Question

Question #2e787

Answer 1

`n_i = 8`

Explanation:

Your tool of choice here will be the Rydberg equation, which looks like this

`1/(lamda) = R * (1/n_f^2 - 1/n_i^2)`

Here

• `lamda` is the wavelength of the photon
• `R` is the Rydberg constant, equal to `1.097 * 10^(7)` `"m"^(-1)`
• `n_I` is the initial energy level of the transition
• `n_f` is the final energy level of the transition

Now, you know that the electron starts on an initial energy level `n_i` and ends up on the third energy level, so

`n_f = 3`

Moreover, you know that this transition is accompanied by the emission of a photon of wavelength

`lamda = 9.54 * 10^(-7)` `"m"`

Rearrange the Rydberg equation to solve for `n_i`

`1/(lamda) = R * (n_i^2 - n_f^2)/(n_i^2 * n_f^2)`

This is equivalent to

`n_i^2 * n_f^2 = lamda * R * n_i^2 - lamda * R * n_f^2`

`lamda * R * n_i^2 - n_i^2 * n_f^2 = lamda * R * n_f^2`

`n_i^2 * (lamda * R - n_f^2) = lamda * R * n_f^2`

Finally, you should end up with

`n_i = sqrt( (lamda * R * n_f^2)/(lamda * R - n_f^2))`

Plug in your values to find

`n_i = sqrt( (9.54 * color(blue)(cancel(color(black)(10^(-7))))color(red)(cancel(color(black)("m"))) * 1.097 * color(blue)(cancel(color(black)(10^7)))color(red)(cancel(color(black)("m"^(-1)))) * 3^2)/(9.54 * color(blue)(cancel(color(black)(10^(-7))))color(red)(cancel(color(black)("m"))) * 1.097 * color(blue)(cancel(color(black)(10^7)))color(red)(cancel(color(black)("m"^(-1)))) - 3^2))`

`n_ i = 8.017 ~~ color(darkgreen)(ul(color(black)(8)))`

Therefore, you can say that this electron underwent a `n_i = 8 -> n_f = 3` transition.