Question

The difference in angular momentum of electron between two successive orbits of hydrogen atom is

• A. \(\frac{h}{2\pi}\)
• B. \(\frac{h}{\pi}\)
• C. \(\frac{h}{2}\)
• D. 2h

Hint:

Remember Bohr's three postulates:
(1) Electrons revolve in stable circular orbits without radiating energy.
(2) Angular momentum is quantized: \(L = n\hbar\).
(3) Energy is radiated/absorbed when an electron jumps between orbits, with frequency \(f = \frac{\Delta E}{h}\).
The difference in angular momentum between any two consecutive orbits is always \(\hbar\) or \(\frac{h}{2\pi}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question relates to Bohr's model of the hydrogen atom, specifically his second postulate, which deals with the quantization of angular momentum. According to Bohr, an electron can only revolve in certain stable orbits where its angular momentum is an integral multiple of a fundamental unit.

Step 2: Key Formula or Approach:
Bohr's quantization condition for the angular momentum (\(L_n\)) of an electron in the \(n^{th}\) orbit is given by: \[ L_n = n \frac{h}{2\pi} \] where \(n\) is the principal quantum number (\(n = 1, 2, 3, \ldots\)) and \(h\) is Planck's constant.

Step 3: Detailed Explanation:
We need to find the difference in angular momentum between two successive orbits. Let's consider two consecutive orbits with principal quantum numbers \(n\) and \(n+1\).
The angular momentum in the \((n+1)^{th}\) orbit is: \[ L_{n+1} = (n+1) \frac{h}{2\pi} \] The angular momentum in the \(n^{th}\) orbit is: \[ L_n = n \frac{h}{2\pi} \] The difference (\(\Delta L\)) between them is: \[ \Delta L = L_{n+1} - L_n = (n+1) \frac{h}{2\pi} - n \frac{h}{2\pi} \] \[ \Delta L = \left( (n+1) - n \right) \frac{h}{2\pi} \] \[ \Delta L = (1) \frac{h}{2\pi} = \frac{h}{2\pi} \] The difference is constant and independent of \(n\).

Step 4: Final Answer:
The difference in angular momentum between any two successive orbits is \(\frac{h}{2\pi}\). This is also often written as \(\hbar\) (h-bar). Therefore, option (A) is correct.