Atomic State of Titanium (Ti, Z = 22) The ground state of Ti has S = 1 and L = 3. (A) Write down the ground state electronic configuration of Ti. (B) What are the possible values of J? Write down the spectroscopic notation for these values of S, L and J. (C) Which on of the states in part (B) represent the ground state of Ti. Briefly justify your answer.
The Correct Answer and Explanation is:
(A) Ground State Electronic Configuration of Ti (Z = 22):
The electron configuration of titanium in its ground state is:
1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d²
Titanium has 22 electrons. The electrons fill up the lower energy orbitals first:
- 1s² (2 electrons)
- 2s² (2 electrons)
- 2p⁶ (6 electrons)
- 3s² (2 electrons)
- 3p⁶ (6 electrons)
- 4s² (2 electrons)
- 3d² (2 electrons)
This gives a total of 22 electrons, matching the atomic number of Ti.
(B) Possible values of J and Spectroscopic Notation:
Given:
- Total spin quantum number S = 1 (which corresponds to two unpaired electrons with parallel spins)
- Total orbital angular momentum quantum number L = 3 (which corresponds to the letter F in spectroscopic notation)
The total angular momentum quantum number J can take the values:
J = L + S, L + S – 1, …, |L − S|
So, for L = 3 and S = 1, the possible values of J are:
- J = 4
- J = 3
- J = 2
Therefore, the spectroscopic terms are:
- ⁵F₄
- ⁵F₃
- ⁵F₂
(C) Which state represents the ground state of Ti and why:
Among the possible terms, the ground state is the one with the lowest energy. According to Hund’s rules:
- The term with the maximum multiplicity (that is, highest value of 2S + 1) is lowest in energy.
- For terms with the same multiplicity, the term with the highest value of L is lower in energy.
- For subshells that are less than half-filled, the lowest J value has the lowest energy.
Titanium has two electrons in the 3d subshell, which is less than half-filled. Therefore, the lowest energy state (ground state) corresponds to the lowest J value, which is:
- ⁵F₂
So, the ground state term symbol for titanium is ⁵F₂.
