Figure 39.19 a shows the energy levels of the sodium atom....

Figure $39.19 \mathrm{a}$ shows the energy levels of the sodium atom. The two lowest excited levels are shown in columns labeled ${ }^{2} P_{3 / 2}$ and ${ }^{2} P_{1 / 2}$. Find the ratio of the number of atoms in a ${ }^{2} P_{3 / 2}$ state to the number in a ${ }^{2} P_{1 / 2}$ state for a sodium gas in thermal equilibrium at $500 \mathrm{~K}$. In which state are more atoms found?

Key Concepts

Thermal Equilibrium in Statistical Mechanics

In thermal equilibrium, the distribution of particles among various energy states is fixed and determined solely by the overall temperature and the properties of the energy levels (their energies and degeneracies). This equilibrium condition is essential to applying statistical mechanics principles, where the relative number of particles in each state can be computed using the Boltzmann distribution.

Atomic Energy Level Splitting

Atomic energy level splitting occurs due to interactions such as spin-orbit coupling, which causes degenerate states to split into sublevels with different total angular momentum values. This splitting leads to states with distinct energies and degeneracies, which must be taken into account when determining the distribution of atoms among these states in a thermal ensemble.

Boltzmann Distribution

The Boltzmann distribution describes how particles in a system at thermal equilibrium are distributed among available energy states. It provides the relative population for each state, factoring in both the energy of the state and the temperature of the system by using the exponential term exp(?E/(kT)). This concept is fundamental in predicting the occupancy of different energy levels in thermal systems.

Degeneracy of Energy Levels

Degeneracy refers to the number of quantum states that have the same energy level. In systems where the energy level is determined by quantum numbers such as angular momentum, the degeneracy is given by factors such as (2J+1). The relative populations of states must account for these degeneracies; states with higher degeneracy can have a higher population even if they are at the same energy, or may outweigh differences in energy when combined with the Boltzmann factor.

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