Proving Internal Pressure of Ideal & VdW Gases

[Jennifer Lyn]

Like in the other problem I posted- This is the other question that I missed and just can't find a solution for.

Homework Statement

Prove the internal pressure is 0 for an ideal gas and ((n^2)a)/(v^2) for a Van der Waals gas.

Homework Equations

1. VdQ Eqn: p= (nRT)/(v-b) - ((n^2)a)/(v^2)
2. (partial S/partial V) for constant T = (partial p/partial T) for contant V.
3. dU = TdS - pdV
4. pi sub t (internal pressure) = (partial U/partial V) for constant T

The Attempt at a Solution

a) Ideal Gas
0 = (partial U/partial V) for const T
int 0 dv = int du
0 = int (TdS - pdV)
int p dv = int T ds
int (nRT/v) dv = int (Pv/nR) dS
nRT x int(1/V) dv = pv/nR x int 1 dS
... and I get kind of lost here, though I know that what I've already done is wrong.. :(

b) VdW gas
I actual have to get going to school, but I'll come back and type up what I've done (incorrectly :( for this part afterwards).

 

[афк]

Hello Jennifer,

I suppose you are given the so called thermal equation of state $p=p(T,V,n)$ for both

the ideal gas

$$p=\frac{nRT}{V}$$

and the Van der Waals gas

$$p=\frac{nRT}{V-nb}-\frac{n^2a}{V^2}$$

The inner pressure $\left(\frac{\partial U}{\partial V}\right)_T$ can be calculated after finding the so called caloric equation of state $U=U(T,V,n)$ for both cases.

Another straightforward method would be to use the following identity which shows that the caloric and thermal equations of state are not independent of each other:

$$\left(\frac{\partial U}{\partial V}\right)_T=T\left(\frac{\partial p}{\partial T}\right)_V-p$$

Do you know how to derive this identity?

 

[Jennifer Lyn]

I think so..
$$\Pi$$t = $$\partial$$u/$$\partial$$v for constant t
= ( 1/$$\partial$$v$$\times$$(Tds - pdv) )
= T $$\times$$ ($$\partial$$p/$$\partial$$t) - p

I think that's right. I still don't know how to get from that Maxwell relation to the ideal gas and Van der Waals eqn, though.

 

[Jennifer Lyn]

Ok, I think I figured it out, from my previous post (sorry- I am still getting used to using the tools for math on this board) I replace the vanderwaals eqn into P in my partial p and then just solve from there.

 

[Jennifer Lyn]

Thanks everyone!