466
10 The Statistical Basis of Bioenergetics
and the Helmholtz free energy A = E −T S in terms of the grand canonical partition
function is
A = γ k B T N − k B T ln .
(10.115)
We now wish, as we did for β, to connect the Lagrange multiplier γ to the
measurable properties of the system at equilibrium. First, we use the fact that a
Lagrange multiplier is determined by the corresponding constraint. In this case, the
constraint is in Eq. (10.107), which, near equilibrium, reads
N =
k N k exp (γ N k − βE k )
k exp (γ N k − βE k )
=
∂ ln
∂γ
.
(10.116)
As with β constraint, an explicit solution of this constraint relation is possible in
terms of standard functions only for special cases, such as an ideal gas. However,
we know that a functional relationship,
γ = γ (β, V , N) ,
(10.117)
exists, i.e. the constraint makes γ a function of the system temperature, its volume,
and its average number of particles. The quantity λ(β, V , N) ≡ exp (γ ) is called
the ‘absolute activity’ a. 38
When all three constraints are applied, the grand canonical partition function
becomes a function of T , V , and N : = (β, V , γ (β, V , N )). Note that the only
dependence of the grand canonical partition function on the average particle number
is through γ , and the only dependence on the ensemble member volume is through
the energies.
Now consider the chemical potential for the system, and apply Eqs. (9.95), (10.115),
and (10.116):
μ =
∂A
∂N
T ,V
= k B T
∂
∂N
Nγ − ln
T ,V
(10.118)
= k B T γ + k B T N
∂ γ
∂N
− k B T
∂ ln
∂γ
∂γ
∂N
(10.119)
= k B T γ .
(10.120)
Thus, γ determines the chemical potential for the system:
38 We introduced the idea of activity in connection to chemical reactions in Sect. 9.21.2.