# Mathematically calculate heat capacity in varying pressure and volume conditions

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 ?

• Are you referring to the heat capacity at constant volume or the heat capacity at constant pressure? – Chet Miller May 24 '18 at 13:47
• @ChesterMiller I interpreted the question as a request for a formula which defines specific heat capacity in terms of quantities like pressure and volume conditions. The questions says "...wherein both - the pressure and the volume...". – user191954 May 24 '18 at 14:53
• @Chair Thanks. You are certainly entitled to your own interpretation. But I would like some clarification directly from the OP before expending any effort composing an answer. – Chet Miller May 24 '18 at 15:08
• @ChesterMiller. I would like to know the behavior of both specific heat capacity at constant volume as well as heat capacity in general. I believe that these quantities for a liquid (say) doesn't have the same value in conditions where I have varying pressure ,volume., temperature. Please correct me if I am wrong. I would like to calculate the new values of these quantities when in such a dynamic environment. Hence, the request for a formula . Thanks ! – Dexter May 25 '18 at 4:15

The starting point for determining the volume dependence of $C_v$ is the equation: $$dU=C_vdT+\left[T\left(\frac{\partial P}{\partial T}\right)_V-P\right]dV$$ This equation appears in every thermodynamics textbook, so I'm not going to derive it. From this equation, it follows that $$\frac{\partial ^2 U}{\partial V\partial T}=\left(\frac{\partial C_v}{\partial V}\right)_T=\left[\frac{\partial\left[T\left(\frac{\partial P}{\partial T}\right)_V-P\right] }{\partial T}\right]_V=T\left(\frac{\partial^2P}{\partial T^2}\right)_V$$
A similar relationship can derived for the partial derivative of $C_p$ with respect to P at constant T starting with the equation: $$dH=C_pdT+\left[V-T\left(\frac{\partial V}{\partial T}\right)_P\right]dP$$