Thermodynamics Work, Heat, First Law Question

[Honger]

Homework Statement

Consider two pistons of equal cross sectional area joined as illustrated in the
following sketch (my rendition):
[A|----|B]
• Each piston contains n moles of an ideal gas of specific heat cV=(3/2) R.
• Each piston's base is fixed, so when the rod joining the piston moves one
must expand for the other to compress.
• Piston A is thermally isolated and is initially at a pressure PA0 and a
temperature T0.
• Piston B is held in thermal equilibrium with a bath at temperature T0 and
is initially at pressure PB0, which is higher than the initial pressure in
piston A.
• Piston B is allowed to expand quasi‐statically (reversibly) until there is no
more net force on the rod joining the pistons. Piston B is frictionless, so
during the expansion Pex=PB.
A. Derive an expression that relates the following dimensionless
quantities:
π =P0B/P0A ,τ =TfA/T0A

Homework Equations

Ideal Gas: PVm=RT, dW=-PdV=CvdT

The Attempt at a Solution

Since the pistons are connected, when the piston stops moving, the work done in A will be equal to the work done by B?
P0AdV=P0BdV
-nRT0A/V0A = nRT0B/V0B
How do I relate to pressure from here?
Also tried using Heat Capacity:
Wadiabatic = Cv(delta)T, which should equal the W of the reversible adiabatic B undergoes:
=-PdV = -nRT/V dV
Again, I can't see how to proceed.

Thank you!

 

[KataKoniK]

Thank you for your question. The problem you are facing is a common one when dealing with thermodynamic systems. In order to relate the pressure and temperature of the two pistons, we can use the ideal gas law, PV = nRT. We can rearrange this equation to solve for pressure:

P = nRT/V

Since the pistons are of equal cross-sectional area, we can assume that they have equal volumes as well. This means that the volume of piston A (V0A) is equal to the volume of piston B (V0B). Therefore, we can write the equation for the pressure in terms of the initial pressure (P0) and temperature (T0) for both pistons:

PA = nRT0A/V0A

PB = nRT0B/V0B

We can now use the given information to write an expression for the ratio of the pressures (π):

π = PB/PA = (nRT0B/V0B)/(nRT0A/V0A)

Since the volumes are equal, they cancel out and we are left with:

π = T0B/T0A

Next, we can use the ideal gas law to relate the temperatures of the two pistons:

T0A = PA0V0A/(nR)

T0B = PB0V0B/(nR)

Substituting these values into our expression for π, we get:

π = (PB0V0B/(nR))/(PA0V0A/(nR))

π = PB0/PA0

Finally, we can use the given information about the temperature ratio (τ) to write an expression for the initial temperature of piston A (T0A) in terms of the final temperature (TfA):

τ = TfA/T0A

T0A = TfA/τ

Substituting this into our previous expression for PB0/PA0, we get:

π = PB0/(TfA/τ)

π = (PB0/τ)/(TfA)

This is the desired expression for relating the dimensionless quantities π and τ. I hope this helps you in your problem solving process. Good luck!