Question

According to Bohr's atomic theory, which of following statement(s) is/are correct?

• A. Kinetic energy of electron $\propto \frac{Z^2}{n^2}$
• B. Frequency of revolution of electron in a Bohr orbit $\propto \frac{n^3}{Z^2}$
• C. Angular momentum of electron $\propto \left(\frac{n}{Z}\right)$
• D. Product of velocity of electron and principal quantum number $n$ $\propto Z$

Answer 1

Answer: (A), (D)

Kinetic energy of electron $\propto \frac{Z^2}{n^2}$
Product of velocity of electron and principal quantum number $n$ $\propto Z$

The statements (A) and (D) are correct.
(A) Kinetic energy of electron $\propto \frac{Z^2}{n^2}$
$K.E=\frac{1}{2}m{u}^2=\frac{1}{2}\frac{kZ{e}^2}{r}=\frac{kZ{e}^2}{2}\frac{4{\pi }^{2}mZ{e}^{2}k}{n^2{h}^2}=k^2{Z}^2{e}^4\frac{2k^2{Z}^2{e}^4{\pi }^{2}m}{n^2{h}^2}$
(B) Frequency of revolution of electron in a Bohr orbit
$\propto \frac{Z^2}{n^3}$
$=\frac{u}{2\pi r}=\frac{Z}{n}\left[\frac{2\pi e^2}{h}\right]\times \frac{4{\pi }^{2}mZ{e}^{2}k}{2\pi n^2{h}^2}=\frac{Z^2}{n^3}\times \frac{4{\pi }^{2}mk{e}^4}{h^3}$
(C) Angular momentum of electron
$\propto n$
Angular momentum of electron $=n\left(\frac{h}{2\pi }\right)$
(D) Product of velocity of electron and principal quantum number $n\propto Z$.
The velocity of electron $u=\frac{Z}{n}\left[\frac{2\pi e^2}{h}\right]$
Product of velocity of electron and principal quantum number $u\times n=Z\left[\frac{2\pi e^2}{h}\right]$