Question 248: Atoms & Nuclei

Question

Muon ($$\mu $$
$$-$$
) is a negatively charged (|q| = |e|) particle with a mass m
$$\mu $$
= 200 m
e
, where m
e
is the mass of the electron and e is the electronic charge. If $$\mu $$
$$-$$
is bond to a proton to form a hydrogen like atom,  identify the correct statements.
(A)   Radis of the muonic orbit is 200 times smaller than that of the electron.
(B)   The speed of the $$\mu $$
$$-$$
in the n
th
orbit is $${1 \over {200}}$$ times that of the electron in the n
th
orbit.
(C)   The ionization energy of muonic atom is 200 timesmore than of an hydroen atom.
(D)   The momentum of the muon in the n
th
orbit is 200 times more than that of the electron.

Accepted Answer

(i) Radius of the orbit is given by
$$r = {{{h^2}} \over {4\pi m{e^2}}} 	imes {{{n^2}} \over Z}$$
Only, m
$$\mu$$
= 200 m
e
rest are same. So, statement (A) is correct: The radius of muonic orbit is 200 times smaller than that of the electron.
(ii) Velocity of particle in an orbit is given by
$$v = {{nh} \over {2\pi mr}} = {{nh 	imes 4\pi m{e^2} 	imes Z} \over {2\pi m 	imes {h^2} 	imes {n^2}}} = {{2{e^2}Z} \over {hn}}$$
Therefore, speed does not change as it does not depend on mass. So, statement (B) is incorrect.
(iii) Ionisation energy is given by
$$\Delta {E_n} =  - {{2{\pi ^2}m{e^4}{Z^2}} \over {{h^2}{{(4\pi {\varepsilon _0})}^2}}} 	imes \left( {{1 \over {n_2^2}} - {1 \over {n_1^2}}} ight)$$
Only, m
$$\mu$$
= 200 m
e
rest are same. So, statement (C) is correct: The ionisation energy of muonic atom is 200 times more than that of an hydrogen atom.
(iv) Since Momentum $$\propto$$ Energy. So, statement (D) is correct: The momentum of the muon in he nth orbit is 200 times more than that of the electron.

Answer

(A),    (B),    (D)

Answer

(A),    (C),    (D)

Answer

(B),    (D)

Answer

(C),    (D)

Atoms & Nuclei

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