Page 35 - Thermodynamics of Biochemical Reactions

2.5 Legendre Transforms for the Definition of Additional Thermodynamic Potentials 29

Since G is additive in terms of pi (equation 2.5-12), S and H are also additive
in terms of partial molar entropies and partial molar enthalpies, respectively:

(2.5-20)

The partial molar entropy Smi was defined in equation 2.5-14. The additivity of
the enthalpy can be seen by substituting equation 2.5-12 in equation 2.5-18:

(2.5-2 1)

where Hmi is the molar enthalpy of species i. Thus the chemical potential of a
species is given by

pi = H,i - TS,, (2.5-22)
When a system at constant pressure is heated, it absorbs heat, and the heat
capacity at constant pressure C, is defined by
c, = (g)p (2.5-23)

Substituting equation 2.5-7 in H = G - TS yields
H = G - T (g) (2.5-24)

P
Substituting this into the definition of C, yields

Thus we have seen that all the thermodynamic properties of a one-phase
nonreaction system can be calculated from G( 7; P,(ni}).
Although Legendre transforms introducing chemical potentials of species as
natural variables are not discussed until Chapter 3, there is one Legendre
transform involving chemical potentials of species that needs to be given here, and
that is the complete Legendre transform U' of the internal energy defined by

NS
U' = u - TS + PV- c ,kini = 0 (2.5-26)
i=l
This transformed internal energy is equal to zero, as indicated by equation 2.2-8.
The total differential of U' is

A', N,
0 = dU - TdS + SdT+ PdV+ VdP - pldnl - 1 n,dpl (2.5-27)
1= 1 1=1
Substituting the fundamental equation for U yields the Gibbs-Duhem equation
2.4-2.
Now we are in a position to generalize on the number of different ther-
modynamic potentials there are for a system. The number of ways to subtract
products of conjugate variables, zero-at-a-time, one-at-a-time, and two-at-a-time,
is 2k, where k is the number of conjugate pairs involved. In probability theory the
number 2k of ways is referred to as the number of sets of k elements. For a
one-phase system involving PV work but no chemical reactions, the number of
natural variables is D and the number of different thermodynamic potentials is
2O. If D = 2 (as in dU = TdS - PdV), the number of different thermodynamic
potentials is 22 = 4, as we have seen with U, H, A, and G. If D = 3 (as in
dU = TdS - P dV + p dn), the number of different thermodynamic potentials is
23 = 8.