In the books I have seen two descriptions of what the heat capacity does across a first-order phase transition. The diagram below seems to indicate that it decreases whilst in other places I have seen it say that it increases. Which of these is right and why? Or are they both right in different circumstances and what circumstances?
The various solid/liquid/gas transitions are classified as first-order transitions because they involve a discontinuous change in density, which is the (inverse of the) first derivative of the free energy $G$ with respect to pressure.
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another one by heat transfer. The term is most commonly used to describe transitions between solid, liquid and gaseous states of matter, and, in rare cases, plasma.
More simply put, the two most types of phase transition are solid to liquid ($S \to L$, i.e. melting) and liquid to gas ($L \to G$, i.e. boiling) are accompanied by heat transfer called Latent Heat:
- $S \to L$ requires Latent Heat of fusion, $L_F$. The reverse transition (solidifying) releases $-L_F$.
- $L \to G$ requires Latent Heat of evaporation, $L_V$. The reverse transition (condensation) releases $-L_V$.
These signs explain what you observe with regards to the Heat Capacity: how it changes during the transition depends on the sense of the transition: 'forward' or 'reverse'. Conventionally melting and boiling requires positive heat inputs, their reverses give negative heat inputs (heat releases).