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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:

1. 2.56 × 10⁻¹³ m, −13.6 eV
2. 0.53 × 10⁻¹³ m, −3.6 eV
3. 25.6 × 10⁻¹³ m, −2.8 eV
4. 2.56 × 10⁻¹³ m, −2.8 eV

Correct answer: 2.56 × 10⁻¹³ m, −2.8 eV

Solution

The Bohr radius is inversely proportional to the mass of the orbiting particle. Since the muon is 207 times heavier than the electron, the first Bohr radius will be reduced by a factor of 207, giving approximately 2.56 × 10⁻¹³ m. The energy is inversely proportional to the radius, so the ground state energy will increase by a factor of 207, resulting in −2.8 eV.

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