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  • 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^– ) of mass about 207 m_e orbits around a proton].

Q 12.17   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 (\mu^{-}) of mass about 207me orbits around a proton].

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As per  Bohr's quantization postulate

m_{\mu ^{-}}v_{n}r_{n}=\frac{nh}{2\pi }

Similarly, like the case in a simple hydrogen atom, the electrostatic force acts centripetally

\frac{m_{\mu ^{-}}v_{n}^{2}}{r_{n}}=\frac{e^{2}}{4\pi\epsilon _{0}r_{n}^{2} }

From the above relations, we can see that in Bohr's model the Radius is inversely proportional to the mass of the orbiting body and Energy is directly proportional to the mass of the orbiting body.

In case of hydrogen, atom r1 is 5.3\times10-11 m

Therefore in case of a muonic hydrogen atom 

r_{1}=\frac{5.3\times 10^{-11}}{207}

r1 = 2.56\times10-13 m

In case of the hydrogen atom, E1 is -13.6 eV

Therefore in case of a muonic hydrogen atom 

E1=207\times(-13.6)

E1=2.81 keV

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Sayak

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