Question

If n and l are respectively the principal and azimuthal quantum numbers, then the expressions for calculating the total number of electrons in any shell is:

• A. $\sum _{l=1}^{l=n}$ $2(2l+1)$
• B. $\sum _{l=1}^{l=n-1}$ $2(2l+1)$
• C. $\sum _{l=0}^{l=n+1}$ $2(2l+1)$
• D. $\sum _{l=0}^{l=n-1}$ $2(2l+1)$

Answer 1

The correct option is D $\sum _{l=0}^{l=n-1}$ $2(2l+1)$

Principal quantum number = n
Azimuthal quantum number '$l$' ranges from 0 to n - 1.
maximum number of electrons in a shell = $2n^2$
Like, for n = 1, l = 0 and m = $-1,0,+1$ i.e 3 orbitals and each orbital contain 2 electrons.
Number of orbitals for a subshell $=$ $2l+1$,
Number of electrons in an orbital $=$ 2
Therefore, total electrons in any orbital $=$ $\sum _{l=0}^{n-1}$ $2(2l+1)$