Step 1: Understanding the Concept:
The question requires us to identify which set of four quantum numbers (n, l, m, s) violates the rules that govern their possible values. These rules define the allowed states for an electron in an atom.
Step 2: Key Formula or Approach:
The rules for the quantum numbers are:
Principal quantum number (n): Can be any positive integer (n = 1, 2, 3, ...).
Azimuthal or angular momentum quantum number (l): Can be any integer from 0 to n-1.
Magnetic quantum number (m or $m_l$): Can be any integer from -l to +l, including 0.
Spin quantum number (s or $m_s$): Can be either +1/2 or -1/2.
We must check each option against these rules.
Step 3: Detailed Explanation:
Let's analyze each set of quantum numbers:
(A) n=2, l=0, m=-1, s=+1/2:
n=2 is valid.
For n=2, l can be 0 or 1. So, l=0 is valid.
For l=0, m can only be 0. The value m=-1 is not allowed.
Therefore, this set is impossible.
(B) n=3, l=0, m=0, s=-1/2:
n=3 is valid.
For n=3, l can be 0, 1, or 2. So, l=0 is valid.
For l=0, m must be 0. So, m=0 is valid.
s=-1/2 is valid.
This set is possible (it describes an electron in a 3s orbital).
(C) n=4, l=1, m=+1, s=+1/2:
n=4 is valid.
For n=4, l can be 0, 1, 2, or 3. So, l=1 is valid.
For l=1, m can be -1, 0, or +1. So, m=+1 is valid.
s=+1/2 is valid.
This set is possible (it describes an electron in a 4p orbital).
(D) n=3, l=2, m=-1, s=-1/2:
n=3 is valid.
For n=3, l can be 0, 1, or 2. So, l=2 is valid.
For l=2, m can be -2, -1, 0, +1, or +2. So, m=-1 is valid.
s=-1/2 is valid.
This set is possible (it describes an electron in a 3d orbital).
Step 4: Final Answer:
The set of quantum numbers (2, 0, -1, +1/2) is impossible because the magnetic quantum number 'm' cannot be -1 when the azimuthal quantum number 'l' is 0. Therefore, option (A) is correct.