Question

Explain giving reasons which of the following sets of quantum numbers are not possible.
a.) n = 0, l = 0, ${m_l}$= 0, ${m_s}$= $+ \dfrac{1}{2}$
b.) n = 1, l = 0, ${m_l}$= 0, ${m_s}$= $- \dfrac{1}{2}$
c.) n = 1, l = 1, ${m_l}$= 0, ${m_s}$= $+ \dfrac{1}{2}$
d.) n = 2, l = 1, ${m_l}$= 0, ${m_s}$= $- \dfrac{1}{2}$
e.) n = 2, l = 3, ${m_l}$= - 3, ${m_s}$= $+ \dfrac{1}{2}$
f.) n = 3, l = 1, ${m_l}$= 0, ${m_s}$= $+ \dfrac{1}{2}$

Answer 1

The value ‘n’ designates the principal shell in which the electron is entering, The value of ‘l’ can be calculated from (n - 1). And further, the value of ‘${m_l}$’ which is a magnetic quantum number can be from +l to -l. The ‘${m_s}$’ can be +$\dfrac{1}{2}$or - $\dfrac{1}{2}$.

Complete step by step answer:
First, let us understand what is n, l, ${m_l}$and ${m_s}$ values. The ‘n’ alphabet is used to describe the Principal quantum number. These designate the main shell in which the electron entered in an atom. Its value can be any integer with a positive value starting from 1.
-The ‘l’ alphabet is used to describe the Azimuthal quantum number. These describe the shape of the given orbital. The value of azimuthal quantum number is obtained from principal quantum number. Its value can be obtained as-
‘l’ = n - 1
-The ‘${m_l}$’ alphabet is used to describe the magnetic quantum number. These describe the total number of orbitals in a sub shell and their orientation. Its value can be obtained from the azimuthal quantum number. The value can be from +l to -l.
-The ‘${m_s}$’ alphabet is used to describe the spin quantum number. It describes the spin of the electron. The spin can be +$\dfrac{1}{2}$or - $\dfrac{1}{2}$.
-Let us see the options given to us and find out, which of these is not possible.
-The first option is n = 0, l = 0, ${m_l}$= 0, ${m_s}$= $+ \dfrac{1}{2}$. This is not possible because ‘n’ can not be zero.
-The second option is n = 1, l = 0, ${m_l}$= 0, ${m_s}$= $- \dfrac{1}{2}$. This is possible because it follows the above rules.
-The third is n = 1, l = 1, ${m_l}$= 0, ${m_s}$= $+ \dfrac{1}{2}$. This is not possible because for n = 1,
‘l’ = n - 1
‘l’ = 1 - 1
‘l’ = 0
-The next option is n = 2, l = 1, ${m_l}$= 0, ${m_s}$= $- \dfrac{1}{2}$. This is possible as it abides all rules.
-The other one is. n = 2, l = 3, ${m_l}$= - 3, ${m_s}$= $+ \dfrac{1}{2}$. This is not possible because for n = 2,
‘l’ = n - 1
‘l’ = 2 - 1
‘l’ = 1
-The last option is n = 3, l = 1, ${m_l}$= 0, ${m_s}$= $+ \dfrac{1}{2}$. This is possible if it follows the above rules.

Thus, the options with sets of quantum numbers are not possible are option a.), c.) and e.).

Note: We know that ‘l’ = n - 1. For any value of ‘n’, l can not be more than or equal to n.
Further, the value of spin magnetic quantum number will be +$\dfrac{1}{2}$or - $\dfrac{1}{2}$. The value can not be any else. In one orbitals, no two electrons can have all values of all quantum numbers the same. If three values are the same, then they will be different in spin quantum numbers.