What is the ionization energy required to raise a hydrogen atom from the n=2 state to the n = infinity state, using the Bohr model?

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The ionization energy is simply the difference in energy between two states, in this case the `n=2` state and the `n=infinity` state.

In the Bohr model (which is a simple but very good approximation for the hydrogen atom), all electron states are modeled as having angular momentum that is some whole number `n` times Planck's reduced constant `h/{2pi}`:

`L = n h/{2 pi}`

This results in energy levels defined by n, such that the energy of each is inversely proportional to `n^2` , with a constant derived from more fundamental constants (but we can just take it as given at -13.6 eV):

`E = - {13.6 eV}/{n^2}`

Then, the n = 2 state has this energy:

`E = - {13.6 ev}/{2^2} = - 3.4 eV`

And there is in fact an n = infinity state, the limit at which the electron's energy reaches zero:

`E = - {13.6 eV}/{infty^2} = 0`

The ionization energy is the difference between these two, which is 3.4 eV. But we are asked for the energy in kJ/mol, so we need to do a unit conversion. There are `1.602*10^{-22}` kilojoules per electron-volt, and `6.02*10^23 ` electrons per mole.

`3.4 eV (1.602*10^{-22} {kJ}/{eV}) (6.02*10^{23} /{mol}) = 327.9 {kJ}/{mol}`
This rounds to 328 kJ/mol, which is answer D.

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