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9 Bioenergetics
H ≡ E + pV ;
the Helmholtz free energy
A ≡ E − T S ;
and the Gibbs free energy
G ≡ E + pV − T S = H − T S =
μ k N k .
From the relation
dH = T dS + V dp +
k
μ k dN k .
(9.108)
we have already observed that the ‘natural variables’ in which to express the
enthalpy are entropy, pressure, and particle numbers. It follows from the smoothness
of H (S, p, {N k }) that
∂T
∂p
S
=
∂V
∂S
p
.
(9.109)
The Helmholtz free energy, with the differential:
dA = −SdT − pdV +
k
μ k dN k ,
(9.110)
is useful for describing changes of a system at constant temperature and volume.
Being that the differential dA of the Helmholtz free energy is naturally expressed in
terms of dT , dV , and the dN k , the ‘natural variables; of A are temperature, volume,
and particle numbers. The smoothness of A(T , V , {N k }) gives
∂S
∂V
T
=
∂p
∂T
V
.
(9.111)
Finally, the Gibbs free energy reversible changes are given by
dG = −SdT + V dp +
k
μ k dN k .
(9.112)
showing that G = G(T , p, {N k }). The smoothness of G(T , p, {N k }) gives
∂S
∂p
T
= −
∂V
∂T
p
.
(9.113)