Question

how can I determine state equation if I have Gibbs potential. I know how to determine other potentials.
Problem says:
Gibbs free energy is G(p,T)=T^2ln(a*p/T), p is a preassure, T tempreature, an a constant.
Determine equation of state f(p,V,T). V is a volume.
I have no idea where to start.
Please help

Answer 1

First you have to recall, that

$$V = \left(\frac {\partial G}{\partial p}\right) _ {T}$$

So, by differentiating $G(p,T)$ with respect to $p$ we will find equation of state:

$$V = \left(\frac {\partial T ^ {2} \ln (a \cdot p / T)}{\partial p}\right) _ {T} = T ^ {2} \frac {1}{a \cdot p / T} \cdot a / T = \frac {T ^ {2}}{p} = V$$

Hence, equation of state is

$$p = \frac {T ^ {2}}{V}$$