Pressure change with constant volume

[SadStudent]

Homework Statement

Given two moles of some gas with two atoms per molecule which has energy $U=aK_{B}T$.
Assume the initial pressure is $P_{i}= 2$ Atm and the initial volume is $P_{i}= 0.001\ m^{3}$.
Now we heat it in an isochoric process to $2P_{i}$.

What would be $\Delta T$?
What would be the work required to do that?
What would be the change in the gas particles' internal energy?

Homework Equations

$U=aK_{B}T$
$dU=PdS-TdV$

The Attempt at a Solution

Well using the first law of thermodynamics, I got that under constant volume $dV=0$ so $\frac{dQ}{dT}=\frac{\partial U}{\partial T}$ so the entire energy that we spent on heating would be transferred into the gas.
What's a tricky to me is to formulate the change in temperature as a function of pressure because $dV=0$ so I'd really welcome a hint.

Thanks!

 

[TSny]

Homework Equations

$dU=PdS-TdV$

Oops, check that.

What's a tricky to me is to formulate the change in temperature as a function of pressure because $dV=0$ so I'd really welcome a hint.

Can you assume the gas obeys the ideal gas law?

 

[SadStudent]

Oops, check that.Can you assume the gas obeys the ideal gas law?

And yes of course it was a typo :) it's supposed to be $dU=TdS-PdV$

I'm really not sure whether I can, is there a way of solving this without this assumption?