- #1

patrickmoloney

- 94

- 4

## Homework Statement

Find the expression for [itex]c_p - c_v[/itex] for a van-der-waals gas, with the equation of state

[tex]\Bigg{(}p+\dfrac{a}{V^2}\Bigg{)}(V-b)=RT[/tex]

## Homework Equations

## The Attempt at a Solution

Basically I've proved

[tex]c_p - c_v = \Bigg{[} p + \Bigg{(}\dfrac{\partial E}{\partial V}\Bigg{)}_T \Bigg{]}\Bigg{(}\dfrac{\partial V}{\partial T}\Bigg{)}_p[/tex]

([itex]E[/itex] and [itex]V[/itex] are the energy and volume of one mole).

The question states that

[tex]p + \Bigg{(}\dfrac{\partial E}{\partial V}\Bigg{)}_T = T \Bigg{(}\dfrac{\partial p}{\partial T}\Bigg{)}[/tex]

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 [itex]p[/itex] which gives

[tex]p = \dfrac{RT}{(V-b)}-\dfrac{a}{V^2}[/tex]

Then

[tex]\Bigg{(}\dfrac{\partial p}{\partial T}\Bigg{)}_V = \dfrac{R}{(V-b)}[/tex]

Which can be substituted into the equation [itex]c_p - c_v[/itex] but rearranging the van der waals formula for [itex]V[/itex] 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