Predict the ground-state term symbol for the tetravalent cation of manganese, Mn4+. Assume that its electron configuration has a d3 valence subshell. Use Table 15.1 for the partial term symbols.
Table 15.1
Term symbols for partially filled subshells
The Correct Answer and Explanation is :
To predict the ground-state term symbol for the tetravalent cation of manganese (( \text{Mn}^{4+} )) with a ( d^3 ) electron configuration, we follow Hund’s rules and consult Table 15.1 for term symbols.
Step-by-Step Solution:
- Determine the electronic configuration:
- Manganese (( \text{Mn} )) has an atomic number of 25, so its neutral configuration is ( [\text{Ar}]3d^54s^2 ).
- For ( \text{Mn}^{4+} ), four electrons are removed: two from the 4s orbital and two from the 3d orbital, resulting in a ( 3d^3 ) configuration.
- Identify the possible term symbols for a ( d^3 ) configuration:
- Table 15.1 provides the term symbols for a partially filled ( d^n ) subshell.
- For a ( d^3 ) configuration, the possible term symbols are derived based on the maximum spin and orbital angular momentum quantum numbers:
- ( S = \frac{3}{2} ) (maximum spin ( S = \frac{1}{2} \times 3 ) since the three electrons are unpaired).
- ( L = 3 ) (the sum of orbital angular momentum ( m_l ) values: ( +2, +1, 0 )).
- From this combination (( S = \frac{3}{2} ) and ( L = 3 )), the ground term symbol is ( ^4F ) (multiplicity ( 2S + 1 = 4 )).
- Apply Hund’s rules:
- Hund’s rules prioritize the highest spin multiplicity, followed by the highest ( L )-value.
- The term ( ^4F ) satisfies these criteria, making it the ground-state term symbol.
Final Answer:
The ground-state term symbol for ( \text{Mn}^{4+} ) is ( ^4F ).
Explanation (300 words):
The term symbol ( ^4F ) reflects the quantum mechanical properties of the ( d^3 ) configuration. Each electron in the ( d^3 ) subshell occupies an orbital to maximize spin multiplicity (( S = \frac{3}{2} )), which minimizes repulsion via exchange energy. The orbital angular momentum (( L = 3 )) arises from the sum of ( m_l ) values of the three electrons, considering their possible arrangements while obeying Pauli’s exclusion principle.
The superscript “4” in ( ^4F ) corresponds to the multiplicity ( 2S + 1 ), indicating a quartet state (( S = \frac{3}{2} )). The letter “F” signifies ( L = 3 ), which is derived from the combination of the electrons’ ( m_l ) values.
Hund’s rules guide us to select the term symbol with the highest multiplicity and highest ( L )-value as the ground state. Thus, ( ^4F ), which has both the maximum multiplicity and ( L )-value among the possible terms, is the correct ground-state term symbol for ( \text{Mn}^{4+} ).