J. Chew. Soc., Faraday Trans. I , 1989, 85(6), 1199-1206 Modelling the Effect of Pressure on the Rates of Ionic and Polar Reactions Benjamin Gavish Biochemistry Department, The Hebrew University and Hadassah Medical School, P.O. Box I 172, Jerusalem, Israel The rate k, of ionic and polar reactions in solutions is known to be sensitive to the applied pressure P. In the present work, structural and dielectric contributions to k, are expressed explicitly as functions of P. The pressure dependence of the volume and the dielectric constant of the solvent are described by the Tait equation. The model predicts the relation ln(k,/k,) = -vP/B-pln(1 + P / B ) where v, p and B are adjustable parameters. This expression fits remarkably well the kinetic data of 11 ionic and polar reactions up to 45 kbar.The resulting activation volume is AV: = AVz+(AV:-AVz)/(l +P/B) where the limiting values of A V,' at zero and infinite pressures are expressed in terms of intrinsic volume changes in the reactants and in the surrounding solvent, and by the strength of the electrostatic interaction. The results suggest that for ionic and polar reactions, the structural and the dielectric contributions to the activation volume can be separated objectively, using a single solvent. The meaning and the significance of the various parameters are discussed. High-pressure kinetics provide a powerful tool for studying the transition states of chemical reactions in the liquid state, in terms of activation volumes.1* Reactions at the transition state of which ions or dipoles are formed are of particular interest for being strikingly accelerated by pressure. The effect has been attributed by Buchanan and Hamann3 to the increase in solvation free energy of ionic charges.The dielectric constant of the solvent seems to dominate this phenomenon due to its strong pressure dependen~e.~~' On the other hand, the contribution to the activation volume of structural origin cannot be overlooked. Various expressions for the pressure dependence of reaction rates have been investigated in attempt to best-fit the kinetic data.9-13 However, the interpretation of the adjustable parameters in terms of structural and dielectric elements, and the uncertainty involved in neglecting the structural one, are still open problems to which the present paper is dedicated. Theory The effect of pressure on the kinetic coefficients k , of a chemical reaction is described (1) where AG: is the change in the free activation energy at pressure P and temperature T, and R is the gas constant.Hereby, atmospheric pressure will be taken as P = 0. This does not introduce any appreciable error in dealing with high-pressure effects. However, it does simplify considerably relevant mathematical expressions. We propose that In k,/k, = - (AG; - AG:)/RT by AG: - AG: = SU; + 6 W; 1199 (2) 41-21200 Eflect of Pressure on Reaction Rates SU:, the dielectric term, stems from the pressure dependence of the dielectric constant of the solvent. 6 W:, the structural term, corresponds to the compression work. This work is associated with volume changes of reactants and solvent molecules, that take place along the reaction pathway without any change in their ionic or polar state.For example, the formation of a bond might alter both the reactants volume and surface area. A solvent molecule, at the reactants surface, might occupy a different volume from one at the bulk. Thus, changes in the solvent volume are coupled, in this case, to the process of bond formation. In a case of a more ionic or polar transition state, a further increase in the volume difference between a solvent molecule in the bulk, and at the vicinity of the reactants, is caused by local compression which increases with increasing the electric fields.l4, l5 Structural Contribution Let us perform the following thought experiment.A cavity of volume & is generated inside a liquid under pressure P. The required energy for the process is I( P. The cavity is then filled with material (under atmospheric pressure) and becomes free to collapse under the external pressure. As the result, the pressure inside the cavity iyreases from 1 atmJf to P. The energy released curing the compression process is -1, P'd V,,.. The total free-energy change is & P+Jo P d Vp.. If the above procedure is applied separately to the transition state and the initial state of the reactants, the free-energy difference between these two processes defines the structural component as follows : 6 w; = A V; P + 1; P dP' d(A V;J/dP' (3) where AV: is the volume change in the system per mole of reactants (including the solvent).In general A V: = A VpfI + A Vps. A VpfI is the intrinsic activation volume, usually referred to as the volume change of the reacting molecules themselves. A V:s corresponds to possible variations in the packing of non-reacting solvent molecules, that follow structural rearrangements in the reactants, as explained before. The significance of this factor in interpreting activation and reaction volumes has already been pointed out by Asano and le Noble.'' It is proposed that: (i) the intrinsic volume of the reacting molecules is incompressible, i.e. d(A V,f,)/aP = 0 or, alternatively, A VpfI = A V& ; (ii) the pressure dependence of A V;ky is given by the Tait equation - d(A Vpfs)/aP = Do A V&/( 1 + P / B ) (4) where Do is the solvent compressibility at atmospheric pressure and B is a constant characteristic of the solvent and probably of the reaction.Assumptions (i) and/or (ii) are currently applied in studying ion-solvent interaction^^^-^' and activation or reaction volumes.1'* 1 3 9 21 Eqn (4) can be inserted into the integral of eqn (3) to yield 6 W: = (A V:I + A V,',) P -Do A VtS B2[P/B - In ( 1 + P/ B)]. A similar expression has been obtained by Owen and Brinkley22 using Gibson's expression for volume changes in electrolyte at the limit of infinite dilution. For P + B eqn (5) is simplified into SWpf =(AV:I+AV:s)P-/30AV:sP2/2. The second term is the elastic energy stored in the solvent. Eqn ( 5 ) and (6) are used in high-pressure kinetics5* 7, ' 9 lo and in studies of electrolyte ~olutions.'~ t 1 atm = 101 325 Pa.B.Gavish 1201 Dielectric Contribution We assume that SU: = Up- Uo, where Uo is part of the electrostatic energy that depends upon the value of the dielectric constant D, of the solvent around the reactants. We further propose that (iii) U , is proportional to I/D,. Thus, U p = Uo Do/D, and (7) For a single sphere of an effective radius r with initial charge zero and final charge Ze, U p is given by the modified Born expression NZ2e2/2rDp,3, l8 where N is the Avogadro number. Here r is taken to be pressure independent, in accordance with assumption (i). Let us consider the reaction M .+ (A + B)* in which the charge on the reactants varies. It can be shown that (8 a) Another case of interest is the occurrence of a dipole moment at the transition state.The energy required for generating a single dipole p at the centre of a spherical cavity of radius r is - (p2/r3) (D,- 1)/(2D, + l).23 Since 2(D, - 1)/(20, + 1) = 1 - (3/2)/(Dp ++), eqn (7) approximates this expression reasonably well for D, + 1/2 with Uo = 3p2/4r3D,. For M +(A+B)* we find SU: = - Uo( 1 - D,/D,). Uo = Ne2(Z2,/r, + PB/r, - ZL/rM)/2D0. uo = (3/4) N O X + P X - PL/r;)/Do; (8 b) We now assume that (iv) the pressure dependence of l/Dp follows that of pure liquids, (9) where A and B are constants characteristic of the solvent and probably of the reaction, and satisfy A / B = 8 In D,/aP at P + 0. Eqn (9) has been applied to studies of reaction kinetic^.'*^^*^^ Eqn (7) and (9) yield and is given by the Tait equation 1 - Do/D, = A In (1 + P / B ) SU; = -AU,ln(l + P I S ) .(10) Eqn (2), (5) and (10) can be combined into AG: -AG: = [AV,',+(l -a)AV,f,]P+aAV&Bln(l +P/B)-AU,ln(I +PI#) (1 1) where a = a, B. Our fifth assumption is (v) B and B' can be taken as equal without aflecting appreciably the results. This is justified by the linear relation observed between volume and l/Dp changes under pressure in pure liquids,26 and by the known insensitivity of Tait-type curves to the value of B (or B'). Under assumption (v) eqn (1 I ) is simplified into the following form: In kp/ko = - v(P/B) - p In (1 + P / B ) (12) where v and ,u are dimensionless quantities given by v = B[AV,S+(l -a)AV,f,]/RT and p = B(-AU,/B+aAV,f,)/RT. (13) Expressions bearing mathematical similarity to eqn (12) have been used by other investigators.l2.21, 27 Activation volume is defined by AV: = -RT(dln k,/aP), = (8AG;/W),. (14) AVZ = AV:+(AV,'-AVZ)/(I + P / B ) (15a) By substituting eqn (1 2) and (1 3) into (14) we obtain the following expression for A V: :1202 Effect of Pressure on Reaction Rates 8 7 6 5 h 0 % S g 4 3 2 1 I I I I I I 1 I 1 Plkbar Fig. 1. Shows a comparison between the model prediction, as expressed by eqn (12) (continuous curves), and high-pressure kinetic data. The curves' numbers correspond to the reaction numbers of table 1 and the corresponding list of parameters in table 2. 10 20 30 40 50 where AV: = AV:I+AV,',-AUo/B= RT(/i+V)/B AVZ = AV;,,+(l -a)AV& = RTv/B AV,+-AVZ =aAV,Sls-AAo/B= RTpIB. A V: and A V; are, respectively, the P -+ 0 (1 atm) and the p + co limits of A V:.Eqn (1 5 aH15 d ) show that the pressure dependence of D, contributes a volume change by the amount of -AUo/B (at I atm). This contribution vanishes gradually with increasing the pressure. The solvent component of AVZ of structural origin is an 'intrinsic volume change' (I -a) A V&. It appears that a is the compressible fraction of the solvent volume, which constitutes the term aAV& in AV: -AVZ. It should be mentioned that a similar interpretation to the pressure dependence of fluid volume was given by Asano.21 For many liquids the value of a varies in the range of 0.09-0.105, with the exception of 0.15 for water."' 28-30 Eqn (15) contains another feature of interest. It defines a limiting pressure P , ; for P 4 141 the dielectric contribution dominates the structural one.By equating the two contributions we obtain (16) If Pc is positive the activation volume changes sign at P = Pc and the reaction rate passes through a maximum. Uo can be evaluated when the dielectric term in eqn (I5d) is dominant. This is generally accepted to be the case in ionic and polar reaction^.^^ Assuming that AUo/B % alAV,f,l, and using the P -+ 0 limit of eqn (9) we obtain pC = B(AV:-AV$)/AV: = - B p / v . Uo = - Do(A Vz - A Vz)/(dD0/dP),. (17) Comparison with Experiments In order to compare eqn (12) with experimental data we shall rewrite it in the following form : Y = A0-vX1-pX2 (18)B. Gavish 1203 Table 1. Detail of analysed reactions change in number of Pm,, reaction ionic charges solvent T/"C/kbar ref. Do" (1) S , 1 solvolysis C(CH,),CI (2) S,2 solvolysis C,H,Br (3) S,2 solvolysis C,H,Br (4) S,2 solvolysis C,HJ (5) C,H5Br + NaOCH, (6) NH,'+NCO- (7) S , 1 solvolysis C,H,Cl (8) C,H5N(CH3)2 + CH,l (9) C6H5N(CH3)2 + 'ZHsBr (lo) C6H,N(CH&2 + C2H51 (1 1) S , 1 solvolysis CH,C,H,Br increase increase increase increase no change decrease increase increase increase increase increase 80% ethanol 80% ethanol methanol methanol methanol water methanol methanol methanol methanol methanol 25 15 29 27.7 55 15 29 23.0 65 15 29 26.3 65 15 29 26.3 30 15 29 30.7 60 15 29 66.6 65 31 30 26.3 25 15 31 32.6 25 15 31 32.6 25 15 31 32.6 23 45 31 33.4 a Calculated using literature data.34 where Xl = P / B and X , = In (1 + P / B ) .A , is expected to be zero. Using multilinear regression analysis eqn (1 8) has been best-fitted to kinetic data of a specific reaction for a series of B' values. The parameters A,, v, p, S.D.(the standard deviation), AVZ, AV: -AVZ and the correlation coefficient were evaluated as functions of B. S.D.(B) reaches a minimum at B = B,, giving ' the ' best-fitted curve. However, the B values in the range B, < B < B,, for which S.D.(B,) < S.D.(B) < S.D.(B,) = S.D.(B,) = (I +u) S.D.(B,) (u > 0), span a family of 'acceptable' curves that are hardly distinguishable from each other by eye. We can define an uncertainty in a best-fitted parameter Z(B,) by Z = lZ(B2)-2(BJl/2. u = 0.1 was found to be appropriate for most cases. Table 1 specifies 11 reactions for which high-pressure kinetic data have been reported, provided that seven, or more, data points were taken in a pressure range that exceeded at least 15 kbar.Table 2 shows the results of the above analysis. U was evaluated using eqn (7). The values of i3Do/i3P were taken or extrapolated from literature data. i3Do/i3P was taken as 3.6 kbar-' for water,32 2.4 kbar-l for ethano122.32 and 3.6 kbar-' for For 80% ethanol we obtain 2.6 kbar-', using the known additivity of Do in mixtures (with molar fractions). Discussion We have derived simple expressions for the pressure dependence of the reaction rate (eqn (12)] and the activation volume [eqn (15)] for ionic and polar reactions. We have assumed that the pressure dependence of the solvent volume and dielectric constant (reciprocal value), in the vicinity of the reactants, follows similar Tait equations, having the same B parameter.The predicted pressure dependence of the reaction rate fits remarkably well the data of polar and ionic reactions up to 45 kbar, with an average correlation coefficient of 0.999. Intrinsic Volume Change Analysis shows (table 2) that the intrinsic part of the activation volume (AV:) constitutes a small fraction of the activation volume at atmospheric pressure (A V,'). The observed range of A VZ (0 to - 5 cm3 mol-') overlies that found by Asano and le Nobletd 0 P Table 2. Shows the results of fitting eqn (12) to the high-pressure kinetic data of the reactions listed in table 1, using multilinear regression [eqn (1 8)] u,' AV: -AV:" B"/kbar AOb V b Pb S.D." A Vzd/cm3 mol-' /cm3 rno1-l 1.6 (0.7) 2.1 (0.5) 1.7 (1.0) 0.5 (0.1) 0.9 (1 .O) 3.2 (1.7) 1.5 (0.6) 0.5 (0.3) 0.3 (0.2) 0.6 (0.3) 1.8 (0.5) - 0.00 (0.02) -0.02 (0.01) 0.00 (0.01) 0.02 (0.04) 0.00 (0.04) 0.00 (0.02) 0.00 (0.02) 0.01 (0.05) - 0.06 (0.02) -0.01 (0.3) 0.00 (0.05) 0.01 (0.04) - 0.06 (0.03) - 0.02 (0.12) -0.01 (0.00) - 0.03 (0.01) - 0.32 (0.05) - 0.08 (0.05) - 0.09 (0.03) - 0.06 (0.03) -0.1 1 (0.04) 0.00 (0.02) - 1.46 (0.46) - 1.46 (0.28) - 1.73 (0.79) -0.71 (0.09) - 0.58 (0.38) 1.86 (0.93) - 1.75 (0.36) - 0.89 (0.22) - 0.85 (0.26) - 1.02 (0.29) -2.35 (0.33) 0.063 0.034 0.077 0.039 0.056 0.058 0.08 1 0.073 0.152 0.083 0.088 0.1 (0.9) - 0.7 (0.5) - 0.3 (1.5) - 0.7 (0.2) - 0.8 (0.6) - 2.5 (1.3) - 1.3 (0.4) -4.2 (0.6) -4.9 (0.8) -4.5 (0.8) 0.0 (0.3) - 22.4 (2.9) - 17.1 (0.9) -25.1 (3.9) -36.7 (4.5) - 15.8 (4.5) -28.8 (5.6) -43.7 (1 1) - 69.8 (29) -41.7 (1 1) 14.3 (1.1) -32.1 (4.8) 0.9991 5.8 0.9997 3.7 2 0.9990 4.4 2 0.9995 6.5 tl 0.9985 3.3 f 0.9994 5.1 0.9996 9.6 0.9984 15.3 8 0.9995 9.1 $' 0.9974 -7.7 2 0.9994 7.2 3 r?.S.D. is the standard deviation of the best-fitted curve; 2 U, uncertainty a B is the Tait parameter; is the strength of the electrostatic interaction [eqn (17)]. The uncertainty in the values of the best-fitted parameters is given in parentheses; is defined here with 5% S.D. change. A,, v and p are defined by theoretical curves [eqn (12) and (18)]; A V: and A V: are the activation volume at 1 atm and at infinite pressure, respectively [eqn (15 b) and (1 5 c)] ; R is the correlation coefficient;B. Gavish 1205 for van der Waals volume changes in pure liquids, for nearly non-polar reactions." However, what part of AV: is contributed by the intrinsic volume change of the solvent [eqn (lSc)] is still an open question.Dielectric vs. Structural Contributions Eqn (156) to (15d) show that the approximation AVZ -AVZ = -AUo/B, made in calculating the strength of the electrostatic interaction Uo, is justified if IAVZI + 1AV;l. This turns out to be the case in the studied reactions. The calculated values of Uo are found to be comparable with a typical strength of electrostatic interaction in the range of a few Angstroms. We may conclude that in ionic and polar reactions, for which [A V:l IA V:/, the dielectric and structural contributions to the activation volume can be separated. However, in order to obtain AVZ in a reasonable accuracy the kinetic measurements should be extended over a wide enough pressure range.Using eqn (16) table 2 shows that in reactions (SHlO) a few kbar are sufficient for obtaining AVZ, while for (1)-(7) tenths of kbar are required for this purpose. In reaction (1 1) A V: seems to vanish. None of these reactions has been found to satisfy the condition for maximum rate. Related Studies The above analysis suggests that under relatively low pressures the dielectric contribution dominates the reaction rate. This explains why the expression In k,/ko cc A VO+ P/( 1 + cP),'O that has been criticised for giving A V z = 0,l2 but behaves very similarly to In (1 + P/B), still fits very well high-pressure kinetic data.g Recently, Basilevsky et derived, from first principles, an expression to Ink,/k, using the Morse potential, and applied it to the same data used here.Their expression seems to fit the data very well. Unfortunately, that paper does not contain a systematic study of best-fitted parameters, and does not relate them to the possible role played by the dielectric constant. Tait Parameter Excluding reaction (6), the mean (S.D.) values of B were found to be 1.15 (0.66) kbar, which does not differ significantly from the bulk values of the solvents 0.86 (0.18) kbar. In reaction (6) the high value of B fits that of the bulk. A closer look at table 2 reveals that large deviations from the bulk values do occur. Theoretical consideration~~~ suggest that B is related to the excluded volume of a liquid. Asano and le Noble'' have shown that the excluded volume of the liquid at the reactants surface is an essential factor in the interpretation of reaction and activation volumes, for nearly non-polar reactions.We may conclude that the best-fitted value of B could be an important probe for characterising the state of the solvent in the vicinity of the reactants. However, a good estimation of B requires a relatively large pressure range. I thank the referees for their most helpful comments. References 1 M. G. Evans and M. Polyani, Trans. Faraday SOC., 1935, 31, 875. 2 S. D. Hamann, Physico-Chemical Efect of Pressure (Butterworth, London, 1957). 3 J. Buchanan and S . D. Hamann, Trans. Faraday Soc., 1953, 49, 1425. 4 E. Whalley, Adv. Phys. Org. Chem., 1964, 2, 93. 5 W. J. le Noble, Prog.Phys. Org. Chern., 1967, 5, 207. 6 G. Kohnstam, Prog. Reac. Kinet., 1970, 5, 335. 7 C . A. Eckert, Ann. Rev. Phys. 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