Mathematically calculate heat capacity in varying pressure and volume conditions

Question

I understand that the heat capacity depends on depends upon both microscopic (molecular/atomic) and macroscopic (phase, temperature, pressure). If I have an environment wherein both - the pressure and volume are subject to change, how can I estimate the instantaneous heat capacity of a substance (liquid) in that case ?

Chet Miller · Accepted Answer

The starting point for determining the volume dependence of $C_v$ is the equation: $$dU=C_vdT+left[Tleft(frac{partial P}{partial T}right)_V-Pright]dV$$ This equation appears in every thermodynamics textbook, so I'm not going to derive it. From this equation, it follows that $$frac{partial ^2 U}{partial Vpartial T}=left(frac{partial C_v}{partial V}right)_T=left[frac{partialleft[Tleft(frac{partial P}{partial T}right)_V-Pright] }{partial T}right]_V=Tleft(frac{partial^2P}{partial T^2}right)_V$$

A similar relationship can derived for the partial derivative of $C_p$ with respect to P at constant T starting with the equation: $$dH=C_pdT+left[V-Tleft(frac{partial V}{partial T}right)_Pright]dP$$

A similar relationship can derived for the partial derivative of $C_p$ with respect to P at constant T starting with the equation:
$$dH=C_pdT+left[V-Tleft(frac{partial V}{partial T}right)_Pright]dP$$

Mathematically calculate heat capacity in varying pressure and volume conditions

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