A large number of neon atoms are in thermal equilibrium....

A large number of neon atoms are in thermal equilibrium. What is the ratio of the number of atoms in a 5 state to the number in a 3$p$ state at (a) $300 \mathrm{K} ;$ (b) $600 \mathrm{K} ;$ (c) 1200 $\mathrm{K} ?$ The energies of these states, relative to the ground state, are $E_{5 s}=20.66 \mathrm{eV}$ and $E_{3 p}=18.70 \mathrm{eV} .$ (d) At any of these temperatures, the rate at which a neon gas will spontaneously emit 632.8 -nm radiation is quite low. Explain why.

A large number of neon atoms are in thermal equilibrium. What is the ratio of the number of atoms in a 5 state to the number in a 3$p$ state at (a) $300 \mathrm{K} ;$ (b) $600 \mathrm{K} ;$ (c) 1200 $\mathrm{K} ?$ The energies of these states, relative to the ground state, are $E_{5 s}=20.66 \mathrm{eV}$ and $E_{3 p}=18.70 \mathrm{eV} .$ (d) At any of these temperatures, the rate at which
a neon gas will spontaneously emit 632.8 -nm radiation is quite low. Explain why.

Key Concepts

Spontaneous Emission

Spontaneous emission refers to the process where an excited atom drops to a lower energy state and emits a photon without any external influence. The rate of spontaneous emission depends on the population of atoms in the excited state and the intrinsic properties of the atomic transition. In cases where the excited state is sparsely populated—often because the energy required to access that state is much higher than what thermal energy provides—the overall emission rate is consequently very low.

Boltzmann Distribution

The Boltzmann distribution describes the relative populations of particles in different energy states in thermal equilibrium. It states that the number of atoms in a state with energy E is proportional to exp(-E/kT), where k is Boltzmann’s constant and T is the temperature. This principle is fundamental for predicting the ratio of populations in atomic states with different energies.

Atomic Energy Levels

Atoms have quantized energy levels, and each state (such as 5s and 3p) corresponds to a specific energy relative to the ground state. The energy difference between these levels is critical in determining the likelihood of transitions and the thermal population of each state. A larger energy difference compared to kT typically results in significantly fewer atoms being found in the higher-energy state.

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