Question

Determine the wavelength, in nanometers, of the line in the Balmer series corresponding to `n_2` = 5? Express your answer using five significant figures.

Information given

"Use the Balmer equation

`nu = 3.2881 xx10^(15)"s"^(−1) * (1/2^2−1/n^2)`

to answer following questions."

Correct answer

`lambda=434.16` nm

What I got

I got `~~ 434.47` nm

Answer 1

`lamda = "434.17 nm"`

Explanation:

This is pretty much a plug-n-play problem in which all you have to do is plug in the value for `n_2` in the given equation.

The problem provides you with the equation

`color(blue)(ul(color(black)(nu = 3.2881 * 10^(15)"s"^(-1) * (1/2^2 - 1/n_2^2))))`

The first thing to do here is to rearrange this equation to work with wavelength, `lamda`. As you know, frequency and wavelength have an inverse relationship described by the equation

`color(blue)(ul(color(black)(lamda * nu = c)))`

Here

`c` - the speed of light in a vacuum, equal to `"299,792,458 m s"^(-1)`

This means that you have

`nu = c/(lamda)`

Plug this into the Balmer equation to get

`c/(lamda) = 3.2881 * 10^(15)"s"^(-1) * (1/4 - 1/(n_2^2))`

`lamda = c/(3.2881 * 10^(15)"s"^(-1) * (1/4 - 1/(n_2^2)))`

Plug in your values to find

`lamda = ("299,792,458 m" color(red)(cancel(color(black)("s"^(-1)))))/(3.2881 * 10^(15) color(red)(cancel(color(black)("s"^(-1)))) * (1/4 - 1/25))`

`lamda = 4.341666 * 10^(-7)"m"`

Expressed in nanometers and rounded to five sig figs, the answer will be

`color(darkgreen)(ul(color(black)(lamda = 434.17 * 10^(-9)"m" = "434.17 nm"))) ->` close enough