A given metal crystalline out with a cubic structure having edge length of `361` pm .if there are four metal atoms in one unit cell, what is the radius of metal atom?

Options

A. 40 pm
B. 127 pm
C. 80 pm
D. 108 pm

Explanation

To find the radius of a metal atom in a cubic crystal structure with a given edge length and number of atoms in the unit cell, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Type of Cubic Structure:**
- Given that there are 4 metal atoms in one unit cell, we can conclude that the structure is Face-Centered Cubic (FCC). In FCC, there are 8 corner atoms (1/8 contribution each) and 6 face atoms (1/2 contribution each), which totals to 4 atoms:
\[ \text{Total atoms} = \left(8 \times \frac{1}{8}\right) + \left(6 \times \frac{1}{2}\right) = 1 + 3 = 4 \]

2. **Given Data:**
- Edge length (A) = 361 pm (picometers)

3. **Relate Edge Length to Atomic Radius:**
- In an FCC structure, the relationship between the edge length (A) and the atomic radius (R) is given by:
\[ A = 2\sqrt{2}R \]
- This relationship arises from the geometry of the FCC structure, where the diagonal of the face of the cube equals four atomic radii (2R).

4. **Rearranging the Formula:**
- To find the radius (R), we rearrange the formula:
\[ R = \frac{A}{2\sqrt{2}} \]

5. **Substituting the Given Edge Length:**
- Now substitute the value of A into the formula:
\[ R = \frac{361 \text{ pm}}{2\sqrt{2}} \]

6. **Calculating the Radius:**
- First, calculate \(2\sqrt{2}\):
\[ 2\sqrt{2} \approx 2 \times 1.414 = 2.828 \]
- Now divide:
\[ R \approx \frac{361}{2.828} \approx 127.0 \text{ pm} \]

7. **Final Answer:**
- The radius of the metal atom is approximately 127 pm.

### Summary:
The radius of the metal atom in the given FCC structure with an edge length of 361 pm is approximately **127 pm**.