Solution to Set 2

Part 1

First one is to find the isothermal compressibility of a Van der Waals gas for $T>T_{c}\;$.

The Van der Waals equation of state is: $\left(P+{N^{2}a \over V^{2}}\right)\left(V-Nb\right)=Nk_{B}T$

Solving this for P gives: $P=\left({Nk_{B}T \over V-Nb}\right)-{N^{2}a \over V^{2}}$

Then taking the partial derivative with respect to V at constant T: $\left({\partial P \over \partial V}\right)_{T}=-{Nk_{B}T \over \left({V-Nb}\right)^{2}}+{2N^{2}a \over V^{3}}$

Bringing the terms over a common denominator looks like: $\left({\partial P \over \partial V}\right)_{T}={2N^{2}a\left({V-Nb}\right)^{2}-Nk_{B}TV^{3} \over V^{3}\left({V-Nb}\right)^{2}}$

Then finding the negative reciprocal of this function gives the isothermal compressibility: $\kappa _{T}=-\left({\partial V \over \partial P}\right)_{T}=-{1 \over {\left({\partial P \over \partial V}\right)_{T}}}={V^{3}\left({V-Nb}\right)^{2} \over Nk_{B}TV^{3}-2N^{2}a\left({V-Nb}\right)^{2}}$

For those of you wondering why the 1/V is missing in the isothermal compressibility equation (it was added to the homework around 5 PM the day the homework was due and is there now), the answer is because it depends on the result desired. The formula used here to solve for the isothermal compressibility gives the total volume change per change in pressure. However, should that 1/V be kept it would give the fractional change in volume per change in pressure. For completeness, the result for the fractional isothermal compressibility is:

$\kappa _{T}=-{1 \over V}\left({\partial V \over \partial P}\right)_{T}=-{1 \over V}{1 \over {\left({\partial P \over \partial V}\right)_{T}}}={V^{2}\left({V-Nb}\right)^{2} \over Nk_{B}TV^{3}-2N^{2}a\left({V-Nb}\right)^{2}}$

Suggestion (Vlad): plot $\kappa _{T}(V)\;$ for sevral tempratues, approaching $T_{c}\;$. This will make it clear how it diverges as the critical point is approached. Then put this plot in the solution here.

Part 2

I thought we would show how to arrive at the solutions for Vc Tc and Pc.

Part 3

First remember the identities for the critical volume and temperature:

$V_{c}$