A particular hydrogen - like ion emits the radiation of frequency $3 \times 10^{15} \mathrm{~Hz}$ when it makes transition from $\mathrm{n}=2$ to $\mathrm{n}=1$. The frequency of radiation emitted in transition from $\mathrm{n}=3$ to $\mathrm{n}=1$ is $\frac{\mathrm{x}}{9} \times 10^{15} \mathrm{~Hz}$, when $\mathrm{x}=$

JEE Main · 2024 · 01 Feb 2024 · Shift 2 · Modern Physics

A particular hydrogen - like ion emits the radiation of frequency

3 \times 1 0^{15} Hz

when it makes transition from

n = 2

n = 1

. The frequency of radiation emitted in transition from

n = 3

n = 1

\frac{x}{9} \times 1 0^{15} Hz

, when

x =

Answer: 32

Solution

For a hydrogen-like ion, the energy of a transition is

E = 13.6 Z^{2} (\frac{1}{n _{f}^{2}} - \frac{1}{n _{i}^{2}})

, and since

E = h ν

, the frequencies of the

2 \to 1

and

3 \to 1

transitions are in the ratio

\frac{ν _{1}}{ν _{2}} = \frac{1 - \frac{1}{4}}{1 - \frac{1}{9}} = \frac{3/4}{8/9} = \frac{27}{32}

. With

ν_{1} = 3 \times 1 0^{15}

Hz,

ν_{2} = \frac{32}{27} \times 3 \times 1 0^{15} = \frac{32}{9} \times 1 0^{15}

Hz, so

x = 32

JEE Main · 2024 · 01 Feb 2024 · Shift 2 · Integer answer

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