 ## Hydrogen Spectrum

Subject: Physics

#### Overview

When an electron jumps from the higher energy state to the lower energy state, the difference of energies of two states is emitted as the radiation of definite frequency which is called spectral line. This note provides us an information on Bohr’s interpretation of the hydrogen spectrum, spectral series of hydrogen atom and energy level diagram of hydrogen atom.

#### Bohr’s Interpretation of the Hydrogen Spectrum

If an electron jumps from an outer orbit n2 of higher energy level to an inner orbit n1 of lower energy level, the frequency of the radiation emitted is given by

\begin{align*} f &= \frac {En_2 – En_1}{h} \\ \text {where} \: En_1 \: \text {and} \: En_2\: \\ \text {are energies of the electro in the stationary orbits}\\ \text {then} \\ En_1&= -\frac {me^4}{8\epsilon_0^2n_1^2h^2} \\ En_2 &= -\frac {me^4}{8\epsilon_0^2n_2^2h^2} \\ \end{align*}therefore, the frequency of radiation emitted is given by\begin{align*}\\ f &= \frac {En_2 – En_1}{h} \\ hf &=En_2 – En_1 \\ hf &= -\frac {me^4}{8\epsilon_0^2n_2^2h^2} - \left (\frac {-me^4}{8\epsilon_0^2n_1^2h^2} \right )\\ &= -\frac {me^4}{8\epsilon_0^2n_2^2h^2} + \frac {me^4}{8\epsilon_0^2n_1^2h^2} \\\ &= \frac {me^4}{8\epsilon_0^2h^2} \left (\frac {1}{n_1^2} - \frac {1}{n_2^2}\right )\\ f &= \frac {me^4}{8\epsilon_0^2h^3} \left (\frac {1}{n_1^2} - \frac {1}{n_2^2}\right ) \dots (x)\\ \end{align*} \begin{align*} \text {i.e.} \: \vec f &= \frac {1}{\lambda } = \frac fc \\ \therefore \vec f &= \frac fc= \frac {me^4}{8\epsilon_0^2ch^3} \left (\frac {1}{n_1^2} - \frac {1}{n_2^2}\right )\dots (xi)\\ \text {the factor} \: \frac {me^4}{8\epsilon_0^2 ch^3} = R \\ \text {R is known as Rydberg constant} \\ \text {Therefore equation,}\: (xi) \: \text {may be written as} \\ \vec f \\&= \frac {1}{\lambda } = R\left (\frac {1}{n_1^2} - \frac {1}{n_2^2}\right ) \dots (xii)\\ \end{align*}

#### Spectral Series of Hydrogen Atom

When an electron jumps from the higher energy state to the lower energy state, the difference of energies of two states is emitted as the radiation of definite frequency. It is called spectral line.

1. ##### Lyman Series:
The spectral lines of this series correspond to the transition of an electron from some higher energy state to the innermost orbit. Therefore, for Lyman series, n1=1 and n2 = 2, 3, 4,… The wave numbers and the wave lengths of the spectral lines constituting the Lyman series are given by
\begin{align*}\vec f = \frac{1}{\lambda }=R\left (\frac {1}{1^2} - \frac{1}{n_2^2}\right ) \end{align*}
This series lies in the ultraviolet region of the spectrum.

2. ##### Balmer Series:
The spectral lines of this series correspond to the transition of an electron from some higher energy state to an orbit having n = 2. Therefore, for Balmer series, n1=2 and n2 = 3, 4,5,… The wave numbers and the wave lengths of the spectral lines constituting the Balmer series are given by
\begin{align*}\vec f = \frac{1}{\lambda }=R\left (\frac {1}{2^2} - \frac{1}{n_2^2}\right ) \end{align*}
This series lies in the infrared region of the spectrum and is invisible.
3. ##### Paschen Series:
The spectral lines of this series correspond to the transition of an electron from some higher energy state to an orbit having n = 3. Therefore, for Paschen series, n1=3 and n2 = 4,5,… The wave numbers and the wave lengths of the spectral lines constituting the Paschen series are given by
\begin{align*}\vec f = \frac{1}{\lambda }=R\left (\frac {1}{3^2} - \frac{1}{n_2^2}\right ) \end{align*}
This series lies in the infrared region of the spectrum.

4. ##### Brackett Series:
The spectral lines of this series correspond to the transition of an electron from some higher energy state to an orbit having n = 4. Therefore, for Brackett series, n1=4 and n2 = 5, 6, 7,… The wave numbers and the wave lengths of the spectral lines constituting the Brackett series are given by
\begin{align*}\vec f = \frac{1}{\lambda }=R\left (\frac {1}{4^2} - \frac{1}{n_2^2}\right ) \end{align*}
This series lies in the infrared region of the spectrum.

5. ##### P-fund Series:
The spectral lines of this series correspond to the transition of an electron from some higher energy state to an orbit having n = 5. Therefore, for P-fund series, n1=4 and n2 = 6, 7,… The wave numbers and the wave lengths of the spectral lines constituting the P-fund series are given by
\begin{align*}\vec f = \frac{1}{\lambda }=R\left (\frac {1}{5^2} - \frac{1}{n_2^2}\right ) \end{align*}
This series lies in the infrared region of the spectrum.

#### Energy Level Diagram of Hydrogen Atom

\begin{align*}\text {The diagrammatic representation of} \\ E_n &= -\frac {me^4}{8\epsilon_0^2h^2n^2} \\ \text { it is called the energy level diagram.} \\ \text {substituting the value of m,e,}\: \epsilon_0\: \text {and h, we get} \\ E_n &= \frac {me^4}{8\epsilon_0^2h^2n^2} \\ &= \frac {(9.11\times 10^{-31} \times (1.6\times 10^{-19})^4}{8(8.854\times 10^{-12})^2n^2(6.625 \times 10^{-34})^2} \\ &= \frac {21.76 \times 10^{-19}}{n^2}j \\ E_n &= \frac {21.76 \times 10^{-19}}{n^2(1.6 \times 10^{-19})}eV \\ \text {or,}\: E_n &= -\frac {13.6}{n^2}eV \\ \end{align*}

reference

Manu Kumar Khatry, Manoj Kumar Thapa, et al.. Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.

##### Things to remember

When an electron jumps from the higher energy state to the lower energy state, the difference of energies of two states is emitted as the radiation of definite frequency which is called spectral line.

The wave numbers and the wave lengths of the spectral lines constituting the lyman series are given by
\begin{align*}\vec f = \frac{1}{\lambda }=R\left (\frac {1}{1^2} - \frac{1}{n_2^2}\right ) \end{align*}

The wave numbers and the wave lengths of the spectral lines constituting the Balmer series are given by
\begin{align*}\vec f = \frac{1}{\lambda }=R\left (\frac {1}{2^2} - \frac{1}{n_2^2}\right ) \end{align*}

The wave numbers and the wave lengths of the spectral lines constituting the Paschen series are given by
\begin{align*}\vec f = \frac{1}{\lambda }=R\left (\frac {1}{3^2} - \frac{1}{n_2^2}\right ) \end{align*}

The wave numbers and the wave lengths of the spectral lines constituting the Brackett series are given by
\begin{align*}\vec f = \frac{1}{\lambda }=R\left (\frac {1}{4^2} - \frac{1}{n_2^2}\right ) \end{align*}

The wave numbers and the wave lengths of the spectral lines constituting the P-fund series are given by
\begin{align*}\vec f = \frac{1}{\lambda }=R\left (\frac {1}{5^2} - \frac{1}{n_2^2}\right ) \end{align*}

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