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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 207me Orbits Around a Proton]. - CBSE (Science) Class 12 - Physics

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Question

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 207me orbits around a proton].

Solution

Mass of a negatively charged muon, `m_mu = 207 m_e`

According to Bohr’s model,

Bohr radius,  `r_e prop (1/m_e)`

And, energy of a ground state electronic hydrogen atom, `E_e prop m_e`

Also energy of a ground state muonic hydrogen atom `E_mu prop m_mu`

We have the value of the first Bohr orbit,  `r_e = 0.53 "Å" = 0.53 xx 10`^(-10) m`

Let rμ be the radius of muonic hydrogen atom.

At equilibrium, we can write the relation as:

`m_mu r_mu = m_e r_e`

`207 m_e  xxr_mu = m_er_e`

`:. r_mu = (0.53 xx 10^(-10))/207  = 2.56 xx 10^(-13) m`

Hence, the value of the first Bohr radius of a muonic hydrogen atom is

2.56 × 10−13 m.

We have,

Ee= − 13.6 eV

Take the ratio of these energies as:

`E_e/E_mu = m_e/m_mu = m_e/(207 m_e)`

`E_mu = 207 E_e`

`= 207 xx (-13.6)  = -2.81 keV`

Hence, the ground state energy of a muonic hydrogen atom is −2.81 keV.

  Is there an error in this question or solution?

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Solution 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 207me Orbits Around a Proton]. Concept: Energy Levels.
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