$C_v$, $C_p$ and $C_m$ at constant temperature

Question

My teacher told that at constant temperature $C_m$ tends to infinity but $C_v$ and $C_p$ do not.

The context was calculating ΔH and ΔU in an isothermal reversible process for ideal gases. The explanation he gave was:

$$\Delta U=nC_v\Delta T=0$$

(since $\Delta T=0$ for isothermal process) and:

$$\Delta H=nC_p\Delta T=0$$

(since $\Delta T=0$ for isothermal process). Therefore:

$$\Delta U=q+w=0$$

or:

$$q=-w=nC_m\Delta T$$

Here too, $\Delta T=0$ but $q=-w\ne 0$ (since $C_m\to\infty$). Therefore we calculate $w$ by:

$$w=\int P_{ext}dV$$

He gave the reason of molar heat capacity to be ∞ as the heat required to raise the temperature of one mole of the substance by 1K at a constant temperature. (Well this sounds funny.) Now I wish to know the explanation of this different behavior of different heat capacities at a constant temperature. (All symbols have their standard meaning. Cm means molar heat capacity)

Answer 1

As pointed out in the comments, $C_m$ is not a thermodynamic property like $C_P$ and $C_V$. The mathematical quantity $$ \frac{dq}{dT}, $$ which you call $C_m$, is ill defined and can have any value between $-\infty$ and $\infty$ for the same substance at the same pressure and temperature. For an isothermal process it is equal to $C_m=+\infty$ if you are adding heat, or to $-\infty$ if you are removing it. If the process is not isothermal, then you can get any value in between.

No standard textbook of thermodynamics talks about this derivative. Your teacher is confusing the subject by introducing it.

Answer 2

It took me some time to understand what is $C_m$. The reason is that usually, one introduces the heat capacity for processes where the experimental conditions are fixed pressure, fixed volume, or states on the coexistence line.

Sometimes, working at a fixed pressure, at a first-order phase transition, it may happen that the temperature remains constant while the system is exchanging heat with the environment. However, apart from this special case, evaluating heat capacity at constant temperature really sounds funny. But from the formal point of view is not wrong. It is just artificial.

Now, you are wondering about the meaning of the different behavior of different head capacities. The obvious reason is that they refer to the heat exchanged in different processes, i.e., under different external conditions. Therefore there is no reason they should provide the same result. Indeed, the change of temperature resulting from a fixed amount of exchanged heat depends on the process because heat is a process and not a state-dependent quantity.