patrickmoloney
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Homework Statement
Find the expression for c_p - c_v for a van-der-waals gas, with the equation of state
\Bigg{(}p+\dfrac{a}{V^2}\Bigg{)}(V-b)=RT
Homework Equations
The Attempt at a Solution
Basically I've proved
c_p - c_v = \Bigg{[} p + \Bigg{(}\dfrac{\partial E}{\partial V}\Bigg{)}_T \Bigg{]}\Bigg{(}\dfrac{\partial V}{\partial T}\Bigg{)}_p
(E and V are the energy and volume of one mole).
The question states that
p + \Bigg{(}\dfrac{\partial E}{\partial V}\Bigg{)}_T = T \Bigg{(}\dfrac{\partial p}{\partial T}\Bigg{)}
I don't need to prove this I just need to use it.
So far the only thing that I can come up with is rearranging the van der waals equation in terms of p which gives
p = \dfrac{RT}{(V-b)}-\dfrac{a}{V^2}
Then
\Bigg{(}\dfrac{\partial p}{\partial T}\Bigg{)}_V = \dfrac{R}{(V-b)}
Which can be substituted into the equation c_p - c_v but rearranging the van der waals formula for V is very tedious. Which leads to believe there is a an easier step. It's an exam question so there much be a neat way to solve this. I want to ask here rather than look up someone else's long winded solution. I'm not looking for the solution I genuinely want to be able to tackle this problem or another problem like it should it come up in the exam