The Estimation of Activation Parameters Corrections and Incorrections S. D. Hamann CSlRO Institute of Industrial Technology, G.P.O. Box 4331, Melbourne, Victoria 3001, Australia W. J. le Noble State University of New York at Stony Brook, Long Island, NY 11794
It is a matter of some regret that a great deal of the current exoerimental work on chemical eauilibria and reaction rates insolution is still reported and discussed in terms of the volume concentrations of the reactants: that is. in terms of the number of moles of each reactant J present in one liter (cubic decimeter) of the solution-the molaritv. - . c.7-.. of J. The use of this volum'etric scale is convenient in analytical chemistry and is unexceptionable as long as we concerned with results a t only a single temperature or pressure. However, since liquids undemo thermal ex~ansionand hvdrostatic comoression. its use canand does lead to complications if changes occur in the temperature or pressure, and hence in the volume of the solution-and such changes are often deliberately made in exoerimental work on eauilibria and kinetics. The orohlem does not arise if the concentrations are expressed i n i n i t s that do not involve volume, such as mole fractiom, XJ, and molalities, mJ (moles of J present in one kilogram of the solvent). Harned and Owen ( I ) pointed this out 40 years ago and added that "the confusion which can he produced. . .makes it undesirable to use the c-scale except a t constant temperature and pressure." However. the use of molarities continues and so does the confusion. Nowhere are the misunderstandings and errors chemistw and, more common than in the field of high - oressure . especially, in estimations of volumes of activation from the results of reaction rate measurements made under pressure. Our aims in this paper are twofold. First, we wish to alert unwarv chemists enterine this field to the oroblem and to suggesiasimple way of avoiding it. ~ e c o n d l ~ , w wish e to warn students of the hieh oressure literature that manv of the activation volumes "already published have heen incorrectly calculated from the experimental results, and that it is best in each case not merely to quote from reviews and other listings, hut t o examine carefully the original papers as well. Where the authors have drawn mechanistic conclusions based on the sign of the activation volume, these can probably be trusted; hut where they have based fine distinctions on the magnitude of this quantity, reservation is in order--especially
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where comparisons have been madehetween values from more than one lahoratory. Nature of the Problem In the countless literature reports of activation energies, enthaloies. . . and entrooies based on the temoerature denendence of rate contants, k , in solution, one virtually never encounters concern on the Dart of the authors about the fact that roncentrations c.1 given in units of moles per liter (molarities) are tcmurrature deoendent. Hut after all the liter is a unit of vo1ume;and since liquids expand when they are heated, a liter of solution will have a smaller number of moles of a solute J a t high temperatures than a t low ones. The temperature ranges employed, of course, are generally small, as are the coefficients of thermal expansion, and it seems probable that any such changes are quickly dismissed by the practicing chemist as being, a t most, of the same order as the precision of the experiments. The same remarks apply to overall reaction enthalpies and entropies as determined from the variation of equilibrium constants, K, with temperature. The story is different in the literature dealing with activation and reaction volumes as deduced from the effect of pressure on the rate or equilibrium constants for reaction: one more often than not encounters vague statements to the effect that the values reported in terms of CJ were "corrected for the effect of compressibility." The reason for this difference is prohably related to the fact that the volume changes engendered by the pressures required, say, to halve or double k or K (commonly about 1 kbar or 100 MPa) are an order of magnitude greater than those caused by the temperature changes necessary to do this (by rule of thumb: 10°C). Whatever the reasons, the situation is the same, and although we here discuss especially the pressure case, our remarks are equally applicable to studies concerned with temperature effects. Clearly, the molarity of J measured a t atmospheric pressure, which we shall write as CJ*, must be corrected for the compression of the mixture in order to obtain the corresponding molarity, cj, at a higher pressure. So, if that is done, where
does the problem lie? Simply in the fact that CJ is not an independent thermodynamic variable. I t depends not only on the chemical composition of the mixture hut also on the total volume of the mixture, and this volume dependence is often wrongly ignored when the molar rate and equilibrium constants k, and Kcare partially differentiated with respect t o pressure to derive "volumes" of activation and reaction. We shall now discuss the kind of error that this neglect causes.
where AV stands for the change of partial molar volume that accom~aniesthe reaction. Exactlv analoeous relationships hold i i the chemical potentials h e expressed in terms bf molalities, by (2). We then have
But for the molarity scale, eqns. (3), (7); and (9) give
Thermodynamic Relationships Eauilibrium Constants In an ideal dilute solution, thr chemical potential, p.), o f a solute species .I can he defined in the following- alternative ways (1-3):
+
p~ = p~O(x) R T In x~
(1)
where R is the gas constant and T is the absolute temperature. The potential u.1 is the same for the three conrentration scales, hut (he standard molar chemical potentials ~ J vary O with the scales and this fact is indicated by the bracketed labels. Here and throughout this paper it is to be understood that the arguments of the logarithms are dimensionless numbers given by dividing dimensional quantities hy their units; for example, in eqn. (3), C J stands for the (pure) number of moles of J present in one cubic decimeter of the solution. The partial derivative of PJ with respect to pressure, P, a t constant temperature and constant composition (constant numbers of moles of all the components) is equal to the partial molar volume, VJ of J in the mixture (4). Since the mole fraction XJ and the molality mJ are independent of pressure a t constant composition, eqns. (1) and (2) yield the derivatives
On the other hand, the molarity CJ varies with the pressure a t constant composition: it is proportional to the density p of the solution, so that (3) gives m - d p ~RT- d i n p ap dp dp
where the term in brackets is the increase in the number of moles that accompanies the reaction. Formula (13) was first derived by Planck (5) in 1887, and (15) was published by Williams (6) in 1921. The difference between them was later emphasized by Guggenheim (7) and Hamann (8). Relationships analogous to (12)-(15) hold also for the temperature dependence of In K (4, 7). For instance, corresponding to (151, we have: d In Kc RT2-=AF-RT2a(r+s+ dT
a-b-
...)
(16)
where AH denotes the change of partial molar enthalpy for the reaction and a is the coefficient of thermal expansion of the solution. The term involving a does not occur in the temperature derivatives of In K, and in K, [cf., eqns. (13) and (14)l. The ahove expressions have been derived for ideal solutions, hut their extension to non-ideal ones is straightforward and has been published elsewhere (9). Reaction Rate Constants In the transition state theory of rates (10,ll) it is assumed that a reaction such as (8)proceeds through a high energy transition state Xtarhich exists in chemical eauilihrium with the initial species A, B, . . . ,with an equilihri&n constant Kt that ohevs the normal thermodvnamic relationships given ahove. 1iis usually further assumed that the rate c o k t i n t k is proportional to KTT,which involves an assumption (10) that the transmission coefficient for passage of Xf into the products R, S, .. . is independent of the pressure and temperature. With those assumptions, and hearing in mind that k has the same concentration units as Kf, we can insert k in place of K in (12)-(15) and replace AV by the "activation volume," AVt, which is the amount by which the partial molar volume of the transition state differs from that of the initial reactants A, B, . . . . For rate constants expressed in mole fraction units, (13) therefore gives
where K = a In p l a P is the compressibility of the solution. For a general chemical reaction in solution, of the type the condition for equilibrium is that the total chemical potential of the products should equal that of the initial reactants, that is, If the solution is ideal, then from (9) and (1) we have
+ . ..- aI(a0(x) - bp&)
-RT In Kx = r p ~ O ( x+ ) spsO(x)
and from (14) a similar relationship holds f o r k , , in molality units. But for volume concentration units, (15) gives d in k
-RT-2=A~t-~T~(l-a-b-...) aP = AVt + R T R ( ~ 1)
(18) (19)
where n is the kinetic order of the reaction. -
. .. (10)
where K, is the mole fraction equilibrium constant
I t follows from (10) and (5) that the pressure dependence of In K, is
Other Concentration Scales The treatment given ahove can he extended easily to other concentration scales. Any units that are independent of volume, such as weiaht fractions, aive relationships like (12-14) and (17). But units that involve the volume of the solution yield more complex relationships, of which (15) and (16) are just two examples. The volume fraction scale, for example, adds terms involving the separate compressibilities and coefficients of expansion of all the components. Practical Applications The ahove results show that the error caused by neglecting Volume 61 Number 8 August 1984
659
the volume dependence of molarities c, and wrongly supposing that a relationship such as - RJ In Th =~~ v t (20) JP
holds, is proportional to the ~ C ~ O I R T K Values . for this factor for some typical solvents a t atmospheric pressure are given in the table Values of RTK ' Solvent Temperature ('C) RTx(cm3 mol-') water
mehnol carbon telrachloride diethyl ether bhexane
25 25 25 25 60
5.0 6.1
For comparison, the value of IAV'l for simple reactions is typically in the range 10-30 cm3 mol-', so that in measurements of any acceptable accuracy, the term involving x is certainly not negliyihle [except, of course, for first order reactions in which the factor i n - 11, in I151 . is zmol. Eouation (19), not (20), therefore must be used in order to dkrive values of AV* from hieh nressure measurements of k,. Gueeenheim pointed this out in 1937 (71,and i t has been emphasized frequently in later reviews (8,14,15). Eyring became persuaded of i t in 1938 (16) and suggested then that the adjective "apparent" should be applied to any values of AVt obtained from the false equality (20). Nevertheless, he continued to favor the use of (20) and similar relationshi~s"because of the simwle way they follow from the data" (16). But from what data? The data from high nressure exweriments verv seldom include directly measured values of iolarities CJ under pressure: these are usuallv estimated indirectly hv correcting values CJ* that have been measured a t atmospheric pressure-before and after the high pressure experiment. T o take an example, let us suppose that we set out to measure AVt for asimple reaction A B R, which obeys the following second-order rate equation and its integrated forms:
-
We decide to DreDare and analvze our solutions volumetricdv a t atmospheric pressure and to use molarity units for all thk concentrations and rate constants. We can now proceed bv the following steps. a) We first measure k, at atmospheric pressure in the usual way. h) We then run the reaction under an applied high pressure, but again prepare and analyze our solutions at atmospheric pressure, so that we measure CJ*. C) We calculate an uncorrected rate constant kc* using those concentrations. d) To obtain the true value of k, under pressure we correct kc* by converting each CJ* to CJ, by multiplying it by plp*, where p is the density of the compressed solution (approximatelythat of thesolvent)and p* is itsdensity at atmospheric pressure. From (211, we see that this amounts to writing h, = h,*p*lp (22) e) If the value of plp*, needed in (dl, is not known, we measure it. fl We repeat (b) at a sufficient number of differentpressures to allow us to estimate the derivative J in k,lJP at atmospheric pressure. [The best methods of making this estimate, since plots of In k, against P are decidedly curved, have be& discussed elsewhere (14).] g) We insert the derivative J In h,lJP in (19)and take x either from theliterature or from our measurementsin (e).toarrive at the . .. values of AVY at atmospheric pressure. ' They should, of course, be precisely and explicitly defined as being. for example, the number of moles of a solute J present in one liter of a solution at a pressure of 1 bar (lo5Pa) and a temperature of 2S°C (298.15 K).
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Journal of Chemical Education
JP
JP
+
or
1.1 3.1 2.7
'Based rn mmprersibllities x measwed by Eduljee et al. (Rhexane) (laand Tyrer (all the others) (19.
+
Some authors have followed that wrocedure and obtained correct values for A V I . Rut wc can easily see that it is more complicatrd and trnuhlesome than it need be, esprciullv if the measurements (el are required. If we insert (221 in (19), and ' K ,we find remember that P' is constant and that d In ~ 1 3 1 = that J In k Jlnk * -RT = -L RT RTK =L AVt + RTK (23)
AVt = -RT-
J In he* JP
(24)
In other words, the correction applied in the step (d) is exactly cancelled bv the comwressihilitv term introduced in the sten (g) (that is true fur reuctionsof any order) and wecould have ohrained A\': directlv and &imvlvfrom the uncorrected rate constants k,* based'on the koiarities CJ* a t atmospheric pressure. That follows, of course, from the fact that a t constant composition CJ*is independent of any high pressure that may be applied to the solution, and so i t yields relationships like those for mole fractions and molalities: (5),(13), and (17). A number of authors have recognized this and correctlv " annlied .. eqn. (24) to their results. However, errors and confusion have arisen because other authors have failed to follow the full procedure (a)-(g) or to take the convenient shortcut indicated in (24). Some have gone as far as step (f), but then used eqn. (20) instead of (19)-result: wrong. Others have unconsciouslv omitted the correction (d) and then used equation (20)-result: right, by accident. Others, still, have listed high pressure rate constants in molarity units but have not saidwhether these were corrected as in (d), or how AVt was derived-result: wrobablv wrong. And some have shown no units a t all and merely listed or plotted quantities such as "log k"-result: who knows? All thesemistakes can be avoided easily by using the mole fraction or molality scales, and many workers have indeed done so. But since molarities are convenient and still widely used, we suggest that authors who do employ them in high pressure work should adopt the convention of listing and using molar concentrations determined a t atmospheric pressure,' which we have here written as CJ*,and should derive their activation volumes from the corresponding uncorrected rate constants kc*. I t is those that are related simolv to AV'. hv (24), and thkir use means that there is no need a t all to know and "correct" for the nressure-volume behavior of a solution-nor should it be measured specially, as some have done! Finally, we reiterate that the above arguments apply equally to eouilibrium constants. and to the influence of temwerature as well as pressure on chemical reaction rates and equilibria. Literature Cited (1) Harned. H. S., and Owen. B. B.."The Physical Chemistry of Elcctrolyfe Solutions? Reinhold, New York. 1943, p. 11. (2) Clastone, S., "Textbmk 01Physi~sIChemistly?Van Noatrand. New York, 1946. p. 665. (3) Robinson, R. A . and Stokea, R. H.. "Electrolyte Solutians? Buttenuonhs, London,
Cuggenheim. E. A,, Trans. ~ o m d n y ~ o c33,607 .. (19371. Hamann. S. D..(a1 "Physieo-Chemical Effects of Pressurn." Butteworths, London, 1951; lh1 in "High Pressure Physicsand Chemistry," (Editor Brad1ey.R. S.) AcadamicPrens, New York. 1963; Vol. 2, pp. 136,165: le) in "ModornAspe~tsofEle~truchemietrv." IEdifors: Conwav. B. E.. and Bockris.J. O'MI Plenum Press. New
..... ....., .....
Tyrer,D.,J. Chem Soe.. 105,2534 (19141. Eduljee,H.E.,Newitt,D.M.,and Weale, K. E..J Chem. Soc.,3066 (1951). le Noble, W. J., Pmg. Phys. Org. Chem., 5,207 (1967);Asano. T., and le Noble, W . J., Chrm. Re"., 78,407 (1978). Kohnstam, G..Prog. R~poef.Kin.S,335(19701. Eyring, H., Tmns. Forodoy Sor., 34. (1 (1958).
