Proc. Natl. Acad. Sci. USA Vol. 93, pp. 7452-7453, July 1996 Chemistry

Persistent confusion of total entropy and chemical system entropy in chemical thermodynamics (van't Hoff equation/model dependent thermodynamics) GREGORIO WEBER University of Illinois School of Chemical Sciences, 600 South Mathews Avenue, Urbana, IL 61801 Contributed by Gregorio Weber, February 29, 1996

ABSTRACT The change in free energy with temperature (dG) dH dSj T d at constant pressure of a chemical reaction is determined by dT dT -Si. [5] the sum (dS) of changes in entropy of the system of reagents, dT dS;, and the additional entropy change of the surroundings, Eq. 4 refers the changes in free energy to the entropy of system dSH, that results from the enthalpy change, dH. A faulty and identification of the total entropy change on reaction with dS; surroundings, as demanded by the second law of thermo- has been dynamics, whereas Eq. 5 expresses the free energy change in responsible for the attribution of general validity to terms of properties intrinsic to the system. If we neglect this the expressions (dAG/dT)p = -AS; and d(AG/1)/d(1/T) = AH, which are found in most textbooks and in innumerable difference and simply write Eq. 5 in the form papers. dG dH dS [5'] The free energy associated to each reagent entering in chem- dT) dT dTS ical reaction under constant temperature and pressure is given we risk concluding from Eqs. 4 and 5' that by the familiar Gibbs function (dG/dT)p= -Si [6] G = H - TSi. [1] and inadvertently also that SH + Si = Si. Yet the simple mistake Eq. 1 is a statement of the heat content of the reagent, part of embodied in Eq. 6 has been exhibited by textbooks and is which is actual and determined by its intrinsic entropy Si, and current in the literature for the last 70 and perhaps more years. partly potential, determined by H, the energy locked in the That dSi does not represent the total entropy change is brought bonds that partake in the chemical reaction. In an isothermal out by the many cases in which the standard change in Si in the reaction reaction is negative. These instances do not contradict the dG = dH - TdSi, [2] second law because of the larger entropy increase of the surroundings that follows the decrease in H. In fact, an the actual change in the composition of the system does not immediate corollary of the second law is that: If the entropy provide a measure of the enthalpy change dH, which can only of any part of a system decreases in the course of an isothermal be appraised by the corresponding change in heat content of process, there must be a concomitant and larger increase in the surroundings of magnitude dQH = -dH. Similarly, TdSj another part of the same system. provides for an additional change in the heat content of the The erroneous identification of the entropy change of the surroundings of magnitude dQi. Calorimetric measurements chemical system dSi with the total entropy change dSH + dSj do not separate these two interdependent but distinct heat leads immediately to a fixed relation of the changes in enthalpy changes and only their sum dQ = - (dQH + dQj) is measurable. and entropy with temperature: IfdG/dT = -Si, it follows from Neglecting the very small contribution to the external work, Eq. 5' that pdV, which is the case in practically all reactions in solution, the change in free energy of a reagent in an isothermal reaction dH/dT = T(dSj/dT) [7] may be expressed as a change in entropy dS of its surroundings, the sum of changes dSH and dSj is valid for each reagent in a chemical reaction, and then for all reactions dG = - T(dSH + dSi) = - TdS. [3] dAH/dT = T(dASj/dT), [8] As such, Eq. 3 satisfies Planck's criterion (1, *) that the decrease in free energy in a spontaneous process is determined where AH and ASi are the standard changes in enthalpy and by the change in entropy of all the bodies in which the heat intrinsic entropy in the reaction, respectively. content changes in the reaction, a condition essentially re- It also follows that quired by the second law (2). It follows from Eq. 3 that the free 1 dAH dASj energy change of the reagent with temperature at constant - [8'] pressure equals T d(l1/T) d(l1/T) (dG/dT)p = - S. [4] *Planck establishes the distinction between the entropy of the envi- ronment, 40, and that of the reagents, 4, states the application of the From Eq. 1, we have second law to reactions as d(4 + 40) - 0, and consistently uses 4) in the same way as I use Si. He notes that dU = dH - pdVoften exceeds Tdo, thus justifying Berthelot's rule. Unfortunately, he did not The publication costs of this article were defrayed in part by page charge explicitly state that in these cases one can have do - 0, which should payment. This article must therefore be hereby marked "advertisement" in have precluded the systematic erroneous identification of S = (A + (A. accordance with 18 U.S.C. §1734 solely to indicate this fact. with Si = (A. 7452 Downloaded by guest on September 24, 2021 Chemistry: Weber Proc. Natl. Acad. Sci. USA 93 (1996) 7453

Starting with the Gibbs relation AG = Al TASi and dividing It may be asked why the van't Hoff equation has proven by T, useful in spite of its spurious derivation. It is simply that most chemical reactions, as already recognized by Berthelot (4) in AG/T= AH/T-AS, [9] 1897, are driven by the change in enthalpy. Any error owing to the neglect of the much smaller entropy contribution would and pass unnoticed because we have no other means of indepen- dently evaluating either AHl or ASi. The consideration of the d(AG/T) 1 dAH dASi association of protein subunits, which are typical entropy = AH + -___ d(l/T) Td(1/T) d(1/T) [10] driven reactions (5, 6), led me to observe the extremely different results obtained from Eqs. 10 and 11 when specific but, according to Eq. 8', because the second and third term relations of H and S of the reagents were assumed (7, 8) to cancel each other permit modeling the reaction. Eventually, I was able to trace the origin of Eq. 11 to the erroneous Eqs. 8 and 8', and the d(AG/T) = AH [1] original confusion between Si and S = SH + Si. This long- d(1/T) standing mistake has clearly arisen from inability to distinguish between the variational conditions that define the chemical The last is the celebrated van't Hoff equation that has been equilibrium, originally stated by Gibbs (9), and the application used for so long in the determination of enthalpy changes in of the laws of thermodynamics to actual processes like those chemical reactions that all papers and most textbooks consider that result from finite changes in temperature or pressure. It it unnecessary to enter into a detailed discussion of its deri- provides a striking example of Truesdell's dictum (10): In vation. Actual examples of the mistaken derivation of Eq. 11 thermodynamics, " Confusion of the nature of the equilibrium encountered in classical textbooks of chemical thermodynam- of a large class ofbodies with the effect ofprocesses undergone ics are given elsewhere (3). No textbook that I have consulted by members of a small class of bodies is nearly universal." writes explicitly Eq. 10, which could have prompted somebody I thank Drs. H. G. Drickamer, D. M. Jameson, G. U. Nienhaus, and to ask what happens when AH = 0 and ASi > 0. Instead they G. D. Reinhart for their comments and suggestions. This work was omit all differences in notation necessary to distinguish Si from supported by U.S. Public Health Service Grant GM11223. S of Eq. 3, and then from Eqs. 4 and 5' derive Eq. 8' and with this in hand declare the universal worth of the van t'Hoff 1. Planck, M. (1922) Treatise on Thermodynamics (Dover, New relation of Eq. 11. The most important consequence of the York), pp. 103-119. dismissal of the relations in Eqs. 8 and 8' is that in a chemical 2. Serrin, J. (1977) Arch. Rational Mech. Anal. 70, 355-371. reaction, there are no fixed relations between the standard 3. Weber, G. (1995) J. Phys. Chem. 99, 13051. 4. Berthelot, M. (1897) Thermochimie (Gauthier-Villars, Paris), changes in enthalpy and entropy deducible from the laws of Vol. 1, p. 13. thermodynamics. Consequently, calculation of the effects of 5. Lauffer, M. (1975) Entropy Driven Reactions in Biology (Springer, pressure and temperature on the chemical equilibria require New York). suitable thermodynamic models that describe the relations of 6. Silva, J. L. & Weber, G. (1993) Annu. Rev. Phys. Chem. 44, H and S expected for each reagent. While success of these ad 89-113. hoc models will not confer on the results the certainty wrongly 7. Weber, G. (1993) J. Phys. Chem. 97, 7105-7115. 8. Weber, G. (1993) J. Phys. Chem. 99, 1052-1059. attributed in the past to the standard changes in enthalpy and 9. Gibbs, J. W. (1993) Scientific Papers I: Thermodynamics (Ox Bow, entropy determined by the van't Hoff equation, they provide Woodbridge, CT), pp. 56-62. us with an opportunity to establish meaningful relations between 10. Truesdell C. (1984) Rational Thermodynamics (Springer, New macroscopic thermodynamics and microscopic chemistry. York), 2nd Ed., p. 31. Downloaded by guest on September 24, 2021