Thermodynamics: Heating gas in two separate containers

[2DGamer]

1. A cylinder with thermally insulated walls contains a movable frictionless thermally insulated piston. On each side of the piston are n moles of an ideal gas. The initial pressure (P_0), volume (V_0), and temperature (T_0) are the same on both sides of the piston. The value for (gamma) is 1.5, and c_v is independent of temperature. By the means of a heating coil in the gas on the left side of the piston, heat is supplied slowly to the gas on this side. It expands and compresses the gas on the right side until its pressure has increased to (27*P0)/8.

1) How much work is done on the gas on the right side in terms of n, c_v, and T_0?
2) What is the final temperature of the gas on the right? (this is the only one I can solve)
3) What is the final temperature of the gas on the left?
4) How much heat flows into the gas on the left?



2. PV = nRT



3. For the first part where I need to find the work on the right side I noted that c_p - c_v = R. Also, since (gamma) = c_p/c_v I put these two together to get:
1.5 = (c_v + R)/c_v which equals: 1.5 = 1 + R/c_v which equals: .5 = R/c_v. Using the equation PV = nRT I replaced R with PV/nT, then solved for PV and got nTc_v/2 which is the correct answer in the book, but I'm wondering if this is even the right way to do the problem. Aren't I supposed to start with the definition of work and go from there? I've tried that, but keep getting stuck.

For the second one I included my work in the attachment. I feel pretty good about that one.

For the third and fourth one I really have no idea how to even get started. If you can just give me a hint, that would be great.

 

[Mapes]

The reason your solution to part (1) doesn't feel right is that work doesn't equal PV, and your answer $nTc_v/2$ doesn't match the solution $nT_0c_v/2$. Try relating the energy gained by the gas on the right to its change in temperature, and working from there.