J . Chem. SOC., Faraday Trans. I , 1985, 81, 875-884 Pressure Dependence of Activation and Reaction Volumes BY M. V. BASILEVSKY, N. N. WEINBERG* AND V. M. ZHULIN Karpov Institute of Physical Chemistry, Obukha 10, Moscow 107120, U.S.S.R. and Zelinsky Institute of Organic Chemistry, Academy of Sciences of the U.S.S.R., Leninsky Prospect 47, Moscow 117913, U.S.S.R. Received 20th Mzrch, 1984 A simple theoretical model of high-pressure kinetic effects is proposed based on the assumption that characteristic (initial, transition or product) states of a reacting system should be identified with stationary points of an enthalpy surface rather than those of a regular potential-energy surface (p.e.s.). An explicit treatment of the deformation of a p.e.s. promoted by pressure enables one to improve the simple cylinder Stearn-Eyring model of activation volumes so as to describe the pressure dependence of activation and reaction volumes. The dependence thus obtained agrees well with experimental data in the wide pressure range from 0 to 45 kbar.The pressure behaviour of activation enthalpies and entropies is also discussed. The kinetic effects of pressure are usually discussed in terms of the transition-state In the regular thermodynamic formulation the rate constant k at a (t.s.) given pressure p is (1) kT k = exp (ASIIR) exp ( - A H t / R T ) AHJ = AUt+pAVJ. (2) The activation enthalpy, AH$, energy, AUJ, and entropy, AS$, are essentially macroscopic quantities. For the gas phase, however, they can have a microscopic interpretation, assuming that the same quantities represent the characteristics of separated microscopic reaction systems, i.e.neglecting the interaction between the systems. Then the first term in eqn (2) is usually identified with the potential barrier height on a potential-energy surface. The second term in eqn (2) is practically inessential for gas-phase processes and contains no useful information, being merely nRT (n is an integer, e.g. n = 1 for a bimolecular reaction). For reactions in a condensed medium interpretation of AUJ as a potential barrier height on a p.e.s. remains valid. For this case A V t , the activation volume, is identified with the difference of volumes between an initial state and a t.s. Its experimental determination provides some indirect information concerning the t.s. structure and reaction mechanism.Stearn and Eyring4 have attempted to relate the activation volume to a t.s. structure in terms of a simple ‘cylinder’ model. Note, however, that there are no fundamental reasons underlying the application of macroscopic thermo- dynamical concepts at a microscopic level. In particular, the volume of a microscopic system in a disordered condensed medium cannot be introduced entirely consistently. This is probably why further attempts to refine the Stearn-Eyring model by detailing the shape of a reacting system5 proved to be unsuccessful.6 The numerous ambiguities and uncertainties in the interpretation of experimental activation volumes seem to have the same origin. 875876 PRESSURE DEPENDENCE OF ACTIVATION AND REACTION VOLUMES Activation volumes decrease, in absolute terms, with pres~ure.~ This dependence is n~n-linear.~* Of all the empirical attempts3* 7-15 to describe it, the El’yanov- Vasylvitskaya equation7? * is the most successful.It reproduces the experimental activation volumes well over a wide pressure range from 0 to 12 kbar. However, because of the empirical nature of this equation its validity must be verified in each particular case when extrapolating to higher pressures. Note also that all empirical correlations, although reproducing the pressure dependence of rate constants, state nothing about the pressure dependence of activation parameters,* which makes the extrapolation of high-pressure data to other temperatures difficult. Here we propose a simple phenomenological model extending the Stearn-Eyring treatment so as to derive the pressure dependences of the activation volumes and enthalpies over a wide pressure range.This model also applies to reaction volumes and enthalpies. THE MODEL When dealing with the canonical relations eqn (1) and (2), it is always implied that configurations of reactants and t .s. are associated with respective stationary points on a p.e.s. The potential energy, U, of a system is considered to be a function of internal coordinates qi, and its extremum points (minima of reactants and products or saddle point of a t.s.) obey the stationary condition - = 0. The volumes of the respective configurations are used to obtain activation (AVt) or reaction (A V) volumes. Distortion of a p.e.s.under pressure is inevitably neglected. In a consistent microscopic treatment the volume of a reaction system (like its potential energy, U ) should be considered as a function of the internal variables qi. Then, the enthalpy, H, of the system can be introduced as a microscopic function of these variables representing the enthalpy surface of the system. This enthalpy surface should be used instead of the p.e.s. to determine the configurations of the reactants, products or t.s. This provides a natural description of the pressure dependence of AVt and AV as a result of the relative shift of the respective stationary points of the enthalpy surface. From the condition au aqi aH = O __ aqi aV - = - p - aU we obtain the main equation aqi %i (3) obeyed for the stationary points of an enthalpy surface, qi = qi. Given an explicit form of the functions U(q,) and V(qi), eqn (3) may be applied to find the coordinates qi(p) of the stationary points, from which the respective volumetric effects can be evaluated.In what follows we adopt the Stearn-Eyring model4 and treat a reaction system as a cylinder of radius r directed along the reaction coordinate q : * The only exception should, however, be mentioned : simple thermodynamical reasoning made it to describe the pressure behaviour of activation enthalpies and entropies at relatively low (ca. 5 kbar) pressures.M. V. BASILEVSKY, N . N . WINBERG AND V. M. ZHULIN 877 In this case eqn (3) takes the form au - = -2nrqp. ar THE HARMONIC APPROXIMATION Consider now the case of relatively low pressures and small p.e.s.distortions. Points (q, F ) lie in the vicinity of the parent stationary points (go, r,) of an initial p.e.s., and thereby a harmonic approximation can be used to describe the function U(q, r). We accept in addition that the cross-derivative vanishes. Then Substituting eqn (6) in eqn ( 5 ) we obtain v(4 - 40) = - nr2p O(r - To) = - 2nrqp r3-3ar+2P = 0 which then reduces to (7) Solving eqn (7) we obtain the radius of the cylinder as a function of pressure r(p) = (22/a) sin [+ arcsin (/?/da")] P 27 nr, 2 4 a = J- 2 - d(v@ - 1 (1 +-J. 2WOP 3 7 1 P The parameters of systems of interest are such that /?/2/a3 < 0.4. Therefore, this expression reduces with an error < 2-3% to the simple formuloh The cross-section of the Stearn-Eyring cylinder is then Substituting eqn (8) in eqn (7) we obtain Eqn (8) and (9) may be used to describe the pressure dependence of the activation and reaction volumes at relatively low pressures.Let q1 be the value of the reaction878 PRESSURE DEPENDENCE OF ACTIVATION AND REACTION VOLUMES 1.0 0.8 0 *4 0.2 t 0 2 4 6 8 10 12 plkbar Fig. 1. Pressure dependence of relative reaction volumes AV/AV,: 1, eqn (10) with K = 0.04 kbar-l and A = 0.10 kbar-’ ; 2, El’yanov-Hamann” equation, A V/A V, = (1 + O.O92p)-l; 3, El’yanov-Vasylvitskaya7* equation, A V/A V, = 1 - 0.17 In (1 + 3.9 1 p ) - 4, eqn (13) with o/o, calculated numerically from eqn (12) and (15) and with Q = 3.3 &l, a = 0.15, j3 = 70 kbar-l and 9 = 0.1 mydn A-l; 5, eqn (13) with constant o/o,. Curves 5 join curve 4 at; (a) pcr = 5 kbar (ocr/oo = 0.72), (b) pcr = 6 kbar (oc!Jo0 = 0.68), (c) pcr = 7 kbar (ocr/oo = 0.64) and ( d ) pcr = 8 kbar (oc,/oo = 0.60).Experimental points are:’ 0, isoprene dimerization; ., pentene+ pentan014 diamyl ether; A, pentene +propano1 -+ 2-methyl- heptan-3-01. coordinate corresponding to a pre-reaction (van der Waals) complex and q2 be its value for a t.s. or for the products. Then, combination of eqn (4), (8) and (9) results in the following equation for the respective volumetric effect, A V = V , - : where ql0 and q20 are the coordinates and ql and r2 are the force constants of the respective stationary points of a p.e.s. (q2 is negative in the case of a t.s.) and A& is the activation or reaction volume at zero (atmospheric) pressure.Eqn (10) represents the pressure dependence of AV and requires the p.e.s. parameters qlo, q20, ql, q2, 8 and oo to be known. If, however, these are unavailable, K and A in eqn (10) may be considered as adjustable empirical parameters. The test of calculations using eqn (10) (in its empirical variant) is presented in fig. 1 ; eqn (10) agrees satisfactorily with the experimental data and the empirical correlation curves7~ *, l1 in the pressure range 0-5 kbar. At higher pressures the model predicts a too rapid decrease of AV. Such non-physical behaviour of AV(p) is caused as the harmonic approximation becomes a poor model at high pressures. Estimations using eqn (8) and (9) with K = 0.04 kbar-l and 3, = 0.10 kbar-l (the values used in the A V calculations) show that at 5-7 kbar lateral compression of the Stearn-Eyring cylinderM. V.BASILEVSKY, N. N. WEINBERG AND V. M. ZHULIN 879 is as high as a/ao = 0.7-0.6; the longitudinal compression is qo-q = 0.7-0.85 A. Further lateral compression of the cylinder seems unlikely, since it would require deformation of the valence 'core' of a molecule (bond lengths and valence angles). It may continue only along the softest reaction mode. However, even for this the harmonic approximation fails to be valid at such large distortions of the initial p.e.s. Therefore, the harmonic potential eqn (6) should be replaced by a more realistic one reproducing the exponential behaviour of the short-range repulsion. THE MORSE POTENTIAL Consider the simple expression U(q, 4 = Udq) + U2(r).In order to proceed with the analytical treatment we approximate the function U,(q) by the Morse potential In this case eqn (5) takes the form U,(q) = D{ 1 - exp b(q0 - q)112. (1 1) p=- 0 0 2Da J and J Unlike eqn (lo), eqn (1 3) neglects the contribution caused by the shift of the product mininum or t.s. saddle point. This approximation should not influence the calculated results, since, as seen from eqn (lo), the relative contribution of the term neglected is proportional (at relatively low pressures when it is largest) to the ratio I qJq2 I of the force constants at the corresponding stationary points of a p.e.s. For example, in the case of dimerization of cylcopentadiene (CPD) discussed below, this contribution is < 10% for the activation volume ( I qreact/qt I w 1/10) and is completely negligible for the reaction volume (qprod 9 qreaCt).To apply eqn (13), the function a(p) = z[i;(p)l2 must be specified. For this purpose eqn (1 1) should be supplemented by an explicit expression for U2(r). In accordance with the above discussion, we approximate the function U2(r) by a harmonic potential truncated at r = rcr: J Since r = r((p), then a certain 'critical' pressure value pcr corresponds to rcr = t(pcr). At higher pressures the function ~ ( p ) becomes constant. This makes possible the direct use of eqn (12) and (13) at p > pcr: a/ao = (rCr/ro)2 = const in this case.880 PRESSURE DEPENDENCE OF ACTIVATION AND REACTION VOLUMES At 0 < p < pcr the value of a(p) can be found by solving numerically eqn (12) (15) together with e(r - ro) = - 27trqp which is obtained by substituting eqn (14) in eqn (5).The results of calculations using a = 3.3 A-l, a = 0.15, p = 70 kbar-1, 8 = 0.1 mdyn A-l and pcr = 5,6,7 and 8 kbar are presented in fig. 1. The numerical curve agrees well with the experimental data and the empirical El’yanov-Vasylvitskaya equation for pressures from 0 to 8 kbar. At higher pressures eqn (1 3) with a constant cross-section a = acr should be used. Otherwise, the degree of lateral compression of the cylinder becomes too large. We chose the junction point pcr so as, first, to fit the high-pressure experimental data and, secondly, to obtain physically reasonable value of the lateral compression acr/aO. From this reasoning the value of pcr = 8 kbar (ac,/ao = 0.60 and rcr/ro = 0.77) appears to be optimal.The joined curve reproduces the experimental data well for pressures from 0 to 12 kbar, providing a fit as good as that of the El’yanov-Vasylvitskaya equation. Thus we anticipate that eqn (13) will also remain applicable for higher pressures. Some evidence that such an extrapolation is indeed plausible is presented later by a comparison with experiment and by analysis of the results at microscopic level. PRESSURE EFFECTS ON RATE AND EQUILIBRIUM CONSTANTS El’yanov and Gonikberggg l6 have proposed that the pressure dependence of the rate and equilibrium constants are treated using I In (&/KO) = - ( A V,/RT) @ @ = [AG(p) - AGo]/A V,, AGO = AG(0) where Kp and KO correspond to pressure p and atmospheric pressure, AG(p) and AGO are the respective activation and reaction free enthalpies, A& is the volumetric effect at atmospheric pressure and @ is the so-called ‘correlation function’, usually evaluated from empirical expression^.^-^^ If the pressure dependence of A Y is known, the function @(p) can be obtained theoretically from @(p) = Jop 9 dp.AV, Integrating eqn (13) over the pressure interval from 8 to p kbar (where a/ao is independent of pressure) gives for @(p) the following expression predicted by our model for p > per: @(PI - @(Per) = acr[A(p) - A(~cr>I/ao a c r B y = - . The values of @ ( p ) for 0 < p < 8 kbar can only be obtained numerically. However, as seen from fig. 1, the empirical El’yanov-Vasylvitskaya equation7* 0 0 AV/AV, = 1 --a ln(1 +bp)M.V. BASILEVSKY, N. N. WEINBERG AND V. M. ZHULIN 88 1 is almost indistinguishable at these pressures from our numerical curve. Therefore, the corresponding analytical expression a(1 +bp) ln(1 +bp) b @,(PI = (1 +a)P- can be used to estimate @(p) at p < 8 kbar without numerical computations. One can use eqn (16) and (17) to calculate reaction rates at high pressures and, comparing the results with experimental data, to estimate the validity range of our model. Such a comparison is presented below for a number of high-pressure reactions. COMPARISON WITH EXPERIMENT CYCLOPENTADIENE DIMERIZATION AT 40 kbar Diels-Alder reactions with their high activation volumes are very sensitive to pressure effects.179 la Unfortunately, we were unable to find any report on the static studies of Diels-Alder reactions at pressures > 10 kbar.There exist, however, data on the solid-state dimerization of cylcopentadiene (CPD) at high pressure (40 kbar), combined with the shear def0rmati0n.l~ Although the interpretation of such experi- ments is rather ambiguous (the role of the shear deformation is not clear), the results of ref. (19) can be used to estimate the lower bound of the rate constant at p = 40 kbar and T = 153 K: k(40 kbar, 153 K) 2 4 x lo-* dm3 mol-1 s-l. (19) To rederive the same quantity in terms of our model, we started with an experimental rate constant k( 1 atm, 353 K) = 9 x dm3 mol-l s-l and A VJ = - 30.5 cm3 m0l-V9 The value obtained using eqn (16)-(18) and the parameters used above (a = 0.15 and y = 42 kbar-l) is k(40 kbar, 353 K) = 1.2 x lo3 dm3 mol-l s-l.It corresponds to T = 353 K. The final extrapolation to T = 153 K was based on the activation enthalpy AH (40 kbar) = 6.5 kcal mol-1 evaluated below. The result: kCa1,(40 kbar, 153 K) = 6.3 x lou3 dm3 mol-l s-l. is in reasonable agreement with experimental estimate.l9 The extrapolation using the empirical eqn (18) with a = 0.17 and b = 4.98 kbar-l recommended for Diels-Alder reactions’9 leads to k(40 kbar, 353 K) = 5.66 dm3 mol-1 s-l. Further extrapolation to T = 153 K appears impossible in the framework of the empirical treatment with no Ahrrenius parameters. OTHER HIGH-PRESSURE REACTIONS The function @(p)17 was parametrized using the value pcr = 8 kbar adopted to reproduce the high-pressure constantlg for the dimerization of CPD.However, the experience of working with empirical correlation functions points to their relatively universal nat~re.~-~v l6 That is why it seemed interesting to apply eqn (17) with the same parametrization to other high-pressure reactions. Fig. 2(a) presents the respective correlations with kinetic data for a number of high-pressure bimolecular r e a ~ t i o n s . ~ ~ - ~ ~ The linear plots obtained demonstrate a good fit of eqn (17) to the experimental data. Only the solvolysis of ally1 bromide in methanol shows poor agreement. Recalculation with changed parameters a = 0.4 and y = 5 kbar-’, however, allows us to improve the correlation considerably [cf fig. 2(b)].882 PRESSURE DEPENDENCE OF ACTIVATION AND REACTION VOLUMES - - +/kbar 4 6 8 10 t , l , I , I I I I I 8 8 4 + 3 / 6 9 12 15 10 20 30 40 I .. I . . I . . I I I I I I l l PlkbX 7 5 4 Fig. 2. Correlations between experimental In kp/ko and @(p):l7 1 , solvolysis of C(CH,),Cl in 80% EtOH;zo 2, solvolysis of C,H,Br in MeOH;20 3, solvolysis of CH,CHCH,Br in MeOH;,l 4, solvolysis of C,H,Cl in MeOH;22 5, ionization of piperidine in MeOH;23 6, esterification of pivalic acid with EtOH;24 7, ethyl pivalate hydrolysi~.~~ The following values of parameters were used in the calculations : (a) a = 0.15 and y = 42 kbar-l ; (b) a = 0.40 and y = 5 kbar-l. DEFORMATION OF A P.E.S. BY PRESSURE ACTIVATION (REACTION) ENTHALPIES AND ENTROPIES Pressure effects on the rate and equilibrium constants, as well as on the respective volumetric effects, i.e. on the macroscopic characteristics of a reaction system, have been described above.One can also estimate how pressure affects the microscopic parameters (activation energy, reaction heat, shift of stationary points), i.e. to describe p.e.s. distortion under pressure. An inherent feature of the present model is that the variation of activation enthalpy, AH2 (or reaction heat, AH), with pressure involves, except for thepAVterm, the contribution 6Ucoming from the increase of the potential energy of a system as a result of its shift along the reaction coordinate from qo (stationary point of a parent p.e.s.) to q (stationary point of an enthalpy surface): AHS(p) = AHJ-dU(p)+pAVS(p) AH$ = AH$(o). The values of AHt(p) (including the 6U and P A V terms), Aq = qo - q and A VS/A VJ for the dimerization of CPD are presented in table 1 for the pressure range 0-50 kbar.Calculations were carried out using eqn (1 1)-( 13) and (20) with a = 0.15, = 70 kbar-l, CT,,/CT~ = 0.60, a = 3.3 A-l and experimental AVJ = - 30.5 cm3 mo1-l. AHJ = AUJ = 16.5 kcal mol-l estimated from the kinetic data of ref. (19). The t.s. shift along the reduction coordinate was neglected (in accordance with the discussion above). The main contribution to AHt(p) variations is from the pAVS(p) term. The contribution of 6U is negligible and is only 0.6 kcal mol-l at 50 kbar. Note that the most essential changes in the activation volume take place in the region of relativelyM. V. BASILEVSKY, N. N . WEINBERG AND V. M. ZHULIN 883 Table 1. Some characteristics of the distorted p.e.s. of the cylcopentadiene dimerization reaction ~~ AS$ - AS,$ AH$ 6U -pA VI AGf, - AGI /cal mol-l p/kbar /kcal mol-1 /kcal mol-1 /kcal mol-l /kcal mol-' K-l A q / A AVI/AVd 0 5 10 15 20 25 30 35 40 45 50 16.5 14.7 13.7 12.3 1 1 .1 9.9 8.7 7.6 6.5 5.4 4.3 0.00 0.07 0.12 0.18 0.24 0.3 1 0.37 0.44 0.50 0.56 0.63 0.00 1.78 2.82 4.04 5.20 6.32 7.4 1 8.47 9.51 10.53 11.54 0.00 2.56 3.84 5.21 6.53 7.82 9.07 10.29 11.49 12.67 13.83 0.00 2.38 3.02 3.32 3.66 3.99 4.33 4.63 4.97 5.30 5.57 0.00 0.64 0.72 0.78 0.82 0.85 0.88 0.90 0.92 0.94 0.96 1 .ooo 0.486 0.386 0.369 0.356 0.347 0.339 0.332 0.326 0.321 0.316 low pressures, where the potential energy is almost constant (6U w 0). This is in accord with the concept6? 25 that the main contribution to A V $ comes from the free activation volume. The same result can be obtained by considering the Aq values.Comparing them with the results of quantum-chemical computations26 one can see that the Aq values listed in table 1 correspond to a pre-reaction stage when chemical bonds are almost unaltered. Therefore, the simple Morse potentialll seems to provide a reasonable approximation. At greater shifts along the reaction coordinate it fails to apply. The pressure region, where the barrier AH$ dissappears, presents a natural bound of the validity of our model. As seen from table 1, it lies near 50 kbar. Even at 40 kbar the activation enthalpy reduces to only 6.5 kcal mol-l. This result is supported by the experimental observation that the temperature dependence of the rate constants corresponding to high pressures combined with the shear deformation is weak.27 A similar effect was reported for high-pressure reactions carried out in a static regime :2s at 90 kbar acynaphthylene polymerizes at room temperature, which points to a considerable decrease of the respective activation barrier or, probably, to its complete disappearance. The values of AGS and AH$ can be obtained independently in our model [eqn (1 6)-( 17) and (20), respectively].That allows estimating the activation entropy as AS$(p) - AS2 = [AH$(p) - AH&/ T - [AG$(p) - AGi] T ASJ = ASt(0). The values of AS$(p)-ASJ calculated using this equation are presented in table 1. Unfortunately, we were unable to reproduce the sharp drop in activation entropies observed in the pressure range 0-5 kbar.29 This is because our model predicts a slight decrease of activation enthalpy AH$ in this range, whereas experiment predicts an increase.The disagreement at relatively low pressures seems to be caused by the fact that we have identified the activation enthalpy with the difference of values taken by the function H(q) at respective stationary points. The temperature dependence of the volume of a reacting system is thus neglected [term 6U in eqn (20) is of minor importance]. As might be anticipated, the corresponding error decreases as the pressure increases.884 PRESSURE DEPENDENCE OF ACTIVATION AND REACTION VOLUMES We now consider possible ways of improving our model. As presented here, it fails to work at large p.e.s. distortions, when the stationary points of the t.s.and reactants appear too close to each other and the Morse potentialll gives an incorrect description of the reaction profile. Nevertheless, the present approach is restricted by neither the applicability of the Morse potential nor the Stearn-Eyring model. One can, in principle, use a more realistic potential U(qi), refine the function V(qi) and solve eqn (3) numerically. A more advanced approach, however, is expected to work without introducing a microscopic quantity, such as volume, into the microscopic description of an elementary chemical process. A non-phenomenological treatment of pressure effects on reaction kinetics and equilibrium should be based entirely on quantum-chemical calculations of the intermolecular potential, operating a chemical system.We thank Prof. B. S. El’yanov and Dr E. M. Vasylvitskaya for helpful discussions. M. G. Evans and M. Polanyi, Trans. 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