why do the atomic shells have certain number of electrons in them?

Answer 1

In fact the number of electrons in a shell is limited by these laws

Each electron in a number is characterized by 4 four numbers called quantic numbers

First the principle of Pauli states that 2 electrons in an atom must differ at least by one quantic number or expressed differently in an atom 2 electrons can not have the same 4 quantic numbers

secondly if n is the number of the shell (n is a positive integer n=1, 2, 3, and so on), the second quantic number is positive, lower than n with the number zero possible (l = 0,1,.., n-1)

the third quantic number is also an integer with values lying between -l and +l with zero possible

the fourth quantic number spin has only 2 values +1/2 or -1/2

these considerations limit the number of electrons in a given shell

example n=1 so l=0 since l must be lower than n , m is also equal to zero and only the spin can differ the two electrons