April, 1952
CALCULATION OF ENTROPY AND ENTHALPY CHANGES FROM FREE-ENERGY DATA
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ON T H E CALCULATION OF ENTROPY AND ENTHALPY CHANGES FROM FREE-ENERGY DATA BY REINOW. HAKALA Department of ChemistTy, Syracuse University, Syracuse, New York Received March 81, 1961
Klotz and Fiessl employed the function ( b A F / b T ) p = -AS to determine the entropy of formation of copper(I1)-albumin complexes using measurements made a t two temperatures 25' apart. With AS thus found they then calculated the enthalpy of formation. It is here pointed out that this procedure leads to exact results only if AH is sensibly constant in the temperature interval under consideration. To ascertain this, measurements a t more than two temperatures are needed. However, the data of Klota and Fiess, though insufficient for the exact calculation of AS and AH, provide ample support for their major conclusion that electrostatic factors play a dominant role in the copper( 11)-albumin interaction. Formulas are derived which are convenient in the determination of AS and AH from AF data a t various temperatures.
I n their very interesting work on %opper(II)albumin complexes, Irving M. Klotz and Harold A. Fiessl employed the general thermodynamic function
(s)p = -AS
apparently taking a ratio of differences instead of the true derivative-or using the integrated form, assuming A S to be constant-to determine the entropy of formation of the complexes a t a given pH over a temperature range of 25O, and then calculated the enthalpy of formation from this value of A S and the general relationship AF = AH
- TAS
(2)
At first glance, it would appear that this procedure can lead only to approximate results, because the temperature range that was taken is far from infinitesimal or because A S may not be temperature independent. It therefore seems pertinent t o inquire, under what conditions, if any, does this procedure lead to exact rather than merely approximate results; i.e., when is A S actually constant over the temperature range taken, and when is the calculated value of A S only an average value? It will be shown in what follows that the procedure of Klotz and Fiess leads to exact results if, and only if, the enthalpy of formation is sensibly constant in the temperature range under consideration; i.e., that the expression
where AFT1 and AFTZ are the free energies of formation at the temperatures T1and Tz,respectively, is exactly true. This is an important point which, though readily derived, is rather likely to be overlooked and so is presented here. A convenient test for the constancy of AH and a convenient procedure for calculatifig AH and A S at various temperatures when AH is not sensibly constant, will also be given. Integration of the Gibbs-Helmholtz equation (4)
assuming AH to be temperature independent, gives AF
=
AH
T + constant
(5)
(1) I. M. Klotz and H,A, Fiess, T ~ r JOURNAL s 55, 108 (1951).
Comparison of this result with equation 2 shows that the constant of integration is the negative of the entropy change, whence the entropy change is also independent of temperature. Integration of equation 1 then leads directly to equation 3. Klotz and Fiess made measurements a t just two temperatures, whence there is no way of determining whether the enthalpy of formation of the complexes is actually constant over the temperature range taken. If the exact values of the entropies and enthalpiee of formation are desired, it is necessary to repeat the measurements at a minimum of one additional temperature. However, their major conclusion, that electrostatic factors play a dominant pa,rt in the copper(I1)-albumin interaction, receives adequate support from their present data. Should measurements be made a t other temperatures, it would be necessary to calculate AH, to determine whether it is constant over the temperature range taken, before calculating AS, for, should AH prove to be appreciably temperature dependent, A S might be. Values of A S a t the various temperatures would then be most conveniently calculated using equation 2. If it is known that AH is appreciably temperature dependent but it is not desired to calculate AH for every temperature a t which A S is to be calculated, average values of A S should be calculated using equation 3, employing the smallest possible temperature intervals, and the values of the true derivative (equation 1) at the various temperatures can then be found by the chord-area method. Integration of equation 4 between limits, assuming that AH is constant over the temperature interval of the integration, leads to the expresqion
which is more convenient t o use than the van't Hoff isochore in those cases where the free energy of formation is found more directly than the equilibrium constant. If A H is sensibly constant for the two extreme temperature intervals in the temperature range under consideration (preferably employing small temperature intervals), then AH is sensibly constant over the entire temperature range. Otherwise it is not, and the chord-area method, employing average values of AH calculated with equation 6, found with the smallest possible temperature intervals, or the van't Hoff isochore, must be
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GEORGEK. SCHWEITZER AND JOHN L.
used to determine AH at the various temperatures if these values are desired. (Equation 6 can obviously also be derived by integrating the van't Hoff isochore between limits
ROSE, JR.
Vol. 56
and solving the result simultaneously with AF' =
- RYIn K
(7)
The derivation from the Gibbe-Helmholtz equation, however, is the most fundamental.)
RACEMIZATION RATE STUDIES ON POTASSIUM TRIS-(OXALAT0)CHROMATE(II1) BY GEORGEI
