If metallic atoms of mass 197 and radius 166 pm are arranged in ABCABC fashion then what is the surface area of each unit cell?

Options

A. 1.32 ×` 10^6`pm2
B. 1.32 ×` 10^-18`pm2
C. 2.20 ×` 10^5`pm2
D. 2.12 ×` 10^-19`pm2

Answer

To solve the problem of finding the surface area of each unit cell for metallic atoms arranged in an ABCABC fashion (which indicates a close-packed structure), we can follow these steps:

### Step 1: Identify the type of packing
The ABCABC arrangement indicates a face-centered cubic (FCC) structure. In this arrangement, each unit cell contains atoms at each corner and the centers of the faces.

### Step 2: Use the relationship between edge length (a) and atomic radius (R)
For a face-centered cubic (FCC) structure, the relationship between the edge length (a) of the unit cell and the atomic radius (R) is given by the formula:
\[ a = 2\sqrt{2}R \]

### Step 3: Substitute the given radius into the formula
Given that the radius \( R \) is 166 pm (picometers), we can substitute this value into the formula:
\[ a = 2\sqrt{2} \times 166 \, \text{pm} \]

### Step 4: Calculate the edge length (a)
First, calculate \( 2\sqrt{2} \):
\[ 2\sqrt{2} \approx 2.828 \]
Now, calculate \( a \):
\[ a = 2.828 \times 166 \approx 469.08 \, \text{pm} \]

### Step 5: Calculate the surface area of the unit cell
The surface area (SA) of a cube is given by the formula:
\[ SA = 6a^2 \]
Now we need to calculate \( a^2 \):
\[ a^2 = (469.08 \, \text{pm})^2 \approx 219,000 \, \text{pm}^2 \]
Now, substitute \( a^2 \) into the surface area formula:
\[ SA = 6 \times 219,000 \, \text{pm}^2 \approx 1,314,000 \, \text{pm}^2 \]

### Step 6: Convert to scientific notation
Finally, convert the surface area to scientific notation:
\[ SA \approx 1.314 \times 10^6 \, \text{pm}^2 \]

### Final Answer
The surface area of each unit cell is approximately:
\[ \text{Surface Area} \approx 1.314 \times 10^6 \, \text{pm}^2 \]

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