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

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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 $A \rightarrow B$?

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  • $\begingroup$ 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. $\endgroup$
    – hyportnex
    Commented Jun 25, 2023 at 23:18
  • $\begingroup$ It can’t be justified because it is not necessarily the case. $\endgroup$ Commented Jun 25, 2023 at 23:30
  • $\begingroup$ 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. $\endgroup$ Commented Jun 26, 2023 at 4:39

1 Answer 1

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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.

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