Temperature vs Volume in an expanding gas

[Fwahm]

Homework Statement

The energy of a gas is given by: E = 3/2*NkT-aN²/V, where E, N, k, and a are held constant (or are just constants). Volume V₁ with temperature T₁ expands adiabatically into V₂. Determine T₂.

Homework Equations

All in part 1.

The Attempt at a Solution

I'm not exactly sure how to start on this problem. I'm not asking for an answer, but some tips on how to proceed would be appreciated.

 

[jdwood983]

What do you know about adiabatic processes?

 

[Fwahm]

The system is naturally undergoing Adiabatic Cooling (as the Volume increases, the aN2/V term decreases, which means the 3/2*NkT term (and thus T) must also decrease for E to remain constant.

However, I don't know how to translate this into an equation for delta T/delta V, based on the other constants.

 

[jdwood983]

So if the only things that can vary are temperature and volume, it seems to me that if you just use

$$\Delta E=0=\frac{3}{2}Nk\Delta T-\frac{aN^2}{\Delta V}$$

you can solve for $\Delta T=T_2-T_1$.

 

[Fwahm]

Thank you very much.

I really need some extra sleep tonight, I can't believe I missed such an easy solution.