Question

What is the total energy needed to see all 15 spectral lines for the electronic transitions involving `n = 1` up to and including `n = 6`, assuming every transition occurs only once? What wavelength is needed to cause this in angstroms?

Answer 1

Well, if you want 15 spectral lines, then you're going to need 15 transitions. That occurs with 6 energy levels involved.

`n = 1->6,5,4,3,2`
`n = 2->6,5,4,3`
`n = 3->6,5,4`
`n = 4->6,5`
`n = 5->6`

If we assume that SOMEHOW, all the energy that went in managed to accomplish ALL of these transitions WITHOUT ANY WEIGHTED BLEEDING ONTO ANY GIVEN TRANSITION, then we need to calculate `DeltaE` for all `15` transitions...

`DeltaE_16 = -"13.61 eV" cdot (1/1^2-1/6^2) = ???`
`DeltaE_15 = -"13.61 eV" cdot (1/1^2-1/5^2) = ???`
`DeltaE_14 = -"13.61 eV" cdot (1/1^2-1/4^2) = ???`
`DeltaE_13 = -"13.61 eV" cdot (1/1^2-1/3^2) = ???`
`DeltaE_12 = -"13.61 eV" cdot (1/1^2-1/2^2) = ???`
`DeltaE_26 = -"13.61 eV" cdot (1/2^2-1/6^2) = ???`
`DeltaE_25 = -"13.61 eV" cdot (1/2^2-1/5^2) = ???`
`DeltaE_24 = -"13.61 eV" cdot (1/2^2-1/4^2) = ???`
`DeltaE_23 = -"13.61 eV" cdot (1/2^2-1/3^2) = ???`
`DeltaE_36 = -"13.61 eV" cdot (1/3^2-1/6^2) = ???`
`DeltaE_35 = -"13.61 eV" cdot (1/3^2-1/5^2) = ???`
`DeltaE_34 = -"13.61 eV" cdot (1/3^2-1/4^2) = ???`
`DeltaE_46 = -"13.61 eV" cdot (1/4^2-1/6^2) = ???`
`DeltaE_45 = -"13.61 eV" cdot (1/4^2-1/5^2) = ???`
`DeltaE_56 = -"13.61 eV" cdot (1/5^2-1/6^2) = ???`

I'll leave the mathematical slog for you to work out, and when you do that,

`DeltaE = DeltaE_16 + . . . + DeltaE_56` (15 terms)

`= E_"photon" = hnu = (hc)/lambda`

Therefore,

`lambda = (hc)/(DeltaE) " m"`

And since `"1 m" = 10^(10) "angstroms"`,

`color(blue)(lambda = (10^10 hc)/(DeltaE) " angstroms")`