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I was challenged to answer this question. I could solve it indirectly, by proving that $C_v$ is always positive (and since $C_p > C_v$, it holds for all cases).

Using the 2nd and 3rd postulates of thermodynamics (from "Thermodynamics and an Introduction to Thermostatistics", by Herbert Callen), we know that entropy is a crescent function of the energy and it always reach a maximum, then we can write:

$\left(\frac{\partial^2S}{\partial U^2}\right)_{V,N} < 0$

$\frac{\partial}{\partial U} \left(\frac{\partial S}{\partial U}\right)_{V,N} = \frac{\partial}{\partial U} \left(\frac{1}{T}\right)_{V,N} = \frac{-1}{T^2} \left(\frac{\partial T}{\partial U}\right)_{V,N} = \frac{-1}{T^2 C_V} < 0 \therefore C_V > 0$

I would like to know whether there is another simple way to show the same result, not using Callen's postulate approach.

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  • $\begingroup$ When adding energy to spin-systems, this will drive them to negative temperatures (ok, pathological case). $\endgroup$ – Pieter Dec 6 '17 at 18:13
  • $\begingroup$ @Pieter Wouldn't this transition to a negative temperature still be consistent with a positive $C_V$, as the temperature is increasing past $\infty$ to $-\infty$ and beyond? (Or, less weirdly, $\beta\propto 1/T$ is constantly decreasing as energy is added, which is exactly what we expect from more conventional systems and a positive $C_V$?) $\endgroup$ – Chemomechanics Dec 6 '17 at 23:54
  • $\begingroup$ Indeed @Chemomechanics the thermodynamic beta is decreasing past zero. $\endgroup$ – Pieter Dec 7 '17 at 1:11
  • $\begingroup$ You've got a sign error somewhere, because your last line implies $C_V < 0$. $\endgroup$ – Chris Dec 8 '17 at 0:34
  • $\begingroup$ Indeed. I will edit it now. $\endgroup$ – David Sousa Dec 8 '17 at 2:13
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The statement is not true in general, so you'll have a hard time proving it.

You have proven that temperature always increases with added heat in isochoric and isobaric processes, but those are not the only valid thermodynamic processes. Heat can be added or removed in a isothermic process, for instance, without changing the temperature.

An example of a real system with a negative heat capacity is a gravitationally bound cloud of gas. As heat is added, the cloud expands, and it can be shown via the virial theorem that the average kinetic energy (and thus the temperature) decreases.

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