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Step-by-Step Solution
Step 1: Understand the Context
We need to compare the ionic radii of two trivalent lanthanide ions: La³⁺ (atomic number 57) and Lu³⁺ (atomic number 71). The provided radius of La³⁺ is 1.06 Å, and we want to estimate the radius of Lu³⁺.
Step 2: Recognize the Trend (Lanthanide Contraction)
As we move from lanthanum (La) to lutetium (Lu) across the lanthanide series, there is a steady decrease in ionic radius. This phenomenon is known as the “lanthanide contraction.” It occurs primarily because the nuclear charge (Z) increases while the additional electrons go into f-orbitals, which are relatively poor at shielding. Thus, the effective nuclear charge felt by the outer electrons increases, leading to a reduction in ionic radius.
Step 3: Proportional Relationship with Nuclear Charge
A simplified proportional approach suggests the ionic radius is approximately inversely related to the effective nuclear charge, $Z_{\text{eff}}$. Hence, if $r \propto \tfrac{1}{Z_{\text{eff}}}$, then we can compare the ionic radii using an inverse ratio of the nuclear charges (or atomic numbers as a basic approximation).
Step 4: Set up the Proportion
We can write:
$ \displaystyle
\frac{r_{(\text{La}^{3+})}}{r_{(\text{Lu}^{3+})}} \approx \frac{Z_{\text{Lu}}}{Z_{\text{La}}}
$
Given:
$ \displaystyle r_{(\text{La}^{3+})} = 1.06 \,\text{Å}, \quad Z_{\text{La}} = 57, \quad Z_{\text{Lu}} = 71
$
So,
$ \displaystyle
\frac{1.06}{r_{(\text{Lu}^{3+})}} = \frac{71}{57}
$
Step 5: Calculate the Radius of Lu³⁺
Solving for $r_{(\text{Lu}^{3+})}$:
$ \displaystyle
r_{(\text{Lu}^{3+})} = 1.06 \times \frac{57}{71}
$
Now performing the multiplication and division:
$ \displaystyle
r_{(\text{Lu}^{3+})} \approx 1.06 \times 0.8028 \approx 0.85 \,\text{Å}
$
Step 6: Final Answer
The radius of Lu³⁺ is closest to 0.85 Å among the given options.
