The radius of the first permitted Bohr orbit, for the electron, in a hydrogen atom equals 0.51 Å and its ground state energy equals -13.6 eV. If the electron in the hydrogen atom is replaced by muon ( μ⁻) [charge same as electron and mass 207 me ], the first Bohr radius and ground state energy will be, (a) 2.56 × 10⁻¹³ m, -13.6 eV (b) 0.53 × 10⁻¹³ m, -3.6 eV (c) 25.6 × 10⁻¹³ m, -2.8 eV (d) 2.56 × 10⁻¹³ m, -2.8 keV

Answer: (d). Bohr radius r ∝ 1/m and energy E ∝ m. rμ = 0.53 × 10⁻¹⁰207 ≈ 2.56 × 10⁻¹³ m Eμ = -13.6 × 207 ≈ -2815 eV ≈ -2.8 keV So the answer is (d) 2.56 × 10⁻¹³ m, -2.8 keV.

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Bohr Model: Orbits, Speed & Angular Momentum

The radius of the first permitted Bohr orbit, for the electron, in a hydrogen atom equals 0.51 Å and its ground state energy equals $-13.6\text{ eV}$ . If the electron in the hydrogen atom is replaced by muon ( $\mu^-$ ) [charge same as electron and mass $207\, m_e$ ], the first Bohr radius and ground state energy will be,

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Answer: (d)

Bohr radius $r \propto \frac{1}{m}$ and energy $E \propto m$ .
$r_\mu = \frac{0.53 \times 10^{-10}}{207} \approx 2.56 \times 10^{-13}\text{ m}$
$E_\mu = -13.6 \times 207 \approx -2815\text{ eV} \approx -2.8\text{ keV}$
So the answer is (d) $2.56 \times 10^{-13}\text{ m}, -2.8\text{ keV}$ .

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