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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)

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    $\begingroup$ What is the standard meaning of $C_m$? $\endgroup$ Jul 28, 2022 at 3:42
  • $\begingroup$ If $C_m$ is meant to denote the molar heat capacity, it's certainly not infinity for an ideal gas. $\endgroup$ Jul 28, 2022 at 4:07
  • $\begingroup$ 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. $\endgroup$ Jul 28, 2022 at 5:29
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    $\begingroup$ 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. $\endgroup$ Jul 28, 2022 at 10:29
  • $\begingroup$ 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. $\endgroup$
    – RC_23
    Jul 31, 2022 at 13:55

2 Answers 2

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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.

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  • $\begingroup$ The OP did not write $C_m$ as a derivative. $\endgroup$ Aug 10, 2022 at 16:32
  • $\begingroup$ 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 $T$. $\endgroup$
    – Themis
    Aug 10, 2022 at 20:03
  • $\begingroup$ 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. $\endgroup$ Aug 10, 2022 at 20:14
  • $\begingroup$ @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 $T$. $\endgroup$
    – Themis
    Aug 10, 2022 at 20:36
  • $\begingroup$ Whatever quantity is used, derivative with respect to temperature at constant temperature is pure mathematical nonsense. Instead, a relation where a difference in temperature appears may have a well definite meaning. $\endgroup$ Aug 10, 2022 at 21:56
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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.

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