A negatively charged muon (mass ) can replace the electron in a hydrogen atom to form…

CHEMISTRY · Multi-part · CBSE · Chemistry · Structure of Atom · Structure of Atom

Problem

A negatively charged muon (mass mμ≈207 mem_{\mu} \approx 207\,m_e) can replace the electron in a hydrogen atom to form a muonic atom. Assume that the characteristic orbital radius scales inversely with the particle’s mass. (i) Taking the radius of the lowest muonic orbit to be rμ=a0/207r_{\mu} = a_0/207, where a0=5.3×10−11 ma_0 = 5.3 \times 10^{-11}\,\text{m}, estimate the minimum kinetic energy of the bound muon. Use h=6.63×10−34 J sh = 6.63 \times 10^{-34}\,\text{J\,s} and me=9.11×10−31 kgm_e = 9.11 \times 10^{-31}\,\text{kg}. (ii) The ground–state energy predicted by the Bohr model for muonic hydrogen is approximately −2.8 keV-2.8\,\text{keV}. Compare your estimate from (i) with this value and state whether the Bohr picture of well-defined circular orbits is compatible with the required kinetic energy.

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