Problem 17.1 (R.H. Swendsen): Properties of the van der Walls gas

1. Just for practice, starting from the expression for the Helmholtz free energy, derive the pressure, $P = P(T, V,N)$.
2. Again starting from the expression for the Helmholtz free energy for the van der Waals fluid, derive the energy as a function of temperature, volume, and number of particles, $U = U(T, V,N)$.
3. Using the conditions for the critical point,

$$\left(\frac{\partial P}{\partial V}\right)_{T,N}=0$$ and $$\left(\frac{\partial^2 P}{\partial V^2}\right)_{T,N}=0$$ derive expressions for $P_c$, $T_c$, and $V_c$ as functions of $a$ and $b$.

1. Derive the value of the ratio $$ \frac{P_c V_c}{Nk_B T_c} $$ as a pure number, independent of the values of $a$ and $b$.