Achieve Week 2 Question 8

[Noah_Amancio_1J]

Hi, I had a question about the achieve homework for week 2. The question is:

Calculate the wavelength, in nanometers, of the spectral line produced when an electron in a hydrogen atom undergoes the transition from the energy level n=7
to the level n=1.

I understand that you have to use the En = - (hR)/(n^2) formula but I am not sure what to do after this step. For E7 I got -4.45x10^-20 J and for E1 I got -2.18 x 10^-18 J.

[Brian Huynh 2L]

The energy (E) of the photon emitted is equivalent to the difference between the energy levels, therefore, Efinal - Einitial = E. And because E = hv (Planck's Constant x frequency) and c(speed of light) = lambda (wavelength) x v (frequency) we can find the relationship between the energy of the photon it's wave length;
Efinal - Einitial = hv
c= (lambda)(v)
rearrange:
v = c / lambda
Plug in c/lambda to represent v in the equation:
Efinal - Einitial = hc/ lambda
Isolate lambda:
lambda (Efinal - Einitial) = hc
lambda = hc / (Efinal - Einitial)

Efinal; -(hR)/(1^2)
Einitial; -(hR)/(7^2)

You should get a negative number, but remember the energy of a photon cannot be negative; this number is negative because when an eletron transitions to a lower energy level it loses energy by emitting it as a photon (here we are calculating how much energy an electron loses when transition between n=7 to n=1). Therefore, the energy of the photon will be the absolute value of the number you get.

[Erika Patel 3I]

Yes, you are on the right track! From there you need to find the change in energy (E_final-E_initial -- i.e. E₁-E₇) when going from energy level n=7 to n=1. Then you can combine the equations E=hv and c = v to form . Rearrange for wavelength, plug in your energy value, planck's constant and speed of light and you will get wavelength. Then convert to nm.

[Joseph Fisher 2K]

You can use this equation to solve it: 1/λ = (1.0974 x 10^7)(1/(n1)^2 - 1/(n2)^2)

With this equation, you should be able to plug in your numbers and solve for λ. But make sure to note units and the request for the answer in nm.