Question

Why are the electronic configuration numbers the way they are? $(2,\,8,\,18,\,32,\,...,\,{\rm{so on)}}$ How were they determined?

Answer 1

Bohr in $1915$ modified the atomic model given by Rutherford and suggested that the electrons in an atom revolve around the nucleus in a fixed orbit and these orbits have fixed energy levels. Bohr proposed the model whose postulates were based on quantum mechanics.

Complete answer:
Bohr proposed important postulates which are as follows:
Each orbit in which electrons are revolving have a fixed energy level and are termed as orbital shells.
The energy levels of the orbit are represented by principal quantum numbers i.e., $n = 1,2,3,\,...$.
The orbits are assigned shells which are named as $K,\,L,\,M,\,...$ shells.
When an electron is at lowest energy level, then it is termed as the ground state of the atom.
When an electron absorbs energy, it gets excited and moves from lower energy level to the higher energy level.
When an electron releases energy, then it moves from higher energy level toward lower energy level.
According to Bohr’s theory, the maximum number of electrons that can be filled in a shell $= 2{n^2}$
Where, $n$ is the principal quantum number.
Therefore, for different shells the maximum number of electrons that can be filled are as follows:
For K- shell: $n = 1$
Highest number of electrons that can be filled $= 2{(1)^2} \Rightarrow 2$
For L- shell: $n = 2$
Highest number of electrons that can be filled $= 2{(2)^2} \Rightarrow 8$
For M- shell: $n = 3$
Highest number of electrons that can be filled $= 2{(3)^2} \Rightarrow 18$
For N- shell: $n = 4$
Highest number of electrons that can be filled $= 2{(4)^2} \Rightarrow 32$
Hence, the electronic configuration numbers are as $2,\,8,\,18,\,32,\,...,\,{\rm{so on}}$, depending on the value of the principal quantum number.

Note:
Many modifications were made to Bohr’s atomic model due to some of its limitations. The Zeeman effect and Stark effect were not explained through Bohr’s atomic model. It also violated Heisenberg’s uncertainty principle. Hence Sommerfeld modified Bohr’s theory by introducing the concepts of additional energy levels which correspond to fine spectral lines and it is called Bohr-Sommerfeld Theory.