How Accurate Is the Estimated Ionization Energy Formula for Hydrogen?

[CatWoman]

Homework Statement

Estimate the energy of an electron in a hydrogen atom and hence deduce an approximate formula for the ionization energy of hydrogen. How accurate is your formula?

Homework Equations

Don't know but it is in the field of quantum mechanics

The Attempt at a Solution

Ionisation Energy is energy required to remove an electron.

Electrostatic Potential (energy per unit charge) V=q/(4πε_0 r)= e/(4πε_0 r)

For a Bohr atom (Z=1), energy required to remove an electron from the atom with a nucleus of charge e and a radius r0 is the Electrostatic Potential Energy:

PE=qV = e^2/(4πε_0 r)×1/e
=(1.6×10^(-19))/(4π×8.85×10^(-12)×10^(-15) )
=1.4MeV

The equation estimates the distance of the electron from the proton in the nucleus as a precise value – this is not the case. An electron’s position is given by a wave-function probability density and it is in effect occupying the whole of the atom at once.

IS ANY OF THIS RIGHT?

 

[Matterwave]

1.4MeV is about 100,000 times too large.

I think it's because your radius appears to be 100,000 times too small.

You are right that this semi-classical model of the electron is wrong, but you can get a good estimate of the ionization energy using this. And I think that's what the question asks.