Maxwell Relation, Gibbs Free Energy, Thermal Expansion Coefficient

[Tian En]

Homework Statement

By means of a Maxwell relation derived from the Gibbs free energy and making use of the third law of thermodynamics, prove that the thermal expansion coefficient β must be zero at T = 0. I tried but I got something funny.

Homework Equations

$$G=U-TS+PV$$
$$dG=\mu dN-SdT+VdP$$
$$S=Nk_B[\ln(\frac{V}{N}(\frac{4\pi mU}{3Nh^2})^{3/2})+\frac{5}{2}]$$
$$PV=Nk_B T$$
$$\beta = \frac{1}{V}\frac{\partial V}{\partial T} \Bigg| _{N,P}$$

The Attempt at a Solution

 

[TSny]

I believe you have essentially shown that an ideal gas is not compatible with the third law.

I think you should be able to finish the proof using your result

without assuming the ideal gas law.