The ratio of wavelength of spectral lines $\mathrm{H}_{\alpha}$ and $\mathrm{H}_{\beta}$ in the Balmer series is $\frac{\mathrm{x}}{20}$. The value of x is

JEE Main · 2023 · 08 Apr 2023 · Shift 2 · Modern Physics

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

Answer: 27

Solution

For the Balmer series of hydrogen, the wavelength is given by:

\frac{1}{λ} = R [\frac{1}{2 ^{2}} - \frac{1}{n ^{2}}]

For

H_{α}

(transition

n = 3 \to 2

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

For

H_{β}

(transition

n = 4 \to 2

\frac{1}{λ _{H_{β}}} = R [\frac{1}{4} - \frac{1}{16}] = R [\frac{16 - 4}{64}] = \frac{3 R}{16}

Taking the ratio:

\frac{λ _{H_{α}}}{λ _{H_{β}}} = \frac{\frac{3 R}{16}}{\frac{5 R}{36}} = \frac{3 R}{16} \times \frac{36}{5 R} = \frac{108}{80} = \frac{27}{20}

Therefore,

x = 27

JEE Main · 2023 · 08 Apr 2023 · Shift 2 · Integer answer

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