + u2(y + U - yi) + 2piU(y + U - yi)] S! =_ _ 3_=_ = '. 2 AM? = Z [pi + u2(y-yi)2 + u4p2/B2 + 2n p(y-yi)/B + Zpiuy + ' i 1,] 2_ _ + 2piu p/B - Zpiuyi] 139 su = 12.L(A) + (B) + (CH I(D L2 7 . =2 su = .21 yij - LMy + LM([32u4/BZ + ‘ p2u2/B)/A 19.] “\ f 1 I, ) 9) 494., Q) 9 1r,- 4b ' 2:- +M/AL(-/ép:1+2puy1 -29uy-}éppu/B-2puy+2puy+2uyy1- ® WKQ’.‘ 19‘ '7k ‘76 -Zu2y: + 2u3pyi /VB - 2u/2pp11 /B - 2pu y/B - 2u4 pZ/BZ) :1 (a) Q‘ 7%) :72L ,,2@ q@ 1® ,, “FM/AZEp: +u(y/-y) +up22/B +2u3p}yq -/yii)/B+2puy+ /7C9 a '7 2 - + 2piu p/B - 2piuyi '. Su = 12):. - LM‘?2 + M/AEE-p: + 2(p137. - p1y)u+u2 (1': +y1-fig'pi/Bfl 1,3 J 1 = 23 yij - LM§2 - WAX“: - 2u(p1-P)(y1y) + u2[(y1-y) 213+L2/B1} i,j which reduces to Equation (2-21) if we substitute in the expressions for A and B. _,,, \ l 1 . H. . . ._::H.... ..d. —».».“h.:... . , .H“.r...\«... . x...\ ,. R .fl . . . y. . _ _. . . .... . _ A _ . ._ _ . . . u . .2 a .. R . . . . . , B 7 . . . . .x. . . LII-4 . . . .. , _ . ,. . . _ ._ .. Yllle . . . . \ ~ . x. . . T 1| . . f . . . . . \... . .... Ell I 4 , . . , . . ,. T ., .. V 1 . , _ . _ . ,. : . . m 3 , I . . _ , Ulllluluo . . , .. , . . ._ , my 3 . . , . . . . , . . C ‘x ,, All-9 . . . , I _ 2 TI , , ,. ,\. . . A _ . _ . _._., , ,. . : . . 7 WM . _ .. .. . . . H . . . . _. .. . . 2 . , M .. , ~ . . , . _. .. . . . .. \ . , , _. ...............