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Othin

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## Homework Statement

Two subsystems within a 20 l cylinder are separated by an internal piston. Each of them is initially composed of 1 mole of component 1 and one mole of component 2, both of which will be treated as a monatomic ideal gas. The cylinder has diathermal walls and is in contact with a thermal reservoir (meaning its temperature should be constant and equal to [itex]T_r = 373 K[/itex] .

**The piston is permeable to component 1 but impermeable to component 2.**The piston is in the center. We're to prove that the Helmholtz potential for this system is:

[itex]F=N\frac{T}{T_o} - \frac{3}{2}NRTln\frac{T}{T_0} - N_1RTln(\frac{VN_o}{V_0N_1}) - N_2^1RTln(\frac{V^1N_0}{V_0N_2^1}) - N_2^2RTln(\frac{V^2N_0}{V_0N_2^2})[/itex]

Where [itex] T_o, f_o, V_o , N_0 [/itex] are attributes of a standard state. N is the total mole number and superscripts represent the subsystem, while subscripts refer to components ([itex] N_2^1 [/itex] is the mole number of component2 at subsystem 1 and so on).

Afterwards, we must calculate the work required to push the piston to a position such that [itex]V^1=5 [/itex] and [itex]V^2=15 [/itex] in two ways : first by direct integration (the relation dW= PdV is valid, for the process is assumed to be carried out quasi-statically)

The answer is W=893J

## Homework Equations

We know that the Helmholtz potential of a mixture of simple ideal gases is the is the sum of the individual potentials. The fundamental equation of a monatomic gas in the Helmholtz potential is:

[itex]F=NRT[\frac{F_0}{N_0RT_0} - ln(\frac{T}{T_0}^{3/2} \frac{V}{V_0} (\frac{N}{N_0})^{-1})][/itex]

## The Attempt at a Solution

The Helmhholtz potential is additive, so I wrote both F as a sum of the potentials of each subsystem. Namely:

[itex] F^1=N_1^1RT[\frac{F_{011}}{RN_{01}T_0} - \frac{3}{2}ln\frac{T}{T_0} -ln(\frac{V^1}{V_{01}}\frac{N_{011}}{N_1^1})] + N_2^1RT[\frac{F_{012}}{RN_{021}T_0} - \frac{3}{2}ln\frac{T}{T_0} - ln(\frac{V^1}{V_{01}}\frac{N_{012}}{N_2^1})] [/itex]

And

[itex] F^2=N_1^2RT[\frac{F_{021}}{RN_{02}T_0} - \frac{3}{2}ln\frac{T}{T_0} -ln(\frac{V^2}{V_{01}}\frac{N_{021}}{N_1^2})] + N_2^2RT[\frac{F_{022}}{RN_{022}T_0} - \frac{3}{2}ln\frac{T}{T_0} - ln(\frac{V^2}{V_{01}}\frac{N_{02}}{N_2^2})] [/itex]

Summing then, I find :

[itex] F-Nfo\frac{T}{T_0} - \frac{3}{2}RTln\frac{T}{T_0} - N_1^1RTln(\frac{V^1N_0}{V_0N_1^1}) - N_1^1RTln(\frac{V^2N_0}{V_0N_1^2}) - N_1^2RTln(\frac{V^1N_0}{V_0N_2^1}) - N_1^2RTln(\frac{V^2N_0}{V_0N_2^2}) [/itex]

Where I've grupped all constants together. That's still different from the result I'm expected to prove, and the only different part seems to regard component one, which oddly enough is the component the piston is permeable to. Can't that be a mistake from the book? Even if it is, my expression wouldn't agree with the one I need, so I'd still need help.

As for the second part, the variation of F is vanishing identically for me. If there's a mistake and the membrane is actually permeable to 1 instead of 2, I get an expression which makes sense, but comes in terms of [itex] N_2^1 and N_2^2 [/itex] . I wasn't able to eliminate both of them to get a number for the work

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