J. Phys. Chem. 1995, 99, 13051
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Reply to the Comment “van’t Hoff Revisited: Enthalpy of Association of Protein Subunits”
dG = -S d T V dp; (dGIdT), = -S (2) and assuming that the S in (2) is the same as the S in G = H TS, one concludes that
Gregorio Weber
dGIdT = (dH1dT) - T(dS1dT) - S; (dHldT) - T(dS1dT) = 0 (3)
Department of Biochemistry, University of Illinois at Urbana-Champaign, 415 Roger Adams Lab, 600 South Matthews Avenue, Urbana, Illinois 61801
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dHldT = T dS/dT
However, if following both Gibbs and Planck, and the second law of thermodynamics, we recognize that the total entropy S in ( 2 ) is the sum of S, and Si, the last relation in (2) should read (dG/dT)p = -(Sx Si), which is not compatible with dG/ dT = Si of the textbooks. Type 2 is favored by Glasstone,’ and it was favored in ref 5 before I realized the error involving the identification of the total entropy change dS of eq 2 with the entropy change exclusive to the system of reagents dSi. The demonstration of (1) offered by Dr. Holtzer is a very belabored example of type 2. Evidently ( 2 ) demands either experimental means of separating S, from Si or some additional assumption as to their relation, before we go any further. The need for an additional assumption is a fundamental premise in my two papers on the association of protein subunit^.^-^ In summary, the pretended demonstrations of the relation dH1 dT = T dS/dT depend on (a) setting the specific heat as equal to each member of the last equality rather than to their algebraic sum or (b) equating the change in free energy with temperature to the entropy of the system of reagents upon reaction, although by definition the free energy change equals the sum of the entropy in reagents and surroundings. Finally, I leave as an exercise for the reader the analysis of the application of relation 1 to an entropy-driven reaction (dG/ dT < 0) when AH = 0.
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Received: June 9, I995
What I consider to be relevant in the comments on my paper by Dr. Holtzer and by Drs. Ragone et al. is a difference of opinion about the relation dAHldT = T dAS/dT. They assert that this relation is valid under all circumstances, and I see it as a useful approximation that breaks down badly in entropydriven reactions when the changes in enthalpy and entropy with temperature cannot be disregarded. In my paper I followed P l a n ~ k in ~ .explicitly ~ recognizing that the entropy change in any chemical reaction includes changes in the entropy of the surroundings dS,and in the system of reagents dSi, although the former does not appear as such in the Gibbs equation, where T dS, is replaced by the enthalpy change dH. The mentioned entropy changes correspond respectively to separate and distinct heat exchanges qx and qi that follow respectively from the changes in internal energy and thermal energy of the reagents. The specific heat change is therefore Cp = (dqx/dT) (dqiIdT), while the change in entropy is dS = dS, dSi. Keeping in mind these two relations, we can analyze the general validity of the equation dAHIdT = T dASldT which faithfully appears in so many textbooks. For it to be valid for all reactions it is necessary that for all reagents
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13051
(1)
The supposed demonstrations of the last relation fall into two categories: Type 1 is found in Lewis and Randall.* On page 102 there appears that Cp = dHldT, from Kirchoff‘s law, and on page 132 Cp = T(dS/dT). The separate identification of the enthalpy and entropy changes with the same heat exchange is not compatible with the identification of Cp as equal to the sum of both changes. A variant of this type that neglects the contribution of T dSi to the heat exchange altogether (e.g. Zemansky*) gives good service when T dSi
