A metal crystallizes in a ccp structure. Its metallic radius is 141.5 pm. The number of unit cells in `64 cm^(3)` of the metal are

Options

A. `2 xx 10^(32)`
B. `1.5 xx 10^(23)`
C. `1 xx 10^(24)`
D. `1.5 xx 10^(22)`

Explanation

To find the number of unit cells in 64 cm³ of a metal that crystallizes in a cubic close-packed (ccp) structure, we can follow these steps:

### Step 1: Determine the relationship between the metallic radius and the edge length of the unit cell.
In a ccp structure (also known as face-centered cubic or fcc), the relationship between the edge length \( a \) and the metallic radius \( r \) is given by the formula:
\[ a = \frac{4r}{\sqrt{2}} \]

### Step 2: Calculate the edge length \( a \).
Given that the metallic radius \( r = 141.5 \) pm (picometers), we can substitute this value into the formula:
\[ a = \frac{4 \times 141.5 \, \text{pm}}{\sqrt{2}} = \frac{566}{1.414} \approx 400 \, \text{pm} \]

### Step 3: Convert the edge length from picometers to centimeters.
To convert picometers to centimeters, we use the conversion factor \( 1 \, \text{pm} = 10^{-12} \, \text{m} = 10^{-10} \, \text{cm} \):
\[ a = 400 \, \text{pm} = 400 \times 10^{-10} \, \text{cm} = 4 \times 10^{-8} \, \text{cm} \]

### Step 4: Calculate the volume of the unit cell.
The volume \( V \) of a cube is given by:
\[ V = a^3 \]
Substituting the value of \( a \):
\[ V = (4 \times 10^{-8} \, \text{cm})^3 = 64 \times 10^{-24} \, \text{cm}^3 \]

### Step 5: Calculate the number of unit cells in 64 cm³ of the metal.
To find the number of unit cells \( N \), we divide the total volume of the metal by the volume of one unit cell:
\[ N = \frac{\text{Total Volume}}{\text{Volume of one unit cell}} = \frac{64 \, \text{cm}^3}{64 \times 10^{-24} \, \text{cm}^3} \]
Simplifying this gives:
\[ N = \frac{64}{64 \times 10^{-24}} = 10^{24} \]

### Final Answer:
The number of unit cells in 64 cm³ of the metal is \( 10^{24} \).

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