Spectral Series

The Spectral Series refers to the set of wavelengths of light emitted or absorbed by electrons in atoms as they transition between energy levels. Each element has its own unique spectral lines, forming distinct series such as the Lyman, Balmer, and Paschen series in hydrogen. These lines appear as bright or dark lines in a spectrum and are key to understanding atomic structure, identifying elements in stars, and exploring quantum mechanics. The study of spectral series has played a crucial role in the development of modern physics and continues to be essential in fields like astronomy, chemistry, and spectroscopy.

1.0Definition of Spectral Series

• A spectral series of the hydrogen atom is a group of spectral lines corresponding to electronic transitions where electrons fall from higher energy levels to a fixed lower energy level, emitting photons of specific wavelengths characteristic of hydrogen.
• It has been shown that the energy of the outer orbit is greater than the energy of the inner ones. When the Hydrogen atom is subjected to external energy, the electron jumps from lower energy State i.e. the hydrogen atom is excited. The excited state is not stable hence the electron returns to its ground state in about 10^{-8} seconds. The excess of energy is now radiated in the form of radiations of different wavelength.The different wavelengths constitute spectral series. Which are characteristic of atom emitting, then the wavelength of different members of series can be found from the following relations,

$\bar{\nu }=\frac{1}{\lambda }=R\left[\frac{1}{n_1^2}-\frac{1}{n_2^2}\right]$

This relation explains the complete spectrum of hydrogen. A detailed account of the important radiations are listed below.

2.0Classification of Spectral Series

1.Lyman Series

The series consists of wavelengths which are emitted when an electron jumps from an outer orbit to the first orbit i. e. the electron jumps to K orbit giving rise to the Lyman series.

Here $n_1=1$ and $n_2=2,3,4,\dots ,\infty$.

The wavelengths of different members of Lyman series are:

1. First Member

In this case $n_1=1$ and $n_2=2$ hence

$\frac{1}{\lambda }=R\left[\frac{1}{1^2}-\frac{1}{2^2}\right]=\frac{3R}{4}$

Or

$\lambda =\frac{4}{3R}$

$\lambda =\frac{4}{3\times 10.97\times 10^6}=1216\times 10^{-10}\text{ m}=1216\text{ A˚}$

2. Second Member

Here $n_1=1$ and $n_2=3$ hence

$\frac{1}{\lambda }=R\left[\frac{1}{1^2}-\frac{1}{3^2}\right]=\frac{8R}{9}\text{ or }\lambda =\frac{9}{8R}$

$\lambda =\frac{9}{8\times 10.97\times 10^6}=1026\times 10^{-10}\text{ m}=1026\text{ A˚}$

3. Limiting Member

Here $n_1=1$ and $n_2=\infty$ hence

$\frac{1}{\lambda }=R\left[\frac{1}{1^2}-\frac{1}{{\infty }^2}\right]=R\text{ or }\lambda =\frac{1}{R}$

$\lambda =\frac{1}{10.97\times 10^6}=912\times 10^{-10}\text{ m}=912\text{ A˚}$

This series lies in ultraviolet region

2.Balmer Series

This series consists of all wavelengths which are emitted when an electron jumps from an outer orbit to the second orbit i. e. the electron jumps to L orbit give rise to the Balmer series. $n_1=2$ and $n_2=3,4,5\dots \infty$

The wavelength of different members of the Balmer series.

1. First Member

In this case $n_1=2$ and $n_2=3$ hence

$\frac{1}{\lambda }=R\left[\frac{1}{2^2}-\frac{1}{3^2}\right]=\frac{5R}{36}\text{ or }\lambda =\frac{36}{5R}$

$\lambda =\frac{36}{5\times 10.97\times 10^6}=6563\times 10^{-10}\text{ m}=6563\text{ A˚}$

2. Second Member

Here $n_1=2$ and $n_2=4$ hence

$\frac{1}{\lambda }=R\left[\frac{1}{2^2}-\frac{1}{4^2}\right]=\frac{3R}{16}\text{ or }\lambda =\frac{16}{3R}$

$\lambda =\frac{16}{3\times 10.97\times 10^6}=4861\times 10^{-10}\text{ m}=4861\text{ A˚}$

3. Limiting Member

Here $n_1=2$ and $n_2=\infty$ hence

$\frac{1}{\lambda }=R\left[\frac{1}{2^2}-\frac{1}{{\infty }^2}\right]=\frac{R}{4}\text{ or }\lambda =\frac{4}{R}=3646\text{ A˚}$

This series lies in the visible and near ultraviolet region.

3.Paschen Series

This series consists of all wavelengths emitted when an electron jumps from an outer orbit to the third orbit i. e. the electron jumps to M orbit give rise to paschen series.Here

$n_1=3$ and $n_2=4,5,6\dots \infty$

The different wavelengths of this series can be obtained from the formula

$\frac{1}{\lambda }=R\left[\frac{1}{3^2}-\frac{1}{n_2^2}\right]\text{ where }{n}_2=4,5,6\dots \infty$

For the first member, the wavelength is 18750Å. This series lies in the infra-red region.

4.Brackett Series

This series consists of all wavelengths which are emitted when an electron jumps from an outer orbit to the fourth orbit i. e. the electron jumps to N orbit giving rise to the Brackett series.Here $n_1=4$ and $n_2=5,6,7\dots \infty$

The different wavelengths of this series can be obtained from the formula

$\frac{1}{\lambda }=R\left[\frac{1}{4^2}-\frac{1}{n_2^2}\right]\text{ where }{n}_2=5,6,7\dots \infty$

This series lies in the infra-red region of the spectrum.

5.Pfund series

The series consists of all wavelengths which are emitted when an electron jumps from an outer orbit to the fifth orbit i. e. the electron jumps to O orbit give right to Pfund series.Here $n_1=5$ and $n_2=6,7,8\dots \infty$

The different wavelengths of this series can be obtained from the formula

$\frac{1}{\lambda }=R\left[\frac{1}{5^2}-\frac{1}{n_2^2}\right]\text{ where }{n}_2=6,7,8\dots \infty$

This series lies in the infra-red region of the spectrum.

Illustration-1.Find the maximum wavelength of the Brackett series of hydrogen atoms.

Solution: $n_1=4and{n}_2=5$

$\therefore \frac{1}{{\lambda }_{max}}=R\left[\frac{1}{4^2}-\frac{1}{5^2}\right]$

${\lambda }_{max}=\frac{25\times 16\times 10^{10}}{9\times 10.97\times 10^6}=40400\text{ A˚}$

Q-2.For the given transitions of electrons, obtain the relation between ${\lambda }_1,{\lambda }_{2}and{\lambda }_3$.

Solution:For given condition

$E_3-E_1=(E_3-E_2)+(E_2-E_1)$

$\Rightarrow \frac{hc}{{\lambda }_3}=\frac{hc}{{\lambda }_2}+\frac{hc}{{\lambda }_1}\Rightarrow \frac{1}{{\lambda }_3}=\frac{1}{{\lambda }_2}+\frac{1}{{\lambda }_1}$

${\lambda }_3=\frac{{\lambda }_2{\lambda }_1}{{\lambda }_2+{\lambda }_1}$