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Standard XII

Physics

Characteristics of Bohr's Atomic Model

Obtain the fi...

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 207m_e orbits around a proton].

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Solution

Mass of a negatively charged muon,

According to Bohr’s model,

Bohr radius,

And, energy of a ground state electronic hydrogen atom,

We have the value of the first Bohr orbit,

Let r_μ be the radius of muonic hydrogen atom.

At equilibrium, we can write the relation as:

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

2.56 × 10⁻¹³ m.

We have,

E_e= − 13.6 eV

Take the ratio of these energies as:

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

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Similar questions

Q. Muon

(μ^{- 1})

is negatively charged

(| q | = | e |)

with a mass

m_{μ} = 200 m_{e}

, where

m_{e}

is the mass of the electron and

e

is the electronic charge. If

μ^{- 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}

in the nth orbit is

\frac{1}{200}

times that of the electron in the nth orbit
(C) The ionization 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

Q. A muonic atom contains a nucleus of charge Ze and a negative muon,

μ^{-}

, moving about it. The

μ^{-}

is an elementary charge with -e and a mass that is approximately 200 times as large as an electron mass.
Calculate the binding energy of a muonic atom(in eV) with

Z = 1

A muonic atom contains a nucleus of charge $Z e$ and a negative muon, $μ^{-}$ , moving about it. The $μ^{-}$ is an elementary charge with $- e$ and a mass that is approximately $200$ times as large as an electron mass. Then

Q. A muonic atom contains a nucleus of charge Ze and a negative muon,

μ^{-}

, moving about it. The

μ^{-}

is an elementary charge with -e and a mass that is approximately 200 times as large as an electron mass.

The first Bohr orbit of the muonic atom with

Z = 1

is found out to be

k \times 10^{- 13} m

. Then the value of

k

Q. The radius of the first orbit of hydrogen is

r_{H}

, and the energy in the ground state is

- 13.6

eV. Considering a

μ^{-}

-particle with a mass

207

m_{e}

revolving round a proton as in Hydrogen atom, the energy and radius of proton and

μ^{-}

combination respectively in the first orbit are:

[Assume nucleus to be stationary]

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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].

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 207m_e orbits around a proton].