As per given figure A, B and C are the first, second and third excited energy level of hydrogen atom respectively. If the ratio of the two wavelengths (i.e., ) is , then the value of will be _______. [2023]

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Solution

Q.
As per given figure A, B and C are the first, second and third excited energy level of hydrogen atom respectively. If the ratio of the two wavelengths (i.e., $\frac{λ_{1}}{λ_{2}}$ ) is $\frac{7}{4 n}$ , then the value of $n$ will be _______. [2023]

Ans.
(5)

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

$\frac{1}{λ_{1}} = R {(1)}^{2} [\frac{1}{{(2)}^{2}} - \frac{1}{{(3)}^{2}}] = R (\frac{5}{36}) \dots (i)$

$\frac{1}{λ_{2}} = R {(1)}^{2} [\frac{1}{{(3)}^{2}} - \frac{1}{{(4)}^{2}}] = R (\frac{7}{144}) \dots (i i)$

(ii) + (i) gives

$\frac{λ_{1}}{λ_{2}} = \frac{\frac{7}{144}}{\frac{5}{36}} = \frac{7}{20} = \frac{7}{4 \times 5}$

$∴$ $n = 5$

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