# Is the specific heat capacity the same between real and ideal gases?

From Section 2.4.1 of this link where the change in the specific internal energy of a gas is given as

$$d u = \frac{\partial u}{\partial T} dT + \frac{\partial u}{\partial v} dv$$

They define ideal gas as a gas where $$\frac{\partial u}{\partial v} dv = 0$$ and define $$C_v(T) = \frac{\partial u}{\partial T}$$. I am wondering, for real gases, where $$\frac{\partial u}{\partial v} dv$$ is not zero, does the construction of $$C_v(T)$$ change (for example, is it still only a function of temperature even though the internal energy is not)? Or are the $$C_v$$ values in tables such as this valid for calculating the $$\frac{\partial u}{\partial T} dT$$ component of the $$du$$ equation for both ideal and real gases (even though the table specifies that it is for ideal gases)?

As a followup question, how are the tables of specific heats such as the one I linked above generated, is it from experiments or based off of theoretical calculations?

## 1 Answer

If u=u(T,V) and $$C_V=\left(\frac{\partial u}{\partial T}\right)_V$$, then $$\left(\frac{\partial C_v}{\partial V}\right)_T=\frac{\partial^2 u}{\partial V\partial T}\neq 0$$So, for real gases, Cv can, in most cases, be expected to be a function of specific volume.

The values of Cv in your table are for the real gas in the limit of ideal gas behavior (i.e., low pressures). They are functions only of temperature, but do not describe the behavior at higher pressures, nor can their derivatives with respect to temperature be applied in the region of high pressures.

To answer your final question, the values in the table are based on experimental data.

It is possible, however, knowing the PVT behavior of a real gas in conjunction with its Cv behavior in the ideal gas limit, to calculate the change in internal energy of a real gas between high pressure states.