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

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)

• What is the standard meaning of $C_m$? Commented Jul 28, 2022 at 3:42
• If $C_m$ is meant to denote the molar heat capacity, it's certainly not infinity for an ideal gas. Commented Jul 28, 2022 at 4:07
• Are you defining $C_m = dq/dT$? If so it can be infinite but this has no special significance. For example when we melt ice we add heat but the temperature stays fixed at 0°C until all the ice has melted. So we also get $dq/dT = \infty$ in this case. Commented Jul 28, 2022 at 5:29
• Cm is a bogus quantity. Heat capacity is supposed to be a physical property of a material, independent of process, and Cm (defined in terms of heat, which depends on path) isn't. Commented Jul 28, 2022 at 10:29
• I've done a lot with thermodynamics, and I've never seen a term called $C_m$. What you are saying "molar heat capacity" is what $C_v$ and $C_p$ are – "Joules per mol per Kelvin." I've never seen a heat capacity other than these two. Commented Jul 31, 2022 at 13:55

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.

• The OP did not write $C_m$ as a derivative. Commented Aug 10, 2022 at 16:32
• The OP introduced $C_m$ via $q = n C_m \Delta$ from which we obtain $C_m = q/(n \Delta T)$. I take this to define $C_m$ as the derivative of heat with respect to $T$. Commented Aug 10, 2022 at 20:03
• Although the formula with a $\Delta T$ could suggest a derivative, this is not the case because $q$ in general is not the difference of values of any function. What can be done is to consider a quasistatic transformation where $q=T\Delta S$. Only in such a case, we can recover a derivative but that is the derivative of the entropy, i.e., of a function of the state. Commented Aug 10, 2022 at 20:14
• @GiorgioP $q$ is a proper function of $P$ and $T$ if the path is specified. The OP has specified the path to be isothermal. The statement "But that's the derivative of entropy" is misleading: it is the derivative of the quantity $T \Delta S\equiv q$ under constant $T$. Commented Aug 10, 2022 at 20:36
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.