Light shines through atomic hydrogen gas. It is seen that the gas absorbs light readily at a wavelength of 91.63 nm. What is the value of n at the level to which the hydrogen is being excited by the absorption of light of this wavelength? Assume that most of the atoms in the gas are at the lowest level. (h = 6.626 x 10^-34 J • s, c = 3.00 x 10^8 m/s, 1 eV = 1.60 x 10^-19 J, the Rydberg constant is R=1.097 x 10^7 m^-1)

The Rydberg equation should be used to solve this problem:
1/lambda = R(1/n_i^2 -1/n_f^2)
where lambda is the wavelength, R is the Rydberg constant, n-subf is the final n-level, and n-subi is the initial n-level.
We know that lambda = 91.63nm, R = 1.097 x 10e7, and n-subi = 1.
 So, 1/(91.63x10^-9) = (1.097x10^7) (1 - 1/n_f^2)
(91.63x10^-9) x (1.097x10^7) = 1.005
1/1.005 = .995
.995 = 1 - 1/n_f^2
1/n_fe2 = .005
1/.005 = n_fe2
sqrt (200) = n_f
The square root of 200 is closer to 14.14, so there's some rounding in the final answer, since n must be an integer. This error is mostly due to the inconsistent significant figures and abbreviated forms of the values for R and lambda.