In a metal M having BCC arrangement edge length of the unit cell is 400 pm. The atomic radius of the metal is:

Options

A. 100 pm
B. 141 pm
C. 173 pm
D. 200 pm

Explanation

To find the atomic radius of a metal M with a BCC (Body-Centered Cubic) arrangement and an edge length of 400 pm, we can follow these steps:

### Step 1: Understand the BCC Structure
In a BCC structure, there are atoms located at each of the eight corners of the cube and one atom at the center of the cube.

### Step 2: Identify the Relationship Between Edge Length and Atomic Radius
In a BCC unit cell, the body diagonal can be expressed in terms of the edge length (a) of the cube. The body diagonal (d) of a cube can be calculated using the Pythagorean theorem:
\[ d = \sqrt{3}a \]

### Step 3: Relate the Body Diagonal to Atomic Radius
In a BCC unit cell, the body diagonal is equal to the sum of the diameters of the atoms along that diagonal. Since there are two corner atoms and one body-centered atom along the body diagonal, the relationship can be expressed as:
\[ d = 4r \]
where \( r \) is the atomic radius.

### Step 4: Set Up the Equation
From the above relationships, we can set up the equation:
\[ \sqrt{3}a = 4r \]

### Step 5: Solve for Atomic Radius
Rearranging the equation to solve for the atomic radius \( r \):
\[ r = \frac{\sqrt{3}a}{4} \]

### Step 6: Substitute the Given Edge Length
Now, we substitute the given edge length \( a = 400 \) pm into the equation:
\[ r = \frac{\sqrt{3} \times 400 \text{ pm}}{4} \]

### Step 7: Calculate the Atomic Radius
Calculating the value:
1. Calculate \( \sqrt{3} \approx 1.732 \)
2. Substitute and calculate:
\[ r = \frac{1.732 \times 400}{4} \]
\[ r = \frac{692.8}{4} \]
\[ r = 173.2 \text{ pm} \]

### Final Answer
The atomic radius of the metal M is **173.2 pm**.

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