Question

An electron in an hydrogen atom makes a transition from the ground state, `n=1`, to the excited state, `n=5`. Calculate the energy in Joules, frequency in Hertz `(1/s)`, and wavelength in `nm` of the photon?

Answer 1

The energy, frequency, and wavelength are, respectively, `2.092 × 10^"-19"color(white)(l) "J", 3.157 × 10^15color(white)(l) "s"^"-1"`, and `"94.96 nm"`.

Explanation:

Wavelength

You use the Rydberg formula to calculate the wavelength, λ:

`color(blue)(bar(ul(|color(white)(a/a) 1/λ = RZ^2(1/n_2^2 -1/n_1^2)color(white)(a/a)|)))" "`

where

`R =` the Rydberg constant (`1.097 × 10^7color(white)(l) "m"^"-1"`)
`Z =` the atomic number of the atom
`n_1` and `n_2` are the initial and final energy levels

In this problem,

`Z = 1`
`n_1 = 1`
`n_2 = 5`

`1/λ = 1.097 × 10^7color(white)(l) "m"^"-1" × 1^2 (1/5^2 -1/1^2) = 1.097 × 10^7color(white)(l) "m"^"-1" (1/25-1/1)`

`= 1.097 × 10^7color(white)(l) "m"^"-1" × (1 - 25)/25 = "-1.097" × 10^7color(white)(l) "m"^"-1" × 24/25 = "-1.053" × 10^7 color(white)(l)"m"^"-1"`

The negative sign shows that energy is absorbed.

#λ = 1/(1.0531 × 10^7 color(white)(l)"m"^"-1") = 9.496 × 10^"-8" "m" = "94.96 nm"#

Frequency

The formula relating frequency `f` and wavelength `λ` is

`color(blue)(bar(ul(|color(white)(a/a)fλ = c color(white)(a/a)|)))" "`

where `c` is the speed of light (`2.998 × 10^8color(white)(l) "m·s"^"-1"`).

We can rearrange the formula to get

`f = c/λ`

∴ `f = (2.998×10^8 color(red)(cancel(color(black)("m")))"·s"^"-1")/(94.96 × 10^"-9" color(red)(cancel(color(black)("m")))) = 3.157 × 10^15color(white)(l) "s"^"-1"`

The formula relating energy `E` and frequency `f` is

`color(blue)(bar(ul(|color(white)(a/a)E = hfcolor(white)(a/a)|)))" "`

where `h` is Planck's constant `6.626 × 10^"-34"color(white)(l) "J·s"`

∴ `E = 6.626 × 10^"-34"color(white)(l) "J"·color(red)(cancel(color(black)("s"))) × 3.157 × 10^"15" color(red)(cancel(color(black)("s"^"-1"))) = 2.092 × 10^"-19"color(white)(l) "J"`