Question

A wavelength of $4.653\textrm{ }\mu\textrm{m}$ is observed in a hydrogen spectrum for a transition that ends in the $n_f = 5$ level. What was $n_i$ for the initial level of the electron?

Final Answer

7

### Solution video

# OpenStax College Physics for AP® Courses, Chapter 30, Problem 19 (Problems & Exercises)

vote with a rating of
votes with an average rating of
.

### Calculator Screenshots

Video Transcript

This is College Physics Answers with Shaun Dychko. A hydrogen atom emits a wavelength of 4.653 micrometers when an electron transitions from some initial state that we don't know to a final state of 5. Now equation [30.13] tells us what the wavelength is that's emitted as a function of the final and initial states of the electron and we need to solve this for

*n i*. So let's first multiply through by Rydberg's constant and then get the*R*over*n i*squared term on the left side and that's gonna give us*R*over*n i*squared equals*R*over*n f*squared minus 1 over*λ*. And then we'll take the reciprocal of both sides but let's also get a common denominator here; we'll multiply top and bottom of this by*λ*and top of bottom of this by*n f*squared over*n f*squared and this makes*Rλ*minus*n f*squared all over*λn f*squared. And then we'll take the reciprocal of both sides which means flip these two fractions so that we get our unknown*n i*in the numerator. So it's*n i*squared at the moment divided by*R*and that equals*λn f*squared over*Rλ*minus*n f*squared. And then multiply both sides by*R*and then take the square root of both sides as well and we end up with*n f*to the power of 1 which because it's squared we can take it out of the square root sign and call it*n f*to the power of 1 times square root of*λ*times Rydberg's constant divided by*Rλ*minus*n f*squared. And so we have the final energy level we are told is 5 and we'll multiply that by the square root of 4.653 times 10 to the minus 6 meters times Rydberg's constant divided by the Rydberg's constant times the wavelength of the emitted photon minus*n f*squared so that's 5 squared and this works out to an initial energy level of 7.