In a planar tetra - atomic molecule, `XY_(3)`,X is at the centroid of the equilateral triangle formed by the atoms , Y. If the X-Y bond distance is `1Å`, what is the distance between the centres of any two Y atoms ?
a. `1//sqrt3 Å`
b. `sqrt2 Å`
c. ` sqrt3 Å`
d. `1//sqrt2 Å`

Options

A. `1//sqrt3 Å`
B. `sqrt2 Å`
C. ` sqrt3 Å`
D. `1//sqrt2 Å`

Answer

To solve the problem, we need to find the distance between the centers of any two Y atoms in the molecule XY₃, where X is at the centroid of an equilateral triangle formed by the Y atoms. Given that the X-Y bond distance is 1 Å, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- The molecule XY₃ has X at the centroid of an equilateral triangle formed by the three Y atoms. Let's denote the Y atoms as A, B, and C.
- The distance from X (the centroid) to any Y atom (A, B, or C) is given as 1 Å.

2. **Properties of the Equilateral Triangle**:
- In an equilateral triangle, all sides are equal, and the centroid divides each median in the ratio 2:1.
- If we denote the side length of the equilateral triangle as 's', the distance from the centroid to any vertex (Y atom) is given by the formula:
\[ \text{Distance from centroid to vertex} = \frac{s \sqrt{3}}{3} \]
- Given that this distance is 1 Å, we can set up the equation:
\[ \frac{s \sqrt{3}}{3} = 1 \]

3. **Solving for the Side Length (s)**:
- Rearranging the equation gives:
\[ s \sqrt{3} = 3 \implies s = \frac{3}{\sqrt{3}} = \sqrt{3} \text{ Å} \]

4. **Finding the Distance Between Two Y Atoms**:
- The distance between any two Y atoms (e.g., A and B) is simply the side length of the triangle, which we have found to be \( s = \sqrt{3} \) Å.

5. **Conclusion**:
- Therefore, the distance between the centers of any two Y atoms is \( \sqrt{3} \) Å.

### Final Answer:
- The distance between the centers of any two Y atoms is **c. \( \sqrt{3} \) Å**.

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