- Rules for quantum numbers state \(n \ge 1\), \(0 \le l < n\), and \(-l \le m_l \le l\).
- Set 1: \(m_l=-3\) violates \(-l \le m_l \le l\) since \(-2 \le -3 \le 2\) is false.
- Set 2 and Set 4: \(l=2\) violates \(l < n\) since \(2 < 2\) is false.
- Set 3: \(n=4, l=3, m_l=-2\) satisfies all rules, so the valid set is \(\boxed{n=4, l=3, m_l=-2}\).
Explanation
- Defining the Rules for Valid Quantum Numbers
To determine which set of quantum numbers represents a valid orbital, we evaluate each set against the fundamental rules of quantum mechanics:
- The principal quantum number \(n\) must be a positive integer (\(n = 1, 2, 3, \dots\)).
- The azimuthal quantum number \(l\) must be an integer such that \(0 \le l < n\).
- The magnetic quantum number \(m_l\) must be an integer such that \(-l \le m_l \le l\).
- Evaluating Each Set of Quantum Numbers
We examine each set provided:
- Set 1: \(n=3, l=2, m_l=-3\). Here, \(n=3\) and \(l=2\), which is valid since \(0 \le 2 < 3\). However, for \(m_l\), we require \(-2 \le m_l \le 2\). Since \(m_l = -3\) is outside this range, this set is invalid.
- Set 2: \(n=2, l=2, m_l=0\). Here, we require \(0 \le l < n\), but \(l=2\) is not less than \(n=2\). Thus, this set is invalid.
- Set 3: \(n=4, l=3, m_l=-2\). Here, \(n=4\) and \(l=3\), which satisfies \(0 \le 3 < 4\). For \(m_l\), we require \(-3 \le m_l \le 3\). Since \(m_l = -2\) falls within this range, this set is valid.
- Set 4: \(n=2, l=2, m_l=-1\). Similar to Set 2, we require \(0 \le l < n\), but \(l=2\) is not less than \(n=2\). Thus, this set is invalid.
- Conclusion
Based on the evaluation of all four sets, only the third set (\(n=4, l=3, m_l=-2\)) satisfies all the requirements for a valid quantum mechanical orbital.
The final answer is \(\boxed{n=4, l=3, m_l=-2}\).
Examples
In chemistry, these rules are essential for understanding the arrangement of electrons in atoms. For instance, if you were studying the electron configuration of a heavy element, knowing these rules helps you determine the capacity of shells and subshells to hold electrons, which in turn predicts the element's chemical reactivity and bonding characteristics. These principles form the foundation for organizing the periodic table and explaining the behavior of matter.