Question

Calculate the wavelength of the first line in the Lyman series of the hydrogen spectrum ($\mathrm{R}=109677{\mathrm{cm}}^{-1}$) how to do this?

Answer 1

Step: 1 Introducing the formula: The Rydberg's Equation

$\frac{1}{\mathrm{\lambda }}=109677\left(\frac{1}{{\mathrm{n}}_1^2}-\frac{1}{{\mathrm{n}}_2^2}\right){\mathrm{z}}^2{\mathrm{cm}}^{-1}$

Step: 2 Gathering data about the first line of the Lyman series:

For the Lyman series, $n_1=1$
For the shortest wavelength in the Lyman series the ${\mathrm{n}}_2=2$

Step: 3 Putting all the data in the Rydberg's equation

$\frac{1}{\mathrm{\lambda }}=109677\left(\frac{1}{1^2}-\frac{1}{2^2}\right)1^2{\mathrm{cm}}^{-1}$

$$\begin{gathered} \frac{1}{\mathrm{\lambda }}=109677\left(\frac{1}{1^2}-\frac{1}{4}\right)1^2{\mathrm{cm}}^{-1} \\ \frac{1}{\mathrm{\lambda }}=109677\left(\frac{4-1}{4}\right){\mathrm{cm}}^{-1} \\ \frac{1}{\mathrm{\lambda }}=109677\left(\frac{3}{4}\right){\mathrm{cm}}^{-1} \\ \mathrm{\lambda }=\frac{4}{3\times 109677}\mathrm{cm}=\frac{4}{329031}\mathrm{cm} \\ \mathrm{\lambda }=1.21\times {10}^{-7}\mathrm{m}=121\times {10}^{-9}\mathrm{m} \\ \mathrm{\lambda }=121\mathrm{nm} \end{gathered}$$

Therefore, the wavelength of the first line of the Lyman Series is 121nm.