Atoms & Nuclei

JEE Main level, single correct, medium

20 of 239

Question 20 of 239

Muon (μ−1)\left(\mu^{-1}\right) is negatively charged (∣q∣=∣e∣)(|\mathrm{q}|=|\mathrm{e}|) with a mass mμ=200 me\mathrm{m}_\mu=200 \mathrm{~m}_{\mathrm{e}}, where me\mathrm{m}_{\mathrm{e}} is the mass of the electron and e is the electronic charge. If μ−1\mu^{-1} is bound to a proton to form a hydrogen like atom, identify the correct statements
(A) Radius of the muonic orbit is 200 times smaller than that of the electron

(B) the speed of the
μ−1\mu^{-1} in the nnth orbit is 1200\frac{1}{200} times that of the election in the nth orbit
(C) The lonization energy of muonic atom is 200 times more than that of an hydrogen atom

(D) The momentum of the muon in the nth orbit is 200 times more than that of the electron

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Answer: (D) (A), (C), (D)

 (A) Radius of muon = Radius of hydrogen 200 Radius of H atom =r=∈0n2h2πme2 Radius of muon =rμ=∈0n2h2π×200me2rμ=r200\begin{aligned} &\text { (A) Radius of muon }=\frac{\text { Radius of hydrogen }}{200} \\ &\text { Radius of } \mathrm{H} \text { atom }=r=\frac{\in_0 n^2 h^2}{\pi m e^2} \\ &\text { Radius of muon }=r_\mu=\frac{\in_0 n^2 h^2}{\pi \times 200 m e^2} \\ &r_\mu=\frac{r}{200} \end{aligned}
(B) Velocity relation given is wrong

(C) Ionization energy in
e−He^{-} \mathrm{H} atom
E=+me48ϵ02n2h2E=\frac{+m e^4}{8 \epsilon_0^2 n^2 h^2}

Eμ=200me48ϵ02n2h2=200EE_\mu=\frac{200 m e^4}{8 \epsilon_0^2 n^2 h^2}=200 E
(D) Momentum of
H\mathrm{H}-atom
mvr=nh2πm v r=\frac{n h}{2 \pi}
Hence (A), (C), (D) are correct.

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