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If an electron jumps from an outer orbit n_{2} of higher energy level to an inner orbit n_{1} 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*}
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.
\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.
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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