# 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!

## Answers and Replies

specific heat or heat capacity are the properties of the substance . the shall remain same under any condition for the same substance.

That explanation isn't compatible with the fact that specific heat is strongly temperature dependent. Why is specific heat temperature dependent and not pressure dependent?