Question

Consider a hydrogen plasma on the surface of a compact star at a temperature of t = 10⁵ K. If the discreet energy levels of hydrogen are given by E _n = -13.6eV/n ² (n - 1,2,3...) Determine the fraction of hydrogen in the ionized state n = infinity, compared to the number of atoms in the ground state. Comment on the level of ionization of hydrogen at this temperature.

Answer 1

Energy of the hydrogen atom in the ground state, Ei = - 13.6 / 1 = - 13.6
Energy when n = infinity, Ef = 0
Change in energy, E = 0 - (- 13.6) = 13.6 eV
E = 13.6 x 1.6 x 10^-19
= 21.76 x 10^-19 J

According to Boltzmann distribution, the ratio of the number of atoms in the ground state to the number of atoms in the excited state is given as
Ni/Nf = e^-[E/kT]
kT = (1.381 x 10^-23) x 105
= 1.45 x 10^-21 J

E/kT = (21.76 x 10^-19) / (1.45 x 10^-21)
= 1500.64
Ni/Nf = e^-1500.64
= 1.912 x 10^-652

ie, the number of excited states at n = infinity is very low, almost nil at this temperature.