15-Consider two gaseous systems interacting mechanically and thermally but not d

Question

15-Consider two gaseous systems interacting mechanically and thermally but not diffusively. They are isolated from the rest of the Universe, and their total volume is fixed at Vo. (a) Show that the total number of accessible states, o, is a very sensitive function of the distribution of volume between them. (See equation 6.11 and the argument preceding it for the dependence of on V) (b) Since =- th, show that asvav, = as/ava when the systems are in equilibrium. (c) From this, what can you conclude about how pi, T.pa, and T2 are related in equilibrium?

Explanation / Answer

a. consider an isolated system in equilibrium whose volume is V

then assume its energy lies in the region E , E + dE

then total energy of the system is the KE of its constituent particles

hence

E = (1/2m) sum of (p^2) [ where p is momentum of individual particles and m is mass of each particle]

hence number of states available between energies E and E + dE is

O(E,V) = k[integrate[d^3r1....d^3rNd^3p1....d^3pN]] from E to E + dE

here

d^3r = dxidyidzi [ volume occupied by ith particle]

d^3p = dpixdpiydpiz [ momentum of particle in that confined volume]

k is a constant

for an ideal gas, E does not dep[end on position of particls]

ans integral dr extgends all over the volume of the container

O(E,V) = k'*V^N

k' is another constant, V is volume of the container, N is number of particles

hence we can see that, N is very large, so number of microstates available O is a very sensitive fuction of V

b. for the two constaiers attached mechanically

dV1 = -dV2

where V1 and V2 are volumes of both the containers respectively

now, entropy is given by

S = k*ln(O) where k is some constant and O is number of microstates available

now, S1 = k*ln(k'*V1^N)

S2 = k*ln(k'*V2^N)

so, dS1/dV1 = k*k'*N(V1^(N-1))/(k'*V1^N)

dS1/dV1 = kN/V1

similarly

dS2/dV2 = k*k'*N(V2^(N-1))/(k'*V2^N)

dS2/dV2 = kN/V2

when systems are in equilibrium

V1 = V2

so ds1/dv1 = ds2/dv2

c. hence at equilibrium

V1 = V2

P1 = P2

T1 = T2

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