Obtain the first Bohr 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 `207m_e` revolves around a proton).

Explanation

A muonic hydrogen is the atom in which a negatively charged muon of mass about `207 m_(e)` revolves around a proton.
In Bohr's atom model as, `r prop (1)/(m)`
`(r_(mu))/(r_(e))=(m_(e))/(m_(mu))=(m_(e))/(207 m_(e))=(1)/(207)`
Here `r_(e)` is radius of first orbit of electron in hydrogen atom `=0.53 Å=0.53 xx10^(-10)m`.
`r_(m)=(r_(e))/(207)=(0.53xx10^(-10))/(207)=2.56xx10^(-13)m`.
Again in Bohr's atomic model
`E propm`
`therefore (E_(mu))/(E_(e))=(m_(mu))/(m_(e))=(207 m_(e))/(m_(e)), E_(mu)=207 E_(e)`
As ground state energy of electron in hydrogen atom is `E_(e)=-13.6 eV`
`E_(mu)=207(-13.6) eV =-2815.2 eV`
`=-2.8152 Ke V.`