A Hint of Quantum Theory — Question 1

Question 1

An energy gap suppresses excited populations at low temperature

An internal molecular mode has two nondegenerate levels, $E_0=0$ and $E_1=\Delta$. Thermal equilibrium gives $P_1/P_0=e^{-\Delta/(k_BT)}$ and $P_0+P_1=1$. Let $\Theta=\Delta/k_B=400$ K.

1. Find the excited fraction at 100 K and 400 K.

2. Find the largest temperature for which the excited fraction is at most 1.00%. Explain why comparing $k_BT$ with the gap is a crossover estimate rather than an abrupt activation threshold.

Question 1 - Solution

An energy gap suppresses excited populations at low temperature

(a) Method and calculation

$$P_1=\frac{e^{-\Theta/T}}{1+e^{-\Theta/T}}=\frac1{e^{\Theta/T}+1}.$$$$P_1(100)=\frac1{e^4+1}\approx.0179862,\quad P_1(400)=\frac1{e+1}\approx.268941.$$

(b) Interpretation and further calculation

$$P_1\le.0100\ \Rightarrow\ e^{400/T}\ge99\ \Rightarrow\ T\le\frac{400}{\ln99}\approx87.0489\,\mathrm K.$$$$0<P_1(T)<\tfrac12\quad\text{for finite }T>0.$$Occupation changes smoothly; a specified small population defines a quantitative freeze-out criterion.

Physical check

An exponentially small excited population is not mathematically zero at finite positive temperature.