Page 39 - Chemical equilibria Volume 4

1.5.1. Heat of transformation at constant pressure and
temperature Physico-Chemical Transformations and Equilibria 15
We can express the elementary heat of a transformation in the form:
δ Q = T d S = T d S T d S [1.57]
−
e
i
By choosing all the variables (T, -P, ξ), we shall have:

S ∂ S ∂ S ∂
=
δ QT dT T+ d P T+ dξ − T d S [1.58]
∂ T ∂ P ξ ∂ i
At constant pressure and temperature, this gives us:

S ∂
δ Q = T dξ − T d S [1.59]
P
ξ ∂ i

However, as the generalized Gibbs energy Γ is a function of state, we
have, according to relation [1.19]:

⎛ S ⎞ ∂ ⎛ ∂ ⎞ A
⎜ ⎟ =− ⎜ ⎟ [1.60]
⎝ ∂ξ ⎠ , PT ⎝ ∂ T , P ⎠ ξ

Thus, we can write relation [1.59] in the form:

⎡ ⎛ ∂ ⎞ A ⎤
δ Q =− ⎢ P T ⎟ + A ⎥ ⎜ dξ [1.61]
⎝ ⎢ ∂ T , P ⎠ ξ ⎥ ⎣ ⎦

By integrating this expression for the whole of the transformation, we
obtain:

ξ =∞ ⎡ ⎛ ∂ ⎞ A ⎤
Q = ∫ δ Q = − ⎢ P T ⎟ + A ⎥ ⎜ [1.62]
P
ξ = 0 ⎝ ⎢ ∂ T , P ⎠ ξ ⎥ ⎣ ⎦

By replacing the affinity with its expression [1.22], we find:

Q = T ∑ ν k ∂ μ k + ∑ νμ k [1.63]
P
k
k ∂ T k