Quantum Numbers and Electron Configuration

Given quantum numbers: magnetic quantum number m = 2 and spin quantum number s = $-\frac{1}{2}$

Step 1: Analyze the magnetic quantum number (m)

The magnetic quantum number m indicates the orientation of the orbital and ranges from -l to +l, where l is the azimuthal quantum number.

Given m = 2, this means l must be at least 2 (since m ranges from -l to +l).

Therefore: $l\ge 2$

Step 2: Determine possible subshells

Azimuthal quantum number l corresponds to subshell types:

• l = 0 → s-subshell
• l = 1 → p-subshell
• l = 2 → d-subshell
• l = 3 → f-subshell

Since l ≥ 2, the possible subshells are d (l=2), f (l=3), and higher.

Step 3: Consider the spin quantum number

The spin quantum number s = $-\frac{1}{2}$ is valid for any electron and doesn't restrict the subshell type.

Step 4: Match with given options

Among the options provided:

• p-subshell (l=1) → m ranges from -1 to +1 → cannot have m=2
• s-subshell (l=0) → m=0 only → cannot have m=2
• d-subshell (l=2) → m ranges from -2 to +2 → can have m=2
• c-subshell → not a valid subshell designation

Final Answer

The electron belongs to the d-subshell.

Related Topics

• Quantum numbers and their significance
• Electron configuration rules
• Orbital shapes and orientations
• Pauli exclusion principle

Key Formulae and Theory

Quantum Number Relationships:

Principal quantum number (n): 1, 2, 3, ...

Azimuthal quantum number (l): 0 to n-1

Magnetic quantum number (m): -l to +l

Spin quantum number (s): $+\frac{1}{2}$ or $-\frac{1}{2}$

Subshell Notation:

l = 0 → s orbital

l = 1 → p orbital

l = 2 → d orbital

l = 3 → f orbital