# 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?

##### 1 Answer

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_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

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