Relationship Between V and T in Adiabatic Expansion

[Chronum]

Homework Statement

The energy and pressure of black body radiation depend on T and V as Eq(1) & Eq(2). Suppose that the temperature and volume of a box of radiation change adiabatically. Find the relation between dE and dT in this process. Next, using Eq(1), show that T ∝V^-1/3

Homework Equations

Eq(1): E = σVT^4;
Eq(2): p = 1/3σT^4;
ΔE = Q - W;

Since Q = 0;
ΔE = -W

The Attempt at a Solution

To begin with we've (a few people working together) have tried what appears to be an overly simple method.
E = σ V T^4
dE/dT = 4 σ V T^3
dE = 4 σ V T^3 dT
V = dE/(4 σ T^3 dT)

∴ V∝T^-1/3

But this seems overly simplistic, especially since volume is changing too. Any formulae/approaches we're missing?

 

[haruspex]

Do you realize that in your final answer V and T are swapped compared with what was given to be proved?
Certainly the method is not valid.
You do not seem to have used Eq 2 at all. I would think you need to use that and some relationship between V, E and p.

 

[Chronum]

Do you realize that in your final answer V and T are swapped compared with what was given to be proved?
Certainly the method is not valid.
You do not seem to have used Eq 2 at all. I would think you need to use that and some relationship between V, E and p.

That is correct. I apologize. It was a mistake of plain anticlimactic proportions.

And yes, I did end up using Eq(2), and we got the answer after some rather petty algebra and a step of differential equations. Problem solved.

 

[haruspex]

That is correct. I apologize. It was a mistake of plain anticlimactic proportions.

And yes, I did end up using Eq(2), and we got the answer after some rather petty algebra and a step of differential equations. Problem solved.

Was that petty or pretty?