Question
Calculate Z(effective) for the f-electron in
Calculate Z(effective) for the f-electron in
The Correct Answer and Explanation is:
To calculate the effective nuclear charge (Zₑff) for an f-electron, we use Slater’s rules. These rules estimate the shielding effect and help us find the Zₑff experienced by an electron in a multi-electron atom.
🔹 General Formula:
Zeff=Z−SZ_{\text{eff}} = Z – S
Where:
- ZZ = atomic number (number of protons in nucleus)
- SS = shielding constant (determined using Slater’s rules)
🔹 Let’s take an example: Cerium (Ce), Z = 58, and calculate Zₑff for an f-electron (e.g., in 4f).
Step 1: Electron configuration of Ce:
[Xe]6s24f15d1[\text{Xe}] 6s^2 4f^1 5d^1
We want Zₑff for the 4f electron.
Step 2: Apply Slater’s Rules (modified version for f-electrons):
For an f-electron, the shielding constant SS is calculated as:
- Electrons in the same f-group (same n and l) = 0.35 each
- Electrons in lower shells (n < 4) = 1.00 each
Here:
- One 4f electron (besides the one we’re focusing on) contributes 0.35
- All electrons from 1s to 4d (i.e., the [Xe] core) are lower shell and contribute 1.00 each
[ Xe configuration = 54 electrons ]
So, S=0.35(samegroup)+54(innercore)=0.35+54=54.35S = 0.35 (same group) + 54 (inner core) = 0.35 + 54 = 54.35
Now, Zeff=Z−S=58−54.35=3.65Z_{\text{eff}} = Z – S = 58 – 54.35 = \boxed{3.65}
✅ Final Answer:
Zeff≈3.65\boxed{Z_{\text{eff}} \approx 3.65}
📘 Explanation
The effective nuclear charge (Zₑff) represents the net positive charge that an electron experiences in a multi-electron atom. Due to shielding by other electrons, an outer electron does not experience the full nuclear charge ZZ from the protons in the nucleus.
Slater’s rules help estimate the amount of shielding, accounting for how electrons in different shells and subshells repel each other. For f-electrons, shielding is generally poor because f-orbitals are more diffused and less effective at penetrating the inner electron cloud. This means f-electrons feel more shielding and thus a lower Zₑff compared to s or p electrons in the same period.
Using Cerium (Ce) as an example, we consider the single 4f electron and apply Slater’s rules. The inner 54 electrons from the [Xe] core significantly shield the 4f electron, contributing 1.00 each to the shielding. The only other electron in the 4f subshell contributes 0.35, following the rule for electrons in the same group.
This results in a shielding constant S=54.35S = 54.35, and a calculated Zₑff of approximately 3.65. This relatively low value reflects the strong shielding and poor penetration of f-orbitals, explaining why f-electrons are less tightly bound and why lanthanides exhibit similar chemical properties—they all have partially filled 4f orbitals with little effect on the outer electrons.
Understanding Zₑff is crucial for interpreting atomic behavior, trends in the periodic table, and the chemistry of transition and inner transition metals.
