Question

If the subsidiary quantum number of a sub-energy level is ${ 4 }$, then maximum and minimum values of spin multiplicities are:
(a) ${ 9 },{ 1 }$
(b) ${ 10 },{ 1 }$
(c) ${ 10 },{ 2 }$
(d) ${ 8 },{ 1 }$

Answer 1

Hint: Subsidiary quantum number is the synonym for azimuthal quantum number.
The Azimuthal quantum number is denoted by ‘l’.
Spin multiplicity:- It indicates the possible quantum states of a system with given principal spin quantum number S.

Complete step-by-step answer:
An azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital.
It has a value from n-1.
where n = principal quantum number
Each value of ‘l’ indicates a subshell, for example,
For s, l = ${ 0 }$
For p, l = ${ 1 }$
For d, l = ${ 2 }$
For f, l = ${ 3 }$
For g, l = ${ 4 }$
It is given that, l = ${ 4 }$. It indicated g-orbital with ${ 9 }$ degenerate orbitals. It contains a maximum of ${ 18 }$ electrons.
The spin multiplicity value is given by the formula = ${ 2s+1 }$
where S = sum of spins of the electrons. The spin of an electron can be ${ +1/2 }$ or ${ -1/2 }$ depending on if the electron spin is clockwise or anticlockwise.
If there is one electron in the g-orbital = ${ +1/2 }$
So, the unpaired electron contributed to spin multiplicity.
So, the number of degenerate orbitals = ${ 2l+1 }$
Putting the value of l, we get
Number of degenerate orbitals = ${ 2×4+1 } = { 9 }$

The maximum value will be given when 9 unpaired electrons are present.
Therefore, maximum spins = ${ 9× 1/2 } = { 5 }$
And maximum multiplicity = ${ 2S+1 } = { 2×9/2 +1 } = { 10 }$

The minimum value will be given when no unpaired electrons are present, S = ${ 0 }$
So, minimum multiplicity = ${ 2×0+1 } = { 1 }$

Hence, If the subsidiary quantum number of a sub-energy level is ${ 4 }$, then maximum and minimum values of spin multiplicities are ${ 10 }$ and ${ 1 }$ respectively.

Additional Information:
Multiplicity shows the various possible quantum states of a system with the principal quantum number and is given by the formula ${ 2S+1 }$

Note: The possibility of making a mistake is that you may confuse between a subsidiary and principal quantum numbers. A subsidiary quantum number is a quantum number that determines its orbital angular momentum while the principal quantum number is the quantum number which describes the electron’s state.