Question

We have been studying the Boltzmann and Saha equations, especially as the pertain to stellar atmospheres. The next set of questions will provide several examples of how to use the equations in practice.

A. Find the temperature at which the number density of H atoms in the fundamental state (n=1) are equal to the of its first excited state (i.e. the Blamer series)

Answer 1

To find the temperature at which the ground state hydrogen atom density equals its first excited state density, the Boltzmann equation is used, considering the statistical weights and the energy difference between the states.

Step-by-step explanation:

To find the temperature at which the number density of H atoms in the fundamental state (n=1) are equal to that of its first excited state, the Boltzmann equation can be utilized. This equation is represented as n1/n2 = (g1/g2) * e-ΔE/kT, where n1 and n2 are the number densities of the two states, g1 and g2 are the statistical weights of these states, ΔE is the difference in energy between the two states, k is the Boltzmann constant, and T is the temperature in Kelvin. For hydrogen, the energy difference between the ground state and first excited state corresponds to the Balmer series. Given that the statistical weights are related to the degeneracy of the energy levels, for hydrogen the ground state (n=1) has a statistical weight g1=2, and the first excited state (n=2) has a statistical weight g2=8. Thus, setting n1 equal to n2 and solving for T, using the known values of ΔE and k, will yield the desired temperature.