According to Bohr’s Model

(A) The radius of the orbiting electron is directly proportional to ’n’.
(B) The speed of the orbiting electron is directly proportional to 1/n.
(C) The magnitude of the total energy of the orbiting electron is directly proportional to 1/n2.
(D) The radius of the orbiting electron is directly proportional to n2.
Choose the correct answer from the options given below

The correct option is (B), (C) and (D) only.

According to Bohr's model of the hydrogen atom (and hydrogen-like ions with atomic number Z), we can derive the relationships for the radius, speed, and energy of an electron in the n-th orbit as follows:

1. Radius of the Orbit (r):
The electrostatic force of attraction between the nucleus and the orbiting electron provides the necessary centripetal force:
$$\frac{m{v}^2}{r}=\frac{kZ{e}^2}{r^2}$$
According to Bohr's quantization condition, the angular momentum of the electron is:
$$mvr=\frac{nh}{2\pi }$$
Solving these equations for the radius r of the n-th orbit yields:
$$r_n=\frac{n^2{h}^2}{4{\pi }^{2}mkZ{e}^2}$$
From this expression, it is clear that for a given atom (constant Z), the radius is directly proportional to n²:
$$r_n\propto n^2$$
Therefore, statement (D) is correct, while statement (A) is incorrect.

2. Speed of the Orbiting Electron (v):
Using the quantization condition, we can substitute the value of r to find the speed v:
$$v_n=\frac{2\pi kZ{e}^2}{nh}$$
From this relation, the speed of the electron is inversely proportional to n, or directly proportional to 1/n:
$$v_n\propto \frac{1}{n}$$
Therefore, statement (B) is correct.

3. Total Energy of the Electron (E):
The total energy E is the sum of the kinetic and potential energies, which is given by:
$$E_n=-\frac{2{\pi }^{2}m{k}^2{Z}^2{e}^4}{n^2{h}^2}$$
The magnitude of the total energy |E_n| is:
$$\left|E_n\right|\propto \frac{1}{n^2}$$
Therefore, statement (C) is correct.

Thus, statements (B), (C), and (D) are correct, matching the correct option.