In higher-temperature sources, sodium atoms emit a doublet...

In higher-temperature sources, sodium atoms emit a doublet with an average wavelength of $1139 \mathrm{nm}$. The transition responsible is from the $4 s$ to $3 p$ state. Calculate the ratio of the number of excited atoms in the 4s level to the number in the ground 3 s level in (a) an acetylene-oxygen flame $\left(3000^{\circ} \mathrm{C}\right)$ (b) the hottest part of an inductively coupled plasma source $\left(\sim 9000^{\circ} \mathrm{C}\right)$

In higher-temperature sources, sodium atoms emit a doublet with an average wavelength of $1139 \mathrm{nm}$. The transition responsible is from the $4 s$ to $3 p$ state. Calculate the ratio of the number of excited atoms in the 4s level to the number in the ground 3 s level in
(a) an acetylene-oxygen flame $\left(3000^{\circ} \mathrm{C}\right)$
(b) the hottest part of an inductively coupled plasma source $\left(\sim 9000^{\circ} \mathrm{C}\right)$

Key Concepts

Temperature Conversion

Accurate calculations in thermal physics require the use of absolute temperature scales, typically Kelvin, instead of Celsius. Converting temperatures from Celsius to Kelvin (by adding 273.15) is essential when applying the Boltzmann factor for computing population ratios in thermal plasmas or flames.

Thermal Excitation in High-Temperature Environments

In environments like acetylene-oxygen flames or inductively coupled plasma sources, high temperatures lead to significant thermal excitation of atoms. This excitation results in a measurable number of atoms occupying higher energy states, making the understanding of these conditions key to interpreting spectral emissions.

Boltzmann Distribution

The Boltzmann distribution governs the relative populations of particles among available energy states at thermal equilibrium. It shows that the number of atoms in an excited state compared to the ground state is proportional to exp(–?E/(kBT)), where ?E is the energy difference between the states, kB is the Boltzmann constant, and T is the absolute temperature. This concept is fundamental for calculating excitation ratios in systems like flames or plasmas.

Atomic Energy Levels

Atomic energy levels are the discrete states that electrons occupy in an atom. Transitions between these levels, such as from a higher to a lower state, result in the emission or absorption of photons with energies corresponding to the difference between the levels. Understanding these levels is crucial for explaining the spectral lines observed in various high?temperature sources.

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