Question

The ratio of the velocities of the electron in the second, third and fourth Bohr’s orbits of the hydrogen atom is:

• A. 3 : 2 : 1
• B. 1 : 2 : 3
• C. 1 : 4 : 9
• D. 6 : 4 : 3
• E. 9 : 4 : 1

Hint:

In Bohr's model, the velocity of the electron is inversely proportional to the principal quantum number \( n \). Thus, the ratio of velocities in different orbits is the inverse ratio of the principal quantum numbers.

Answer 1

The Correct Option is D

The velocity of an electron in a Bohr orbit is given by the formula: \[ v_n = \frac{2 \pi k e^2}{h n} \] where: - \( v_n \) is the velocity of the electron in the \( n \)-th orbit,
- \( k \) is Coulomb's constant,
- \( e \) is the charge of the electron,
- \( h \) is Planck's constant,
- \( n \) is the principal quantum number of the orbit.
The velocity is inversely proportional to the principal quantum number \( n \).
Thus, the ratio of the velocities in the second, third, and fourth orbits is: \[ \frac{v_2}{v_3} = \frac{3}{2}, \quad \frac{v_3}{v_4} = \frac{4}{3} \] Hence, the velocity ratio is: \[ v_2 : v_3 : v_4 = 6 : 4 : 3 \] Therefore, the correct answer is: \[ \boxed{6 : 4 : 3} \]