Quantum Numbers Explanation

Quantum numbers describe the properties of atomic orbitals and the electrons within them. There are four quantum numbers:

• Principal Quantum Number (n): Represents the energy level/shell. n ≥ 1 (n = 1, 2, 3, ...)
• Azimuthal Quantum Number (ℓ): Represents the subshell/orbital shape. For a given n, ℓ can range from 0 to (n-1).
• Magnetic Quantum Number (m_ℓ): Represents the orientation of the orbital. For a given ℓ, m_ℓ can range from -ℓ to +ℓ, including 0.
• Spin Quantum Number (s or m_s): Represents the electron's spin. It can only be +1/2 or -1/2.

For n = 4, the possible values for the other numbers are:

• ℓ can be 0, 1, 2, or 3.
• For each ℓ value, m_ℓ has specific allowed values.
• s is always either +1/2 or -1/2.

Step-by-Step Analysis of Each Option

Step 1: Analyze Option 1

n = 4, ℓ = 3, m = +2, s = +1/2

• Check ℓ: For n=4, ℓ can be 0,1,2,3. ℓ=3 is allowed.
• Check m: For ℓ=3, m can be -3, -2, -1, 0, +1, +2, +3. m=+2 is allowed.
• Check s: s=+1/2 is allowed.

Conclusion: This set is possible.

Step 2: Analyze Option 2

n = 4, ℓ = 1, m = –2, s = –1/2

• Check ℓ: For n=4, ℓ can be 0,1,2,3. ℓ=1 is allowed.
• Check m: For ℓ=1, m can be -1, 0, +1. m=-2 is NOT allowed.

Conclusion: This set is not possible because the magnetic quantum number (m) is outside its allowed range for the given azimuthal quantum number (ℓ).

Step 3: Analyze Option 3

n = 4, ℓ = 2, m = –2, s = +1/2

• Check ℓ: For n=4, ℓ can be 0,1,2,3. ℓ=2 is allowed.
• Check m: For ℓ=2, m can be -2, -1, 0, +1, +2. m=-2 is allowed.
• Check s: s=+1/2 is allowed.

Conclusion: This set is possible.

Step 4: Analyze Option 4

n = 4, ℓ = 2, m = +2, s = –1/2

• Check ℓ: For n=4, ℓ can be 0,1,2,3. ℓ=2 is allowed.
• Check m: For ℓ=2, m can be -2, -1, 0, +1, +2. m=+2 is allowed.
• Check s: s=-1/2 is allowed.

Conclusion: This set is possible.

Final Answer

The set of quantum numbers that is not possible is:

n = 4, ℓ = 1, m = –2, s = –1/2

This is because for ℓ = 1, the magnetic quantum number (m) can only be -1, 0, or +1. The value m = -2 violates this rule.

Related Topics & Formulae

Key Rules for Quantum Numbers:

• Principal Quantum Number: $n\ge 1$
• Azimuthal Quantum Number: $0\le \ell \le n-1$
• Magnetic Quantum Number: $-\ell \le m_{\ell }\le +\ell$
• Spin Quantum Number: $m_s=+\frac{1}{2},-\frac{1}{2}$

Pauli Exclusion Principle: No two electrons in an atom can have the same set of all four quantum numbers. This is why each orbital (defined by n, ℓ, m_ℓ) can hold a maximum of two electrons with opposite spins.

Subshell Notation: The value of ℓ is often represented by a letter. $\ell =0\to s$, $\ell =1\to p$, $\ell =2\to d$, $\ell =3\to f$.