A rigid vessel whose walls are in good thermal contact with a large bath at is…

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 Thermodynamic potentials, Legendre transforms and Maxwell relations

Problem

A rigid vessel whose walls are in good thermal contact with a large bath at

T = 310\ \mathrm{K}

is divided by a partition. One side holds

N = 3.00\times10^{23}

atoms of helium, treated as a monatomic ideal gas; the other side is evacuated. The partition is removed and the gas fills the whole vessel, whose volume is

2.50

times the volume the gas first occupied. The vessel's total volume is fixed and the bath holds the temperature at

310\ \mathrm{K}

. ((a)) Show that

\mathrm{d}z = \dfrac{3Nk_B}{2T}\,\mathrm{d}T + \dfrac{Nk_B}{V}\,\mathrm{d}V

is an exact differential in the variables

T

and

V

at fixed

N

, and name the function of state whose differential it is. ((b)) Find the entropy change of the gas in the free expansion. ((c)) Find

\Delta F

for the gas, and say what its sign tells you about the direction of the change. ((d)) What is the largest work the gas could have delivered in going between the same two states at

310\ \mathrm{K}

, and how much did it deliver here?

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