TBR Atomic Theory Question

[LuminousTruth]

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TBR says the energy level distance between from n=2 to n=1 is greater than n=4 to n=2.

I know the energy circles are closer together with each increasing energy level, but wouldn't that mean the energy released from n=4 to n=2 still be MORE energy than n=2 to n=1? Since there is two energy levels? If it was from like n=4 to n=2 VS n=3 to n=1, I would say more energy is released from n=3 to n=1.

How do we know the energy released from n=2 to n=1 is greater than n=4 to n=2?

 

[cortana431]

The energy released is n2-n1 and n4-n2. The distance between energy levels increases as energy level increases, and n=1 is the lowest energy, so the n2-n1 will result in the largest value. It doesn't matter as much that n=4 -> n=2 changes by 2 energy levels because the distance decreases exponentially. Look at this figure for clarification. http://www.tannerm.com/Quick_atom/series.gif

 

[circulus vitios]

The potential energy of an electron is given by

.

Focus on E∝1/n^2, with n being the principal quantum number, and ignore all of the other stuff.

ΔE = Efinal - Einitial

Dropping from n=2 to n=1 you have: (1/1^2)-(1/2^2) = 3/4 = 9/16.
Energy is reduced by a factor of 9/16.

Dropping from n=4 to n=2 you have: (1/2^2)-(1/4^2) = 3/16.
Energy is reduced by a factor of 3/16.

So clearly there is a bigger change in energy when you're going from n=2 to n=1.