To determine which sets of quantum numbers are valid for an electron in an atom, we must check each set against the three primary quantum number rules:
1. Principal Quantum Number ([tex]\(n\)[/tex]): [tex]\(n\)[/tex] must be a positive integer ([tex]\(n > 0\)[/tex]).
2. Azimuthal Quantum Number ([tex]\(l\)[/tex]): [tex]\(l\)[/tex] can be any integer from 0 to [tex]\(n-1\)[/tex] ([tex]\(0 \leq l < n\)[/tex]).
3. Magnetic Quantum Number ([tex]\(m_e\)[/tex]): [tex]\(m_e\)[/tex] can be any integer ranging from [tex]\(-l\)[/tex] to [tex]\(+l\)[/tex] ([tex]\(-l \leq m_e \leq l\)[/tex]).
Let's evaluate each given set of quantum numbers one by one:
1. [tex]\((-2, 1, 0)\)[/tex]
- [tex]\(n = -2\)[/tex]: Not valid because [tex]\(n\)[/tex] must be positive.
2. [tex]\((3, 2, -3)\)[/tex]
- [tex]\(n = 3\)[/tex]: Valid.
- [tex]\(l = 2\)[/tex]: Valid because [tex]\(0 \leq 2 < 3\)[/tex].
- [tex]\(m_e = -3\)[/tex]: Not valid because [tex]\(-2 \leq m_e \leq 2\)[/tex].
3. [tex]\((4, 3, 4)\)[/tex]
- [tex]\(n = 4\)[/tex]: Valid.
- [tex]\(l = 3\)[/tex]: Valid because [tex]\(0 \leq 3 < 4\)[/tex].
- [tex]\(m_e = 4\)[/tex]: Not valid because [tex]\(-3 \leq m_e \leq 3\)[/tex].
4. [tex]\((2, 2, 2)\)[/tex]
- [tex]\(n = 2\)[/tex]: Valid.
- [tex]\(l = 2\)[/tex]: Not valid because [tex]\(l\)[/tex] must be less than [tex]\(n\)[/tex] (i.e., [tex]\(0 \leq l < 2\)[/tex]).
5. [tex]\((4, 2, -1)\)[/tex]
- [tex]\(n = 4\)[/tex]: Valid.
- [tex]\(l = 2\)[/tex]: Valid because [tex]\(0 \leq 2 < 4\)[/tex].
- [tex]\(m_e = -1\)[/tex]: Valid because [tex]\(-2 \leq m_e \leq 2\)[/tex].
6. [tex]\((3, 2, 0)\)[/tex]
- [tex]\(n = 3\)[/tex]: Valid.
- [tex]\(l = 2\)[/tex]: Valid because [tex]\(0 \leq 2 < 3\)[/tex].
- [tex]\(m_e = 0\)[/tex]: Valid because [tex]\(-2 \leq m_e \leq 2\)[/tex].
7. [tex]\((5, 3, -3)\)[/tex]
- [tex]\(n = 5\)[/tex]: Valid.
- [tex]\(l = 3\)[/tex]: Valid because [tex]\(0 \leq 3 < 5\)[/tex].
- [tex]\(m_e = -3\)[/tex]: Valid because [tex]\(-3 \leq m_e \leq 3\)[/tex].
So, the valid sets of quantum numbers are:
- [tex]\((4, 2, -1)\)[/tex]
- [tex]\((3, 2, 0)\)[/tex]
- [tex]\((5, 3, -3)\)[/tex]
These sets correctly satisfy all quantum number rules.
