Page 34 - Chemical equilibria Volume 4

10 Chemical Equilibria
In the case where that set of variables comprises only intensive variables
Y l, the differential of the affinity is written:
d A = ∑ ∂A dY + ∂A dξ [1.34]
l ∂ Y l l ∂ξ
However, by applying relation [1.33] and the symmetry of the
characteristic matrix, if X l is the conjugate extensive value of Y l , we obtain:

∂ A = ∑ i ν X = Δ X [1.35]
∂ l Y l i i r l
i
This gives us the differential of the affinity in intensive variables (partial
molar values) and extent:
∂A
d A = Δ X dY + dξ [1.36]
l
r
l
ξ ∂

Let us apply these results to chemical systems with variables –P, T, so
then equation [1.35] gives us:
∂A = ∑ S =ν Δ S [1.37]
∂ T i i i r
i

In addition, we have:

∂A = − ∑ Vν = − Δ V [1.38]
∂ P i i i i r

The differential of the affinity then becomes:

2
∂ G
d A = Δ S dT − Δ V d P − d ξ [1.39]
r
r
ξ ∂ 2

NOTE 1.2.– Helmholtz’s second relation gives the derivative of the ratio μ ι /T
with the temperature:
⎡ μ ⎤ ⎛ i ⎞
∂ ⎜ ⎢ ⎟ ⎥
⎢ ⎝ T ⎠ ⎥ = H i i [1.40]
⎢ ∂ T ⎥ T 2
⎢ ⎥
⎣ ⎦ , P ξ