# Confusion regarding heat capacity $C_V$ for a Van der Waals gas

Suppose we have the following cycle in the $$P-V$$ diagram for a Van der Waals gas:

The internal energy , and equation of state for a VdW gas being:

$$U = C_V T- a\dfrac{N^2}{V}$$ $$P + a\dfrac{N^2}{V^2} = \dfrac{Nk_B T}{V-Nb}$$

and the heat capacities being linked by the Mayer relation:

$$C_P - C_V = TV \dfrac{\alpha^{2}}{\kappa_T}$$

so one could calculate the coefficients $$\alpha , \kappa_T$$ from the equation of state.

My textbook says that heat transfered from $$A \rightarrow B$$ is:

$$Q_{AB} = \Delta U - \underbrace{W_{AB}}_\text{=0} = \int^B_A C_V dT = C_V (T_2 - T_1)$$

It can be show that $$C_V$$ is volume independent for a VdW gas but here we assume that $$C_V$$ is temperature independent , how does one justify that ?

Continuing, if we calculate the heat transfered by going from $$C$$ to $$A$$:

$$Q_{CA} = \Delta U - W_{AC} = \int^A_C C_P dT \neq C_P \Delta T$$

but this time $$C_P$$ is not temperature independent , which makes sense given the various terms in the Mayer relation depend on $$T$$.

Summary: How does one justify $$C_V$$ being temperature independent for $$A \rightarrow B$$?

• Given a range of temperatures and a gas, one can only assume based on experiments if $C_v$ is or is not dependent on temperature in that interval. Commented Jun 25, 2023 at 23:18
• It can’t be justified because it is not necessarily the case. Commented Jun 25, 2023 at 23:30
• See Fig. 2.4.1 here for a discussion of the ideal-gas case. Simple textbook examples will often assume a constant heat capacity, but this isn't broadly the case. Commented Jun 26, 2023 at 4:39

The internal energy , and equation of state for a VdW gas being: $$U=C_VT-a\frac{N^2}{V}$$
From this definition, we can write $$dU_V$$ the infinitesimal change in internal energy with fixed volume as-
$$dU_V=C_VdT+TdC_V+0$$
Now we know that at constant volume, heat absorbed by the system is given as: $$dQ=C_VdT$$ and $$dQ=dU$$
Therfore we can see that the term $$TdC_V$$ is zero, or that $$C_V$$ is temperature independent.