Question

A system of particles occupying single-particle levels and obeying Maxwell-Boltzmann statistics is in thermal contact with a heat reservoir at temperature, T. If the population distribution in the non-degenerate energy levels with energies 21.5×10^-3eV, 12.9×10^-3eV and 4.3×10^-3 are 8.5%, 23% and 63% respectively. What is the average temperature of the system?

Answer 1

Answer on Question #83665 - Physics - Molecular Physics - Thermodynamics

A system of particles occupying single-particle levels and obeying Maxwell-Boltzmann statistics is in thermal contact with a heat reservoir at temperature $T$ . If the population distribution in the nondegenerate energy levels with energies $21.5 \times 10^{-3}$ eV, $12.9 \times 10^{-3}$ eV and $4.3 \times 10^{-3}$ eV are $8.5\%$ , $23\%$ and $63\%$ , respectively, what is the average temperature of the system?

Solution:

Denote the energy levels by $E_{1} = 21.5 \times 10^{-3} \mathrm{eV}$ , $E_{2} = 12.9 \times 10^{-3} \mathrm{eV}$ and $E_{3} = 4.3 \times 10^{-3} \mathrm{eV}$ , and the corresponding populations by $P_{1} = 8.5\%$ , $P_{2} = 23\%$ and $P_{3} = 63\%$ . In thermal distribution with Maxwell-Boltzmann statistics, the populations $P_{i}$ and $P_{j}$ on the respective levels $E_{i}$ and $E_{j}$ are related as

$$\frac {P _ {i}}{P _ {j}} = e ^ {\frac {E _ {j} - E _ {i}}{k T}},$$

where $k = 8.6 \times 10^{-5} \, \text{eV/K}$ is the Boltzmann constant. Taking the logarithm of this relation, we obtain $\left( \frac{E_j - E_i}{kT} \right) = \log \left( \frac{P_i}{P_j} \right)$ , whence $T = \left( \frac{E_j - E_i}{k \log \left( \frac{P_i}{P_j} \right)} \right)$ . Substituting for $i$ and $j$ any pair of numbers from $\{1, 2, 3\}$ , we obtain the sought answer. Thus, $T = \left( \frac{E_1 - E_2}{k \log \left( \frac{P_2}{P_1} \right)} \right) \approx 100K$ .

Answer: 100 K.

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