For Na atoms and Mg^+ ions, compare the ratios of the number...

For Na atoms and $\mathrm{Mg}^{+}$ ions, compare the ratios of the number of atoms or ions in the $3 p$ excited state to the number in the ground state in
(a) a natural gas-air flame (1800 K).
(b) a hydrogen-oxygen flame (2950 K).
(c) an inductively coupled plasma source (7250 K).

Key Concepts

Local Thermodynamic Equilibrium (LTE)

Local Thermodynamic Equilibrium is an assumption used in many thermodynamic and spectroscopic analyses where the distribution of particles among energy levels is determined solely by the local temperature. Under LTE, the Boltzmann distribution accurately describes the population ratios of different energy levels, allowing for consistent analysis across systems like flames and plasmas.

Statistical Weights (Degeneracy)

Statistical weights, or degeneracies, represent the number of quantum states that share the same energy level. These weights modify the population ratios predicted by the Boltzmann distribution by effectively scaling the contribution of each energy level based on how many states it comprises.

Excitation Energy

Excitation energy is the amount of energy required to elevate a particle from its ground state to an excited state. It plays a central role in determining state populations because the higher the excitation energy relative to the thermal energy available, the fewer particles will be found in the excited state, as results from the Boltzmann factor.

Temperature Dependence

Temperature critically influences the population of energy states. As temperature increases, more particles have sufficient energy to overcome the energy gap between the ground and excited states, thus increasing the ratio of particles in the excited state relative to the ground state.

Boltzmann Distribution

The Boltzmann distribution is a statistical tool that explains how particles in a thermalized system occupy different energy levels. It provides the ratio of the number of particles in an excited state to those in the ground state through the exponential factor exp(-?E/kT), where ?E is the energy difference between the states, k is Boltzmann’s constant, and T is the absolute temperature.

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