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My thermodynamics textbook (Blundell & Blundell, second edn.) states in equation (16.65) that $$\frac{C_V}{T}=\left(\frac{\partial S}{\partial T}\right)_V \quad and \quad \frac{C_p}{T}=\left(\frac{\partial S}{\partial T}\right)_p. \tag{1}\label{1}$$ but does not specify the condition under which they can be used.

I try to derive these equations as follows. If one starts with the definition of heat capacity at constant volume $$C_V = \left(\frac{\partial Q}{\partial T}\right)_V\ ,$$ or $$dQ=C_V dT \quad \mathrm{at\ constant\ volume}. \tag{2}\label{2}$$ The definition of entropy reads $$dQ_{rev} = T\ dS, \tag{3}\label{3}$$ where $dQ_{rev}$ is the reversible change of heat.

Combining \eqref{2} and \eqref{3} one obtains the first of \eqref{1}, provided only that $dQ=dQ_{rev}$. Similar argument holds for the second one in \eqref{1}. Does this mean \eqref{1} is only true for reversible processes, or am I missing something?

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    $\begingroup$ What does heat capacity at constant temperature mean? $\endgroup$
    – Chet Miller
    Commented Jul 6, 2022 at 10:43
  • $\begingroup$ @ChetMiller Thanks—it was a typo. $\endgroup$
    – user339978
    Commented Jul 6, 2022 at 10:53
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    $\begingroup$ Writing $\partial Q/\partial T$ assumes that $Q$ is a state parameter that depends on $T$. Such a parameter doesn't exist by definition of $Q$, so your definition of $C_V$ can't be correct. $\endgroup$
    – Miyase
    Commented Jul 6, 2022 at 10:54

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The defining equation you used for heat capacity is not the definition we use in thermodynamics. In thermodynamics, we use $$C_V=\left(\frac{\partial U}{\partial T}\right)_V$$where U is the internal energy. For a pure, single phase substance, it matches your definition only if V is constant. And, there is no particular reversible or irreversible process path involved. This just corresponds to the difference in internal energy between to closely neighboring thermodynamic equilibrium states of the material.

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  • $\begingroup$ Is this definition a universal convention? In our course, which is based on the textbook I mentioned in the post, the heat capacity is defined as the derivative of heat even in thermodynamics, if you refer to the passage around equation (16.7) of the said book, which reads ' $\Delta U = \int C_V dT$ is only true for systems held at constant volume. $\endgroup$
    – user339978
    Commented Jul 6, 2022 at 11:09
  • $\begingroup$ Also, are you suggesting at the end of your answer that reversibility does not matter for infinitesimal changes between states? I don't have an intuitive feel for this because, after all, a finite change consists of a series of infinitesimal changes. Can you explain this in more detail? $\endgroup$
    – user339978
    Commented Jul 6, 2022 at 11:10
  • $\begingroup$ You can bring about a change in U between two closely neighboring thermodynamic equilibrium states of a system by an infinite number of either reversible or irreversible paths, all of which have different Q and W between the initial and final states. Even the reversible paths will have different Q's, but all of them will have the same $dU=C_vdT$. This is the reason we define Cv in terms of U rather than Q, since Cv is supposed to be unitque to the material and not the path. $\endgroup$
    – Chet Miller
    Commented Jul 6, 2022 at 12:43

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