Page 197 - Chemical equilibria Volume 4

For the internal energy:
∂ ln Z ∂ ln Z Appendix 2 173
B ∑
() U
UT − (0) = − ∑ C (A) = k T 2 C (A) [A2.38]
A ∂ β A ∂ T
For the entropy:

⎡ 1 ∂ ln Z ⎤
S = k B ⎢ ln Z − C ⎥ [A2.39]
C
⎣ T ∂ ln β ⎦
For the Helmholtz energy function:

∑ ln Z C (A)
B ∑
FT − (0) = − A = − k T ln Z [A2.40]
() F
β A C (A)
For the pressure:

Using equation [A2.40], we calculate:

⎛ ∂ F ⎞ ⎛ ∂ ln Z C (A) ⎞
B ∑
P =− ⎜ ⎟ = k T ⎜ ⎟ [A2.41]
⎝ ∂ V ⎠ , TN A ,.. A ⎝ ∂ V A ⎠ , TN A ,..
For the Gibbs energy:

Equations [A2.40] and [A2.41] enable us to write:
∑ ln Z C (A) ∂ ln Z
−
GG (0) = F − F (0) PV− = − A + V ∑ ⎛ ⎜ C (A) ⎞ ⎟ [A2.42]
β β A ⎝ ∂ V A ⎠ , TN

NOTE ON THE MOLAR PROPERTIES.– in the expressions of the molecular and
canonical partition functions, we have based our discussions on a certain
number of molecules – a number of particles symbolized by N A for the
component A.

In order to obtain the molar values of the thermodynamic functions, for
N A it is prudent to choose Avogadro’s number (N a).