Ionization energy calculations

[fsci]

Why isn't the ionization energy of an electron equal to it's energy level such that:
E(electron)= -13.6(Z^2/n^2) = IP for that electron
But instead it is equal to the energy difference in energy between the atom and its ionized cation:
IP = E(A)-E(A+)

 

[Quantum Defect]

Why isn't the ionization energy of an electron equal to it's energy level such that:
E(electron)= -13.6(Z^2/n^2) = IP for that electron
But instead it is equal to the energy difference in energy between the atom and its ionized cation:
IP = E(A)-E(A+)

For a hydrogen atom, this would be true (ignoring for the moment, things like the Lamb shift, etc.)

The problem with multi-electron atoms is that the "truth" is that the picture of electrons sitting in discrete energy levels given by the equation that you have above is not right. The primary problem is that electron-electron repulsions are present and are not insignificant. Imagine that you have a two-electron atom. How many electron-electron repulsions do you have to consider? You could apportion this interaction energy (50:50) to the two electrons to calculate an effective energy level for each electron. But when you ionize one electron, what happens to this interaction? The "energy level" of the remaining electron changes, too, no?

Chemists play all sorts of games to take into account the effect of electron-electron interactions. You will see things like Zeff (an effective nuclear charge) discussed. In other contexts, you will see fudge factors on "n" called a "quantum defect" -- where have I seen that before... no matter.

The ionization energy is, by definition, equal lto the energy required to ionize the atom, which is the Delta E for:

A ----> A+ + e-

 

[fsci]

For a hydrogen atom, this would be true (ignoring for the moment, things like the Lamb shift, etc.)

The problem with multi-electron atoms is that the "truth" is that the picture of electrons sitting in discrete energy levels given by the equation that you have above is not right. The primary problem is that electron-electron repulsions are present and are not insignificant. Imagine that you have a two-electron atom. How many electron-electron repulsions do you have to consider? You could apportion this interaction energy (50:50) to the two electrons to calculate an effective energy level for each electron. But when you ionize one electron, what happens to this interaction? The "energy level" of the remaining electron changes, too, no?

Chemists play all sorts of games to take into account the effect of electron-electron interactions. You will see things like Zeff (an effective nuclear charge) discussed. In other contexts, you will see fudge factors on "n" called a "quantum defect" -- where have I seen that before... no matter.

The ionization energy is, by definition, equal lto the energy required to ionize the atom, which is the Delta E for:

A ----> A+ + e-

Ahhhhhhh, that makes more sense now, thanks a ton!
So if I were to calculate E(A+) for the cation using the new Zeff values for the remaining electrons I am basically accounting for the decrease in electron-electron repulsion due to one less electron? And using IP= -13.6eV (Zeff^2/n^2) for the ionized electron does not make sense because it assumes there is no change in energy in the cation even though electron-electron repulsion has decreased?

 

[Quantum Defect]

Ahhhhhhh, that makes more sense now, thanks a ton!
So if I were to calculate E(A+) for the cation using the new Zeff values for the remaining electrons I am basically accounting for the decrease in electron-electron repulsion due to one less electron? And using IP= -13.6eV (Zeff^2/n^2) for the ionized electron does not make sense because it assumes there is no change in energy in the cation even though electron-electron repulsion has decreased?

I think that sometimes people calculate a Zeff from the IP, but remember that all of this is a fudge to take into account electron-electron repulsions. It is useful, to some extent, to compare the "shielding" provided by electrons -- Inorganic Chemistry textbooks sometimes talk about ways to estimate what Zeff is based upon the electron configuration. These are all pretty crude approximations.