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NCERT Class XII Chapter
Atoms
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Question : 17 of 18
Marks: +1, -0
Obtain the first Bohr’s radius and the ground state energy of a muonic hydrogen atom [i.e., an atom in which a negatively charged muon (μ\mu^{-}) of mass about 207 me orbits around a proton].
Solution:  
In Bohr’s model, the radius of nth orbit, r = n2h2ε0πZme2\frac{n^2 h^2 \varepsilon_0}{\pi Z m_e^2}
In the given muonic hydrogen atom, a negatively charged muon (μ\mu^{-}) of mass 207 mem_e revolve around a proton.
Therefore radius of electron and muon can be written as
rμre\frac{r_\mu}{r_e} = memμ\frac{m_e}{m_\mu} = me207me\frac{m_e}{207 m_e}
rμr_\mu = re207\frac{r_e}{207} = 0.53×1010207\frac{0.53 \times 10^{-10}}{207} m = 2.5 × 101210^{-12} m
Energy of electron in the orbit, E = - me48ε0n2h2\frac{m e^4}{8 \varepsilon_0 n^2 h^2}EeEμ\frac{E_e}{E_\mu} = memμ\frac{m_e}{m_\mu} = me207me\frac{m_e}{207 m_e}
EμE_\mu = 207 EeE_e = 207 [- 13.6 eV] = - 2.8 keV
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