Pressure of piston in a cylinder containing gas, find heat (Q)

[Violagirl]

Homework Statement

A cylinder of cross-sectional A is filled with N ideal gas molecules at temperature T and pressure p, and a piston of mass m seals the gas in the cylinder with a frictionless seal, as shown on the figure attached. The trapped column of gas has an initial height h. The piston and cylinder are surronded by air at pressure patem. A) How much heat Q does the gas absorb if it ends up at temperature 2T and pressure p after expanding and pushing the piston to a final height H? B) What is the final height H of the piston as some multiple of h. Note that the piston is at rest when the heating starts and is at rest when it reaches its final height H. Your answer to (A) should be expressed in terms of the quantities h, g, A, m, and patm. Please be careful when calculating the work done by the expanding gas on its surrondings.

Homework Equations

Change in internal energy (U) equation: ΔU=U_f-U_i

Internal energy equation to heat (Q) and work (W): U = Q-W

Force of a gas equation to pressure (p) and cross-sectional area (A): F=P*A

Work equation to pressure (p) times the change in volume (ΔV): W = PΔV

The Attempt at a Solution

I think I have an answer for part A:

I drew in the forces on the attached document.

For finding U:

U = 3/2NK_BT

Initial T=T

Final T=2T

So for U, we have:

ΔU=U_f-U_i =

3/2NK_B2T-3/2NKT = 3/2NK_BT = U

F=P*A: -mg+p=p_atm*A

V=Δh(?) = H-h

So:

Q=U+W:

Q= (3/2NK_BT+mg+p-p_atm*A*(H-h)

I'm guessing this isn't completely correct but if someone could let me know, that would be great. Otherwise, I have no idea how to do the second part to this problem.

 

[Violagirl]

Oops, forgot to include the attachment! It is now attached to this post.

 

[Violagirl]

Ok here, much bigger.