The radius of `La^(3+)(Z=57)` is 106 pm. Which one of the following given values will be closest to the radius of `Lu^(3+)(Z=71)`?

Options

A. 160 pm
B. 140 pm
C. 106 pm
D. 85 pm

Explanation

To determine the radius of `Lu^(3+)` (Lutetium) based on the radius of `La^(3+)` (Lanthanum), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Lanthanide Contraction**:
- The lanthanide series consists of elements from Lanthanum (La) to Lutetium (Lu). As we move from La to Lu, there is a gradual decrease in atomic radii due to the increase in nuclear charge and the poor shielding effect of the f-electrons.

2. **Identify the Given Data**:
- The radius of `La^(3+)` is given as 106 pm.

3. **Consider the Trend in the Lanthanide Series**:
- As we move from La to Lu, the atomic radius decreases. This is primarily due to the increase in the positive charge of the nucleus (more protons) which pulls the electrons closer to the nucleus, resulting in a smaller radius.

4. **Estimate the Radius of `Lu^(3+)`**:
- Since `Lu` is the last element in the lanthanide series, we can infer that its radius will be smaller than that of `La`.
- The decrease in radius is not uniform but tends to be around 10-15 pm for each element in the series.

5. **Calculate the Approximate Radius**:
- If we assume a decrease of about 15 pm from `La^(3+)` to `Lu^(3+)`, we can calculate:
\[ \text{Radius of } Lu^{3+} \approx 106 \text{ pm} - 15 \text{ pm} = 91 \text{ pm} \]
- However, based on the trend, the radius is likely to be slightly larger than this estimate due to the gradual decrease.

6. **Select the Closest Value**:
- Among the given options, we look for a value that is closest to our estimated radius of `Lu^(3+)`. If the options include values like 85 pm, 90 pm, and 95 pm, we would select 85 pm as the closest value based on the trend of decreasing radii.

### Conclusion:
The closest value to the radius of `Lu^(3+)` is **85 pm**.