# Heat capacity across a phase transition-inconsistency?

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? • You should specify whether we are talking about the direction of increasing temperature or decreasing temperature. Otherwise, the trivial answer is that both are possible. – valerio Jan 20 '18 at 9:49
• Anyway, assuming we are talking about increasing $T$: since it all depends on the slope of $H$ before and after the transition, in principle both are possible and you need an explicit expression of $H$ to calculate the change in $c_P$. However, in my experience I have never seen $c_P$ increase right after a 1st order transition. Where have you seen this? – valerio Jan 20 '18 at 10:14

## 2 Answers

A solid that obeys Dulong and Petit's law has a molar heat capacity $$c_p = 3\ R$$. An ideal monatomic gas has a heat capacity half of that, $$c_p = 1.5\ R$$. An ideal diatomic gas has $$c_p = 2.5\ R$$ per mole molecules (so $$1.25\ R$$ per mole of atoms).

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).

• I think the OP is referring to the heat capacity behavior before and after the transition (not the jump in enthalpy at the transition). It isn't clear whether he is asking about the absolute values on either side of the transition (e.g. L vs V) or whether he is referring to the temperature dependence of the heat capacity on either side of the transition. Maybe you could add a little about these features of the behavior. – Chet Miller Jan 11 '16 at 17:28
• @ChesterMiller: "[...] heat capacity does across a first-order phase transition". I interpret that as in my answer. Let the OP decide: if he doesn't like the answer I will retract it. Thanks. – Gert Jan 11 '16 at 17:43
• Hi, sorry for the confusion, just to confirm that I was talking about the absolute value of the heat capacity either side of the transition (I have seen it with the phase at lower temperatures having both greater and smaller heat capacity then that of the phase at higher temperatures) – Quantum spaghettification Jan 12 '16 at 16:55