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

##### 1 Answer

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. **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

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

#((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 *natural variables* of the **Helmholtz free energy**,

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

Using the fact that the Helmholtz free energy is a state function (just like

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

Plugging into

#((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 *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

#((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

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

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

Plugging this back into

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

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

#= color(blue)(0)#