Principle Quantum Number Question

[wilson_chem90]

Homework Statement

The hydrogen spectrum contains a blue line with a wavelength of 434 nm. Photons of blue light are emitted when hydrogens electron drops from the fifth energy level to a lover energy level. What is the lower energy level?

Homework Equations

En = 13.6 eV/n^2
E = hc/lambda

The Attempt at a Solution

First I'm going to find E.

E = hc/lambda
= (6.63 x 10^-34 J s)(3.00 x 10^8 m/s) / (4.34 x 10^-7 m)
= 4.583 x 10^-19 J

(1.60 x 10^-19 eV/J) x (4.583 x 10^-19 J)
= 2.86 eV

Next i tried 2 different approaches, I am not sure either is correct though. I used En = -(13.6eV/n^2) - (-13.6eV/n^2) and i just used En = -(13.6eV/n^2)

1st approach:
En = -(13.6eV/n^2) - (-13.6eV/n^2)

I rearranged this equation to be n^2 = 2(13.6eV)/En5^2 (i'm pretty sure that's incorrect)
so n^2 = 2(13.6 eV)/2.86(25)
n = sqrt(0.38041958)
n = 0.62

2nd approach:
En = -(13.6eV/n^2)
Now i rearranged this equation to be n^2 = En + 13.6eV
= 2.86eV + 13.6eV
n = sqrt16.46 eV
= 4.06

So I am pretty sure my problem lies within rearranging the equations. Any suggestions? thanks

 

[wilson_chem90]

I also just realized i can rearrange En = -(13.6eV/n^2) to be n^2 = 13.6eV/En

so using that equation it would be n^2 = 13.6eV/2.86eV
n = sqrt4.75
= 2.2

this one makes a lot more sense, seeing how the units cancel

 

[SmashtheVan]

it looks like your problem is with the relationship between the energy of the photon and the change in energy levels.

E_{photon}= E_{f}- E_{i}

where the initial state is the 5th energy level, E of the photon you figured out was 2.86, and the final state is what we're solving for.

so,
E_{n}= E_{photon} + E_{5}

and

\frac{13.6}{n^{2}} = 2.86 + \frac{13.6}{5^{2}}

now its just algebra to find out n

 

[wilson_chem90]

thanks, i thought it was that, but i might've messed up the rearrangement, I am not very strong when it comes to that... i got n^2 = 13.6eV - 0.544eV/2.86eV (the 0.544 is 13.6/5^2)
I worked it out and got an answer of 2.14

 

[SmashtheVan]

thanks, i thought it was that, but i might've messed up the rearrangement, I am not very strong when it comes to that... i got n^2 = 13.6eV - 0.544eV/2.86eV (the 0.544 is 13.6/5^2)
I worked it out and got an answer of 2.14

i still think you're doing something a bit off here. the .544 and 2.86 should be summed, not divided. and then divide 13.6 by the new sum for your answer. it should give you n^2=(approx.)4, and n=2

remember that n is a discrete energy level, which are integers, and if your decimal is off by more than a tenth or two from the whole, something might be awry in your calculations.

 

[wilson_chem90]

but when i do that i get a decimal, not a whole number. and its 0.544