# 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 me orbits around a proton]. - Physics

Numerical

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 me orbits around a proton].

#### Solution

Mass of a negatively charged muon, mμ = 207 me

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"_μ prop "m"_μ.

We have the value of the first Bohr orbit, "r"_"e" = 0.53 Å = 0.53 × 10−10 m

Let rμ be the radius of muonic hydrogen atom.

At equilibrium, we can write the relation as:

"m"_μ "r"_μ = "m"_"e""r"_"e"

207  "m"_"e" xx "r"_μ = "m"_"e""r"_"e"

∴ "r"_μ = (0.53 xx 10^(-10))/207

= 2.56 × 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"_μ = "m"_"e"/"m"_μ = "m"_"e"/(207  "m"_"e")

"E"_μ = 207  "E"_"e"

= 207 × (−13.6)

= −2.81 keV

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

Concept: Energy Levels
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#### APPEARS IN

NCERT Physics Part 1 and 2 Class 12
Chapter 12 Atoms
Exercise | Q 12.17 | Page 437
NCERT Class 12 Physics Textbook
Chapter 12 Atoms
Exercise | Q 17 | Page 437

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