Textbook Mistake: Ionization Energy of He+

[p.tryon]

This is a question from a chem textbook but it seems relevant to electrodynamics/quantum mechanics. The energy to ionize a hydrogen atom in its ground state is given by:

En = -Rhc / n²

Ei = Rhc (because the electron is being taken to infinity) = 1312kJ/mol

The book claims that to ionize a He+ ion (which has 2 protons in its nucleus) would require 4 times the amount of energy. However the potential energy of two charges separated by distance r is given by

P.E. = kq₁q₂/r

Since the charge of an electron is the same in both cases (the H atom and the He ion) and the nuclear charge is doubled in the case of the He+ ion- shouldn't the P.E. at infinity (therefore ionization energy) also double? (not quadruple as the book claims!)

 

[Doc Al]

The book is correct. To a rough approximation, the energy depends on the square of the number of protons.

 

[p.tryon]

Thanks for your reply. Why does it depend on the square of the number of protons? Doesn't that contradict the equation below?

P.E. = kq₁q₂/r

 

[Physics Monkey]

Hi p.tryon,

You may find it useful to think about how the effective value of $$r$$ in your equation depends on the number of protons.

 

[Doc Al]

Thanks for your reply. Why does it depend on the square of the number of protons? Doesn't that contradict the equation below?

P.E. = kq₁q₂/r

No, there's no contradiction. In the Bohr model of the atom, realize that r is inversely proportional to Z (the number of protons) so the total energy ends up being proportional to Z². (You might want to review how energy levels are derived in the Bohr model, which is admittedly only an approximation for multi-electron atoms. But it's OK for this purpose with helium, since there are only two electrons.)

 

[p.tryon]

Hello! Wow I see thank you! That makes sense