Suppose that an atom has two energy levels separated by an energy difference $\Delta E$. When such atoms are in thermal contact with a reservoir at temperature $T$, there is roughly 1 atom in the higher state for every 4 in the lower state. What is the approximate ratio of $\Delta E$ to $k_{B} T$ ?
A) $\frac{1}{4}$
B) 4
C) In 4
D) $-\ln 4$
E) $e^{-1 / 4}$
F) $e^{-4}$
G) Other (specify)

Question

Quizplus · Accepted Answer

The ratio of populations in two energy levels at thermal equilibrium is given by the Boltzmann distribution, which is $\frac{N_2}{N_1} = e^{-\Delta E / k_B T}$. Given that $\frac{N_2}{N_1} = \frac{1}{4}$, solving for $\Delta E / k_B T$ gives $\ln 4$.

Suppose That an Atom Has Two Energy Levels Separated by an Energy