Solve Asymmetric Piston Problem for SL

[SchroedingersLion]

Greetings!

Can you help me understand what this text book problem asks of me?

The situation was considered in the text: In equilibrium, the forces on both sides of the piston are equal: $A_1P_1 = A_2P_2$.
This is the first equation. It should also answer part 3 of the question. The piston allows heat exchange, so that in equilibrium I have $T_1=T_2$. This should answer part 2. To get a second equation to answer part one, I can just apply the ideal gas law to the first equation to get
$$
A_j\frac{N_j}{V_j} = A_k\frac{N_k}{V_k}.
$$

I feel like this is way too easy and I am missing something... I don't know what the author wants me to do. I should note that he derived the entropy function S(E,V,N) of the ideal gas, and also the three partial derivatives such as $\left(\frac{\partial S}{\partial V}\right)_{E,N}=\frac {P} {T}$, so maybe he expects some more wizardry with them?SL

 

[anuttarasammyak]

The problem says the piston is diathermal so $T_1 \neq T_2$.

EDIT I confused it with adiabatic.

 

[Chestermiller]

The problem says the piston is diathermal so $T_1 \neq T_2$.

Diathermal means that the temperature ARE equal in the end.

 

[Chestermiller]

I don't know what the author had in mind, but here are some thoughts on how I would approach this problem (assuming the same gas species is in both chambers). For the final temperature, I would have $$T=\frac{(E_1+E_2)}{C_v(N_1+N_2)}$$
For the final volumes, I would have: $$V_1=V_{10}+A_1\delta$$$$V_2=V_{20}-A_2\delta$$where $\delta$ is the displacement of the piston. So, for the final pressures, we would have: $$P_1=\frac{N_1RT}{V_1}$$$$P_2=\frac{N_2RT}{V_2}$$subject to $$P_1A_1=P_2A_2$$This provides sufficient information to determine the piston displacement $\delta$.

 

[SchroedingersLion]

Thanks Chestermiller!

The author did not ask for the displacement, so it is really unclear what he even expects. Glad to know that I did not miss something obvious.

 

[Chestermiller]

To get the final pressure, you need to first solve for the displacement.