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Dual Nature of Radiation question

2021 · 26 Feb · Shift 2 · Q54
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  5. /2021 · 26 Feb · Shift 2 · Q54

Dual Nature of Radiation question

2021 · 26 Feb · Shift 2 · Q54

JEE MainPhysicsDual Nature of RadiationMCQ+4 / −1
The recoil speed of a hydrogen atom after it emits a photon in going from n = 5 state to n = 1 state will be :
  1. A
    4.34 m/s
  2. B
    2.19 m/s
  3. C
    3.25 m/s
  4. D
    4.17 m/s
View written solutionFree

Correct answer: D

  1. Energy of emitted photon

For hydrogen, the energy levels are En=−13.6n2 eVE_n=-\frac{13.6}{n^2}\ \text{eV}En​=−n213.6​ eV

For the transition n=5→n=1n=5 \to n=1n=5→n=1, ΔE=13.6(1−125)=13.6⋅2425=13.056 eV\Delta E=13.6\left(1-\frac{1}{25}\right)=13.6\cdot \frac{24}{25}=13.056\ \text{eV}ΔE=13.6(1−251​)=13.6⋅2524​=13.056 eV

So the emitted photon has energy Eγ=13.056 eVE_\gamma=13.056\ \text{eV}Eγ​=13.056 eV

  1. Photon momentum

For a photon, pγ=Eγcp_\gamma=\frac{E_\gamma}{c}pγ​=cEγ​​

Convert energy into joules: Eγ=13.056×1.6×10−19≈2.089×10−18 JE_\gamma=13.056\times 1.6\times 10^{-19}\approx 2.089\times 10^{-18}\ \text{J}Eγ​=13.056×1.6×10−19≈2.089×10−18 J

Thus,

\approx 6.96\times 10^{-27}\ \text{kg m/s}$$ 3. **Recoil momentum of hydrogen atom** By conservation of momentum, the hydrogen atom recoils with equal and opposite momentum: $$p_H=p_\gamma$$ If $M$ is the mass of hydrogen atom, then $$Mv=p_\gamma$$ Using $$M\approx 1.67\times 10^{-27}\ \text{kg}$$ we get $$v=\frac{6.96\times 10^{-27}}{1.67\times 10^{-27}}\approx 4.17\ \text{m/s}$$ 4. **Match with options** The recoil speed is $$\boxed{4.17\ \text{m/s}}$$ So the correct option is **D**.
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