Question

According to Bohr's atomic theory : (A) Kinetic energy of electron is $\propto Z^2 / n^2$. (B) The product of velocity (v) of electron and principal quantum number (n), 'vn' $\propto Z^2$. (C) Frequency of revolution of electron in an orbit is $\propto Z^2 / n^3$. (D) Coulombic force of attraction on the electron is $\propto Z^3 / n^4$. Choose the most appropriate answer :

• A. (A), (C) and (D) only
• B. (A) and (D) only
• C. (C) only
• D. (A) only

Hint:

Remember the core proportionalities: $r \propto n^2/Z$, $v \propto Z/n$, and $E \propto Z^2/n^2$. Most other properties like force, frequency, and momentum can be derived from these three.

Answer 1

The Correct Option is B

Bohr's atomic theory describes the behavior of electrons in atoms, most notably in hydrogen atoms. Let's analyze each part of the question to identify the correct statements:

1. Kinetic Energy of Electron: According to Bohr's theory, the kinetic energy of an electron in an orbit is given by KE = \frac{m e^4 Z^2}{8 \varepsilon_0^2 h^2 n^2}, where:
   • Z is the atomic number,
   • n is the principal quantum number.
   Thus, kinetic energy is proportional to \frac{Z^2}{n^2}. Statement (A) is correct.
2. Product of Velocity and Principal Quantum Number: The velocity of an electron in a Bohr orbit is given by v = \frac{e^2 Z}{2 \varepsilon_0 h n}. Hence, vn = \frac{e^2 Z}{2 \varepsilon_0 h} which does not depend on Z^2. This implies the product of velocity and principal quantum number is proportional to Z, not Z^2. This invalidates statement (B).
3. Frequency of Revolution: The frequency of revolution is given by f = \frac{v}{2 \pi r}. Substituting the equations for v and r, we find that the frequency is proportional to \frac{Z^2}{n^3}, as stated by Bohr's formula: f = \frac{Z^2 e^4 m}{4 \pi \varepsilon_0^2 h^3 n^3}. This makes statement (C) correct.
4. Coulombic Force of Attraction: The Coulombic force is given by F = \frac{Z e^2}{4 \pi \varepsilon_0 r^2} where r = \frac{n^2 h^2}{4 \pi^2 m Z e^2}. Simplifying this relation gives F \propto \frac{Z^3}{n^4}, corroborating statement (D).

From the analysis, both statement (A) and (D) are correct. Therefore, the most appropriate answer is (A) and (D) only.