Energy level diagram and spectra of hydrogen atom

Energy levle diagram :

A diagram in which horizontal lines represents values of energy of electron and vertical line with arrow indicates the transition of electron from one energy level to another energy level is known as energy level diagram.. The energy level are not equally spaced.

Origination of spectral series in H-atom:

Different spectral series is emitted due to transition of electron from higher energy level \(t_{n_2}\) to lower energy level \(t_{n_1}\).

From, Bohr's assumption, $$h\nu= E_{n_2}-E_{n_1}\;\;\;\;[E_{n_2}>E_{n_1}]$$

\begin{align*} =\frac{-Me^4Z^2}{8\epsilon_{\circ}^2h^2}\frac{1}{n_2^2}-\biggl[\frac{-me^4Z^2}{8\epsilon_{\circ}^2h^2}\biggr]\frac{1}{n_2^2}\end{align*} \begin{align*}\nu =\frac{Me^4Z^2}{8\epsilon_{\circ}^2h^3}\biggl[\frac{1}{n_1^2}-\frac{1}{n_2^2}\biggr]\dotsm(1)\end{align*}

Equation (1) gives the frequency of radiation emitted when electron jumps from \(n_2\) level to \(n_1\) level.

Since, \(\nu=\frac{c}{\lambda}\) $$\therefore\;\; \frac{c}{\lambda}= \frac{Me^4Z^2}{8\epsilon_{\circ}^2h^3}\biggl(\frac{1}{n_1^2}-\frac{1}{n_2^2}\biggr)$$ $$or,\;\; \frac{1}{\lambda}=\frac{Me^4Z^2}{8\epsilon_{\circ}^2h^3c}\biggl(\frac{1}{n_1^2}-\frac{1}{n_2^2}\biggr)$$ $$\therefore\;\; \frac{1}{\lambda}= R_{\infty}Z^2\biggl(\frac{1}{n_1^2}-\frac{1}{n_2^2}\biggr)\dotsm(2)$$

Where, $$R_\infty=\frac{-Me^4Z^2}{8\epsilon_{\circ}^2h^3c}= \frac{9.1 X 10^{-31} (1.602 X 10^{-19})^4}{8 X ( 8.854 X 10^{-12})^2 3 X 10^8 ( 6.625 X 10^{-34})^3}$$ $$ R_\infty= 1.097 X 10^7 m^{-1}$$

This is known as Rydberg's constant when mass of nucleus is assume to be \(\infty\) ( infinity).

1. Lymen series:

A series of spectrical line originated due to transition of electron from higer energy level (i.e \(n_2\) = 2,3,4,5,6,...) to ground state (\(n_1=1\)) is known as lyman series.

For lyman series. $$ \frac{1}{\lambda}= R_\infty Z^2\biggl[ \frac{1}{1^2}-\frac{1}{n_\infty ^2}\biggr]$$

The longes wavelength of series is obtained when \(n_2\)=2 $$\therefore\;\;\; \lambda_longest= R_\infty Z^2 \bigg[1= \frac{1}{2^2}\biggr]\;\;\; ( 2=1 )$$ $$or\;\;\; \frac{1}{\lambda_longest}= R_\infty\cdot\frac34$$ $$\therefore\;\; \lambda_longest= \frac{4}{3R_\infty}$$ $$\lambda_longest= \frac{4}{3 X 1.09 X 10^7}$$ $$=1.215 X 10^-7$$

Thus shortest wavelength of lyman series is also known as ( seresix limit ). This is possible when \(n_2=\infty\)

$$\frac{1}{\lambda shortest}= \frac{1}{R_\infty} \biggl(\frac{1}{12}-\frac{1}{\infty})$$

$$\lambda shortest= \frac{1}{R_\infty}= \frac{1}{1.097 X 10^7}$$

Balmar series:

A series of spectral line originated due to transition of electron from higer energy level ( i.e \(n_2\)= 3,4,5,6,... to the level with n=2 is known as balmar series.

For Balmar series $$\frac{1}{\lambda}= R_\infty Z^2 \biggl( \frac{1}{2^2}-\frac{1}{n_2^2}\biggr)$$

\(\frac{1}{\lambda}= R_\infty\biggl(\frac14-\frac{1}{n_2^2}\biggr)\) \(n_2\)= 3,4,5,..

This series of spectrum lines in visible rigion of spectrum. The first line of this series is known as \(H_\alpha\) -line.

For \(H_\alpha\)-line \(n_2\)= 2, n=2 $$\frac{1}{{\lambda}H_\alpha}=R_\alpha\biggl(\frac14-\frac{1}{3^2}\biggr)$$

$$=R_\infty\biggl(\frac{9-4}{36}\biggr)$$

$$=\frac{36}{5}R_\infty$$

Similarly 2nd, 3rd, 4th etc lines are known as \(H_\beta\), \(H_\alpha\), \(H_\gamma\) etc respectively.

Calculation of wavelength of \(H_\beta\) -lines of H -atom:

For \(H_\beta\)-line \(n_1\)=2, \(n_2\)= 4 $$ \therefore\;\;\; \frac{1}{\lambda_{H\beta}}=R_\infty\biggl(\frac14-\frac{1}{16}\biggr)$$

$$= R_\infty\biggl(\frac{4-1}{16}\bigg)$$ $$\lambda_{H\beta}=\frac{16}{3R_\infty}$$

Paschen series:

A series of spectral line originated due to the transition of an electron from higher energy level ( i.e \(n_2\)= 4,5,6,7,.. ) to the level with ( \(n_1\)= 3) is called paschen series.

For paschen series,

$$\frac{1}{\lambda}= R_\infty\biggl(\frac{1}{3^2}-\frac{1}{n_2^2}\biggr);\;\;\; n_2= 4,5,6,7,...$$

This series of spectrum lies on infra-red region.

Brackett Series:

A series of spectral line originated due to the transition of an electron from higher energy level i.e ( \(n_2\)= 5,6,7,8,..) to the level ( \(n_1\)=4 ) is called Brackett series. For brackett series,

$$\frac{1}{\lambda}= R_\infty\biggl(\frac{1}{4^2}-\frac{1}{n^2}\biggr);\;\; n_2=5,6,7,8,...$$

P-fund series:

A series of spectral line originated due to the transmittion of electron from higher energy level ( i.e \(n_2\)= 6,7,8,9,..) to the level with (\(n_1\)=5) is called P-fund series.

$$\frac{1}{\lambda}= R_\infty\biggl(\frac{1}{5^2}-\frac{1}{n_2^2}\biggr);\;\;\; n_2= 6,7,8,9,...$$

Reference:

Reviews of Modern Physics. Lancaster, P.A.: Published for the American Physical Society by the American Institute of Physics, 1952. Print.

Wehr, M. Russell, and James A. Richards. Physics of the Atom. Reading, MA: Addison-Wesley Pub., 1984. Print.

Young, Hugh D., and Roger A. Freedman. University Physics. Boston, MA: Pearson Custom, 2008. Print.

Adhikari, P.B. A Textbook of Physics. 2070 ed. Vol. II. Kathmandu: Sukunda Publication, 2070. Print.