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Question: If muonic hydrogen atom is an atom in which a negatively charged muon \((\mu)\) of mass about \(207\...

If muonic hydrogen atom is an atom in which a negatively charged muon (μ)(\mu) of mass about 207 me207\text{ }\text{m}_{e} revolves around a proton, then first Bohr radius of this atom is (re=0.53×10−10 m)(r_{e} = 0.53 \times 10^{- 10}\text{ m})

A

2.56×10−10 m2.56 \times 10^{- 10}\text{ m}

B

2.56×10−11 m2.56 \times 10^{- 11}\text{ m}

C

2.56×10−12 m2.56 \times 10^{- 12}\text{ m}

D

2.56×10−13 m2.56 \times 10^{- 13}\text{ m}

Answer

2.56×10−13 m2.56 \times 10^{- 13}\text{ m}

Explanation

Solution

According to Bohr’s atomic model,

r∝1mr \propto \frac{1}{m}

⇒rμre=memμ\Rightarrow \frac{r_{\mu}}{r_{e}} = \frac{m_{e}}{m_{\mu}} … (i)

Here re=0.53×10−10mr_{e} = 0.53 \times 10^{- 10}m

mμ=207mem_{\mu} = 207m_{e}

∴rμ=me207me×0.53×10−10\therefore r_{\mu} = \frac{m_{e}}{207m_{e}} \times 0.53 \times 10^{- 10} (using (i))

=2.56×10−13m= 2.56 \times 10^{- 13}m