Quantum numbers arise naturally from the mathematics used to describe the possible states of an electron in an atom.

Quantum numbers arise naturally from the mathematics used to describe the possible states of an electron in an atom. The four quantum numbers, the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l), and the spin quantum number (m_s) have strict rules which govern the possible values. Identify allowable combinations of quantum numbers for an electron. Select all that apply. n = 2, l = 1, m_l = -1, m_s = 0 n = 3, l = -2, m_l = 1, m_s = +1/2 n = 3, l = 2, m_l = 1, m_s = +1/2 n = 5, l = 5, m_l = -1, m_s = +1/2 n = 5, l = 3, m_l = 3, m_s = +1/2 n = 4, l = 3, m_l = 4, m_s = +1/2

The Correct Answer and Explanation is:

Let’s analyze each combination of quantum numbers based on the rules for quantum numbers:

1. n = 2, l = 1, m_l = -1, m_s = 0
   • The principal quantum number nnn must be a positive integer (2 is fine).
   • The angular momentum quantum number lll can range from 0 to n−1n-1n−1. Here, l=1l = 1l=1 is valid because 0≤l≤n−10 \leq l \leq n-10≤l≤n−1, where n=2n = 2n=2, so lll can be 0 or 1.
   • The magnetic quantum number mlm_lml​ ranges from −l-l−l to +l+l+l, so for l=1l = 1l=1, mlm_lml​ can be -1, 0, or +1. Therefore, ml=−1m_l = -1ml​=−1 is valid.
   • The spin quantum number msm_sms​ can only be +1/2 or -1/2, so ms=0m_s = 0ms​=0 is not valid.
   Conclusion: This combination is invalid because msm_sms​ cannot be 0.
2. n = 3, l = -2, m_l = 1, m_s = +1/2
   • The angular momentum quantum number lll must be between 0 and n−1n-1n−1, so lll can range from 0 to 2 when n=3n = 3n=3. Therefore, l=−2l = -2l=−2 is not valid.
   Conclusion: This combination is invalid because lll cannot be negative.
3. n = 3, l = 2, m_l = 1, m_s = +1/2
   • For n=3n = 3n=3, lll can be 0, 1, or 2, so l=2l = 2l=2 is valid.
   • The magnetic quantum number mlm_lml​ ranges from −l-l−l to +l+l+l, so for l=2l = 2l=2, mlm_lml​ can be -2, -1, 0, +1, or +2. Therefore, ml=1m_l = 1ml​=1 is valid.
   • The spin quantum number ms=+1/2m_s = +1/2ms​=+1/2 is valid.
   Conclusion: This combination is valid.
4. n = 5, l = 5, m_l = -1, m_s = +1/2
   • The angular momentum quantum number lll must be between 0 and n−1n-1n−1. Here, n=5n = 5n=5, so lll can range from 0 to 4. Therefore, l=5l = 5l=5 is not valid because it exceeds n−1n-1n−1.
   Conclusion: This combination is invalid because lll must be less than nnn.
5. n = 5, l = 3, m_l = 3, m_s = +1/2
   • For n=5n = 5n=5, l=3l = 3l=3 is valid (since lll ranges from 0 to 4).
   • The magnetic quantum number mlm_lml​ must be between −l-l−l and +l+l+l, so for l=3l = 3l=3, mlm_lml​ can range from -3 to +3. Therefore, ml=3m_l = 3ml​=3 is valid.
   • The spin quantum number ms=+1/2m_s = +1/2ms​=+1/2 is valid.
   Conclusion: This combination is valid.
6. n = 4, l = 3, m_l = 4, m_s = +1/2
   • For n=4n = 4n=4, l=3l = 3l=3 is valid.
   • The magnetic quantum number mlm_lml​ can range from −l-l−l to +l+l+l, so for l=3l = 3l=3, mlm_lml​ can be -3, -2, -1, 0, 1, 2, or 3. Therefore, ml=4m_l = 4ml​=4 is not valid because it exceeds lll.
   Conclusion: This combination is invalid because mlm_lml​ must be within the range of −l-l−l to +l+l+l.

Final Allowable Combinations:

• Valid combinations:
  • n = 3, l = 2, m_l = 1, m_s = +1/2
  • n = 5, l = 3, m_l = 3, m_s = +1/2