Ratio of volumes in a vertical cylinder with a piston

[danut]

Homework Statement

A vertical cylinder closed at both ends is separated into two compartments by a movable piston of negligible volume. In the two compartments there are equal masses of the same ideal gas, at the same temperature T₁. At equilibrium, the ratio of the volumes of the two compartments is k = 3.

Relevant Equations

What will be the ratio of the two volumes, if the temperature rises to 4T₁/3?

First, I thought of the forces which are acting upon the piston.
F1 + G = F2, where F1 = p1 * S and F2 = p2 * S
p1 + mg/S = p2

I figured that before and after the gas' temperature rises, the piston has to be at equilibrium, so p2 - p1 = p2' - p1'.

p1V1 = niu * R * T1
p2V2 = niu * R * T1 => p1V1 = p2V2, but V1/V2 = k = 3. so p1/p2 = 1/3, so p2 = 3p1.

Nothing that I think of adds up to anything, the correct answer is: sqrt(2) + 1.

 

[TSny]

First, I thought of the forces which are acting upon the piston.
F1 + G = F2, where F1 = p1 * S and F2 = p2 * S
p1 + mg/S = p2

We can probably guess what G and S represent, but you should always define your notation.

I figured that before and after the gas' temperature rises, the piston has to be at equilibrium, so p2 - p1 = p2' - p1'.

OK.

Hint: Let $V_0$ be the total volume of the cylinder. Can you express $V_1$ in terms of $V_0$ and $k$? Likewise for $V_2$.

 

[danut]

We can probably guess what G and S represent, but you should always define your notation.

I apologize and thank you, will do that from now on!!
So V₁ = V₀*k/(k+1) and V₂ = V₀/(k+1).

I wrote the equation p₂ - p₁ = p₂' - p₁' in terms of ν, R, T and the corresponding volumes and finally got the correct answer!! Thank you so much, I've struggled with this problem for the longest time.

 

[TSny]

Great! Good work.