Physics Problems & Solutions: Thermal Physics - Maxwell-Boltzmann Statistics

Consider a system of N non-interacting particles confined in a volume V at a temperature such that the particles obey classical Boltzmann statistics. If the temperature is lowered to the point at which quantum effects become important, the pressure of the gas may differ depending on whether the particles are fermions or bosons. Let P_F be the pressure exerted by the particles if they are fermions. P_B be the pressure if they are bosons and, and P_C be the pressure the particles would exert if quantum effects are ignored. Which of the following is true?

A. $P_F = P_B = P_C$
B. $P_F > P_C > P_B$
C. $P_F > P_B > P_C$
D. $P_F < P_B < P_C$
E. $P_F < P_C < P_B$

(GR8677 #88)

Solution:

Ideal gas: $PV=NkT \rightarrow P=\frac{N}{V}kT$

Classical (Maxwell-Boltzmann Statistics): $E=\frac{3}{2}kT$

$P=\frac{N}{V}kT\rightarrow P_C = \frac{2}{3} \frac{N}{V} \frac{3}{2} kT=\frac{2}{3} \frac{NE}{V}$

For Bosons (photon): $E=3kT$

$P=\frac{N}{V}kT\rightarrow P_B = \frac{1}{3} \frac{N}{V} 3 kT=\frac{1}{3} \frac{NE}{V}$

→ $P_C>P_B$

For Fermion: $E=\frac{3}{5}E_F$
$E_F$ = Fermi Energy, $E_F \gg kT$

For the same temperature: $P_F>P_C>P_B$

Answer: B

Thermal Physics - Maxwell-Boltzmann Statistics

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