# Pressure dependence of the Heat Capacity

I am trying to understand the nature of the dependence of heat capacity/specific heat on pressure.

I understand that one may give the the following relations:

$\frac{C_p}{C_v}=1+\alpha\gamma T$
where $C_p,C_v,\alpha,\gamma, and T$ are isoberic specific heat, isochoric specific heat, thermal expansivity, Anderson-Gruneisen parameter, and Temperature, respectively.
$C_v=\frac{\alpha V K_T}{\gamma}$
where V is volume and K_T is the isothermal bulk modulus.

What I do not think I fully grasp from these relationships is how the specific heat is related to the pressure. Is appears as though the heat capacity of a material depends on the pressure only because the density depends on the pressure.

In other words, and more specifically related to my own challenges; If I wanted to model heat transport with both the density and specific heat depending on pressure and temperature in a thermodynamically consistent way (lets ignore conductivity, with which I have no problems), I could use temperature-dependent data for $C_p$ and temperature-pressure dependent data for the density and the model would be fully consistent (lets also ignore phase changes)?

If so, am I correct to say that it must be true that the specific heat, having the dimensions of J kg-1 K-1 actually does NOT have a dependence on pressure? And also, that the volumetric specific heat having dimensions J m-3 K-1 is actually just $\rho C_p$.

Thank you!