The ratio of wavelength of spectral lines and in the Balmer series is . The value of is ________. [2023]

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Solution

Q.
The ratio of wavelength of spectral lines $H_{α}$ and $H_{β}$ in the Balmer series is $\frac{x}{20}$ . The value of $x$ is ________. [2023]

Ans.
(27)

$\frac{1}{λ} = R [\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}] for H-atom$

$For Balmer series, n_{1} = 2$

$\frac{1}{λ} = R [\frac{1}{4} - \frac{1}{n_{2}^{2}}]$

$For H_{α}, n_{2} = 3 and H_{β}, n_{2} = 4$

$\frac{1}{λ_{H_{α}}} = R [\frac{1}{4} - \frac{1}{9}] = \frac{5 R}{36}$

$\frac{1}{λ_{H_{β}}} = R [\frac{1}{4} - \frac{1}{16}] = \frac{3 R}{16}$

$\frac{\frac{1}{λ_{H_{α}}}}{\frac{1}{λ_{H_{β}}}} = \frac{\frac{5 R}{36}}{\frac{3 R}{16}}$

$\Rightarrow \frac{λ_{H_{α}}}{λ_{H_{β}}} = \frac{27}{20} = \frac{x}{20} \Rightarrow x = 27$

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