Page 35 - Chemical equilibria Volume 4

By coupling this relation with expression [1.23], we obtain:
⎡ ⎛ A ⎤ ⎞ Physico-Chemical Transformations and Equilibria 11
⎢ ∂ ⎜ ⎟ ⎥ ∑ν i H i i Δ H
⎢ ⎝ T ⎠ ⎥ = i = r [1.41]
⎢ ∂T ⎥ T 2 T 2
⎢ ⎥
⎣ , P ⎦ ξ

Δ H is the enthalpy associated with the transformation studied.
r

1.3.6. Derivatives of the affinity in relation to the extent and the
chemical potentials

By deriving equation [1.23], we obtain:

∂A =− ν ∂μ i [1.42]
∑
∂ξ i i ∂ξ

However, by taking account of relation [1.3], we can write:

∂ μ i = ∑ N ∂ μ k n ∂ k = ∑ ν ∂ μ i [1.43]
ξ ∂ k= 1 n ∂ k ξ ∂ k k n ∂ k

Thus:

∂A =− ∑∑ νν ∂μ i [1.44]
∂ξ k i ki n ∂ k

which can be expressed in the form:

2
∂A =− ∑∑ νν ∂μ i + ∑ ν 2 n ∂μ i [1.45]
i
2
∂ξ k k i ≠ ki n ∂ k i i n ∂ n i
i
However, according to the Gibbs-Duhem relation, we have:
∂ μ ∂ μ
n i =− ∑ n k [1.46a]
i
n ∂ i ki ≠ k n ∂ i