Hydrogen atoms:

$($a$)$ What is the minimum kinetic energy in electron volts that an electron must have to be able to ionize a hydrogen atom that is in its first excited state $($that is$,$ remove the electron from being bound to the proton$)?(5$ points$)$

$($b$)$ If electrons of energy $13\mathrm{.}15$ eV are incident on a gas of hydrogen atoms in their ground state, what are the energies of photons that can be emitted by the excited gas?

$(8$ points$)$

$($c$)$ The Boltzmann factor is $.$ If we have hydrogen gas at a temperature of $69,600K($twelve times hotter than the surface of the sun$),$ what fraction of the hydrogen atoms $($comparted to the ground state$)$ are in the third excited $($electronic$)$ state? $E$ is the gap in energy between the ground state and the $3$rd excited state. Boltzmann's constant $k_B=1\mathrm{.}38{10}^{-23}\frac{J}{K}.(5$ points$)$