Question

1. A muon is an elementary particle with a negative charge equal to the charge of an electron and a mass approximately 200 times that of the mass of an electron. The muonium atom is formed from a proton and a muon. (a) Calculate the reduced mass, μ, for the muonium atom. (b) Now calculate the Rydberg constant, R, for the muonium atom, the scaling factor for the energy levels: (c) For the hydrogen atom, calculate the Bohr radius, ao, the radius of the circular orbit for the electron in state n = 1 in Bohr's model for the hydrogen atom: e2 (d) Repeat the calculation of the Bohr radius for the muonium atom e) Compare your results in parts (c) and (d); how does changing the mass of the orbiting charge affect the orbit? 2. Calculate the difference in the ionization energies for the 'H and He atoms. This calculation allows you to compare the effect on the ionization energy of changing the nuclear charge Z on an atom with changing the nuclear mass.

Answer 1

mass of Muon = 200 times Mass of e^- 200 times 9. 1 times 10^-31 = 0. 182 times 10^-27 kg Muonium atom is formed from a proton and a Muon Mass of proton = 1. 6 times 10^-27 kg Reduced Mass mu = M_1 M_2/M_1 + M_2 = 1. 6 times 10^-27 times . 182 times 10^-27/ (1. 6 + . 182) times 10^-27 mu = 1. 63 times 10^-28 kg R = e^4 mu /8E_o^2 h^3 e , charge of Muonium ion = -1. 6 times 10^-19 c h = 6. 62 times 10^_34 m^2 kg s^-1 c = 3 times 10^8 Ms^-1 e_o 8. 85 times 10^_12 FM^-1 R = (-1. 6 times 10^-19) ^4 (1. 63 times 10^_26) /8(8. 85 times 10^-12) ^2 (6. 62 times 10^-24) ^3 (3 time 10^8) R = 1. 97 times 10^9 For H atom [Because h = h/2n] a_o = h^2 4 pi E_0/mu e^2 = h^2 elementof_o/pi mu e^2 mu = 9. 1 times 10^-31 a_o = (6. 62 times 10^-34) ^2 (8. 85 times 10^-12) /9 times pi times 9. 1