The shortest distance between two `Na^(+)` ions in rock - salt arrangement having edge length equal to `asqrt2` picmeters is -

Options

A. 2a
B. `2asqrt2`
C. `(a)/(sqrt2)`
D. a

Answer

To find the shortest distance between two `Na^(+)` ions in a rock salt arrangement with an edge length of `a√2` picometers, we can follow these steps:

### Step 1: Understand the Rock Salt Structure
In a rock salt structure, sodium ions (`Na^(+)`) occupy the octahedral voids, while chloride ions (`Cl^(-)`) are located at the corners and face centers of the cubic unit cell. Each unit cell contains 4 sodium ions and 4 chloride ions.

### Step 2: Identify the Edge Length
We are given that the edge length of the unit cell is `a√2` picometers.

### Step 3: Determine the Positions of the Ions
In the rock salt structure:
- The sodium ions are located at the edge centers and the body center of the unit cell.
- The chloride ions are located at the corners and face centers.

### Step 4: Calculate the Shortest Distance
The shortest distance between two sodium ions will be along the edge of the cube. Since the edge length is `a√2`, the distance between two `Na^(+)` ions located at the edge centers can be calculated as follows:

1. The distance between two `Na^(+)` ions at the edge centers (let's denote them as A and B) is half the edge length:
\[ \text{Distance} = \frac{a\sqrt{2}}{2} \]

### Step 5: Simplify the Expression
To find the shortest distance:
\[ \text{Shortest Distance} = \frac{a\sqrt{2}}{2} \]

### Step 6: Final Calculation
Since we are looking for the shortest distance between two `Na^(+)` ions, we can denote the edge length as `a`. Thus, the shortest distance between the two `Na^(+)` ions is:
\[ \text{Shortest Distance} = a \]

### Conclusion
The shortest distance between two `Na^(+)` ions in the rock salt arrangement with an edge length of `a√2` picometers is `a` picometers.

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