Pr³⁺ Ground State Term Symbol Calculation

To find the ground state term symbol for the Pr³⁺ ion, we first need to determine its electronic configuration and then apply Hund's rules.

The atomic number of Praseodymium (Pr) is 59.

• Neutral Pr electronic configuration: [Xe] 4f³ 6s².
• Pr³⁺ ion electronic configuration: Removing three electrons (two from 6s and one from 4f) results in [Xe] 4f².

The Pr³⁺ ion has 2 electrons in the 4f subshell. The f subshell has 7 orbitals, corresponding to m_l values from +3 to -3.

We fill these 2 electrons into the f orbitals according to Hund's rules to determine the ground state.

• Hund's First Rule: Maximize Spin (S). The electrons occupy different orbitals with parallel spins to maximize the total spin angular momentum. With two electrons, the maximum spin is achieved by putting them in two different f orbitals with the same spin direction (e.g., both spin up).
• Hund's Second Rule: Maximize Orbital Angular Momentum (L). For a given maximum spin, electrons occupy orbitals with the highest possible m_l values. The highest m_l values in the f subshell are +3, +2, +1, 0, -1, -2, -3. To maximize L, we place the two electrons in the orbitals with m_l = +3 and m_l = +2.

Let's represent the orbital filling and calculate L and S:

• Total spin angular momentum (S): Each electron contributes s = 1/2. Since spins are parallel, S = 1/2 + 1/2 = 1.
• Spin multiplicity (2S+1): 2(1) + 1 = 3.
• Total orbital angular momentum (L): Sum of m_l values. L = (+3) + (+2) = 5.

The value L=5 corresponds to the letter H in the term symbol notation (L=0 → S, L=1 → P, L=2 → D, L=3 → F, L=4 → G, L=5 → H, L=6 → I, etc.).

The term symbol is in the form \( ^{2S+1}L \). So far, we have \( ^3H \).

• Hund's Third Rule: Determine Total Angular Momentum (J). The value of J depends on whether the subshell is less than, more than, or exactly half-filled. The f subshell can hold up to 14 electrons. Pr³⁺ has 2 electrons, which is less than half-filled (7 electrons).

For a less than half-filled subshell, \( J = |L - S| \).

Using our calculated values L=5 and S=1:

\( J = |5 - 1| = 4 \)

The ground state term symbol for Pr³⁺ is \( ^3H_4 \).

Pr³⁺ g_J Value Calculation

The g_J value (Landé g-factor) for a term symbol \( ^3H_4 \) can be calculated using the formula:

\( g_J = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)} \)

We have the values L=5, S=1, and J=4 from the term symbol \( ^3H_4 \).

Substitute these values into the formula:

\( g_J = 1 + \frac{4(4+1) + 1(1+1) - 5(5+1)}{2(4)(4+1)} \)

\( g_J = 1 + \frac{4(5) + 1(2) - 5(6)}{2(4)(5)} \)

\( g_J = 1 + \frac{20 + 2 - 30}{40} \)

\( g_J = 1 + \frac{22 - 30}{40} \)

\( g_J = 1 + \frac{-8}{40} \)

\( g_J = 1 - \frac{1}{5} \)

\( g_J = \frac{5}{5} - \frac{1}{5} \)

\( g_J = \frac{4}{5} \)

Thus, the g_J value for the ground state of Pr³⁺ is \( \frac{4}{5} \).

The ground state term symbol and the g_J value for Pr³⁺ ion are \( ^3H_4 \) and \( \frac{4}{5} \), respectively.