Question

Show that for a van der Waals gas, `((delC_V)/(delV))_T = 0`, where `C_V = ((delU)/(delT))_V`?

Answer 1

By definition, the constant-volume heat capacity was:

`C_V = ((delU)/(delT))_V`,`" "" "bb((1))`

where `U` is the internal energy, and `T` and `V` are temperature and volume, respectively, as defined in the ideal gas law and other gas laws.

To show that for a van der Waals gas, the constant-volume heat capacity does not change due to a change in volume at a constant temperature, i.e. `((delC_V)/(delV))_T = 0`, we first write out the van der Waals (vdW) equation of state:

`bb(P = (RT)/(barV - b) - a/(barV^2) = (nRT)/(V - nb) - (an^2)/(V^2))`, `" "" "bb((2))`

where `barV = V/n` is the molar volume, `R` is the universal gas constant, `a` and `b` are the vdW constants for intermolecular forces of attraction and excluded volume (respectively), and `P` is the pressure.

This means we'll need to have an expression for `dU` that includes `P` somehow, or even a partial derivative.

We can use the Maxwell relation for the internal energy in a closed system undergoing a reversible process:

`dU = TdS - PdV`,

If we divide by `(delV)_T` for the entire equation, we get:

`((delU)/(delV))_T = T((delS)/(delV))_T - Pcancel(((delV)/(delV))_T)^(1)` `" "" "bb((3))`

This will be the main expression we'll work with. Note that `T` and `V` are the natural variables of the Helmholtz free energy, `A`, whose Maxwell relation is:

`dcolor(red)(A) = -Scolor(red)(dT) - Pcolor(red)(dV)`

Using the fact that the Helmholtz free energy is a state function (just like `G`, `H`, `S`, etc), there is a cyclic relationship we can use:

`-((delS)/(delV))_T = -((delP)/(delT))_V`

Plugging into `bb((3))` gives:

`((delU)/(delV))_T = T((delP)/(delT))_V - P` `" "" "bb((4))`

which is something we can relate back to the vdW equation of state, using `((delP)/(delT))_V`. We're almost there.

Given that `U` is a state function (and this is very important!), the order of partial differentiation does not matter:

`(del)/(delV)[((delU)/(delT))_V]_T = (del)/(delT)[((delU)/(delV))_T]_V`

or

`((del^2U)/(delVdelT))_(V,T) = ((del^2U)/(delTdelV))_(T,V)`

This will become relevant if we take the partial derivative of `C_V` with respect to `V` at constant `T` and plug in the definition of `C_V` from `bb((1))`:

`((delC_V)/(delV))_T = (del)/(delV)[C_V]_T`

`= (del)/(delV)[((delU)/(delT))_V]_T`

`= (del)/(delT)[((delU)/(delV))_T]_V` `" "" "bb((5))`

Now consider the right side of `bb((5))` and plug in `bb((4))`:

`((delC_V)/(delV))_T`

`= (del)/(delT)[T((delP)/(delT))_V - P]_V`

`= (del)/(delT)[T((delP)/(delT))_V]_V - ((delP)/(delT))_V` `" "" "bb((6))`

Now, we should figure out what `((delP)/(delT))_V` is so we can use it! Plugging in the vdW equation of state from `bb((2))`:

`((delP)/(delT))_V = (del)/(delT)[(nRT)/(V - nb) - (an^2)/(V^2)]_V`

`= (nR)/(V - nb)`

Plugging this back into `bb((6))`, we get:

`color(blue)(((delC_V)/(delV))_T) = (del)/(delT)[(nRT)/(V - nb)]_V - (nR)/(V - nb)`

`= (nR)/(V - nb) - (nR)/(V - nb)`

`= color(blue)(0)`