Question

To what series does the spectral lines of atomic hydrogen belong if its wave number is equal number is equal to the difference between the wave numbers of the following two lines of the Balmer series

486.1

and

410.2 nm

? What is the wavelength of this line?

Accepted Answer

Given that $λ_{1} = 486.1 \times 10^{- 9} m = 486.1 \times 10^{- 7} cm$ and $λ_{2} = 410.2 \times 10^{- 9} m = 410.2 \times 10^{- 7} cm$

According to Rydberg's equation,

$\bar{ν} = \frac{1}{λ} = {RZ}^{2} (\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}})$

and $\bar{ν} = {\bar{ν}}_{2} - {\bar{ν}}_{1} = [\frac{1}{λ_{2}} - \frac{1}{λ_{1}}]$

$= R_{H} = [\frac{1}{2^{2}} - \frac{1}{n_{2}^{2}}] - R_{H} [\frac{1}{2^{2}} - \frac{1}{n_{1}^{2}}]$

$\bar{ν} = R_{H} [\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}]$ ......(i)

Balmer series originates due to the transition of electron from higher energy levels to $n = 2$ . So,

For line $I$ of Balmer series:

$\frac{1}{λ_{1}} = R {(1)}^{2} [\frac{1}{2^{2}} - \frac{1}{n_{1}^{2}}] \frac{1}{486.1 \times 10^{- 7}} = 109678 [\frac{1}{2^{2}} - \frac{1}{n_{1}^{2}}] \frac{1}{5.33} = \frac{1}{4} - \frac{1}{n_{1}^{2}} \frac{1}{n_{1}^{2}} = \frac{1}{4} - \frac{1}{5.33} \frac{1}{n_{1}^{2}} = \frac{1.33}{21.33} \frac{1}{n_{1}^{2}} = \frac{1}{16}$

$∴ n_{1} = 4$

For line $II$ of Balmer series:

$\frac{1}{λ_{2}} = R {(1)}^{2} [\frac{1}{2^{2}} - \frac{1}{n_{2}^{2}}] \frac{1}{410.2 \times 10^{- 7}} = 109678 [\frac{1}{2^{2}} - \frac{1}{n_{2}^{2}}] \frac{1}{4.5} = \frac{1}{4} - \frac{1}{n_{2}^{2}} \frac{1}{n_{2}^{2}} = \frac{1}{4} - \frac{1}{4.5} \frac{1}{n_{2}^{2}} = \frac{0.5}{18} \frac{1}{n_{2}^{2}} = \frac{1}{36}$

$∴ n_{2} = 6$

Thus given electronic transition occurs from $6^{th}$ to $4^{th}$ shell. Also by eq. (i)

$\bar{ν} = \frac{1}{λ} = 109678 [\frac{1}{4^{2}} - \frac{1}{6^{2}}]$

$∴ λ = 2.63 \times 10^{- 4} cm$

Embibe Experts Solutions for Chapter: Structure of Atom, Exercise 4: EXERCISE-4

Embibe Experts Chemistry Solutions for Exercise - Embibe Experts Solutions for Chapter: Structure of Atom, Exercise 4: EXERCISE-4

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