A muonic hydrogen atom is a proton orbited by a muon (a particle with the same charge as an electron and 207 times its mass)in which the mass of the muon is significant relative to the mass of the proton.What would be the radius of the smallest muon orbit in a muonic hydrogen atom? A) 1.03 × 10^-8m B) 1.10 × 10^-8m C) 2.56 × 10^-13m D) 2.84 × 10^-13m E) 5.29 × 10^-11m

Question

Quizplus · Accepted Answer

The radius of the smallest electron orbit in a hydrogen atom can be calculated using the Bohr radius formula:
r = (4πε0ħ²)/(me²)
where ε0 is the permittivity of free space, ħ is the reduced Planck constant, and me is the mass of the electron.
For a muonic hydrogen atom, we need to replace me with the reduced mass:
μ = (mp*mμ)/(mp+ mμ)
where mp is the mass of the proton and mμ is the mass of the muon.
Using the same formula, we get:
r = (4πε0ħ²)/(μe²)
Plugging in the values, we get:
r = (4π*8.85×10⁻¹²*1.05×10⁻³⁴/(2.01*1.67×10⁻²⁷*5.69×10⁻28*1.60×10⁻¹⁹)
r = 2.84 × 10⁻¹³ m, which is closest to choice (D).

A Muonic Hydrogen Atom Is a Proton Orbited by a Muon