Question

Choose the correct relation on the basis of Bohr's theory -

• A. velocity of electron $\propto$
• B. frequency of revolution $\propto \frac{Z^2}{n^3}$
• C. radius of orbit $\propto n^2 Z$
• D. force on electron $\propto \frac{Z^3}{n^4}$

Answer 1

Answer: (A), (B), (D)

A, B, D

Answer 2

Bohr's theory provides the following relations for hydrogen-like atoms:

1. Radius of orbit ($r_n$): $r_n = \frac{n^2 a_0}{Z}$, where $a_0$ is Bohr radius. Thus, $r_n \propto \frac{n^2}{Z}$. Option (C) $r_n \propto n^2 Z$ is incorrect.

2. Velocity of electron ($v_n$): $v_n = \frac{Z e^2}{2 \epsilon_0 n h} = \frac{Z v_0}{n}$, where $v_0$ is the velocity in the first Bohr orbit of H. Thus, $v_n \propto \frac{Z}{n}$. Option (A) is correct.

3. Frequency of revolution ($f_n$): $f_n = \frac{v_n}{2\pi r_n}$. Substituting $v_n \propto \frac{Z}{n}$ and $r_n \propto \frac{n^2}{Z}$: $f_n \propto \frac{Z/n}{n^2/Z} = \frac{Z^2}{n^3}$. Option (B) is correct.

4. Force on electron ($F$): This is the electrostatic force between the nucleus and the electron. $F = \frac{k (Ze) e}{r_n^2} = \frac{k Z e^2}{r_n^2}$. Substituting $r_n \propto \frac{n^2}{Z}$: $F \propto \frac{Z}{(n^2/Z)^2} = \frac{Z}{n^4/Z^2} = \frac{Z^3}{n^4}$. Option (D) is correct.