Proc. Natl. Acad. Sci. USA Vol. 77, No. 4, pp. 1728-1731, April 1980 Chemistry

Elementary transition state theory of the Soret and Dufour effects (Onsager reciprocal relationship/ideal mixtures/ dependence of thermal factor) ROBERT G. MORTIMER* AND HENRY EYRINGt *Department of Chemistry, Southwestern at Memphis, Memphis, Tennessee 38112; and tDepartment of Chemistry, University of Utah, Salt Lake City, Utah 84112 Contributed by Henry Eyring, January 2,1980

ABSTRACT The elementary transition state approach has Assume that midway between site a and site b there is a po- been used to obtain a simple model theory for the Soret effect sition of high potential energy through which a must (thermal diffusion) and the Dufour effect. The flow of in pass in jumping from site a to site b or vice versa. The tem- the Dufour effect is identified as the transport of the enthalpy change of activation as diffuse. The theory as now perature at this position is denoted by Tm, and we can write formulated applies only to thermodynamically ideal mixtures Xm - Xa = Xb - Xm = X/2. [2] of substances with molecules of nearly equal size. The results of the theory conform to the Onsager reciprocal relationship. In the elementary transition state theory of isothermal dif- When the results were fit to data on thermal diffusion for two fusion, the probability per unit time that a molecule of com- liquid systems, a close fit was obtained and yielded reasonable ponent i makes a transition from one equilibrium site to the next values of between 2 and 3 kcal mol-I for enthalpy changes of activation and differences between the entropy changes of ac- is given by tivation for the two components of between 0 and 1 cal K-1 kBT mol'-. pi= y( -jr exp(-AGi/RT) (i = 1, 2) [3] The Soret effect (thermal diffusion) is the occurrence of a dif- in which h is called the transmission coefficient, kB is fusion due to a temperature gradient. The Dufour effect Boltzmann's constant, h is Planck's constant, R is the gas con- is the reciprocal phenomenon, the occurrence of a due stant, and AG' is the standard state molal Gibbs free energy to a chemical potential gradient. Both effects have been ex- change for the activation process, in which the molecule moves tensively studied in gases, and the Soret effect has been studied from the initial to the transition state of high potential energy both theoretically and experimentally in liquids. However, between the initial and final states. partly because of the smallness of the effect, accurate mea- We must treat the nonisothermal case, so we assume that Eq. surements of the Dufour effect in liquids have only recently 3 is replaced for motion from site a to site b by been carried out (1). There does not seem to be a model theory of the liquid-state Dufour effect in the literature. Pia = X exp(-AG* [4] This paper presents a simple model theory for both effects Mh /RTa) in liquids, based on elementary transition state theory. The in which AG' is that value of the activation free energy change liquid is represented by a somewhat disordered quasi-lattice pertaining to temperature Ta. The motivation for this as- model in which a molecule can move from one equilibrium sumption is that the pre-exponential factor is derived from the position to another, passing through a transition state of high partition function for motion through the transition state and potential energy. This beginning point has been used in previous should therefore contain Tm and not Ta, whereas the expo- theories of the Soret effect (2, 3). nential factor is derived from the equilibrium constant for molecules moving from their initial state at temperature Ta. The Soret effect We restrict our discussion to the case of small deviations from Consider a liquid two-component mixture that is uniform in equilibrium (the linear case) and expand Eq. 4 in a Taylor se- the y and z directions but has a temperature and composition ries, truncating the expression at the linear term. that depend on x and t, the time. Denote two adjacent equi- Pia = Pim + (OPia/8Ta)Ta = Tm(?Ta/IXa)xa = xm(Xa - Xm). librium molecular positions by a and b, and assume that the two components are similar enough that these positions can be oc- [5] cupied by molecules of component 1 or component 2. Let xa We assume the pressure to be constant, and we apply a ther- be the value of the x coordinate at site a and Xb (presumed modynamic identity. larger than Xa) be the value of x at site b. Let the jump distance, = i X, be given by Pia Pi RR ?T ax_TI-2) [6a] X = Xb - Xa [1] X 6 - AkH*i ." Pim 1 (i = 1, 2). [6b] and assume that this distance is equal for all pairs of adjacent 2 RT2 ajX sites. The temperature at xa is denoted by Ta, and the con- The corresponding equation for motion from site b to site a centration of component 1 at xa is denoted by cla, with similar is definitions for Tb, C2a, Clb, and c2b. i el Pib =Pim [1 + 2 RT. (i = 1, 2). [7] The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "ad- The difference in sign between the two equations is due to the vertisement" in accordance with 18 U. S. C. §1734 solely to indicate difference in sign between xa - xm and Xb - Xm. In both this fact. equations, all quantities are to be evaluated at x = Xm. 1728 Downloaded by guest on October 1, 2021 Chemistry: Mortimer and Eyring Proc. Natl. Acad. Sci. USA 77 (1980) 1729 The net diffusion flux of component i, in moles per unil To describe the Dufour effect, we seek a contribution to the heat per unit time, in the coordinate frame, which is stationary flux that is proportional to a concentration gradient. Such a term respect to the local quasi-lattice, is given by will not occur in J,(), so we focus our attention on J,(particle). J = Xcpi. - Xcibpib The crucial assumption which we now make is that, when in which the concentrations are measured in moles per a particle jumps from one equilibrium position to another, it volume. We expand cqb in a Taylor series about xa, trunc. carries with it not only the enthalpy that it had in its initial the series again at the linear term. position but also the enthalpy of activation that it had to acquire to make the passage through the transition state. This idea was Cib = Cia + (6cib/?Xb)xb = xa(Xb - Xa) mentioned in earlier works on thermal diffusion but was not = Ca + X(bci/bx). exploited to discuss the Dufour effect (3, 6, 7). We first write the particle contribution to the enthalpy Eqs. 6, 7, and 9 are substituted into Eq. 8, and only linear ti flux: are kept. The result is 2 A = - Jqx.(particle) = A (CiapiaHim- cibpibHim ) [19] Jix X2p,(C/X) - CpPX2 RT2 (bT/3x). i=l in which Him is the molal enthalpy of molecules of component Note that subscripts specifying location are no longer neces i in the transition state, including the enthalpy of activation. The mean velocity of molecules of species i is in the x dire( We expand the p and c as before, stopping at the linear and is given by terms:

_~ = J / X2p (ac1 _ 2_-i_ ix=jixi - -Ci o1fxJ A iRT2 aX [11] Jqx(particle) = A cipim 1 - aHim) In order to determine the formulas which our model theeory ( cA (1 XAH*T 1 implies for the diffusion coefficient and the thermal diffu soCian+ J Pim 1 +RT2 ) HimJ [20a] coefficient, we compare our results with the macroscopic 2 equation (4) Jilt(particle) = -X2 EPiHim(bc / x) V2 i=l l-52 = --°O12 [vX, + kTV(ln T)]. [12] - ClC2 PAc RT2JR Hm(oT/6x). [20b] In this equation, °12 is called the mutual diffusion coefficient The heat flux is the enthalpy flux minus the enthalpy of the and kT is called the thermal diffusion ratio. The total concen- transported molecules. We write the particle contribution tration is denoted by c and given by J/(particle) = Jq(particle) - J1H1 - J2H2. [21] C = C1 + C2. [13] Using the fact The mole fraction of component 1 is denoted by XI. Unfortu- nately there is a bewildering variety of equations and symbols AHI= Him-Hi, [22] used in the phenomenology of diffusion and thermal diffusion. we obtain We choose to compare with Eq. 12 because it is not restricted to a particular coordinate system. J, (particle) = -A2 E [piAH'(bci/bx) We now assume that the partial molal volumes of our two i=l components are nearly equal and that the thermal expansion of the liquid can be neglected. In this case, + Pici (he) (bT/bx)J. [23] OCI/k) "t~c(OX11/x) = -c(aX2/1x) [14] For each component, this expression contains one term corre- sponding to the Dufour effect (proportional to the concentration and we can write from Eq. 11 gradients) and one term corresponding to VlX- V2X = -X2(p1/Xl + p2/X2)(aXI/bx) (proportional to the temperature gradient). We compare Eq. 23 with the macroscopic equation (10)

1 I -2 - A ) axn [15] Pi RT P2RTJ ( T/6x). = -AXtvT -cTD''(bjl/8XI)PTVX1 [24] Comparison with Eq. 13 shows that in which Al is the chemical potential (partial molal Gibbs free energy) of component 1, Xt is the , and 012 = X2(plX2 + p2XI) [16] D" is called the Dufour coefficient. and Comparison of Eqs. 23 and 24 shows that the particle con- tribution to the thermal conductivity is given by c012 kT A2(Pi _ 2 [171 X2 ClC2 ~ RT RT I Xt(particle) = RT2 [plcl(\AH')2 + p2c2(AHt)2] [25] Eq. 17 is the result for our elementary theory of the Soret effect. It resembles earlier results (5-7). This differs from previous work (8, 9) but the particle contri- bution to the thermal conductivity is small compared with the The Dufour effect phonon contribution, and it is probably not now possible to test We proceed to write, for the heat flux, an expression analogous Eq. 25 experimentally. to Eq. 10 for the diffusion flux. We assume that the heat flux We now apply Eq. 14 to Eq. 23 and write, by comparison consists of a vibrational (phonon) contribution and a contri- with Eq. 24, bution due to particle motion (8, 9). cl TD"(b~ul/8Xl),T = cX2(pAH' -p2NHA). [26] = J'q J'(phonon)q + J'(particle)q [18] If the mixture is thermodynamically ideal, Downloaded by guest on October 1, 2021 1730 Chemistry: Mortimer and Eyring Proc. Natl. Acad. Sci. USA 77 (1980)

(4/.L1/bX1)p,T = RT/X,, [27] Discussion and comparison with experiment so that Until recently (1), there has been no experimental verification of the Onsager reciprocal relationship for the Soret and Dufour X2 D" = R (pH -p2AHM). [28] effects in liquids, although Rastogi and his coworkers attempted RT2 measurements several years ago (9). One of us (R.G.M.) has also This is the result for the Dufour coefficient in our model liquid carried out preliminary measurements of the Dufour effect in system. liquid systems and obtained reasonable agreement between D' and D' for the carbon tetrachloride/chloroform system and The Onsager reciprocal relationship the carbon tetrachloride/benzene system at 25°C. There are data in the literature on the thermal diffusion of Because the Soret and Dufour effects are reciprocal effects, binary mixtures of nearly special molecules, which should their phenomenological coefficients must be equal if the proper conform fairly nearly to the assumptions of our model theory, choice of forces and is used (11). In the center of mass and these can be used to test the theory. Saxton et al. (12) used coordinate frame, the proper diffusion fluxes are measured in a diaphragm cell to measure a for several mixtures, including mass per unit area per unit time, and the phenomenological carbon tetrabromide/carbon tetrachloride and C2H4C12/ relations are written (10) C2H4Br2 (isomers unspecified), over a range of tempera- tures. j -L VT_ (VA4)T [29a] to compare our we j= Liq~j L11i w29 In order theory with their results, write, from Eqs. 16 and 28 VT (VAi)T TD' pAH'-p2 - HN Jq qqTjj Lqi W2T [29b] [38] 12 RT(X2Pl + X1P2)' in which W2 is the mass fraction of component 2 and ,u is the Using Eq. 3, we obtain chemical potential of component per unit mass (partial specific e-Hi/RTAHf/Re( eAH*/RTAH* Gibbs free energy). The quantity (VA1)T is the equivalent RT[X2e-AHI/RT1 + isothermal chemical potential gradient, and is given by Xje(ASt-ASf)/R2 1 e-AHt/RT]i2 [39] (VU4)T = Vu' + sjvT [30] Notice that the transmission coefficient X has cancelled out, in which sl is the partial specific entropy of component 1. and that a depends only on the difference AS' - AS' rather The Onsager reciprocal relationship implies that than on AS' and AS' separately. Therefore, the thermal dif- fusion factor a depends, as well as on the temperature, on three Llq=Lqi. [31] values: AMH, AHt, and AS* - AS*. If we identify the coordinate frame of Eq. 24 with the center The formula of Eq. 39 was fit to data of Saxton et al. (12) by of mass coordinate frame, the coefficient D" is related to Lqi using nonlinear regression. Unfortunately, the sum of the by squares of the residuals appeared to be a complicated function of the parameters, possibly containing various saddle points and D= Lq [32] several relative extrema. Ordinary iterative numerical proce- pW2W2T2 dures failed to converge, so the minimum in this function was in which p is the density of the system in mass per unit volume. found by trial and error, using a computer program that re- The thermal diffusion coefficient D' is related to peatedly computed the sum of the squares for various values Liq by of the parameters. DI = Lig [33] Table 1 and Fig. 1 show the results for two different mixtures: pW1W2T2 CBr4/CCL4 (mole fraction of CBr4, 0.09) and C2H4Br2/C2H4C12 so that the Onsager reciprocal relationship requires D' and D" (mole fraction of each component, 0.50). Table 1 gives the to be equal. values of the parameters which correspond to the curves shown The thermal diffusion ratio, kT is related to D' by in Fig. 1. The values of the parameters found correspond to true relative minima in the sums of the squares of the residuals, and kT = CCT'[4 no smaller values of these functions were found, but there is no guarantee that a lower minimum does not exist. Also, the data Thermal diffusion results are also often expressed in terms of show considerable scatter, and the expected errors in the pa- rameters are presumably large, although no statistical procedure a, the thermal diffusion factor, given by was found to estimate them. TDI C2 However, the curves in Fig. 1 appear to fit the data reason- a = T kT. [35] 012 C1C2 ably well. The curves are nearly linear but, over a larger tem- This quantity is also apparently called the thermal diffusion perature range, there is a noticeable upward concavity. ratio, as is kT. Using Eq. 34, we obtain from Eq. 28 Table 1. Values of parameters in Eq. 38 to fit data for two-liquid mixtures D = RT2 (p1AHI - [36] P2AH'). CBr4 (1)/ C2H4Br2 (1)/ Comparison with Eq. 28 shows that CC14 (2) C2H4C12 (2) AHf, kcal mol1 (kJ mol-1) 2.29 (9.58) 2.16 (9.04) D =D, [37] AH ,kcal mol-1 (kJ mol-1) 2.29 (9.58) 2.20 (9.21) so our theory conforms to the Onsager reciprocal relation- AS]-ASI, cal K-1 mol-I ship. (JK-1 mol-1) -1.0 (-4.2) -0.40 (-1.7) Downloaded by guest on October 1, 2021 Chemistry: Mortimer and Eyring Proc. Natl. Acad. Sci. USA 77 (1980) 1731

crease in order at the transition state (compared to no molecule in the transition state). At first glance, it might seem that a larger molecule should lower the entropy more in moving to the transition state than would a smaller molecule, but it would also have more of an effect in disordering the liquid quasilattice in - 1.4 the equilibrium site than would a smaller molecule, and the effect in the equilibrium site seems to be more pronounced, 1.2 according to our results. .2 The numerical values for the enthalpy changes of activation seem reasonable. For example, a self-diffusion data on CCI4 (14) 1.0 * indicate an enthalpy change of activation of 2.8 kcal molh; in the more disordered quasi-lattice of the CC14/CBr4 mixture, our value of 2.3 0.8 kcal molh is about what we might expect. The collaboration of the two authors began when R.G.M. spent part of a sabbatical leave at the University of Utah, and he expresses his 0.6 appreciation to H.E. and to the University of Utah for this opportunity. This work was supported in part by a Cottrell College Science Grant to Southwestern at Memphis from the Research Corporation, and by 10 20 30 40 50 a grant from the Research and Creative Activities Committee of Temperature, OC Southwestern at Memphis; this support is gratefully acknowledged. FIG. 1. Thermal diffusion factor, a, for two-liquid mixtures as 1. Rowley, R. L. & Horne, F. H. (1978) Chem. 325- a function of temperature. Upper curve, CBr4/CCl4; lower curve, J. Phys. 68, C2H4Br2/C2H4Cl,. (Data from ref. 12.) 326. 2. Dougherty, E. L., Jr. & Drickamer, H. G. (1955) J. Phys. Chem. 59,443-449. The positive sign of a corresponds to motion of component 3. Prager, S. & Eyring, H. (1953) J. Chem. Phys. 21, 1347-1350. 1 toward the end of the cell in thermal diffusion and to a 4. Hirschfelder, J. O., Curtiss, C. F. & Bird, R. B. (1954) Molecular flow of heat away from the larger concentration of component Theory of Gases and Liquids (Wiley, New York), p. 518. 1 in the Dufour effect. In both of the systems analyzed, we have 5. Prigogine, I., de Broukere, L. & Amand, R. (1956) Physica 16, assigned the heavier molecular weight substance to be com- 577-598. ponent 1, and in all six of the systems studied by Saxton et al. 6. Prigogine, I., de Broukere, L. & Amand, R. (1956) Physica 16, (12) the component with larger and heavier molecules moved 851-860. 7. L. W. S. & H. toward the cold end of the cells. The same true Tichacek, J., Kmak, Drickamer, G. (1956) J. Phys. is of the data of Chem. 660-665. jeener and Thomaes (13), who 60, studied mixtures of CC14 and 8. Horrocks, J. K. & McLaughlin, E. (1960) Trans. Faraday Soc. 56, the other chlorinated methanes. 206-212. It is interesting that, for both mixtures analyzed, the values 9. Lin, S. H., Eyring, H. & Davis, W. J. (1964) J. Phys. Chem. 68, of the enthalpy changes of activation that fit the data are equal 3017-3020. or nearly equal for both components in the mixture. It is also 10. Rastogi, R. P. & Yadava, B. L. S. (1969) J. Chem. Phys. 51, interesting that in both cases the entropy change of activation 2826-2830. is more negative for component 2 (with smaller molecules) than 11. deGroot, S. R. & Mazur, P. (1969) Non-Equilibrium Thermo- for component 1. With enthalpy changes of activation nearly dynamics (North-Holland, Amsterdam), pp. 35 ff. 12. Saxton, R. L., Dougherty, E. L. & Drickamer, H. G. (1954) equal, this must always be true if component 1 is to move to the J. cold end of the cell. Chem. Phys. 22, 1166-1168. 13. Jeener, J. & Thomaes, G. (1954) J. Chem. Phys. 22, 566-567. The entropy change of activation is negative (probably t6 14. Rathbun, R. E. & Babb, A. L. (1961) J. Phys. Chem. 65, 1072- cal K-1 molh for these substances), corresponding to an in- 1074. Downloaded by guest on October 1, 2021