Two bodies, and , are enclosed together in a rigid isolated container. Each body has…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Thermodynamic potentials and the formal structure of thermodynamics · Thermodynamic potentials and the formal structure of thermodynamics

Thermal Physics, Ch Chemical potential, open systems and chemical equilibrium

Problem

Two bodies,

1

and

2

, are enclosed together in a rigid isolated container. Each body has a fixed volume, but they may now exchange both energy and molecules with each other, so that

U_1 + U_2

and

N_1 + N_2

are fixed while

U_1

and

N_1

may vary independently. The coupling is weak, so the total entropy is

S_{\rm tot} = S_1(U_1,V_1,N_1) + S_2(U_2,V_2,N_2)

. The two bodies are not assumed to start at the same temperature. ((a)) Write

\mathrm{d}S_{\rm tot}

in terms of

\mathrm{d}U_1

and

\mathrm{d}N_1

, using the entropy derivatives of the extended identity

\mathrm{d}U = T\,\mathrm{d}S - p\,\mathrm{d}V + \mu\,\mathrm{d}N

. ((b)) Hence show that at equilibrium

T_1 = T_2

and

\mu_1 = \mu_2

both hold.

Browse the problem library