Answer

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**Hint:**By using Azimuthal quantum number it is easy to find the orbital angular momentum and shape of the orbital. The Azimuthal quantum number is going to be denoted with a symbol l.

l value for s, p, d and f orbitals are 0, 1, 2 and 3 respectively.

There is a formula to calculate the spin multiplicity and it is as follows.

Spin multiplicity = 2S+1, here S = spin of the all electrons present in the orbital.

**Complete step by step answer:**

- In the question it is given that azimuthal quantum number is 3 and we have to find the maximum and minimum value of the multiplicities.

- We know that l value is 3 for f-orbital.

- The maximum number of unpaired electrons that can be accommodated in f-orbital are 7.

- Therefore for 7 electrons the spin quantum number S = $(7\times \dfrac{1}{2})=\dfrac{7}{2}$

- Thus the maximum spin multiplicity is as follows.

\[\begin{align}

& =2S+1 \\

& =2\times \dfrac{7}{2}+1 \\

& =7+1 \\

& =8 \\

\end{align}\]

- Therefore the maximum spin multiplicity if f-orbital is 8.

- Now we have to calculate the minimum multiplicity of f-orbital.

- The minimum number of unpaired electrons that can be accommodated in f-orbital is one.

- Therefore for 1 electron the spin quantum number S = $(1\times \dfrac{1}{2})=\dfrac{1}{2}$

- Thus the minimum spin multiplicity is as follows

\[\begin{align}

& =2S+1 \\

& =2\times \dfrac{1}{2}+1 \\

& =2 \\

\end{align}\]

- Therefore the maximum and minimum multiplicity of f- orbital is 8 and 2.

**- So, the correct option is D.**

**Note:**Maximum multiplicity can be calculated by taking the maximum number of unpaired electrons that can accommodate the respective orbital and the minimum multiplicity can be calculated by taking the minimum number of unpaired electrons that can accommodate the respective orbital.

For s- orbital, the maximum number of unpaired electrons is 1 and minimum number of unpaired electrons also 1.

For p- orbital, the maximum number of unpaired electrons is 3 and minimum number of unpaired electrons also 1.

For d- orbital, the maximum number of unpaired electrons is 5 and minimum number of unpaired electrons is also 1.

For f- orbital, the maximum number of unpaired electrons is 7 and minimum number of unpaired electrons also 1.

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