Question

Question #cf90a

Answer 1

`"93.8 nm"`

Explanation:

That formula will not give you the wavelength of the emitted photon, it will give you its energy.

You would then use the Planck - Einestein equation to get the wavelength of the photon.

That value given to you , `2.18 * 10^(-18)"J"`, represents the Rydberg constant expressed in Joules, rather than in `"m"^(-1)`.

So, the energy of a photon emitted when an electron drops from the `n=6` to the `n=1` energy level is

`E = R_E * (1/n_1^2 - 1/n_2^2)" "`, where

`R_E` - the Rydberg constant* expressed in Joules
`n_1` - the final energy level of the electron
`n_2` - the initial energy level of the electron

In your case, the energy of the photon would be

`E = 2.18 * 10^(-18)"J" * (1/1^2 - 1/6^2)`

`E = 2.18 * 10^(-18)"J" * (1 - 1/36) = 2.12 * 10^(-18)"J"`

The Planck - Einstein equation looks like this

`E = h * nu" "`, where

`E` - the energy of the photon
`h` - Planck's constant, equal to `6.626 * 10^(-34)"J s"`
`nu` - the frequency of the photon

You know that frequency and wavelength have an inverse relationship

`nu * lamda = c" "`, where

`lamda` - wavelength
`c` - the speed of light in vacuum, approximately equal to `3.0 * 10^(8)"m s"^(-1)`

This means that the relationship between energy and wavelength looks like this

`E = h * c/(lamda)`

Plug in your values and rearrange to solve for `lamda`

`lamda = (h * c)/E = (6.626 * 10^(-34)color(red)(cancel(color(black)("J"))) color(red)(cancel(color(black)("s"))) * 3.0 * 10^8"m" color(red)(cancel(color(black)("s"^(-1)))))/(2.12 * 10^(-18)color(red)(cancel(color(black)("J"))))`

`lamda = 9.376 * 10^(-8)"m" = "93.8 nm"`

SIDE NOTE You can incorporate the Planck - Einstein equation into the Rydberg equation

`E = R_E * (1/n_1^2 - 1/n_2^2)`

`h * c/(lamda) = R_E * (1/n_1^2 - 1/n_2^2)`

Rearrange to get

`1/(lamda) = R_E/(h * c) * (1/n_1^2 - 1/n_2^2)`

The term `R_E/(h * c)` is another version of Rydberg's constant, this time expressed in `'m"^(-1)`, equal to

`R = (2.18 * 10^(-18)color(red)(cancel(color(black)("J"))))/(6.626 * 10^(-34)color(red)(cancel(color(black)("J"))) color(red)(cancel(color(black)("s"))) * 3.0 * 10^8 "m" color(red)(cancel(color(black)("s"^(-1))))) ~~ 1.9073... * 10^(7)"m"^(-1)`