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

A large number of hydrogen atoms are in thermal equilibrium. Let $n_{2} / n_{1}$ be the ratio of the number of atoms in an $n=2$ excited state to the number of atoms in an $n=1$ ground state. At what temperature is $n_{2} / n_{1}$ equal to $(a) 10^{-12} ;$ (b) $10^{-8} ;(c) 10^{-4} ?$ (d) Like the sun, other stars have continuous spectra with dark absorption lines $($ see Fig. 39.9$) .$ The absorption takes place in the star's atmosphere, which in all stars is composed primarily of hydrogen. Explain why the Balmer absorption lines are relatively weak in stars with low atmospheric temperatures such as the sun (atmosphere temperature 5800 $\mathrm{K}$ ) but strong in stars with higher atmospheric temperatures.

A large number of hydrogen atoms are in thermal equilibrium. Let $n_{2} / n_{1}$ be the ratio of the number of atoms in an $n=2$ excited state to the number of atoms in an $n=1$ ground state. At
what temperature is $n_{2} / n_{1}$ equal to $(a) 10^{-12} ;$ (b) $10^{-8} ;(c) 10^{-4} ?$ (d) Like the sun, other stars have continuous spectra with dark absorption lines $($ see Fig. 39.9$) .$ The absorption takes place in the star's atmosphere, which in all stars is composed primarily of hydrogen. Explain why the Balmer absorption lines are relatively weak in stars with low atmospheric temperatures such as the sun (atmosphere temperature 5800 $\mathrm{K}$ ) but strong in stars with higher atmospheric temperatures.

Key Concepts

Boltzmann Distribution

The Boltzmann distribution describes the statistical distribution of particles over various energy states in thermal equilibrium. It indicates that the ratio of the number of particles in two energy levels is proportional to the ratio of their degeneracy factors multiplied by the exponential of the negative energy difference divided by the product of the Boltzmann constant and temperature, thereby linking excitation levels to temperature.

Atomic Energy Levels

Atomic energy levels, especially in hydrogen, are quantized states where electrons reside. The energy difference between these levels governs the probability of transitions, and in the case of hydrogen the precise energy levels (like n=1 and n=2) determine the absorption and emission of photons at specific wavelengths.

Temperature Dependence of Excitation

The population of excited states in an atom depends heavily on temperature. At low temperatures, thermal energy is insufficient to promote many electrons to higher energy levels, resulting in a low ratio of excited to ground state populations. Increasing the temperature raises the probability of electrons being thermally excited to higher energy levels, affecting spectral line strengths.

Stellar Atmospheres and Absorption Lines

The stellar atmosphere is the outer layer of a star where light is absorbed and emitted. The absorption lines seen in stellar spectra are formed when ions or atoms absorb photons at particular wavelengths corresponding to electron transitions. The temperature of the stellar atmosphere influences the population of electrons in various energy levels, thereby affecting the depth and strength of these absorption lines.

Balmer Series

The Balmer series consists of the set of spectral emission and absorption lines corresponding to electron transitions from higher energy levels to the n=2 level in hydrogen. The strength of the Balmer lines in a star’s spectrum is contingent on the number of hydrogen atoms with electrons in the n=2 state, which is directly influenced by the temperature of the star’s atmosphere.

Transcript

00:01 Okay, in this problem we're looking at hydrogen and we're investigating various ratios of occupancy and between the ground state and the excited state and the first excited state rather.

00:14 So we have this increasing ratio and we're asked to find for each of the cases the temperature that a system in thermal equilibrium would recreate these ratios.

00:26 So the equation that we're going to be using is the boltzman equation, which simply states that the ratio of occupancy for particular states is equal to the e to the energy difference between those states divided by kt, the temperature.

00:47 So the only constant we need for this problem is the boltzman constant 1 .38 times 10 to negative 3 joules per kelvin, and we should be able to solve for t.

00:56 We also, because this is hydrogen, we can use the bore model to find the differences, the different energy levels.

01:03 This would be the balmer series.

01:05 And once we have these occupancies, occupancy ratios, rather, and we can get a corresponding temperature, we can answer some questions about stellar atmospheres.

01:16 So let's jump into it.

01:19 Okay, so doing this calculation once, we can use the, we can use the bore model.

01:31 To find the energy levels.

01:32 Remember that e sub n is equal to negative 13 .6 ev over n squared.

01:41 Thus, for the n equals 1 case, it's simply negative 13 .6 ev.

01:49 For the n equals 2 case, it's 13 .6 divided by 4, which will come out to be negative 3 .4 ev.

01:56 So this delta e will be negative 10 .2 ev.

02:11 So then we can find the occupancy.

02:15 Well, not find it.

02:16 We know it.

02:16 We can relate it to the temperature, which is what we want.

02:22 Delta e over kt.