Step 1: Understanding the Concept:
Quantum numbers provide a unique "address" for every electron in an atom.
1. Principal Quantum Number (\( n \)): Specifies the shell or energy level (\( n = 1, 2, 3, \dots \)).
2. Azimuthal Quantum Number (\( l \)): Defines the shape of the subshell (\( l \) ranges from 0 to \( n-1 \)).
- \( l=0 (s), l=1 (p), l=2 (d), l=3 (f) \).
3. Magnetic Quantum Number (\( m_l \)): Defines the orientation of the orbital in space (\( m_l \) ranges from \( -l \) to \( +l \)).
4. Spin Quantum Number (\( m_s \)): Defines the direction of electron spin (\( +1/2 \) or \( -1/2 \)).
Step 2: Key Formula or Approach:
For a \( 4d \) orbital, we extract the values of \( n \) and \( l \) from the notation:
- The digit "4" gives the principal quantum number, so \( n = 4 \).
- The letter "d" signifies the azimuthal quantum number \( l = 2 \).
- Based on \( l = 2 \), \( m_l \) must be one of the values in the set \( \{-2, -1, 0, 1, 2\} \).
Step 3: Detailed Explanation:
Let's analyze the given options based on these constraints:
- Option (1): \( (4, 3, 2, 1/2) \). Here \( n=4 \) and \( l=3 \). Since \( l=3 \) refers to an \( f \) subshell, this describes a \( 4f \) orbital, not a \( 4d \). Incorrect.
- Option (2): \( (4, 2, 1, 1/2) \). Here \( n=4 \) and \( l=2 \), which is consistent with the \( 4d \) subshell. Additionally, \( m_l=1 \) is within the valid range of \( -2 \) to \( +2 \). Correct.
- Option (3): \( (4, 1, 2, 1/2) \). Here \( n=4 \) and \( l=1 \), which describes a \( 4p \) subshell. Furthermore, \( m_l=2 \) is impossible for \( l=1 \) because \( |m_l| \) cannot exceed \( l \). Incorrect.
Step 4: Final Answer:
By evaluating the allowed ranges for all four quantum numbers, we conclude that only option (B) satisfies the requirements for an electron in a \( 4d \) orbital.