How would you show that the maximum number of electrons that can be accommodated in an orbit with principal quantum number 'n' is `2n^(2)` ?

Explanation

The electrons, revolving around the nucleus in an atom, are described by four quantum numbers [viz., n, l, m and s]. For a given value of (n), (l) can have n values [these are l=0,1,2,. . .(n-1)]. This shows that the number of subshells present in the 'n' th orbit=n. again the number of orbitals preset in any subshell=(2l+1). Moreover, each orbitals may contain a maximum of 2 electrons. based on this concept, the maximum number of electrons that a particular principal energy level can accommodate, can be calculated. thus for the principal quantum
number 'n', values of l=0,1,2, . . . (n-1).
For l=0, number of orbitals present `=2xx0+1=`
For l=1, number of orbitals present`=2xx1+1=3`
For l=2, number of orbitals present`=2xx2+1=5`
For l=(n-1), number of orbitals present `=(2n-1)+1=(2n-1)`
`therefore`Total numbe of orbitals in n-th energy level
`=1+3+5+7+ . .+(2n-1)`
`=(n)/(2){1+(2n-1)}=n^(2)`
Thus total number of electron in 'n'th energy level
`=n^(2)xx2=2n^(2)`.