Which of the following is not a valid set of four quantum numbers? (n, l, m₁, mₛ) a. 4, 0, 0, +1/2 b. 2, 1, 0, -1/2 c. 3, 2, -1, -1/2 d. 1, 0, 0, +1/2 e. 1, 1, 0, +1/2
The Correct Answer and Explanation is:
The correct answer is e. 1, 1, 0, +1/2.
Here’s why:
Quantum numbers describe the unique state of an electron in an atom. There are four quantum numbers:
- Principal quantum number (n): This represents the energy level of the electron. It must be a positive integer (1, 2, 3, …).
- Angular momentum quantum number (l): This defines the shape of the orbital. It ranges from 0 to (n-1), where 0 corresponds to an s orbital, 1 corresponds to a p orbital, 2 corresponds to a d orbital, and so on.
- Magnetic quantum number (m₁): This specifies the orientation of the orbital in space. It can range from -l to +l, including 0.
- Spin quantum number (mₛ): This describes the electron’s spin, which can be either +1/2 or -1/2.
Now, let’s analyze each option:
- Option a (4, 0, 0, +1/2):
- n = 4 (valid)
- l = 0 (valid because l ranges from 0 to n-1, so for n=4, l can be 0, 1, 2, or 3)
- m₁ = 0 (valid because m₁ can range from -l to +l, and since l=0, m₁ must be 0)
- mₛ = +1/2 (valid, spin can be either +1/2 or -1/2)
This is a valid set of quantum numbers.
- Option b (2, 1, 0, -1/2):
- n = 2 (valid)
- l = 1 (valid, l can be 0 or 1 when n=2)
- m₁ = 0 (valid, m₁ ranges from -l to +l, so m₁ can be -1, 0, or +1 for l=1)
- mₛ = -1/2 (valid)
This is a valid set of quantum numbers.
- Option c (3, 2, -1, -1/2):
- n = 3 (valid)
- l = 2 (valid, l can be 0, 1, or 2 when n=3)
- m₁ = -1 (valid, m₁ ranges from -l to +l, so m₁ can be -2, -1, 0, +1, +2 for l=2)
- mₛ = -1/2 (valid)
This is a valid set of quantum numbers.
- Option d (1, 0, 0, +1/2):
- n = 1 (valid)
- l = 0 (valid, l must be less than n, so for n=1, l=0)
- m₁ = 0 (valid, since l=0, m₁ must be 0)
- mₛ = +1/2 (valid)
This is a valid set of quantum numbers.
- Option e (1, 1, 0, +1/2):
- n = 1 (valid)
- l = 1 (invalid because for n=1, l can only be 0)
- m₁ = 0 (this would be valid if l were 1, but l cannot be 1 when n=1)
- mₛ = +1/2 (valid)
This is an invalid set of quantum numbers because l cannot be 1 when n=1.
Thus, option e is the only invalid set.
