J . Chem. SOC., Faraday Trans. 1, 1985, 81, 11-18 Kinetics of the Reaction between Cyanide Ions and Tris(4-methyl- 1 , 10-phenanthroline)iron(n) Cations in Aqueous Solutions Analysis of Kinetic Data for this Reaction and for Solvolysis of Benzyl Chloride in Water in Terms of Isothermal, Isobaric and Related Isochoric Activation Parameters BY MICHAEL J. BLANDAMER,* JOHN BURGESS AND BARBARA CLARK Department of Chemistry, The University, Leicester LEI 7RH AND ROSS E. ROBERTSON Department of Chemistry, University of Calgary, Calgary, Alberta, Canada AND JOHN M. W. SCOTT Department of Chemistry, Memorial University of Newfoundland, St John’s, Newfoundland, Canada Received 13th January, 1984 Kinetic data for the title reaction and for the solvolysis of benzyl chloride in water are analysed to obtain isobaric and isothermal activation parameters.Isochoric parameters are defined and the rate constants described as functions of temperature, pressure and molar volume of the pure solvent, V;”. A quantity Afv/( V:) characterises the dependence of rate constant on temperature at constant V:. We counter claims that isochoric activation parameters necessarily provide an insight into the process of activation in chemical reactions. Isobaric-isothermal functions, within the limitations of transition-state theory, have a sounder basis in terms of thermodynamic analysis of chemical reactions. The dependence on temperature is reported for the rate constant of reaction between cyanide ions and tris(4-methyl- 1 ,lo-phenanthroline)iron(rr)cations in aqueous solution.The data supplement the set reporting the dependence of rate constant on pressure.l The two sets of data are analysed to obtain isobaric and isothermal activation parameters. The latter are used in the calculation of isochoric activation parameters. The significance of activation parameters calculated for isochoric conditions is examined in some detail. Numerous accounts of kinetic data refer to isochoric activation parameters. [For further references see ref. (2).] Nevertheless, their meaning has seemed, at least to us, to be obscure. Previously2 we posed two questions which in effect asked ‘what volume is held constant?’ and ‘what reference states are used in the definition of isochoric activation parameters?’. The latter question was prompted by reports of isochoric heat capacities of activation, AICF [cf.ref. (3), where the symbol ACg is used]. These questions were answered2 in part by reference to the treatment of chemical equilibria in solution. For example, the dependence of an equilibrium quotient Q(s1n : T : p ) on temperature can be characterised under the constraint that the pressure changes to hold the molar volume of the solvent, V,*, 1112 ISOBARIC, ISOTHERMAL AND ISOCHORIC PARAMETERS constant. The condition isochoric ( V,*) is therefore extrinsic to the system characterised by the equilibrium quotient Q(s1n: T : p ) . By way of contrast, the dependences of In Q on temperature at constant pressure and on pressure at constant temperature yield the intrinsic isobaric and isothermal functions (cf.the van’t Hoff equations). Here we extend the analysis to activation parameters characterising chemical reaction in solution. In addition to the well established isothermal and isobaric activation parameters, the dependence of rate constant on temperature can be characterised under the constraint that the pressure changes whereby the molar volume of the solvent remains constant. This dependence yields an isochoric ( V ; ) activation parameter where the isochoric condition is extrinsic to the reacting system. We also comment on a number of related isochoric parameters. Attention is confined to kinetic data for chemical reactions using water as the solvent. The necessary density data for water over extended ranges of temperature and pressure are a~ailable.~? However, the procedures described below can be readily extended in analyses of kinetic data for reactions when the solvent is a binary liquid Data for the dependence of the density of these mixtures on mole-fraction composition, temperature and pressure are becoming available,8* providing the basis of the calculations discussed below. At this stage we conclude that isochoric activation parameters are certainly as complicated as isothermal and isobaric activation parameters, a conclusion which is at odds with that advanced by other authors [cf.ref. (10) and (1 l)]. EXPERIMENTAL MATERIALS These were prepared and purified as described previous1y.l KINETICS The progress of the title reaction was followed spectrophotometrically.lq l2 A first-order rate constant was calculated using a Hewlett-Packard minicomputer which controlled the spectrophotometer and logged the absorbance as a function of time.12 RESULTS The reaction between cyanide ions and Fe(4Me-phen)tS, to give Fe(4Me-phen), (CN), plus free ligand, follows first-order kinetics in aqueous solution when [CN-] % [Fe(4Me-phen)f+] : - d[Fe(4Me-phen)f+]/dt = ko,,[Fe(4Me-phen)t+].Except at very high cyanide concentrations The k, term corresponds to rate-determining dissociation of the c0mp1ex.l~. l4 The k, term corresponds to bimolecular reaction between the complex and cyanide. Whether this occurs by direct attack at the iron or initially by attack at the coordinated ligand is a matter still not ~ett1ed.l~~ l6 Although in one or two special cases there is evidence for two-stage attack involving initial attack at the ligand and subsequent transfer of cyanide (or hydroxide) to the meta1,l7?l8 there is no evidence for two-stage kinetics at iron(IIEphenanthro1ine complexes except when strongly electron-withdrawing nitro or sulphonato groups are present.For the Fe(4Me-phen):+ +cyanide reaction the kinetics conform to a single-step process.M. J. BLANDAMER et al. 13 Table 1. Rate constants for the reaction between Fe(4Me-phen)i+ and cyanide in aqueous solution ([CN-] = 0.5 mol dm-3) and for Fe(4Me-phen)i' dissociation ([H,SO,] = 0.5 mol dmP3) T/K k1/10-5 s-l k,,,/10-3 s-' k2/10P3 dm3 mo1-l s-l 298 3.59 1.84 3.61 30 1 6.1 3a 2.78 5.44 305 1 1.7a 4.29 8.35 308 1 9.0b 6.68 13.0 312 38.2a 9.82 18.9 315 55.1 10.65 24.2 318 89.4" 16.42 31.1 a Interpolated or extrapolated values (cf.text). In good agreement with earlier results [ref. (l)]. We have established k , by monitoring dissociation of the complex in acid solution (0.5 mol dm-3 sulphuric acid) at 298,308 and 315 K (table 1); k, at other temperatures was obtained by interpolation or (3 18 K) extrapolation; the Arrhenius dependence yields an activation energy of 126 kJ mol-l. Rate constants kobs are reported in table 1 ; k, was obtained from (kobs-kl)/[CN-]. All k, and kobs values reported in table 1 represent the means of three independent consistent determinations; they are quoted to a slightly higher precision than their accuracy warrants to avoid any information loss from rounding-off in the statistical analysis. ACTIVATION PARAMETERS According to transition-state theory,lg for reaction in solution reactants are in equilibrium with the transition state.The rate constant for reaction is given by where we have assumed that the transmission coefficient is unity and the solution is ideal. Further, AlG*(T) =-RTlnSKe(T) =pP(sln;T)-Z(j= l ; j = vlvj/pp(sln;T) (4) At Vm(sln; T ; p ) = VF(s1n; T ; p ) - C ( j = 1 ; j = v) I vi 1 Vj"O(s1n; T ; p ) . (5) and Thus the standard equilibrium constant $K*( T ) for the equilibrium between reactants and transition state is dependent on temperature and independent of pressure. Further, K e ( T ) is related through eqn (4) to the chemical potentials of reactants and transition states in their solution standard states at temperature T and standard pressure p e .ISOCHORIC ACTIVATION PARAMETERS According to eqn (3), the dependent variable In ( k / T ) can be expressed as a function of independent variables T and p : In ( k / T ) = In ( k / T ) ( T ; p ) . (6) Similarly the molar volume of the (pure) solvent Vi;" can be defined by the same independent variables : Vi;" = Vi;"(T; p ) . (7)14 ISOBARIC, ISOTHERMAL AND ISOCHORIC PARAMETERS The argument follows that described2 for equilibrium quotients. Consider a given temperature 8 and pressure n, where the molar volume of the pure solvent is V,*(8; n). We assert that there exists a temperature 8 + A 8 at pressure n+An, where the condition holds, Anl being characteristic of the solvent. According to eqn (6) there also exists a quantity In (k/T) at (O+A8, n+An,).In other words we may compare In (k/T) for reaction in solutions under conditions where the molar volume of the solvent is the same, eqn (8). The gradient of the tangent in the ln(k/T)-T-p domain, conforming to the isochoric condition in eqn (8), can be calculated from the isobaric-isothermal v,*(e; n) = v,*(e+Ae; ~ + A Z J (8) gradients The latter equation can be rewritten in terms of the enthalpies and volumes of Here a,*, K,* and #I,* are the thermal expansivity, isothermal compressibility and isochoric thermal pressure coefficient of the pure solvent, respectively. An analogous line of argument can be used in conjunction with eqn (8) such that, following an increase in pressure from n to n+An, there exists a temperature 8+A8, where the following condition holds: (1 1) V,*(e; n) = V,*(6+A8,; n+Alt).Here A8, is characteristic of the solvent. Hence AW~(T;P) 1 A%H~(T;P) +- RT #I,* RT2 . Again the calculated quantity is isochoric (V,*), meaning isochoric with respect to the molar volume of the solvent. NUMERICAL ANALYSIS The dependence of (k/T) on temperature and pressure was fitted to the equation ln(k/T) = ln(k[6; n]/8)+a,(T-8)+a,(p-n)+a,(T-8)2 + a,(p - n)2 + a,(p - n) (T- 8) + a6(T- 8)2 ( p - n) (1 3) about a reference temperature 6 and reference pressure n. In practice the rate constant k was corrected using the density of the pure solvent as indicated in eqn (3). BENZYL CHLORIDE First-order rate constants for the solvolysis of benzyl chloride were taken from published data,209 21 including the data reported by Robertson and Scott.22 Estimates of parameters in eqn (13) are summarized in table 2.Derived activation parameters are set out in table 3. In table 2 VR = V,*(e;n). The negative volume of activation is consistent with that reported by Hyne.lo We also calculate that AjVw(sln:T:p) decreases with pressure, but the dependence is smalllo and on the borderline of statistical significance.M. J. BLANDAMER et al. 15 Table 2. Benzyl chloride in water: numerical analysisa (n/bar = 1380, 8/K = 323.4, VR/m3 mol-' = 17.31 x parameter estimate standard error ~~ ~ ~~ In (k[8; .]/8) - 13.7108 6.9 x 10-3 a,/K-' 0.10059 6.2 x 10-4 a3/K-2 -4.233 x lo-* 8.07 x 10-5 a,/ bar-' 7.242 x 4.7 x 10-9 a,/bar-' K-' 2.339 x 6.1 x 10-7 a, / bar-' 3.519 x lop4 6.41 x lop6 a,/bar-' KP2 2.8 x 6 x a Standard error = 2.37 x (degrees of freedom = 43).Table 3. Benzyl chloride in water: derived parameters property value reference temperature, B/K reference pressure, n/bar V*(H,O: 1: 8: n)/m3 mol-' In (k[8: n]/@ (obs) A: Vm(sln: 8: n)/m3 mol-1 [aAI Vm(sln: 8: n)/ap],/m3 mol-l bar [aAI V"(s1n: 8: n)/aT],/m3 mol-' K-' AjHa(sln:8:n)/kJ mol-l [aAiHa(sln: 8: n)/@]/J mol-l bar-' AlC,"(sln: 8: n)/J K-' mot1 {(a In k$[B: n]/aT) at VR]/K-' A:v/( Vf)/kJ mol-' {[aAJly( Vf)/dT] at VR}/J K-' rno1-l {[aA: Vm(sln: O:n)/dT] at VR>/m3 mol-l K-l 323.4 1380 17.31 x - 13.721 -(9.46+0.12) x lo-' -(3.89f2.5) x lo-, -(2.95+0.05) x 87.47k0.54 0.04 0.08 0.1055+0.0004 - 195 k 143 91.74f0.51 - 143f 122 9.76 Table 4. Kinetics of reaction between cyanide ions and tris(4-methyl- 1,lO-phenanthroline) iron@) in aqueous solutiona (B/K = 298.2, n/bar = 68, Vg/m3 mol-l = 18.02 x lo-,) parameter estimate standard error In [k(e: n)/e] - 7.3274 5.9 x a,/K-' 0.10704 5.8 x 10-3 a,/bar-l -2.8065 x 10-4 9.2 x 10-5 a 11 data points: standard deviation = 1.27 x 10-l.CYANIDE IONS AND METAL COMPLEX The data for this reaction cover smaller pressure and temperature ranges than for the previous example. However, the consistency obtained by the analysis for benzyl chloride gave support to the idea of using the same procedures for the second-order reaction involving cyanide ions (tables 4 and 5). Nevertheless fewer terms were statis- tically significant in the fitting of the kinetic data to eqn (13).16 ISOBARIC, ISOTHERMAL AND ISOCHORIC PARAMETERS Table 5. Derivation parameters for reaction between cyanide ions and tris(4-methyl-1,lO- phenanthroline)iron(Ir) in aqueous solution parameter value AIH"(s1n: 8:z)lkJ mol-l 79.4k4.3 At V"(s1n: 8: z)/m3 mo1-I (6.96 f 2.2) x lop6 [a In (k(8: ~ ) / 8 ) / a T ] at 0.1044 & 0.0052 A$w( V:)/kJ mo1-I 77.22 f 3.84 [a In (k$(O: z)/8)/a V:]/mol m-3 (4.09 & 0.11) x lo6 Both sets of kinetic data are used to calculate a quantity ASv/(VF) defined by eqn (14) which is expressed in J mol-l : An analogous quantity AS@( V,*) calculated using eqn (1 5 ) is expressed in m3 mol-1: As@( V,*> RT * (15) DISCUSSION Relative to the extensive literature concerned with reactions of organic solutes, the literature dealing with the effect of pressure on rates of reactions concerning metal complexes remains small.23-25 Within this general area, isochoric activation parameters have also attracted little attention, whereas these quantities have prompted considerable interest for organic reactions. The claims made for isochoric activation parameters are strong. Hills and Vianall suggest that isobaric are more complicated than isochoric activation parameters. Further, these authors1' suggest that the isobaric heat capacity of activationzs* 27 is particularly complicated. Whalleyz3 has argued that isothermal-isochoric parameters are more fundamentallo, 28 and 29 particularly for understanding the depen- dence of activation parameters on composition of solvent for reactions in binary aqueous mixtures. Holterman and Engberts7 find, however, no compelling evidence, at least for one reaction, for preferring either isobaric or isochroic activation par- ameters.Nevertheless, our concern with the enthusiasm for isochoric activation parameters is, we suggest, more fundamental. From the outset, the term isochoric is not used in the sense normally used in thermodynamics. As far as we can discern from published papers, the rate constant for a given chemical reaction always characterises the approach of a system to a minimum in G under isothermal-isobaric conditions rather than to a minimum in F under isothermal-isochoric conditions. Reported isochoric functions are calculated using, for example, eqn (lo), in which the properties of the pure solvent are used. In other words, isochoric means 'at constant molar volume of the pure solvent'.Hence we have used the term isochoric (V;"). Other isochoric functions have been commented on. Whalley30 has argued that the volume kept constant is 'the volume of an equilibrium mixture of initial and transition state '.M. J . BLANDAMER et al. 17 The volume of the reacting system is given by I/(sln: T : p ) = (l/Ml) Vl(sln: T : p ) + C ( j = 2; j = i)mj y(s1n: T:p)+m$ V&sln: T : p ) . (16) For this system ml < mj and mi for all reactants and products is dependent on time as is F(sln : T : p ) . We might envisage a system containing a pseudochemical equilibrium between reactants and transition state: V(hypothetica1 equilibrium; sln: T : p ) = (l/Ml) Vl(sln: T:p)+mfq Vfq(s1n: T : p ) +C ( j = 2; j = r ) I vj I mFq Vfq(sln: T : p ) . (17) An alternative method considers a solution in 1 kg of solvent comprising 1 mol of transition state and I vj 1 mol of reactants where the partial molar volumes of reactants and transition states equal the volumes at infinite dilution.Hence, by definition, Vm(sln: T : p ) = (l/Ml) VT(1: T : p ) + Vr(sln: T : p ) + C ( j = 2; j = r ) I vj I VY(s1n: T : p ) . (18) For example, in the case of a second-order reaction, with reacting solutes 2 and 3, Vm(sln: T : p ) = (l/Ml) V,*(l: T : p ) + Vr(s1n: T : p ) + VP(s1n: T : p ) + VP(s1n: T : p ) . (19) Unfortunately it is not immediately obvious how one might calculate the corresponding thermal and pressure coefficients for these solutions. Alternatively a set of isochoric (AIVm) parameters may be calculated; a pro- posal along these lines was made by Caldin.31 In effect, the volume of activation At V"(s1n: T : p ) is treated as a dependent variable defined, for a given system, by the independent variables T and p .The analysis described above is repeated where At Vm replaces V,* in eqn (7), (8) and (1 1). Then, for example, we may define the partial differential Irrespective of the isochoric condition, our contention is that these constant volumes are extrinsic to the system undergoing reaction. It may also be useful to calculate AZv/(VT) as defined in eqn (14), but this term is not the thermodynamic energy of activation. Nor is its temperature derivative the molar isochoric heat capacity of activation. Finally we accept that isochoric functions may provide an insight into mechanisms of reaction.However, the precise meaning of these quantities must first be established. We hope that we have achieved at least part of that task. For the moment, however, we conclude that isobaric activation parameters26* 27 are free from the ambiguity associated with isochoric parameters. We thank the S.E.R.C. for a maintenance award to B.C. We thank Dr E. Whalley for providing reprints of his papers. 2 FAR 118 1 2 3 4 6 6 7 8 9 10 11 12 13 14 15 18 17 18 18 20 21 2P 23 24 25 26 27 28 28 30 31 ISOBARIC, ISOTHERMAL AND ISOCHORIC PARAMETERS F. M. Mikhail, P. Askalani, J. Burgess and R. Sherry, Transition Met. Chem., 1981, 6, 51. M. J. Blandamer, J. Burgess, B. Clark and J. M. W. Scott, J. Chem. SOC., Faraday Trans. 1, 1984,80, 3359. B. T. Baliga, R.J. Withey, D. Poulton and E. Whalley, Trans. Faraday SOC., 1965, 61, 517. G. S. Kell and E. Whalley, Philos. Trans. R. SOC. London, Sect. A, 1965, 258, 565. G. S. Kell, J. Chem. Eng. Data, 1970, 15, 1 19. B. T. Baliga and E. Whalley, J. Phys. Chem., 1967, 71, 1 166. H. A. J. Holterman and J. B. F. N. Engberts, J. Am. Chem. 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