Discussion
Consider an ensemble of systems having only two energy levels having energies 0 and ε respectively. The average energy of the system at temperature T is :
Answer: B
The average energy is \(\frac{\epsilon}{e^{\epsilon/kT} + 1}\).
With levels 0 and \(\epsilon\), the partition function is \(Z = 1 + e^{-\epsilon/kT}\). The mean energy is the Boltzmann-weighted average:
\[ \begin{aligned} \langle E\rangle &= \frac{0\cdot 1 + \epsilon\, e^{-\epsilon/kT}}{1 + e^{-\epsilon/kT}} \\ &= \frac{\epsilon}{e^{\epsilon/kT} + 1} \end{aligned} \]
Check the limits: at T = 0 it gives 0, and at high T it tends to \(\epsilon/2\), as both levels become equally occupied. The form with \(-1\) in the denominator is the mean energy of a harmonic oscillator mode (Planck's formula), which has infinitely many levels. The forms with \(e^{-\epsilon/kT}\) give the wrong limits, and the one with \(-1\) even becomes negative.
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