Question

For electron in '2s' and '2p' orbitals, the orbital angular momentum values, respectively are :

• A. \( \sqrt{2} \frac{h}{2\pi} \) and 0
• B. \( \frac{h}{2\pi} \) and \( \sqrt{2} \frac{h}{2\pi} \)
• C. 0 and \( \sqrt{6} \frac{h}{2\pi} \)
• D. 0 and \( \sqrt{2} \frac{h}{2\pi} \)

Hint:

The orbital angular momentum depends only on the azimuthal quantum number \( l \). Remember the values of \( l \) for different orbitals: s (\( l=0 \)), p (\( l=1 \)), d (\( l=2 \)), f (\( l=3 \)), and so on. Use the formula \( L = \sqrt{l(l+1)} \frac{h}{2\pi} \) to calculate the orbital angular momentum for each orbital.

Answer 1

The Correct Option is D

The orbital angular momentum of an electron in an atom is given by the formula: \[ L = \sqrt{l(l+1)} \frac{h}{2\pi} = \sqrt{l(l+1)} \hbar \] where \( l \) is the azimuthal quantum number (also known as the orbital angular momentum quantum number), and \( h \) is Planck's constant, with \( \hbar = \frac{h}{2\pi} \) being the reduced Planck constant. 

For a '2s' orbital, the principal quantum number \( n = 2 \), and for an 's' orbital, the azimuthal quantum number \( l = 0 \). Substituting \( l = 0 \) into the formula for orbital angular momentum: \[ L_{2s} = \sqrt{0(0+1)} \frac{h}{2\pi} = \sqrt{0} \frac{h}{2\pi} = 0 \] So, the orbital angular momentum for an electron in a 2s orbital is 0. For a '2p' orbital, the principal quantum number \( n = 2 \), and for a 'p' orbital, the azimuthal quantum number \( l = 1 \). Substituting \( l = 1 \) into the formula for orbital angular momentum: \[ L_{2p} = \sqrt{1(1+1)} \frac{h}{2\pi} = \sqrt{1(2)} \frac{h}{2\pi} = \sqrt{2} \frac{h}{2\pi} \] So, the orbital angular momentum for an electron in a 2p orbital is \( \sqrt{2} \frac{h}{2\pi} \). The orbital angular momentum values for electrons in '2s' and '2p' orbitals are 0 and \( \sqrt{2} \frac{h}{2\pi} \) respectively. 

This corresponds to option (4).

Answer 2

The orbital angular momentum of an electron in an atomic orbital is given by the formula:

\(L = \sqrt{l(l+1)} \frac{h}{2\pi}\)

where:

• \(L\) is the angular momentum.
• \(l\) is the azimuthal quantum number (orbital quantum number).
• \(h\) is Planck's constant.

The azimuthal quantum number \(l\) determines the subshell of the electron:

• For an \(s\) orbital, \(l = 0\)
• For a \(p\) orbital, \(l = 1\)

Now, calculate the orbital angular momentum for:

1. 2s orbital:

Since \(l = 0\)

\(L = \sqrt{0(0+1)} \frac{h}{2\pi} = 0\)

1. 2p orbital:

Since \(l = 1\)

\(L = \sqrt{1(1+1)} \frac{h}{2\pi} = \sqrt{2} \frac{h}{2\pi}\)

Therefore, for electrons in 2s and 2p orbitals, the orbital angular momentum values are 0 and \(\sqrt{2} \frac{h}{2\pi}\) respectively.

So, the correct answer is: 0 and \(\sqrt{2} \frac{h}{2\pi}\).