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

An ensemble of system is in thermal equilibrium with a reservoir for which kT = 0.025 eV. State A has an energy that is 0.1 eV above that of state B. If it is assumed the systems obey Maxwell-Boltzmann statistics and that the degeneracies of the two states are the same, then the ratio of the number of the systems in state A to the number in state B is

A. e⁺⁴
B. e^+0.25
C. 1
D. e^-0.25
E. e^-4

(GR0177 #77)

Solution:

Maxwell-Boltzmann distribution: $N_i \propto g_i e^{-\varepsilon _i/kT}$

$\\kT=0.025 $ eV$ \\g_A=g_B \\\varepsilon_A=0.1+\varepsilon_B$

$\begin{aligned} \\\frac{N_A}{N_B}&= \frac{g_A e^{-\varepsilon _A/kT}}{g_B e^{-\varepsilon _B/kT}} \\&=e^{-\varepsilon_A/kT}\times e^{\varepsilon_B/kT} \\&=e^{(\varepsilon _B-\varepsilon _A)/kT}=e^{(\varepsilon _B-0.1-\varepsilon_B)/kT} \\&=e^{(-0.1)/(0.025 )}=e^{-4} \end{aligned}$

Answer: E

Thermal Physics - Maxwell-Boltzmann Statistics

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