9. Gibbs \"Parador\": Consider a box of total volume V partitioned into two sub-

Question

9. Gibbs "Parador": Consider a box of total volume V partitioned into two sub-volumes Vi on the left and V2 on the right, containing respectively N1 and N2 molecules which behave as idea gases, at a common initial temperature T and pressure P Bob believes that all of the gas molecules are indistinguishable, so ascribes no entropy of mixing when the partition is removed. (According to him, he can restore the initial macrostate without any external changes, just by re-inserting the partition). Despite the partition, from Bob's perspective the system is in internal thermodynamic equilibrium so no work can be extracted from this system alone However, unbeknownst to Bob, Alice has discovered that all gas molecules on the left are actually a

Explanation / Answer

a) before we remove the partition (Alice) total volume was V=V1+V2 and ttotal number of prticle N=N1+N2

The gas is ideal that means number density d=N1/V1=N2/V2 for the whole system.

Now from the definition of entropy using multiplicity we can write S1=KN1lnV1+3/2N1K (ideal gas) and

S2=KN2lnV2+3/2N2K

so using same formula for total entropy we get S=S1+S2

after removing partition we get total entropy S=(N1+N2)Kln(V1+V2)+3/2(N1+N2)K

so change in entropy= Sfinal-Sinitial=N1Kln(V/V1)+N2Kln(V/V2)

it is clear that entropy change>0

but if we consider entropy change is 0 then we should introduce lnN! part in the definition of S

similarly to use the said we need stirling formula,

lnN!=NlnN-N

then total entropy S=NK(V/Nh^3(2*pie*m/beta)^3/2)+5/2K

which leads us to Sackur-Tetrode formula

using this we get entropy difference =N1KLn(VN1/NV1)+N2Kln(VN2/NV2)

taking number density d= constant we get change in entropy=0

b)now change in entropy is defined by dQ/T

for ideal gas dQ=dU-dW

then if change in entropy=0 then dU-dW=0 which gives dW=dU thaat means work done =change in internal energy

c) according to second law of thermodynamics the change in entropy remain constant for ideal,isolated system.

Now from Bob's perspective we know that equilibrium means no transfer heat/work/energy nothing. But from a we see that change in entropy>0(before using stirling formula) that means it is increasing but at equilibrium it should be constant.So there is violation from Bob's perspective.

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