A solid has a `b.c.c.` structure . If the distance of closest approach between the two atoms is `1.73 Å`. The edge length of the cell is `:`

Options

A. 199 pm
B. `sqrt(3//2)` pm
C. 142.2 pm
D. `sqrt2` pm

Answer

To find the edge length of a body-centered cubic (b.c.c.) structure given the distance of closest approach between two atoms, we can follow these steps:

### Step 1: Understand the b.c.c. structure
In a b.c.c. structure, atoms are located at the corners of the cube and one atom is located at the center of the cube. The atoms at the corners are shared among eight adjacent cubes.

### Step 2: Identify the distance of closest approach
The distance of closest approach between two atoms in a b.c.c. structure is given as \(1.73 \, \text{Å}\). This distance corresponds to the distance between the centers of two atoms that are touching each other.

### Step 3: Relate the edge length to atomic radius
In a b.c.c. structure, the relationship between the edge length \(a\) and the atomic radius \(r\) can be expressed as:
\[ a = 2r \]
This is because the closest approach occurs along the edge of the cube, where two corner atoms touch the atom at the center.

### Step 4: Set up the equation
From the information given, we can set up the equation:
\[ 2r = 1.73 \, \text{Å} \]

### Step 5: Solve for the atomic radius \(r\)
To find the radius \(r\):
\[ r = \frac{1.73 \, \text{Å}}{2} = 0.865 \, \text{Å} \]

### Step 6: Use the relationship for b.c.c. structure
For a b.c.c. structure, the relationship between the edge length \(a\) and the atomic radius \(r\) is also given by:
\[ \sqrt{3}a = 4r \]

### Step 7: Substitute the value of \(r\) into the equation
Substituting the value of \(r\) into the equation:
\[ \sqrt{3}a = 4 \times 0.865 \, \text{Å} \]
\[ \sqrt{3}a = 3.46 \, \text{Å} \]

### Step 8: Solve for the edge length \(a\)
Now, solve for \(a\):
\[ a = \frac{3.46 \, \text{Å}}{\sqrt{3}} \approx \frac{3.46 \, \text{Å}}{1.732} \approx 2.00 \, \text{Å} \]

### Step 9: Convert to picometers
To convert the edge length from angstroms to picometers:
\[ 2.00 \, \text{Å} = 200 \, \text{pm} \]

### Final Answer
The edge length of the cell is approximately \(200 \, \text{pm}\).

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