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A star is composed of 91% H and 9% He. What energy is required to completely ionize this gas mixture?

Energies of ionization are 13.6eV for H and 54.4eV and 24.5eV for the 1st and 2nd ionizations of He.

My method:

Let n be the total number of atoms, then n(H)=.91n and n(He)=.09n.

The energy required to completely ionize the gas is

(13.6 eV)(.91n)+(24.5+54.4 eV)(.09n)=(19.477 eV)n

Equating this with the average kinetic energy from rms speed (kinetic theory of gases),

(3/2)kT*n=(19.477 eV)*n

we find that T=151,000 K. So at 151,000 K the gas mixture is a pure plasma..

I'm unsure of the validity of my first step, where I find that the average energy of each atom would be 19.477 eV/atom. This energy is enough to break the ion potential on hydrogen, but not helium. Perhaps it is the remainder of this energy (kinetic energy of the free electrons, almost 6 eV) that ionizes the He? Also, in the completely ionized gas we could say that H+ is two particles (electron, proton) and He++ is three particles (2 electron, alpha) so perhaps n is multiplied at ionization; should this be taken into account?

Note I didn't use the Saha equation. As far as I know the Saha equation is not fit to predict complete ionization since the number density of neutral atoms would be 0...

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# Energy to completely ionize gas in star

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