Step 1: Understanding the Concept:
An electron's state in an atom is described by four quantum numbers: principal (\(n\)), azimuthal (\(l\)), magnetic (\(m_l\)), and spin (\(s\)).
For any subshell designated as \(nl\), the first number is \(n\) and the letter indicates the value of \(l\).
Step 2: Key Formula or Approach:
1. Principal quantum number (\(n\)) corresponds to the shell number.
2. Azimuthal quantum number (\(l\)): \(s \to 0\), \(p \to 1\), \(d \to 2\), \(f \to 3\).
3. Magnetic quantum number (\(m_l\)) ranges from \(-l\) to \(+l\).
4. Spin quantum number (\(s\)) is either \(+1/2\) or \(-1/2\).
Step 3: Detailed Explanation:
For a 4d orbital:
\(n = 4\) (Principal shell is 4).
\(l = 2\) (The subshell is 'd').
Now check the conditions for \(m_l\):
If \(l = 2\), then \(m_l\) can be \(-2, -1, 0, 1, 2\).
Check the options:
(A) (4, 3, 2, +1/2): Here \(l = 3\), which corresponds to 4f. Incorrect.
(B) (4, 2, 1, +1/2): Here \(n=4, l=2, m_l=1\). Since \(|1| \le 2\), this is valid. Correct.
(C) (4, 1, 2, +1/2): Here \(l = 1\), which corresponds to 4p. Also \(m_l = 2\) is impossible for \(l = 1\). Incorrect.
Step 4: Final Answer:
The correct set is \((4, 2, 1, +1/2)\).