J. Phys. Chem. 1995,99, 13048-13049
13048
COMMENTS Comment on “van’t Hoff Revisited: Enthalpy of Association of Protein Subunits” Alfred Holtzert Department of Chemistry, Washington University, St. Louis, Missouri 63130-4899 Received: March 21, I995 A recent paper (W)’ presents a treatment of protein association equilibria. Its initial pages summarize the thermodynamic underpinnings of the approach. Regrettably, so many of the statements in W are false that the entire approach must be doubted. The first two sentences of the abstract of W read, “The amounts of heat absorbed from and released to the environment in a chemical reaction are not experimentally separable. In consequence, entropy and enthalpy of reaction can only be computed with the help of specific hypotheses that relate them.” This beginning is inauspicious, since both sentences are wrong, taken by themselves; and, in spite of the initial phrase of the second sentence, they are independent. Since, by definition, the heat absorbed by any system from the environment is the negative of the heat released by it to the environment, there is no problem whatever in “separating them”. The first sentence therefore makes no sense. The second deals with a different issue, namely, the possibility of measuring AS and AH of a reaction independently, but its content is palpably false. Calorimetry supplies us with numerous values of enthalpy changes for reactions (at T, P) directly from experiment. Measurement of the corresponding Gibbs energy changes (via galvanic cell or equilibrium constant determinations) then allows immediate calculation of the entropies from AG = AH - TAS, requiring no hypothesis other than thermodynamic laws. Moreover, in many cases, the entropy changes have been directly determined from third law entropies for the participant compounds. Granted, some of these methods (galvanic cells, third law) are rarely feasible for proteins, but extensive calorimetry and equilibrium constant data do indeed exist for them. The formal treatment of the thermodynamic underpinnings begins in W with its first equation (eq Wl), which alleges that the change in Gibbs energy for an isobaric process is dG = -d(TS)
+ P dV
(W1)
This equation is false, and W’s accompanying text, which accepts it, is thereby vitiated. The Gibbs energy of any system is defined as G= U
-+ PV - TS
(1)
wherein U is the intemal energy of the system and other symbols have their usual meaning. Differentiation gives dG = d U
+ P d V + V d P - d(TS)
(2)
All we need to do is substitute the first law expression, dQ P dV, for dU into (2) to see that the P dV terms cancel. Moreover, for constant pressure (which is assumed in W), the V dP term also vanishes, giving
’ Telephone: 3 14-935-6572. FAX: 3 14-935-4481. E-mail: holtzer@ wuchem.wustl.edu 0022-3654/95/2099-13048$09.00/0
dG = dQ - d(TS)
(3)
a result markedly differing from eq W1. We will leave aside here the question of the dubious wisdom of leaving the quantity TS intact and of calling it the “total heat content” of a system (as W does), given that heat is not a state function. Further difficulties appear concerning eq W5, which says that AG = AH - TAS, a well-known result for an isothermal change. However, W’s accompanying text adds that these will all be standard changes if “the composition of the system is maintained constant, at the stable values characteristic of the chemical equilibrium”. The emphasis shown is W’s. However, this text statement is quite false. If the concentrations are equilibrium values, then AG is necessarily 0, that being the thermodynamic condition for equilibrium for a system constrained at constant T, P. To obtain standard changes, the concentrations (actually activities) must each be the standard value, usually 1 M. Equation W15 and W’s attendant discussion also contain serious flaws. The equation reads d(AG/T)/d(l/T) = AH
(WW
which is fine, since P is constant, so we may eschew cumbersome partial derivative notation, thereby also facilitating direct comparison with W. However, W avers that eq W15 is true “& and only 8 AH and AS are independent of temperature.” Again, the emphasis is W’s, and the ensuing text, which makes much of this point, is false. Equation W15 is true whether or not both AH and AS are temperature dependent, as we next show. Calculus provides for the temperature derivative of AG/T d(AG/T)/dT = [(dAG/dT)/T] - [AG/?]
(4)
(recall that P is constant), from which we easily obtain the derivative with respect to UT by the chain rule of calculus: d(AG/T)/d(l/T) = [d(AG/T)/dU[dT/d(l/T)] =
-[d(AG/T)/dq[?] Inserting eq 4 into the rhs of the latter gives d(AG/T)/d( l/T) = AG - T(dAG/dT) = AG
+ TAS = AH ( 5 )
without any assumption as to whether AH and AS depend on T or not. Of course, if AH is temperature dependent, a plot of AGIT vs 1IT will be nonlinear, but its slope at any T is nonetheless AH(T). The source of the difficulty appears subsequently in W, wherein a quantity T(T) is defined without realizing that it is identically 0. It is defined by eq W16’:
r(T)= (l/T)d(AH)/d(l/T) - dAS/d(l/T)
(W16’)
Since d(l/T) = -dT/p, this becomes
T(T) = nT(dAS/dT) - (dAH/dT)]
(6)
However, the last square bracket is easily shown to be 0. Taking the derivative of AG = AH - TAS with respect to T, wherein all depend on T, 0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 34, 1995 13049
Comments dAGIdT = dAHIdT - T(dASId?") - AS
(7)
However, P being constant, we also have the fundamental relation dAGldT = -AS, which when put into (7) shows that dAHldT = T dASldT
(8)
Insertion of eq 8 into eq 6 proves that I' is always 0. The text of W makes much of r, its alleged temperature dependence, and its magnitude compared with AH. It follows from eqs 6 and 8 above that this entire discussion in W is meaningless. A great deal of W suffers from this failure to grasp the fundamental truth embodied in eq 8, namely, that at
given pressure the temperature dependences of AH and AS are linked by thermodynamic laws. In view of these shortcomings in the thermodynamic underpinnings of W, it seems unlikely that the ideas in it can have much validity.
Acknowledgment. This work was supported by Grant GM20064 from the division of General Medical Sciences, U.S. Public Health Service, and a grant from the Muscular Dystrophy Association. References and Notes (1) Weber, G . J . Phys. Chem. 1995, 99, 1052-1059.
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