Question

In any subshell, the maximum number of electrons having same value of spin quantum number is :
a.) $\sqrt {l(l + 1)} $
b.) $l + 2$
c.) $2l + 1$
d.) $4l + 2$

Answer 1

Hint: According to Pauli’s exclusion principle, each orbital can have only one electron with the same spin quantum number. So, the maximum number of electrons having the same value of spin quantum number is equal to the number of orbitals present in any sub shell.

Complete step by step solution:
The number of shells in an atom is denoted by ‘n’ which can have value from 0, 1, 2 etc.
Each shell consists of subshells that are ‘$s$’, ‘$p$’, ‘$d$’ etc. and each subshell has a certain number of orbitals that have electrons in them. The number of orbitals in each subshell may vary but the number of electrons in each orbital is always equal and the value is equal to 2.
The value of spin is $\dfrac{1}{2}$. It can be either positive or negative. This is in accordance with Pauli’s exclusion principle.
Thus, the maximum number of electrons that have the same value of spin is equal to the number of orbitals present in that sub shell because each orbital has one electron with the same spin.
And we know that the number of orbitals in a subshell is given by the formula - $(2l + 1)$
So, even the maximum number of electrons having the same value of spin quantum number will be given by formula - $(2l + 1)$.

The option c.) is the correct answer.

Note: So, in ‘$s$’ subshell, where $l = 0$, the maximum number of electrons having same value of spin quantum number is -
$(2 \times 0 + 1) = 1$
in ‘$p$’ subshell, where $l = 1$, the maximum number of electrons having same value of spin quantum number is -
$(2 \times 1 + 1) = 3$
in ‘$d$’ subshell, where $l = 2$, the maximum number of electrons having same value of spin quantum number is -
$(2 \times 2 + 1) = 5$