Question

Total degeneracy (number of microstates) for a Ti³⁺ ion in spherical symmetry is ____.

Answer 1

Correct Answer: 10

To determine the total degeneracy (number of microstates) for a Ti³⁺ ion in spherical symmetry, we need to understand the electron configuration and apply the formula for calculating microstates. A Ti³⁺ ion is derived from a titanium atom (atomic number 22) by removing three electrons, giving an electron configuration of 1s² 2s² 2p⁶ 3s² 3p⁶ 3d¹. The electron configuration shows that there is one electron in the 3d subshell. The number of microstates for a given number of electrons (n) in a d-orbital is calculated using the formula: total microstates = (2l + 1)! / [n!(2l + 1 - n)!], where l = 2 for d-orbitals. Plugging the values into the formula, we have: total microstates = 5! / [1!(5 - 1)!] = 5! / [1!×4!] = 5 × 4! / 4! = 5. The calculated number of microstates for a single electron in a 3d orbital is 5. A singly filled d-orbital has 10 microstates due to spin multiplicity, as each orbital has two possible spin orientations. Thus, the total degeneracy for Ti³⁺ is 10.