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Can you derive a general form (not keeping either volume or pressure constant) for heat capacity from the first law of thermodynamics? Do you have to make any assumptions to do so?

It sounds trival, but I can't seem to work something out:

$$dU = \delta Q - PdV$$

$$dU = \left( \frac{\partial U}{\partial T} \right) dT + \left( \frac{\partial U}{\partial V} \right) dV + \left( \frac{\partial U}{\partial P} \right) dP$$

$$dV = \left( \frac{\partial V}{\partial T} \right)_P dT + \left( \frac{\partial V}{\partial P} \right)_T dP$$

$$\delta Q = \left( \left( \frac{\partial U}{\partial T} \right) + P \left( \frac{\partial V}{\partial T} \right) \right) dT + \left( \left( \frac{\partial U}{\partial P} \right) + P \left( \frac{\partial V}{\partial P} \right) \right) dP + \left( \frac{\partial U}{\partial T}\right)dV$$

But from there, don't you end up having to hold P and V constant to get $\frac{\partial Q}{\partial T}$ (the heat capacity)?

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There is not a closed form that generally describes the heat capacity of a substance independent of constant pressure or volume. This is mainly because there is not a general way to describe the interrelation between heat and work for an arbitrary physical process. As such, certain assumptions must be made. In a derivation from the first law of thermodynamics, it is necessary to assume that the process occurs through either constant volume or constant pressure. A solid derivation of this is available on the wikipedia page for heat capacity.

It is also possible to approach heat capacities in other contexts based on other assumptions about how the physical processes involved might work (no pun intended). In modeling stars, for example, it is possible to use the equation of state (describing the star based on current values) to analytically approach a value for heat capacities. The details of this particular example are worked out in Stellar Structure and Evolution by R. Kippenhahn and A. Weigart. This approach is certainly prevalent in other areas of study and research. Once one of the heat capacities has been obtained (either at constant volume or constant pressure), one can then obtain relations between these two heat capacity values in a given substance (see again the wikipedia page).

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There are closed formulae for the heat capacities of a Gibbsian substance which is desribed by two equations of the form $T=f(p,V), S=g(p,V) $. They are $C_V=\frac{fg_p}{f_p}$ and $C_p=\frac{fg_V}{f_V}$ where I am using subscripts to indicate partial derivatives. The details can be found in the arXiv paper 1102.1540 (of which I am a co-author) which also displays a simple Mathematica programme which allows one to compute all thermodynamic quantities of a Gibbsian substace (e.g., the van der Waals or the Feynman gas) from the inputs $f$ and $g$.

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