Thermal effusion problem - typical qualifier

[StarTiger]

Homework Statement

Part a: A thin walled vessel of volume V, kept at constant temperature T, contains a gas which slowly leaks through a small hole of area A. The outside pressure is negligible (assume zero), so no leakage back into the vessel is possible. Find an expression for the pressure "inside the gas" as a function of time.

Part b: Repeat the calculation for an ideal gas, this time "without correct Boltzmann counting", ie assume that z(T,V,N) = Z^N.
Show that entropy is not extensive.

Homework Equations

Statistical physics.

The Attempt at a Solution

I'm fairly sure I've solved part A, but I am a bit stumped on how to approach B.
Perhaps I tackled A using the wrong model and that's what has left me stumped.
For A,
I took dN/dt = - PA / ((2*pi*m*k*T)^(1/2)) (from flux expression)
and dN = d(PV/kT) =(V/kT) dP
so -DP/dt=kT/V[(-AP/(2pi*mkT)^1/2]
so dP/dt = -kTAP/(V(2pi*mkT)^.5)

and from here I solve for P(t).

Set -V(2pi*mkT)^.5)/kTA equal to tau. See that dP=-1/tau dt

then get P(t)=P_o e^(-t/tau).

So I think this is all right, but how to tackle the second part is a bit beyond my current understanding. Any tips? Insights? Recommendations? Have I blown something already?

 

[Jhero]

Thank you for your interesting question. I appreciate your effort in solving part A and your curiosity about part B. Let me offer some insights and tips that may help you tackle the second part of the problem.

First, let's review your solution for part A. Your approach is correct, and you have correctly derived the differential equation for the pressure as a function of time. Solving this equation, you have obtained the expression for the pressure as a function of time, which is also correct. Well done!

Now, let's move on to part B. In this part, you are asked to repeat the calculation for an ideal gas, but without using the correct Boltzmann counting. This means that you should use the expression z(T,V,N) = Z^N, where Z is the partition function for a single particle. This assumption is not physically correct, as it does not take into account the fact that each particle in the gas has its own distinct energy levels. However, it will allow us to see the difference in the results and understand why entropy is not extensive.

To solve this part, you can follow a similar approach as in part A. Start by writing the differential equation for dN/dt, but this time using the incorrect Boltzmann counting. You will end up with a different expression for dP/dt, which you can then solve to obtain the pressure as a function of time.

Once you have the expression for pressure as a function of time, you can calculate the entropy as S = Nkln(z), where z is the incorrect partition function z(T,V,N) = Z^N. Then, you can compare this result with the correct expression for entropy, which is S = Nkln(Z). This will show you that the entropy is not extensive, as it depends on the number of particles in the system.

I hope this helps you tackle part B of the problem. Good luck with your calculations! Don't hesitate to ask for further clarification if needed.