I don't understand this Rydberg Equation Question

Question

I found this question in my textbook.

The hydrogen atom emits a photon with a 486 nm wavelength, corresponding to an electron decaying from the n = 4 level to which level?

I first calculated c=lambda*nu, and solved for nu. Then I used E=h*nu for E.

I then used E=-13.6z^2(1/nf^2-1/ni^2) and solved for nf^2. I got nf =4, which sends the equation to 0. The answer is 2. How is this calculated?

Answer 1

I use the wavelength in meters and then the equation is use is:

1/λ = RZ²(1/nf² - 1/ni²)

1/λ = 1.097x10⁷ m^-1 (1/nf² - 1/4²)

1/4.86x10^-7 m = 1.097x10⁷ m^-1 (1/nf² - 1/16)

1/4.86x10^-7 m = 1.097x10⁷ m^-1 (1/nf² - 0.0625)

0.1877 = 1/nf² - 0.0625

1/nf² = 0.125

nf² =8

nf = 2.8 = 3

I think that the equation you are trying to use is actually ∆E = E_f - E_i = R_HZ² (1/nf² - 1/ni²) but here you have 2 unknowns, i.e. nf and ∆E since you don't know the change in energy. You are using the energy of a hydrogen electron in the ground state (13.6 electron volts), but this isn't ∆E. At least this is what I think is happening in your case. If you knew nf and ni and wanted to find the change in energy of the electron, then you might use the equation as you have. Maybe someone else will weigh in.