Identify the concept involved

This question is based on lanthanoid contraction.

As we move from $La$ to $Lu$ in the lanthanoid series, electrons are added to the $4f$-subshell. Since $4f$ electrons shield the nuclear charge poorly, the effective nuclear charge felt by outer electrons increases.

Therefore, the size of the ions decreases gradually from $La^{3+}$ to $Lu^{3+}$.

Given data

$r(La^{3+}) = 1.06\ \text{Å}$

We need the approximate radius of:

$Lu^{3+}$

Use lanthanoid contraction trend

Due to lanthanoid contraction, $Lu^{3+}$ must be smaller than $La^{3+}$.

So the correct option must be less than $1.06$ Å.

Check the options

• A: $1.60$ Å
  This is much larger than $1.06$ Å, so impossible.

• B: $1.40$ Å
  Also larger than $1.06$ Å, so impossible.

• C: $0.85$ Å
  This is smaller than $1.06$ Å and matches the expected decrease due to lanthanoid contraction.

• D: $1.60$ Å
  Same as A, impossible.

Conclusion

The value closest to the radius of $Lu^{3+}$ is:

$0.85\ \text{Å}$

So the correct option is C.