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