Solving Maxwell Relations Homework with Van der Waals Gas

[subzero0137]

Homework Statement

[/B]
I'm stuck on part c of the attached problem:

Homework Equations

$$C_P - C_V = \left[P + \left( \frac {∂U}{∂V} \right)_T \right]\left( \frac {∂V}{∂T} \right)_P$$

$$P + \left( \frac {∂U}{∂V} \right)_T = T \left( \frac {∂P}{∂T} \right)_V$$

$$\left(P + \frac {a}{V^2} \right)(V - b) = RT$$

The Attempt at a Solution

I need to use the bottom two equations to find to find $P + \left( \frac {∂U}{∂V} \right)_T$ and $\left( \frac {∂V}{∂T} \right)_P$ and plug these expressions in the top equation for C_P - C_V.

I've found $$\left (\frac {∂P}{∂T} \right)_V = \frac {R}{V - b}$$
$$∴ P + \left( \frac {∂U}{∂V} \right)_T = \frac {RT}{V - B}$$

But I'm having trouble finding $\left (\frac {∂V}{∂T} \right)_P$ because I can't seem to make V the subject of the Van der Waals gas expression.

 

[Charles Link]

Take the Van der Waal's equation and take a differential of both sides, using the product rule, (or just multiply out the left side), and treating $P$ as a constant. You should get a couple of terms that are multiplied by $dV$. On the right side you will have $R \, dT$.

 

[subzero0137]

Take the Van der Waal's equation and take a differential of both sides, using the product rule, (or just multiply out the left side), and treating $P$ as a constant. You should get a couple of terms that have are multiplied by $dV$. On the right side you will have $R \, dT$.

Oh I see. Thanks, I got it now :)