Discussion
Consider a system with only two energy levels with energies \(E_{0} = 0\) and \(E_{1} = 1.3806 \times 10^{-23}\) J. The level \(E_{0}\) has degeneracy 2 and the level \(E_{1}\) has degeneracy 3. The partition function at T = 1 K is (\(K_{B} = 1.3806 \times 10^{-23}\) JK\(^{-1}\)) :
Answer: A
The partition function is \(z = 2e^{0} + 3e^{-1}\).
The partition function sums the Boltzmann factor of each level, weighted by its degeneracy: \(z = \sum g_{i}e^{-E_{i}/k_{B}T}\). At T = 1 K, \(k_{B}T = 1.3806 \times 10^{-23}\) J, exactly \(E_{1}\), so \(E_{1}/k_{B}T = 1\):
\[ \begin{aligned} z &= 2e^{-0} + 3e^{-1} \\ &= 2 + 3e^{-1} \approx 3.10 \end{aligned} \]
The options with 6 multiply the two degeneracies together, which would be the rule for combining two independent systems, not for summing over the levels of one system.
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