If wavelength of the first line of the Paschen series of hydrogen atom is 720 nm, then the wavelength of the second line of this series is ________ nm. (Nearest integer) [2023]

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Solution

Q.
If wavelength of the first line of the Paschen series of hydrogen atom is 720 nm, then the wavelength of the second line of this series is ________ nm. (Nearest integer)
[2023]

Ans.
(492)

$\frac{1}{λ} = R_{H} \times [\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}] \times {(Z)}^{2}$

$For 1st line of Paschen series: n_{1} = 3, n_{2} = 4$

$\frac{1}{720} = \frac{1}{λ_{1}} = R_{H} \times {(Z)}^{2} \times [\frac{1}{{(3)}^{2}} - \frac{1}{{(4)}^{2}}]$ ...(i)

$\frac{1}{720} = \frac{5}{9 \times 16} \times R_{H} \times {(Z)}^{2}$

$For IInd line of Paschen series: n_{1} = 3, n_{2} = 5$

$\frac{1}{λ_{2}} = R_{H} \times {(Z)}^{2} \times [\frac{1}{{(3)}^{2}} - \frac{1}{{(5)}^{2}}]$ ...(2)

$Dividing (1) by (2)$

$\frac{λ_{2}}{720} = \frac{7}{9 \times 16} \times \frac{9 \times 25}{16}$

$λ_{2} = 492 nm$

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