# Text book mistake?

## Main Question or Discussion Point

This is a question from a chem text book 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!)

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

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
Homework Helper
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
Mentor
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.)

Hello! Wow I see thank you! That makes sense