Question

Derive the thermodynamic identity for $G$ (equation 5.23), and from it the three partial derivative relations 5.24.

Short answer

The expression for change in G is $(\mu dN-SdT+VdP)$ and the relations are

$S=-{\left(\frac{\partial G}{\partial T}\right)}_{P,N},V={\left(\frac{\partial G}{\partial P}\right)}_{T,N}\text{and}\mu ={\left(\frac{\partial G}{\partial N}\right)}_{T,P}.$

Step-by-step solution

Explanation

Write the expression for Gibbs free energy.

$G=U-TS+PV$

Here, G is Gibbs free energy, T is the absolute temperature, S is the entropy, P is the pressure and V is the volume.

Write the expression for the infinitesimal change in G.

$dG=dU-TdS-SdT+PdV+VdP\dots \dots ..\text{(1)}$

Write the expression for the infinitesimal change in U.

$dU=TdS-PdV+\mu dN$

Substitute $(TdS-PdV+\mu dN)$for dU in expression (1).

$dG=\mu dN-SdT+VdP\dots \dots ..\left(2\right)$

Rearrange expression (2) for constant P and constant N.

$dG=-SdT$

Rearrange the above expression.

$S=-{\left(\frac{\partial G}{\partial T}\right)}_{P,N}$

Calculation

Rearrange expression (2) for constant T and constant N.

$dG=VdP$

Rearrange the above expression.

$V={\left(\frac{\partial G}{\partial P}\right)}_{T,N}$

Rearrange expression (2) for constant T and constant P.

$dG=\mu dN$

Rearrange the above expression.

$\mu ={\left(\frac{\partial G}{\partial N}\right)}_{T,P}$

Thus, the expression for change in G is $(\mu dN-SdT+VdP)$ and the relations are

$S=-{\left(\frac{\partial G}{\partial T}\right)}_{P,N},V={\left(\frac{\partial G}{\partial P}\right)}_{T,N}\text{and}\mu ={\left(\frac{\partial G}{\partial N}\right)}_{T,P}.$