Effect of Pressure and Temperature on some Kinetic and Thermodynamic Parameters of Non-ionic Reactions Volume Changes on Activation and Reaction BY BORIS S. EL’YANOV* AND ELENA M. GONIKBERG N. D. Zelinsky Institute of Organic Chemistry of the Academy of Sciences of the U.S.S.R., Leninsky Prosp., 47, Moscow, U.S.S.R. Received 21st February, 1978 The formulae currently used to calculate volume changes on activation, A V,f and reaction, A V,, are shown to be unsuitable even at pressures of m 1 kbar, and calculations are shown to be sensitive to errors, owing to empirical coefficients present in the formulae. The A V’ = A V,[1 -a In (1 + ,6”)] relationship is substantiated and its validity in a broad pressure range shown, as exemplified by 23 non-ionic reactions of various types.The dependence of equili- brium constants on pressure is described by the formula RTln (K’IK,) = -AVo[(l +a)p-(a/ /3)(1+ /3p) In (1 + fip)], where, as a first approximation, CL = 0.170 and /3 = 3.91 x bar-’. A second approximation can be obtained with the help of the linear free energy relationship. The validity of these formulae for kinetic values is confirmed. The accuracy of A Yof and A & calculations can be increased with the help of both these formulae and some criteria of the value of the computa- tional procedure used. The method of calculation is illustrated by 56 Diels-Alder reactions ; the values obtained are a = 0.170 and j3 = 4 . 9 4 ~ As exemplified by 34 reactions of eight types, a linear dependence of A Von temperature has been shown to exist in the range of temperature variation up to 90°C.The values of K = @A V/W)p/A Y are very close for all the reactions. We have formulated the rule that a 1°C increase in temperature results in ]A VI being increased by (0.5+0.1)% (at temperatures close to room temperature). The widespread nature of the compensation effect has been demonstrated, as well as the determining role of the entropy in the change of free energy with pressure. Formulae have been obtained to compute the enthalpies and entropies of reaction or activation under pressure from limited data. bar-l. Investigation of the dependence of reaction rate constant on pressure makes it, in principle, possible to determine the value of A“”, the volume of activation, using the formula of Evans and Polanyi :l (1) Here k is the reaction rate constant expressed in concentration units independent of pressure.Like activation energy and entropy, A V # is a fundamental value character- izing the reaction mechanism and the transition state properties. With the expanding range of problems whose solution makes it necessary to use numerical values of AY#, the requirements on the accuracy of its determination increase. In so far as the Ink = f(p) relationship is not linear, and no sufficiently well- grounded formula for the f ( p ) function has as yet been proposed, serious problems arise in determining the (a In kldp), derivative. Using a comparatively narrow pressure range to determine A V# (by graphical or analytical differentiation) has the disadvantage of the result being highly sensitive to random errors.Attempts to expand the pressure range by using insufficiently well-grounded empirical formulae may result in unpredictable systematic errors. This apprehension is shown to be justified. AV# = -RT@ In klap),. 172B . s. EL’YANOV AND E . M. GONIKBERG 173 The accuracy in determining A V f may be increased as follows, apart from reducing the experimental errors : (1) finding theoretically and experimentally substantiated formulae for f(p) ; (2) decreasing the number of empirical coefficients inf(p), based on the generalization of experimental data or theoretical considerations ; (3) elaborating and applying tests of goodness of fit of the formulae used. In this work an attempt has been made to carry out this programme for non-ionic reactions.Although the emphasis is placed on determining AV#, this is closely related to the solution of the direct problem of describing quantitatively the effect of pressure on the reaction rate constant and other quasithermodynamic activation parameters. The other objective of this work was to investigate the temperature dependence of volumes of activation. Its regularities and quantitative characteristics have not been ascertained up to the present, because of the insufficient accuracy in calculating AV’ and the laboriousness of experiments. Apart from its intrinsic value, this information is needed to solve certain practical problems : calculating the kinetic effects of pressure at different temperatures and comparing the AV# values of several reactions studied at different temperatures. Such information also allows quantitative assessments and calculations of pressure effects on the activation enthalpy and entropy .1. EFFECT OF PRESSURE ON VOLUME CHANGES OF ACTIVATION AND REACTION AND ON RATE AND EQUILIBRIUM CONSTANTS A. FORMULA FOR VOLUME CHANGES Integration of eqn (1) yields RTInk, = RTlnko- rAV’dp. 0 Here subscript 0 refers to the atmospheric pressure.* To find the form of the AV# = [ ( p ) function we shall assume the compressibility of initial reagents as well as that of transition state to adhere to the Tait equation At present this equation is regarded as the best one to describe the effect of high pressure (reaching in some cases 20 kbar) on the compressibility of water and aqueous solutions,2* organic dense gases and even solids7 First obtained empirically, eqn (3) has been subjected to theoretical examination,4u* and has been shown 8b to be close to the theoretically better-grounded equation of Moelwyn- hug he^.^ The assumption that eqn (3) is applicable to the compressibility of the transition state is a serious one.It was first made by Benson and Berson ;5b but doubts as to its validity have been expressed.’O Deviations due to the “abnormal ” compressibility of the transition state, as compared with that of stable molecules, were explained by a shift of the transition state along the reaction coordinate associated with the con- tribution of the pAV term at high pressures,1o and the abnormal compressibility of weak b0nds.l’ In our opinion, however, a possible “ abnormal ” compressibility of the transition state does not necessarily mean that the Tait equation is inapplicable. Moreover, assessments made 1 1 * l 2 indicate the insignificance of the effects in question.* Here and later we disregard the differences in the magnitudes of all values at p = 0 and at atmospheric perssure.174 VOLUME CHANGES ON ACTIVATION A N D REACTION In the subsequent discussion we omit superscript (#), since the required expression We write eqn (3) for the i-th component of reaction in the following form : for [ ( p ) must be equally valid for volume changes on reaction, AV. Vi = V,,[l -ai In (1 +Pip)]. (4) Here ai = AJln 10, and Pi = l/Bp Differentiation of eqn (4) with respect t o p yields Let us now introduce the coefficient P (without a subscript) whose meaning will become clear from subsequent discussion.Adding to and subtracting from the right hand side of eqn (5) the term aiVOiPi/(l +Pp), we get Using eqn (6) we find dAV/dp = Here vi is the stoichiometric coefficient (for equilibria) or the order of reaction with respect to the ith component (for kinetics) taken as positive for reaction products or transition state and negative for reagents : vi(aV,/ap). 1 f 7) or Here Let P = C viaiVoiPi/F viaiVoi* (10) Then y is equal to zero at p = 0, at p p l/Pi * and at any p in the case of the Pi being equal to one another. Therefore, in many cases, especially when the differences between the pi values are not too high, the following relationship will hold true From eqn (8) and (11) it then follows that i IYl @ 1.(1 1) Integration of eqn (12) yields the required relationship, AV = [ ( p ) Here AV' = AVo[l -a In (1 +Pp)J. (13) a = C viaiVoi/?i/AVoP. 1 * pi N" (0.6-6) x bar-' (see below).B. s. EL'YANOV AND E. M. GONIKBERG 175 B. VERIFICATION OF EXISTING EMPIRICAL RELATIONSHIPS AND FORMULA (13) The following relationships are at present most frequently used for calculating AV; Ink, = In ko+ap+bp2 In kp = In ko+ap+bp1-523 In kp = In ko +up/( 1 + bp). (14)10~, 1 3 (15)5b (16)14 Here a = -AVg/RT. Calculations are either performed by the least-squares method (1.s.m.) or, for eqn (14) and (15), by the plotting straight lines of In (kp/ko)/p against p and In (k,/k,)/p against p0.523. The practice of using these relationships has not yet made it possible to either reject or prefer conclusively any one of them.l 3-1 We felt it to be difficult to raise the accuracy of verification using direct kinetic data.It is possible to increase the sensitivity of verification of the goodness of fit of a certain functional dependence by examining a corresponding derivative. We therefore decided to check the following relationships corresponding to eqn (14)-(16) AVp = AVO-b'p (17) AV, = AVo-b'p0*523 (18) AVp = AVo/(l+bp)2. For this purpose, using the values of density and compressibility of liquids at different pressures, taken from literature,16 we calculated their molar volumes, V, and from these the AV of reactions (both real and hypothetical). The list of reactions is given in table 1. Graphical verification of eqn (17) and (19) showed deviations from the straight lines in corresponding coordinates to occur even in the pressure range below 1 kbar.Eqn (18) is better satisfied, but above 1.5 kbar it also no longer holds true. The next stage in the investigation was to verify eqn (13). Calculations were made using a computer. The optimum a, p and AVO(calc.) parameters corresponded to the minimum sum of squares of errors, ZA2 = $(a, p, AVO). Minimization included two stages: ( 1 ) the minimum value of $ at fixed p values was determined by the standard 1.s.m. procedure ; (2) the @ values obtained were minimized by varying 8. Comparison of the " experimental " AVO values and those calculated by this procedure, presented in table 1, showed their excellent agreement ; the standard error for AV does not exceed 0.8 % of the AVO value.Table 2 illustrates the validity of eqn (13) throughout the whole pressure range up to 12 kbar ; a similarly good agreement between the calculated and the experi- mental values was obtained in all the reactions studied. For the above reactions, the a and p values vary within 0.14 < a < 0.27 and 1.2 < p x lo3 bar-' < 7.8 ; the averaged values are a = 0.187+0.033 and p x lo3 = 3.84 + 1.66 bar-' (standard deviations are given). We now compare a and p with the corresponding characteristics of liquids, al and pl. The data presented by Benson and Berson 5 b make it possible to estimate the approximate range of al and p1 values : 0.08 < al < 0.11 and 0.6 < x lo3 < 6. The ranges of p1 and #I values are thus approximately the same.The values of a, as well as those of a,, vary within a comparatively narrow range, although the a values are about twice as high as those of al. Examples of such graphs are shown in fig. 1.reaction b TABLE AN ANALYSIS OF APPLICABILITY OF EQN (13) AND A Yo CALCULATIONS a eqn (1 3 ) ~ eqn (1 3)d eqn (17)e P ~ X number -A VO(exptl lkbar of points /cm3 mol- (i) isoprene dimerization 12 7 (ii) pentene + pentane 3 decane 3 4 (iii) pentene + heptane 3 dodecane 1.5 4 (iv) pentene+ propanol -+ octan-3-01 4 7 (v) pentene+ propanol -+ 2-methylheptan-3-01 5 8 (vi) pentene + propanol -+ 2-methylheptan-5-01 5 8 (vii) pentene + propano1 -+ 3-methylheptan-1-01 5 8 (viii) pentene + propanol -+ 3-methylheptan-4-01 5 8 (ix) pentene+ pentanol -+ diamyl ether 5 11 (x) acetone condensation into mesitylene 3 7 (xi) pentene + water -+ pentanol 5 11 (xii) octenefwater + octan-3-01 4 7 (xiii) octene + water -+ 2-methylheptan-3-01 5 8 (xiv) octene+ water -+ 2-methylheptan-5-01 5 8 (xv) octene+ water -+ 3-methylheptan-1-01 5 8 (xvi) octene+ water -+ 3-methylheptan-4-01 5 8 (xviii) acetone+ pentane 3 octan-3-01 4 5 (xvii) acetaldehyde condensation into ethyl acetate 1.5 5 (xix) acetone + pentane 3 2-methylheptan-3-01 5 6 (xx) acetone + pentane + 2-methylheptan-5-01 5 6 (xxi) acetone+pentane 4 3-methylheptan-1-01 5 6 (xxii) acetone + pentane 4 3-methylheptan-4-01 5 6 (xxiii) ally1 aIcohol+ pentane + octan-3-01 3 4 37.88 29.79 28.66 25.04 25.49 24.45 26.73 26.87 15.59 26.78 19.31 14.42 14.87 13.83 16.11 16.25 14.74 29.26 29.71 29.42 30.95 31.09 24.20 -A Vo(ca1c) /cm3 mol-1 37.89 29.79 28.65 25.07 25.52 24.46 26.76 26.90 15.60 26.72 19.31 14.42 14.87 13.80 16.11 16.25 14.74 29.26 29.72 29.43 30.96 31.10 24.20 a 0.148 0.169 0.251 0.157 0.154 0.160 0.161 0.159 0.144 0.242 0.154 0.195 0.196 0.21 5 0.200 0.204 0.265 0.191 0.186 0.191 0.190 0.190 0.187 ~ ~ 1 0 3 s 4.40 0.19 7.77 0.12 2.87 0.13 5.25 0.16 4.76 0.14 4.64 0.07 4.96 0.14 4.43 0.15 7.09 0.12 1.72 0.17 3.91 0.04 2.47 0.02 1.96 0.07 1.75 0.08 2.40 0.05 1.90 0.03 1.25 0.01 4.16 0.11 4.02 0.17 3.74 0.10 4.20 0.14 3.79 0.12 4.85 0.02 /bar- 1 /cm3 mol- 1 -A Vo(ca~c) 39.39 27.21 27.15 24.70 25.72 24.42 26.36 27.07 15.23 27.19 20.22 14.79 15.79 14.49 16.43 17.14 15.42 27.66 28.44 28.10 29.06 29.77 22.78 /cm3 mol-1 a/ % 4.0 - 8.7 - 5.3 - 1.4 0.9 -0.1 - 1.4 0.7 - 2.3 1.5 4.7 2.6 6.2 4.8 2.0 5.5 4.6 - 5.5 -4.3 - 4.5 - 6.1 - 4.2 - 5.9 -A J’o(cslc) /cm3 mol-1 35.17 27.25 27.55 21.96 22.61 21.62 23.86 23.50 13.46 25.06 17.32 13.10 13.75 12.76 14.64 14.99 14.43 26.34 26.93 26.69 27.91 28.20 22.44 s/ % - 7.2 - 8.5 - 3.9 - 12.3 -11.3 -11.6 -11.2 - 12.1 - 13.7 - 6.4 - 10.3 - 9.2 - 7.5 - 7.7 -9.1 - 7.7 -2.1 - 10.0 - 9.4 - 9.3 - 9.8 - 9.3 - 7.3 0 r d 5 n X 9 Vl 0 Z 9 0 4 9 0 Z 9 =! 2 z tl 9 0 0 Z 2 * L.s.m.-calculations ; S is the standard deviation, S = (A V&alc)-A Vo(,,pt)>/A Vo(e,pt) ; b all the reagents are of normal structure, the double bond or the functional group are in position 1 ; C results of optimization ; d a = 0.170, t3 = 3.91 x bar-’ ; epma, not exceeding 4 kbar.B .S .EL’YANOV A N D E . M . GONIKBERG 177 Benson and B e r s ~ n , ~ ~ when deriving eqn (18), also proceeded from the Tait equation. With the help of an appropriate approximation to the initially complex expression for the AVZ = c(p) function they obtained eqn (18). These authors believed that this formula should be valid within the pressure range of 1-16 kbar. However, it follows from our results that this conclusion is unwarranted, and formula (18) cannot be considered satisfactory. p/kbar 52 3 /barO * 52 3 FIG. 1.178 VOLUME CHANGES ON ACTIVATION AND REACTION 1.0 + I I I I I I 0 1 2 3 2 5 Plkbar FIG. 1.-Verification of eqn (17)-(19) for the reactions (0) (i), (a) (v) and (A) (xi) from table 1; (4 eqn (13, eqn (181, (4 eqn (19). TABLE CO COMPARISON OF CALCULATED a AND EXPERIMENTAL AV VALUES FOR REACTION (i) plkbar 0 0.49 0.98 2.94 5.83 9.81 11.77 -A Vlcm3 mol-1 expt.calc. 37.88 37.89 31.52 31.45 28.49 28.52 22.91 23.10 19.61 19.42 16.79 16.62 15.42 15.62 a Eqn (13), a = 0.148, /3 = 4.40 x bar-'. c. FORMULA FOR THE CONSTANTS. CALCULATION OF VOLUMES OF ACTIVATION FROM KINETIC DATA Using eqn (13) we get from eqn (2) the formula describing the dependence of reaction rate constant on pressure Although formula (20) must conform with experimental results better than eqn (14)- (16), it contains one extra empirical parameter, which increases the sensitivity of AV,f values to random errors. In addition, it prevents the use of the simple 1.s.m. procedure. As a first approximation, we therefore took the " averaged " values of a = 0.170 and p = 3.91 x bar-l, found by optimizing all the data for the reactions presented in table 1 with the help of eqn (21) RTln k, = RTln ko-AV,f[(l +a)p-(a/fi)(l+fip) In (1 +fip)].(20)" W = (AVO-AVp)/AVo = a In (l+/3p). (21) * For equilibrium constants, K, one obtains an expression similar to eqn (20) (with K substituted for k and A Yo for A Vof ) by using, instead of eqn (l), its thermodynamic analogue, Planck's equation : A V = - RT(a In K l a p ) ~ .B . s. EL'YANOV AND E . M. GONIKBERG 179 The optimization procedure is similar to that for eqn (13). The AVO values determined to this approximation with the help of 1.s.m. from eqn (13) and the errors, 6, are given in table 1. Their examination shows that, even under the assumptions made, the value of AVO can in most cases be determined with good accuracy (the [Sl mean value is equal to 3.8 % and the maximum deviation does not exceed 8.7 %).For comparison the same table shows these values calculated using eqn (17). In this case, despite the pressure range decreasing to 4 kbar, for most of the reactions the errors in determining AVO are substantially higher and, more important, are all negative. A second approximation for the a and p values can be obtained by proceeding from the linear free energy relationship : l7 Here @ is a function of pressure, universal for reactions of the same type. The form of the @ function can be found by juxtaposing eqn (20) and (22) : It follows that reactions of the same type have the same a and p values. They can be found by proceeding from the exact data for a standard reaction or by processing statistically the data for a large number of reactions.An example of such an optimization is given below. An important problem is that of tests of goodness of fit of a certain equation for A Vg calculation and the indices characterizing the calculation. This question was previously touched upon by El'yanov et al. ;14c it was, however, not discussed with sufficient comprehensiveness and clarity. In this paper a more detailed and precise examination of the problem is presented, a distinction between tests and indices is introduced and a new test formulated.* The test of goodness of fit of an equation is a characteristic having the properties of the necessary or sufficient condition, as distinct from the index of goodness of calculation, which does not have such a categorical property although in some cases it can be used for the same purpose.Tests based on statistical assessments are, however, of a probabilistic nature. The following tests and indices can be formulated. @ = [(l +a)p-(a//I)(l +pp) In (1 +pp)]/R In 10. (23) TEST AND INDEX OF ACCURACY If the accuracy of experimental log k values is characterized by a standard error, Sexp., and the standard error of calculation performed with the help of agiven equation, by Scale., one can then, using the statistical test of verification of hypotheses,' * regard the equation as not contradicting the experimental data if s&p./%!alc. F0.05 (24) where Fo,05 is Fisher's test at the significance level of 0.05.A doubtful case can be characterized by the relationship : and at S&,JS&,. > FoSol the equation verified can be regarded as not satisfying the requirements placed upon a strict functional dependence. To evaluate the data from literature which do not contain any information on Scxp. we have proposed 14c the index of accuracy based on the following classification * The symbols have also been changed.180 VOLUME CHANGES ON ACTIVATION AND REACTION of the accuracy of kinetic data : excellent accuracy : the maximum deviation of the calculated and the experimental values of the rate constant does not exceed 5 %; good accuracy : 10 % ; satisfactory : 20 %. For reactions where deviations exceed 20 % the accuracy will be regarded as low. In this case we get the following ratings of accuracy : accuracy excellent good satisfactory low S (for log k ) GO.011 G0.022 < 0.044 > 0.044.A low accuracy can be caused both by the equation being incongruous with the experimental data and by the poor quality of the data themselves. TEST OF LINEARITY If the dependence of log k on pressure is presented in the form of eqn (22), this dependence for correct @ values will be graphically expressed as a straight line by plotting log k against a. If the experimental log k values fall on a curve they will be better described by eqn (25) Here b is an empirical coefficient; ko, AV,f and b are determined by 1.s.m. As a test of linearity, indicating the insignificance of the differences between eqn (22) and (25), we can take the relationship Here S,, and S25 are standard errors which result from using eqn (22) and (25).s;2/si5 G F0.05* (26) A doubtful case will be characterized by the relationship : F0.05 < s?2/S:5 I ; b . O 1 7 and at Si2/S,”5 > F0.,, we shall regard eqn (22) as not being satisfied. We thugsee that the test of linearity is a necessary but not sufficient condition to decide on the goodness of fit of eqn (22). TEST OF RANDOMNESS OF ERRORS In the case when great numbers of data are available for reactions of the same type, it becomes possible to verify the goodness of fit of the equation used with the help of a test that can be conventionally defined as the test of error random- ness, as follows. If we process experimental data using eqn (25), then, provided eqn (22) is valid, the deviation of b from zero, with the chosen @ functions, results from random causes. Reactions with b > 0 and with b < 0 must then be equipro- bable.This is also equivalent to the equiprobability of cases satisfying the conditions of (AV,f>,, > (AVg)25 and (AV;),, < (AVz)2s. Here and (AVof)25 are AV; values calculated from eqn (22) and (25), respectively. The confidence interval for j3, the probability of the appearance of b < 0, can be determined by proceeding from the frequency of this condition being fulfilled, 0, in n tests :l Here u1--p0~2 is the quantile of the normal distribution at the significance level of po. For po = 0.05 (assumed in this work), z.iO.975 = 1.96. The test of error random- ness is the falling of the p = 0.5 point within the confidence interval of eqn (27).The fulfilment of this test is a necessary but not sufficient condition to decide on the goodness of fit of the equation verified.B. s. EL'YANOV AND E . M . GONIKBERG 181 INDICES OF ACCURACY OF AvZ The standard error for AV:, which we designate as Sv, can serve as the index of accuracy of its calculation. When the equation used for calculation has the form of eqn (22), Sy is found from formula sv = ST[C Q,Z-(C ~ ~ y 1 ~ 1 - 0 . 5 . (28) i i Here S is the standard error of the function, Tis the temperature, and n is the number of points. If the @ function has been chosen correctly, one can then determine with the help of Sv the confidence interval (at probability p) within which lies the true AVZ value, using the formula where t is Student's coefficient found from tables.18 If, however, the function is incorrect, the true value of AV;, as a result of the systematic error, may lie outside the confidence interval.(gAv,z)m,x. = kSVt(P, n) (29) TABLE 3.-EFFECT OF UPPER PRESSURE LIMIT ON CALCULATION ' OF Avg (REACTION OF ISOPRENE WITH MALEIC ANHYDRIDE IN ETHYL ACETATE AT 35"@ - A V ~ icm3 mol-1 highest plbar 51 7 689 1013 1379 2068 3102 41 36 5169 6204 number of points used in fit 4 5 6 7 8 3 10 11 12 eqn (14) eqn (16)~ eqn (20)d 36.1 37.5 38.1 37.4 35.5 33.3 32.4 31.6 30.3 36.9 36.3 35.2 34.4 33.9 34.0 34.2 34.3 34.8 (1) 39.6 39.5 39.0 38.5 38.2 38.4 38.1 37.8 37.7 (2) 38.1 37.7 38.6 38.2 39.8 42.2 39.6 38.1 40.1 (3) 40.5 40.5 40.2 39.9 39.9 40.2 40.1 40.0 40.0 vd.)inst /cm3 mol-l 7.8 3 .O 1.9 4.5 0.6 a Least squares method ; b ref. (19) ; C b = 9 . 2 0 ~ bar-l [ref. (14c)l; d (1) 01 = 0.170, /3 = 3.91 x bar-l, (2) optimization of all the parameters, (3) a = 0.170, /3 = 4 . 9 4 ~ b a r i . Apart from Sy, another index of the goodness of fit of the AVO calculation can be used. As noted by E ~ k e r t , ~ ~ ~ the AV; value can vary significantly if we vary the number of points, obtained in the same measurement series, used in a fitting. By way of illustration table 3 shows the results of Eckert's calculations,13c expanded by us for a broader pressure range, using eqn (14), as well as the calculations of the same data using eqn (16)14c (at b = 9.20 x bar-I) and eqn (20) [procedures (1)-(3)]. It is seen from the table that, when eqn (14) is used, the difference in AV,f can reach 7.8 cm3 mol-l.This difference, designated as (6AV&st., can be defined as the index of instability. It reflects the effect exerted upon the AV,f calculation both by the random experimental errors and the systematic errors resulting from an inexact formula being used. Generally, the value of (6AVZ)inst. must increase with the increasing number of empirical coefficients in the calculation formula. This can be illustrated by comparing the results of calculations performed with the help of eqn (20) using procedures (2)182 VOLUME CHANGES ON ACTIVATION AND REACTION and (3) (further details will be given in the following section). With procedure (2), as distinct from procedure (3), the a and p parameters are optimized in the course of calculation, and, consequently, the value of (6A Vz)jnst.is substantially higher. Despite the relative nature of the index of instability, its application, together with the other tests and indices, can be useful in characterizing the accuracy of the A V$ values obtained and comparing different calculation procedures. Calculation by eqn (20) [procedure (l)] thus, shows good stability, exceeding that of the calculations by eqn (14) and (16). This result is an additional argument in favour of the recommendations formulated above. The application of the above tests and indices makes it possible to judge with greater confidence whether a certain equation is applicable for calculation. With their help one can eliminate equations not suited to describe experimental data or to identify the data which, for some reason or other, are not described by the given equation.D. APPLICATION TO DIELS-ALDER REACTIONS When sufficiently accurate kinetic data are available, one can attempt to carry out the optimization of all the parameters in eqn (20) using 1.s.m. As seen from table 3, [procedure (2)], the values of AVg obtained by optimizing the data for the cited reaction, have good stability despite the great number of parameters being optimized. For the entire pressure range, S = 0.0074 is an excellent index of accuracy with such a broad interval. These results confirm the high accuracy of the initial data and also the validity of eqn (20) for kinetic values within a broad pressure range. Despite the great number of parameters optimized in this case, (C~AV&~.is smaller than when the optimization is performed using a quadratic equation [eqn (14)]. The constants obtained by optimization in the entire pressure range (a = 0.170 and p = 4.94 x bar-l) proved to be quite close to the " averaged " values previously found for non-kinetic data. The high AV,Z stability obtained in calcula- tions with these constants (table 3, last column) makes it possible to use them with sufficient reliability. Let us now consider the possibility of calculating the volume of activation for a broad variety of Diels-Alder reactions. The first attempt 14c was made to apply an equation similar to eqn (22) to these reactions using the @(p) function found for electrolyte ionization reactions.2o In this case eqn (22) acquires the form of eqn (16) with b = 9.20 x bar-l.Verification of this relationship has shown that out of 34 reactions only one does not satisfy the linearity test and 2 reactions are regarded as doubtful. The equation was satisfied with good indices of AV: accuracy and stability. From this the relationship was concluded to be valid. The test of error randomness was, however, not applied. The calculation of fi from eqn (27) for these reactions gives the interval of 0.179 5 5 0.497. The = 0.5 point, although close to this interval, still does not fall within it. For the verification to be more reliable we increased the number of reactions to 56, using some additional data.2f Calculations gave the interval 0.115 < jj < 0.332, which is now seen to be sufficiently removed from the jj = 0.5 point.The test of randomness of errors is, thus, not fulfilled, and the @(p) function has to be further specified. The next stage consisted of verifying the goodness of fit of eqn (23) at a = 0.170 and #I = 4.94 x bar-l. 3 reactions did not satisfy the test of linearity and 4 fell into the category of doubtful reactions, with 13 reactions (23 %) being characterized as excellent, 22 (39 %) as good, 12 (21 %) as satisfactory and 9 (16 %) as of lowB . s. EL'YANOV AND E. M. GONIKBERG 183 accuracy. For j we obtained the value of @ = 0.41 +0.13, which does not differ significantly from 0.5. Eqn (23) with the constants in question is, thus, with some exceptions, consistent with the tests of goodness of fit, and is satisfied with a good index of accuracy for a large number of reactions.Let us consider the indices of accuracy for AVZ. In the last calculation the values of (C~AV&,~~. for 85 % of the reactions proved to be lower than with the calculation by eqn (14). Comparison of the results for only 35 reactions, studied by Eckert and 21b* 22 in which the maximum pressure range did not exceed 1380 bar, gave almost the same result : 81 % (with the minimum number of points equal to 4). Given below are the rneanvaluesof (SAV~)in,,., SYand(GAV$)max, for these reactions ; the error indicated is the standard deviation : accuracy excellent good satisfactory number of reactions 12 15 8 (SAVt )inst. 1.5k0.7 2.85 1.2 4.852.8 SJ'/cm3 rno1-I 0.6k0.1 l . l k 0 . 2 1.9k0.3 !ciAJqlmax.1.550.3 2.8k0.5 4.9k0.8. These figures give an indication of the accuracy of the AV; calculation. Both indices, Sy and (C5AVgZ)inst.y are not contradictory, since (6AV,+),n,,. does not fall beyond the limits of the confidence interval, equal to ~ ( C ~ A V : ) ~ ~ ~ . In this case the value Of (dA Vgf)inst. equals exactly half the confidence interval at any grade of accuracy. TABLE 4 . 4 VALUES AT VARIOUS a AND /3 VALUES IN EQN (23) a = 0.236 a = 0.170 a = 0.170 /3 =:4.94x 10-3 bar1 = 2 . 5 2 ~ 10-3 bar1 /3 = 3.91 x 10-3 bar1 plkbar @,t/cm-3 K cP/cm-3K @/cm-3K dl%a 0.1 0.5 1 .o 1.5 2.0 3.0 4.0 5 -0 7.5 10.0 0.503 2.279 4.209 5.947 7.554 10.48 13.13 15.56 20.94 25.60 0.508 2.326 4.289 6.026 7.598 10.38 12.79 14.92 19.31 22.70 1 .o 2.1 1.9 1.3 0.6 - 1.0 - 2.6 -4.1 - 7.8 - 11.3 0.507 2.328 4.337 6.163 7.861 10.98 13.82 16.45 22.33 27.49 0.8 2.2 3.0 3.6 4.1 4.8 5.2 5.7 6.7 7.4 Another, independent procedure was used to determine the a and p constants in eqn (23).It consisted of optimizing the data for all the reactions. The optimization procedure was as follows. First, the data of each reaction were optimized in accordance with eqn (20) for all 4 parameters by 1.s.m. In so far as, in most cases, a and p varied within unreasonable limits (3-4 orders of exponent), which resulted in abnormally high A V,f deviations from their reasonable values, a restricted interval of j? values was introduced : 0.6 < p x lo3 bar-l < 15. From the values found for AV; we then calculated the @ values for all pressures, using eqn (22).The next stage consisted of optimizing the a and p values in eqn (22) for the whole array of values for all reactions at all pressures. The values obtained were : a = 0.236 and /3 = 2 . 5 2 ~ bar-l. Table 4 gives a comparison of values, calculated by proceeding from different a and /3 values. Their examination shows that up to 4 kbar the values of @ in the first two columns are quite similar and differ only slightly from those in the third column. At higher pressures the difference begins to increase184 VOLUME CHANGES ON ACTIVATION AND REACTION rapidly. Taking into account that out of 56 reactions only 7 have been studied at pressures above 4 kbar, a small discrepancy between the optimization data and the data for a standard reaction at these pressures is not surprising.The excellent coincidence of @ values obtained by two independent procedures for Diels-Alder reactions and their closeness to the values obtained by a third procedure indicate their reliability at least up to 4 kbar. Even this small difference in <D values above 4 kbar in columns 1 and 2 of table 4 results in the standard reaction not satisfying the test of linearity when its computation is attempted with a = 0.236 and /3 = 2.52 x bar-l in the pressure range of up to 6 kbar. Eqn (20) with a = 0.170 and /3 = 4.94 x bar-l can thus be recom- mended for use in the pressure range of up to 4 kbar. 2. TEMPERATURE DEPENDENCE OF VOLUME CHANGES ON ACTIVATION AND REACTION A. QUANTITATIVE CHARACTERISTICS The starting point of our investigation was an important result obtained by Marani and Talamir~i.~~ Proceeding from quasithermodynamic relationships these authors showed that, within limits of applicability of the Arrhenius equation, a linear dependence of A V on T must hold true.Extensive verification of this dependence for A P values and its quantitative assessment appears at present to be 35 H I '3 30 E s I 25 20 3 d 15 PPI 0 20 40 60 80 T/"C and A, (Gl) (table 5). FIG. 2.-Temperature dependence of volume changes on the reactions x , (A4) ; 0, (A5) ; 0, (E2) practically impossible for the reasons described in the introduction. We assumed that the temperature dependence of AVf can be simulated by a corresponding dependence of A V. This offers possibilities of performing necessary calculations, since the numerous data on the densities of compounds at different temperatures, available in literature, can be used to calculate AV of various reactions.We have€3. S. EL'YANOV AND E . M. GONIKBERG 185 TABLE 5.-TEMPERATURE DEPENDENCE OF VOLUME CHANGE ON REACTION reaction a (1) pentene-2+ pentane (2) pentene-2+ hexane (3) pentene-2+ heptane (4) dodecene+ octane (5) dodecene+ nonane -AVd temperature /cm3 mol-1 rangePC (A) alkylation of alkanes with olefins 28.1 1 60 27.48 40 34.39 40 28.64 90 24.65 60 (B) C-alkylation of alcohols with olefins (1) hexene+ethanol 26.10 30 (2) heptene C+ methanol 19.61 30 (3) heptene C+ propanol 24.04 10 (4) heptene C+ propan-2-01 25.79 10 (5) hexene+ butanol 24.95 10 (6) hexene+ 2-methylpropan-1-01 25.82 10 (7) octenef ethanol 23.65 10 (C) unsaturated alcohol+ alkane + alcohol (1) allyl alcohol+ pentane 26.30 30 (2) allyl alcohol+ 2methylbutane 27.68 30 @> C-alkylation of acetone with alkanes (1) pentane+ acetone (2) heptane+ acetone (1) CH31+ hexene (2) CH31+ heptene (3) C2H5Br+ hexene 28.65 30 28.39 10 (E) addition of alkylhalides to olefins 23.30 30 18.00 30 26.02 30 (F) olefin dimerization (1) hexene + dodecene 28.10 10 (2) heptene -+ tetradecene 27.14 10 (3) octene 3 hexadecene 26.60 10 (4) octene+ heptene 4 pentadecene 26.84 10 (5) octene+ hexene +.tetradecene 27.37 10 (6) hexene+ heptene -+ tridecene 27.62 10 (1) into mesitylene 27.15 35 (G) acetone condensation (2) into 1,2,3-trirnethylbenzene 32.10 15 (H) Tischchenko condensation (1) 2 CH3COH + ethyl acetate 15.41 20 (2) 2 C2HsCOH 4 propyl propionate 13.95 30 (3) 2 CzHsCOH + ethyl butyrate 13.60 30 (4) 2 C2HSCOH + butyl acetate 13.95 30 (5) 2 C3H7COH +.heptyl formate 15.75 30 (6) 2 C3H7COH -+ hexyl acetate 14.79 30 (7) 2 C3H7COH 3 methyl oenanthate 16.10 30 -8AV - ~ T F 0.144- 0.126 0.130 0.125 0.123 0.107 0.080 0.097 0.105 0.107 0.109 0.095 0.110 0.128 0.153 0.129 0.098 0.080 0.121 0.140 0.130 0.109 0.118 0.126 0.130 0.168 0.163 0.060 0.054 0.054 0.059 0.069 0.061 0.066 1 / A V (aA V/aT)p x 103/1(-1 5.12 4.59 3.78 4.36 4.98 4.10 4.08 4.03 4.07 4.29 4.22 4.02 4.18 4.64 4.85 4.54 4.13 4.44 4.66 4.98 4.79 4.10 4.41 4.60 4.73 6.17 5.06 3.89 3.87 3.95 4.23 4.38 4.12 4.07 a Unless otherwise stated, the reagents and products are of normal structure, the double bond or the reaction product is 3-ethyloctadecane ; C 3-ethylpent-2- the functional group are in position 1 ; ene ; d at 20°C.186 VOLUME CHANGES ON ACTIVATION AND REACTION carried out such calculations for 34 reactions (real and hypothetical) presented in table 5, using the data from ref.[16(b)]. In no case have we noted any substantial deviations from a straight line, some of which are shown in fig. 2. With each type of reaction, characterized by different AV and (dAV/aT), values, the values of IC = (aAV/aT),/AV for 20°C show remarkable constancy. The mean value of E~~ thus equals (4.43 f 0.48) x “C-l. The calculated standard error of only 11 % of the mean value is taken as the measure of deviation. The results obtained make it possible to formulate a simple rule : a 1°C increase of temperature results in a (0.5f0.1) % increase of IAV] (at temperatures close to room temperature).Let us derive formulae suitable for A Y and IC calculations at different temperatures, We shall present the linear dependence of AV on T as AV, = AVO +( aAv T),o-TO). Here subscript 0 refers to a certain standard temperature. Equality (30) can be presented as AVT = AVo[l+ KO(T-T*)] which is suitable for the change-over from standard A V values to their values at another temperature and vice versa. Dividing both parts of equality (30) by (aA V/dT), we get 1IICT = l/rco+fT-To). (32) Eqn (32) is more convenient for use as It is appropriate to examine some possible methods of assessing the IC value in a certain reaction, apart from direct calculations based on densities or dilatometric measurements.(1) As a first approximation we can take the mean value of E20 obtained by us, which is equal to 4.4 x deg-I. (2) A very narrow range of K fluctuations for different reactions gives us grounds to assume that IC is constant for the same type of similar reactions. As a more precise IC value we can thus take its mean value for a given type of reaction, or if it is not available, K: for an individual reaction belonging to this type. (3) It can probably be assumed that IC# i2i IC,,,, where IC* is a value characterizing the activation process and K , ~ , is a corresponding equilibrium value for the same reaction. Assessments of this kind will be presented in the following section. B. SOME COROLLARIES: EFFECT OF PRESSURE O N ENTHALPY AND ENTROPY When a great number of bimolecular reactions were being investigated it was noted 24 that the general increase of reaction rate under pressure (AV” < 0) is accompanied by the growth of the pre-exponential factor [(aAS”/dp)T > O].* Laidler 26 arrived at the conclusion that in a series of heterolytic reactions AV” and (aAS* lap), must not only have an opposite sign but also vary in opposite direc- tions.This conclusion follows from consideration of the thermodynamic relationship 8AV’ dASf (2T), = -( dp), * CX, however, ref. (25). (35)B . s. EL'YANOV AND E. M. GONIKBERG 187 and the conclusion of Fajans and Johnson 27 on the approximately linear dependence (with a positive slope) between the partial molar volumes of ions in water solutions and their temperature coefficients. A similar conclusion for non-ionic reactions, based on the assumption that the values of AVX,, and (aAYFfree/aT) run parallel to each other in a series of reactions, was made by G~nikberg.~~ Using the results obtained in the previous section we can estimate the effect of pressure on reaction or activation enthalpy and make some quantitative assessments.Using the notation of the preceding section r$)p = KAV. Taking into account the thermodynamic equality AV = rz) T where AG is the Gibbs free energy, and eqn (35) and (36) we get ( Y ) T = In so far as K > 0, derivatives in both which is in agreement with experiment. Let us denote (37) parts of equality (38) have different signs, and from eqn (38), (39) and (41) Since from eqn (38)-(40) we obtain (39) Let us assess the value of By the meaning of which is that of isokinetic (isoequili- brium) temperature.For this purpose from eqn (41) and (32) we first get J = T O - l / K o . (43) Taking the mean value of go = 4.4 x deg-' obtained by us as corresponding to To = 293 K, we shall get for non-ionic reactions the most probable value of fl = 65 K. Using the extreme values of K from table 5 we get 30 < fl < 130 K. The possible boundary values of fl are seen to differ from the probable value by a factor of ~52. The value of fl is positive, i.e., according to eqn (39), the changes of AH and AS with pressure run parallel to each other and exert an opposite (compensating) effect on the reaction free energy (or rate), which is observed experimentally.From eqn (39) it also follows that the value of (aAH/ap), is from 0.1 to 0.5 of the T(aAS/ap), value at temperatures close to room temperature.188 VOLUME CHANGES ON ACTIVATION AND REACTION This analysis, which does not contain any assumptions and is based on the generalization of many experimental data, thus demonstrates the widespread nature of the compensation effect, as well as the determining role of the entropy component in the change of free energy under the effect of pressure. Further possibilities for quantitative computations are offered by the linear relationships between the enthalpy and the entropy (AH against AS linear relation- ships) with changing pressure. Previously l7 we showed that these relationships must hold true for a certain type of similar reactions, if the CD parameter does not depend on temperature.Up to now this dependence has not been observed. A good agreement for the log k against rP linear relationship was noted 2o for the ioniza- tion of acetic acid in the temperature range 25-225°C at pressures of up to 3 kbar ; with the iD values for ionization reactions having been obtained at 25°C. One can, therefore, expect the AH against A S linear relationship to hold true, at least for some types of reactions. In the case under consideration the AH against A S linear relationship means that p in eqn (39) is not dependent on pressure. From eqn (41) it then follows that K is not dependent on pressure either. In this case integration of eqn (38) and (42) yields linear relations hips (AS,- AS,) = - K(AG,- AGO) (44) (AH,-AHO) = (1 -xT)(AG,--AGO). (45) Having estimated in some way the values of K [e.g., with the help of approximations examined in section 2(A)], one can then, using the values of (AG,-AGO), calculate (AS, -ASo) and (AHp - AHo).We shall illustrate dependences (44) and (45) and K # calculations using as an example the kinetic parameters of two Diels-Alder reactions : isoprene and cyclo- pentadiene dimerizations. The first reaction was studied by Rimmelin and Jenner 21a at pressures of up to 8 kbar in the temperature range of 4O-7O0C, and the second, by Walling and Schugar,28 at pressures of up to 3 kbar and temperatures of 0-40°C. In fig. 3(a) and (b) experimental points corresponding to 40°C are plotted and straight lines are drawn by 1.s.m.in accordance with eqn (44) and (45). Calculated IC# values are presented below. For the first reaction it also proved possible to calculate the IC,,, value for the entire reaction, since the values of AV,,, measured by Rimmelin and Jenner at four temperatures fitted the straight line remarkably well. Finally, the same table also contains the value of E40. Comparison of the IC' values of both reactions shows their closeness, just as we expected for reactions of the same type. The differences between K,, and E40 values on the one hand, and K' on the other shown here are close to the experimental error, in so far as the accuracy of AS' reagent K + ~ ~ 0 3 = K r r n ~ 1 0 3 b 6 / % c ?i40x103' s / % c isoprene 5.84 4.68 20 31 4'04 27 cyclopentadiene 5.50 - Q From experimental data with the help of 1.s.m.; b from the linear dependence of AVr,, on T; determination, as indicated by Rimmelin and Jenner,21a is equal to f 4 cal "C-' mob1, which amounts to 21 % of the maximum (ASp-ASo) value. These results bear witness to the feasibility of quantitative estimates and more precise calculations of enthalpies and entropies for reactions under pressure from a limited number of initial data. When experimental AG, values are not available, the calculation can be performed if the AGO and Qi, values for a given reaction are - C deviation from K+ ; d calculated for 40°C by eqn (33) from &O = 4.4 x OC-'.B. s. EL’YANOV AND E . M . GONIKBERG . .28 29 30 31 32 44 I I I 189 I I I I I 24 25 26 27 AGZ /kcal mol-I 23 2 2 21 20 19 10 AG# /kcal moV FIG.3.- -Linear dependences of (a) A S against AG and (6) AH against AG for the isoprene (0) and cyclopentadiene (0) dimerizations at 40°C.190 VOLUME CHANGES ON ACTIVATION AND REACTION known. As has been previously shown,17 given a linear relationship between AH and AS, the following linear dependences will hold true : ASp = ASo-m@ (46) AHp = AHo +n@. (47) Here m = R In 10 x (aAV0/aT), n = R In 10 x [AVO - T(BAV0/BT),]. The last two equalities are easily transformed into m = R In 10 x AV,K n = R In 10 x AVo(l -TIC). (483 (49) (50) (51) In this case, the calculation of ASp and AHp can thus be carried out from eqn (46) and (47) using eqn (50) and (51). ADDENDUM After the manuscript of this paper had been prepared for publication, the work of Orszagh et aL2’ became known to us.In it formula (13) was also proposed, based on the assumptions that the Tait equation holds true and the ai and pi constants for the reagents and the transition state have similar values. The formula was verified with 4 reactions by straight-line plots of (a In klap), against In (1 +Qp) coordinates, where k is the rate constant. The (a In k/Bp), values were found by graphical differentiation, and those of Q constants by approximate estimation. The authors came to the conclusion that, in all the reactions examined, experimental data cannot be described by one equation valid within the entire pressure range. According to their explanation, there exists a critical pressure at which the mechanical properties of the transition state undergo sharp changes and, therefore, its compressibility before and szfter the critical pressure is described by different Tait equations.Consequently, the change of AV# with pressure is also described by two equations similar to eqn (13).* Another explanation can be offered, uiz., the simplified initial assumptions of the authors are not verified and, therefore, eqn (14) is, in principle, not valid within the entire pressure range, even when the Tait equation holds true for every component. We believe, however, that the conclusion obtained by Orszagh et al., is wrong. It is not corroborated by our data on volume changes both for reaction [section l(B)] and for activation [section l(D)], and sharply contradicts our conclusion on the applicability of eqn (13) within a broad pressure range.This result can be explained by gross, poorly controllable errors which appear during the plotting of a smooth Ink = f ( p ) curve and the graphical differentiation and, at the same time, by the extreme sensitivity of the calculated p values to errors ; it illustrates the unsuitability of this method for the verification of eqn (13). Bearing in mind the work of Orszagh et al. we are, nevertheless, convinced that our work on the substantiation and verification of eqn (13) is useful and, therefore, stand by the conclusions drawn in this paper. * The critical point is approximately in the middle of the pressure range, irrespective of its magni- tude. 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