Question

What are the values of the orbital angular momentum of an electron in the orbitals $ 1s $ , $ 3s $ , $ 3d $ and $ 2p $ ?
A. $ 0,\,0,\,\sqrt 6 \dfrac{h}{{\pi 2}},\,\sqrt 2 \dfrac{h}{{\pi 2}} $
B. $ 1,1,\sqrt 4 \dfrac{h}{{\pi 2}},\sqrt 2 \dfrac{h}{{\pi 2}} $
C. $ 0,1,\sqrt 6 \dfrac{h}{{\pi 2}},\sqrt 3 \dfrac{h}{{\pi 2}} $
D. $ 0,0,\sqrt {20} \dfrac{h}{{\pi 2}},\sqrt 6 \dfrac{h}{{\pi 2}} $

Answer 1

Hint: To answer the question, we must first understand what an electron's orbital angular momentum is by definition. The electron's orbital angular momentum is a rotational motion property that is linked to the structure of its orbital. The orbital is the region around the nucleus where the electron will be discovered if detection is tried.

Complete answer:
The formula for orbital angular momentum is given by $ \dfrac{{\surd l\left( {l + 1} \right)h}}{{2\pi }} $ where $ l $ is called as azimuthal quantum number.
The value of $ 'l' $ is determined only by the sub shell and not by the shell. For various subshells, the value of $ 'l' $ is:
 $ s = 0 $
 $ p = 1 $
 $ d = 2 $
 $ f = 3 $
Putting $ l = 0 $ in the above calculation gives us orbital angular momentum $ = 0 $ for a $ 1s $ and $ 3s $ orbital.
 $ l = 2 $ for $ 3d $ orbital; when $ l = 2 $ is substituted into the an above formula, we get
Orbital angular momentum $ = \sqrt {2\left( {2 + 1} \right)} \dfrac{h}{{2\pi }} = \sqrt 6 \dfrac{h}{{2\pi }} $
 $ l = 1 $ for a $ 2p $ orbital, therefore orbital angular momentum $ = \sqrt {1\left( {1 + 1} \right)} \dfrac{h}{{2\pi }} = \sqrt 2 \dfrac{h}{{2\pi }} $
Therefore, the values of the orbital angular momentum of an electron in the orbitals $ 1s $ , $ 3s $ , $ 3d $ and $ 2p $ is $ 0,\,0,\,\sqrt 6 \dfrac{h}{{\pi 2}},\,\sqrt 2 \dfrac{h}{{\pi 2}} $
So, the correct option is: (A) $ 0,\,0,\,\sqrt 6 \dfrac{h}{{\pi 2}},\,\sqrt 2 \dfrac{h}{{\pi 2}} $ .

Additional Information:
One of the electron's "Quantum Numbers," or important attributes, is orbital angular momentum. The “Azimuthal quantum number” is the name given to orbital angular momentum when it is thought of as a Quantum Number. Physicists frequently refer to the electron's "Quantum Numbers" as four key quantized properties:
1. Principal quantum number (energy level of the electron)
2. Azimuthal quantum number (orbital angular momentum)
3. Magnetic quantum number
4. Spin quantum number
Each of these numbers is quantized, which means that only whole numbers can be used to represent them. Because mass is not quantized, it does not show as a Quantum Number.

Note:
The “Azimuthal quantum number” and “extrinsic angular momentum” are other names for orbital angular momentum. The phrases "spin" and "orbital angular momentum" are sometimes used interchangeably. This can be perplexing because “spin” also refers to another form of subatomic particle rotation.