# Spectral Series of Hydrogen Atom

When an electron jumps from higher energy state to lower energy state a radiation of definite energy is emitted. This is called spectral line. Radiation of emitted energy depends on the orbits in which the transition is taking place. Since there are many orbits in an atom a number of such spectral line are formed which gives rise to spectral series.

We have, the total energy of a hydrogen atom in n^{th} orbit is given by:

$$E_n = -\frac{me^4}{8\varepsilon_\circ n^2h^2}$$

For two orbits n_{1} and n_{2} we have,

$$E_{n_1} = -\frac{me^4}{8\varepsilon_\circ n_1^2h^2}……(i)$$

$$E_{n_2} = -\frac{me^4}{8\varepsilon_\circ n_2^2h^2}……(ii)$$

When electron jumps from n_{2} to n_{1}, the energy emitted in the form of photon (hf) which is given by:

$$\text{hf} = E_{n2} – E_{n1}$$

$$or, \text{hf} = -\frac{me^4}{8\varepsilon_\circ n_1^2h^2} – \left ( -\frac{me^4}{8\varepsilon_\circ n_2^2h^2} \right )$$

$$or, \frac{ch}{\lambda} = \frac{me^4}{8\varepsilon_\circ h^2}\left ( \frac{1}{n_1^2}- \frac{1}{n_2^2} \right )$$

$$or,\frac{1}{\lambda} = \frac{me^4}{8\varepsilon_\circ ch^2} \left ( \frac{1}{n_1^2}- \frac{1}{n_2^2}

\right )$$

$$or,\frac{1}{\lambda} = R \left ( \frac{1}{n_1^2}- \frac{1}{n_2^2} \right )$$

$$\text{Where} R = -\frac{me^4}{8{\varepsilon_\circ}ch^3} \text{called Raydberg Constant. It’s value is 1.09} \times 10^7 m^{-1}.$$

The energy level diagram is given and described below:

**Lymen series:** Lymen series are the series of spectral lines obtained when electron of electron jumps from higher energy level to 1^{st} orbit (i.e. n=1). This series lies within ultraviolet region. The wave numbers and the wavelength of the spectral lines having lymen series is given by:

$$\text{f} = \frac{1}{\lambda} = R \left ( \frac{1}{1^2}- \frac{1}{n_2^2} \right )$$

**Balmer series**: Balmer series is the series of spectral lines obtained when electron jumps from higher energy state to 2^{nd} orbit (i.e. n=2). This series lies within visible region. The wave numbers and the wavelength of the spectral lines having balmer series is given by:

$$\text{f} = \frac{1}{\lambda} = R \left ( \frac{1}{2^2}- \frac{1}{n_2^2} \right )$$

**Paschen series: **Paschen series is the series of spectral lines obtained when electron from higher energy state jump to 3^{rd} orbit (i.e. n=3). This series lies within infrared region. The wave numbers and the wavelength of the spectral lines having paschen series is given by:

$$\text{f} = \frac{1}{\lambda} = R \left ( \frac{1}{3^2}- \frac{1}{n_2^2} \right )$$

**Brackett series: **Bracket series is the series of spectral lines obtained when electron from higher energy state jump to 4^{th} orbit (i.e. n=4). This series lies in far infrared region. The wave numbers and the wavelength of the spectral lines having brackett series is given by:

$$\text{f} = \frac{1}{\lambda} = R \left ( \frac{1}{4^2}- \frac{1}{n_2^2} \right )$$

Other higher spectral series are called p-fund, d-fund respectively.

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