Question

Calculate the frequency of the n = 5 line in the Lyman series of hydrogen.

Answer 1

The Lyman series
The version of the Rydberg formula that generated the Lyman series was:
\[{1 \over \lambda} = R_\text{H} \left( 1 - \frac{1}{n^2} \right) \qquad \left( R_\text{H} = 1.0968{\times}10^7\,\text{m}^{-1} = \frac{13.6\,\text{eV}}{hc} \right)\]
Where n is a natural number greater than or equal to 2 (i.e. n = 2,3,4,...).
Therefore, the lines seen in the image above are the wavelengths corresponding to n=2 on the right, to n=\[\infty\] on the left (there are infinitely many spectral lines, but they become very dense as they approach to n= \[\infty\] (Lyman limit), so only some of the first lines and the last one appear).
The wavelengths (nm) in the Lyman series are all ultraviolet:
2 3 4 5 6 7 8 9 10 11
Wavelength (nm) 121.6 102.6 97.3 95 93.8 93.1 92.6 92.3 92.1 91.9 91.18 (Lyman limit)

In your case for the n=5 line you have to replace "n" in the above formula for 5 and you should get a value of 95 x 10^-9 m for the wavelength.

then you have to use the other equation that convert wavelength to frequency.
\[\nu=\frac{ c }{ \lambda }\]
\[\nu = frequency ;
c= 3 x 10^{8}m/s ;
\lambda= wavelenght\]