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Question: The time of revolution of an electron around a nucleus of charge \(Ze\) in nth Bohr's orbit is direc...

The time of revolution of an electron around a nucleus of charge ZeZe in nth Bohr's orbit is directly proportional to:
A. nn
B. n3Z2\dfrac{{{n^3}}}{{{Z^2}}}
C. n2Z\dfrac{{{n^2}}}{Z}
D. Zn\dfrac{Z}{n}

Explanation

Solution

Hint Electrons revolve around the nucleus because the centripetal force is balanced by the electrostatic force of attraction between electron and positively charged nucleus. To find time of revolution of an electron, first need to find the radius and velocity of revolution of electron around the nucleus. Use the Bohr’s postulate of quantization of angular momentum to find the radius and velocity of revolution.

Complete step-by-step solution :
Coulomb force of attraction between the nucleus of charge
ZeZe and electron revolving in the orbit of radius rn{r_n} id given by
Fn=14π∈0Ze×ern2=Ze24π∈0rn2{F_n} = \dfrac{1}{{4\pi { \in _0}}}\dfrac{{Ze \times e}}{{{r_n}^2}} = \dfrac{{Z{e^2}}}{{4\pi { \in _0}{r_n}^2}} -(i)
This force provides the necessary centripetal force for the electron to move in a circular orbit of radius rn{r_n}with a speed vn{v_n}.
i.e., mvn2rn=Ze24π∈0rn2\dfrac{{m{v_n}^2}}{{{r_n}}} = \dfrac{{Z{e^2}}}{{4\pi { \in _0}{r_n}^2}} or mvn2=Ze24π∈0rnm{v_n}^2 = \dfrac{{Z{e^2}}}{{4\pi { \in _0}{r_n}}} -(ii)
According to Bohr’s postulate of quantization of angular momentum
Ln=mvnrn=nh2π{L_n} = m{v_n}{r_n} = \dfrac{{nh}}{{2\pi }}
i.e., vn=nh2πmrn{v_n} = \dfrac{{nh}}{{2\pi m{r_n}}} -(iii)
Substituting eqn. (iii) in eqn. (ii), we get,
m(nh2πmrn)2=Ze24π∈0rnm{\left( {\dfrac{{nh}}{{2\pi m{r_n}}}} \right)^2} = \dfrac{{Z{e^2}}}{{4\pi { \in _0}{r_n}}} or n2h24π2mrn=Ze24π∈0\dfrac{{{n^2}{h^2}}}{{4{\pi ^2}m{r_n}}} = \dfrac{{Z{e^2}}}{{4\pi { \in _0}}}
i.e., rn=n2h2∈0πmZe2{r_n} = \dfrac{{{n^2}{h^2}{ \in _0}}}{{\pi mZ{e^2}}} -(iv)
Substituting eqn. (iv) in eqn. (iii)
vn=Ze22h∈0n{v_n} = \dfrac{{Z{e^2}}}{{2h{ \in _0}n}} -(v)
Time period of revolution of electron is given by:
T=2πrnvnT = \dfrac{{2\pi {r_n}}}{{{v_n}}}
Hence, TαrnvnT\alpha \dfrac{{{r_n}}}{{{v_n}}}
From eqn. (iv) rnαn2Z{r_n}\alpha \dfrac{{{n^2}}}{Z} and from eqn. (v) vnαZn{v_n}\alpha \dfrac{Z}{n}
Hence Tαn3Z2T\alpha \dfrac{{{n^3}}}{{{Z^2}}}.
So, the correct option is B.

Note:- Bohr’s postulates are only applicable on Hydrogen like atom means atoms with only one electron in their valence shell. Bohr’s gives the postulates about hydrogen atoms but these are also applicable hydrogen like atoms.