A Hint of Quantum Theory — Question 3

Question 3

A two-level mode has vanishing heat capacity at both temperature extremes

One mole of independent nondegenerate two-level modes has internal energy $U=R\Theta/(e^{\Theta/T}+1)$, with constant gap parameter $\Theta>0$ and ground energy zero. Define its molar contribution $C=dU/dT$.

1. Derive $C$ and calculate $C/R$ at $T=\Theta/2$.

2. Find the low- and high-temperature limits of $C$. Explain why the high-temperature limit differs from a classical harmonic mode with heat capacity $R$.

Question 3 - Solution

A two-level mode has vanishing heat capacity at both temperature extremes

(a) Method and calculation

$$x=\Theta/T,\quad\frac{dx}{dT}=-\frac xT,\quad C=R\frac{x^2e^x}{(1+e^x)^2}.$$$$T=\Theta/2:\quad C/R=\frac{4e^2}{(1+e^2)^2}\approx.419974.$$

(b) Interpretation and further calculation

$$x\to\infty:\quad C/R\sim x^2e^{-x}\to0.$$$$x\to0:\quad C/R\sim x^2/4\to0,\quad U\to R\Theta/2.$$Only two levels exist in this stipulated model, so the energy saturates. A harmonic oscillator has an unlimited ladder and can continue storing energy as temperature rises.

Physical check

A mode can be populated but contribute little additional heat capacity when its populations barely change with temperature.