Pressure on system and work done by the system

[leroyjenkens]

A certain insulating material has 5 x 10²² atoms, each having six degrees of freedom. It initially occupies a volume of 10^-6 m³ at a pressure of 10⁵ Pa. The pressure and volume are related by p(V - V₀) = constant, where V₀ = 0.94 x 10^-6 m³

Homework Equations

$$W = -\int p\, dV$$
PV = nRT
$$\bigtriangleup E = \bigtriangleup Q - \bigtriangleup W$$
P_iV_i = P_fV_f

The Attempt at a Solution

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This one is confusing because it gives me one initial volume, and then gives me a V₀ that's almost the same as the initial volume, so I don't know if I only use the V₀ for that equation with the constant, or if I use that as the initial volume for something else. No idea why it gives me two different (but almost identical) initial volumes.
I solved for the constant, using the initial volume and the other initial volume for V, and then tried to use that constant and plugging in what I had in V in for V₀ and solving for the V, which would be the final volume, and then plugging that into the limits of integration. That did nothing good.

I tried solving for the final volume, using P_iV_i = P_fV_f and then plugging that into my limits of integration, and multiplying P times ten. That didn't work.

I tried using PV = nRT and solving for T, but I don't know what to do with T. But the question gives me numbers of atoms, which is why I tried using the ideal gas law. Otherwise it would have been irrelevant information, as far as I can tell. It's an insulating material, so I guess it's not an ideal gas. It's probably a solid.
It gives me the degrees of freedom of the atoms. I have no idea how that's relevant.

 

[leroyjenkens]

I forgot to ask the question and I can't edit. But I figured out that part. The next part asks "Suppose that the potential energy per particle remains constant, and that the pressure increases sufficiently quickly that no heat enters or leaves the system during the process, ie, the process is adiabatic. By how much does the temperature of the insulator rise?"

I tried the ideal gas law, but it's not a gas, so I don't think it applies. I'm getting an answer with that equation that's close to the real answer; it has the same order of magnitude, but it's not close enough. I need something that relates pressure, temperature, moles, and volume, that can be relevant to a solid. I'm assuming it's a solid, and I don't think the ideal gas law applies to solids.