(a) Explain what is meant by a first-order phase transition and a second-order phase transition.

(b) Explain why the (Helmholtz) free energy is the appropriate thermodynamic potential to consider at fixed $T, V$ and $N$.

(c) Consider a ferromagnet with free energy

$$F(T, m)=F_{0}(T)+\frac{a}{2}\left(T-T_{c}\right) m^{2}+\frac{b}{4} m^{4}$$

where $T$ is the temperature, $m$ is the magnetization, and $a, b, T_{c}>0$ are constants.

Find the equilibrium value of $m$ at high and low temperatures. Hence, evaluate the equilibrium thermodynamic free energy as a function of $T$ and compute the entropy and heat capacity. Determine the jump in the heat capacity and identify the order of the phase transition.

(d) Now consider a ferromagnet with free energy

$$F(T, m)=F_{0}(T)+\frac{a}{2}\left(T-T_{c}\right) m^{2}+\frac{b}{4} m^{4}+\frac{c}{6} m^{6}$$

where $a, b, c, T_{c}$ are constants with $a, c, T_{c}>0$, but $b \leqslant 0$.

Find the equilibrium value of $m$ at high and low temperatures. What is the order of the phase transition?

For $b=0$ determine the behaviour of the heat capacity at high and low temperatures.