Question

If n and l are principal and azimuthal quantum numbers respectively, then the expression for calculating the total number of electrons in any energy level is:
(A)$\sum\limits_{l = 0}^{l = n} {2(2l + 1)} $
(B) $\sum\limits_{l = 1}^{l = n - 1} {2(2l + 1)} $
(C) $\sum\limits_{l = 0}^{l = n + 1} {2(2l + 1)} $
(D) $\sum\limits_{l = 0}^{l = n - 1} {2(2l + 1)} $

Answer 1

Hint: If a principal quantum number is n, then the value of azimuthal quantum number can be found from it. The azimuthal quantum number for a shell is from 0 to n-1.

Complete step by step solution:
We know that n is the principal quantum number and it has the values 1,2,3…. for K,L,M… shells respectively. Thus, n shows the shell number. The Azimuthal quantum number indicates the sub-shell of the electron and it is shown by l.
- Now, we need to check all the expressions in order to find the correct answer.
A) $\sum\limits_{l = 0}^{l = n} {2(2l + 1)} $
Here, it is given that for a particular energy level, we need to sum the given function from l=0 to l=n .
- The value of the azimuthal quantum number for a shell cannot be more than n-1 value of that cell. So, the value of the azimuthal quantum number for a shell is from 0 to n-1.
- Here, l=n limit is not correct. Thus, this option is wrong.
B) Here, the limit of the function is from l=1 to l=n-1.
- All the shells have an orbital which has azimuthal quantum number of 0. So, here, l=1 limit is not correct. So, this option is also wrong.
C) Here, the limit of the function if given from l=0 to l=n+1.
The value of azimuthal quantum number of a shell cannot be l=n+1. So, this option is also wrong.
D) $\sum\limits_{l = 0}^{l = n - 1} {2(2l + 1)} $
The given expression is correct for calculating the total number of electrons in an energy level.
Here, the limits are l=o to l=n-1 which will give correct values of the number of electrons.
For n=1 level, we get
$\sum\limits_{l = 0}^{l = n - 1} {2(2l + 1)} = 2(2(0) + 1) = 2$ electrons
For n=2 level, we get
$\sum\limits_{l = 0}^{l = n - 1} {2(2l + 1)} = 2(2(0) + 1) + 2(2(1) + 1) = 2 + 6 = 8$ electrons
Same way, we can obtain 18 electrons for n=3 level.

Thus, the correct answer is (D).

Note: Remember that all electrons present in the same orbital have the same energy. Means electrons with the same azimuthal quantum number have the same amount of energy. The electrons with the same principal quantum number can have different energy as the azimuthal quantum number of them may differ.