Question

What is the maximum number of electrons in an atom that can have the following quantum numbers?

a) n = 5, l = 3, ml = +3
b) n = 3, l = 1, ml = -1

Answer 1

(a) The maximum number of electrons in
`n=5,l=3,m_l=+3` are, 2 electrons.

(b) The maximum number of electrons in
`n=3,l=1,m_l=-1` are, 2 electrons.

Explanation :

Azimuthal Quantum Number : It describes the shape of the orbital. It is represented as 'l'. The value of l ranges from 0 to (n-1). For l = 0,1,2,3... the orbitals are s, p, d, f...

Magnetic Quantum Number : It describes the orientation of the orbitals. It is represented as
`m_l`. The value of this quantum number ranges from
`(-l\text{ to }+l)`. When l = 2, the value of
`m_l` will be -2, -1, 0, +1, +2.

Spin Quantum number : It describes the direction of electron spin. This is represented as
`m_s` The value of this is
`+(1)/(2)` for upward spin and
`-(1)/(2)` for downward spin.

(a)
`n=5,l=3,m_l=+3`

As per above information,
`m_s=+(1)/(2)\text{ and }-(1)/(2)` (For each sub-shell)

From this we conclude that,
`m_l=+3` means there are 1 orbital and each orbital contains 2 electrons. So, the number of electrons held in this will be, 2 electrons
`(m_s=+(1)/(2)\text{ and }-(1)/(2))`.

(b)
`n=3,l=1,m_l=-1`

As per above information,
`m_s=+(1)/(2)\text{ and }-(1)/(2)` (For each sub-shell)

From this we conclude that,
`m_l=-1` means there are 1 orbital and each orbital contains 2 electrons. So, the number of electrons held in this will be, 2 electrons
`(m_s=+(1)/(2)\text{ and }-(1)/(2))`.

Answer 2

We have to get the maximum number of electrons in an atom that can have the given quantum numbers.

(a) The maximum number of electrons in an atom that can have quantum number n = 5, l = 3,
`m_(l)` = +3 is 2 (two).

(b) The maximum number of electrons in an atom that can have quantum number n = 3, l = 1,
`m_(l)` = -1 is 2 (two).

Here, n= principal quantum number, l= azimuthal quantum number and
`m_(l)`= magnetic quantum number.

l= 0 to (n-1) and
`m_(l)`= -l to +l including 0.

(a) n = 5, l = 3,
`m_(l)` = +3, this set of quantum number represents one 5f orbital which can contain maximum two electron of opposite spins.

(b) n = 3, l = 1,
`m_(l)` = -1, this set of quantum number represents one 3p orbital which can contain maximum two electron of opposite spins.