Given length of side of hexagonal unit cell is `(100)/sqrt(2)` pm . The volume of hexagonal unit cell is `("in" "pm"^(3))`:

Options

A. `8xx10^(6)`
B. `1.5 xx 10^(6)`
C. `64 xx 10^(6)`
D. `36 xx 10 ^(6)`

Answer

To find the volume of a hexagonal unit cell given the length of its side, we can follow these steps:

### Step 1: Identify the given values
We are given the length of the side of the hexagonal unit cell, denoted as \( A \):
\[ A = \frac{100}{\sqrt{2}} \text{ pm} \]

### Step 2: Calculate the height (C) of the hexagonal unit cell
For a hexagonal close-packed (HCP) unit cell, the height \( C \) can be calculated using the formula:
\[ C = \frac{\sqrt{8}}{3} A \]
Substituting the value of \( A \):
\[ C = \frac{\sqrt{8}}{3} \left(\frac{100}{\sqrt{2}}\right) = \frac{100 \sqrt{8}}{3 \sqrt{2}} = \frac{100 \cdot 2\sqrt{2}}{3 \sqrt{2}} = \frac{200}{3} \text{ pm} \]

### Step 3: Calculate the area of the basal plane
The area of the basal plane of a hexagonal unit cell can be calculated using the formula:
\[ \text{Area} = \frac{3\sqrt{3}}{2} A^2 \]
Substituting the value of \( A \):
\[ \text{Area} = \frac{3\sqrt{3}}{2} \left(\frac{100}{\sqrt{2}}\right)^2 = \frac{3\sqrt{3}}{2} \cdot \frac{10000}{2} = \frac{3\sqrt{3} \cdot 10000}{4} = 7500\sqrt{3} \text{ pm}^2 \]

### Step 4: Calculate the volume of the hexagonal unit cell
The volume \( V \) of the hexagonal unit cell can be calculated using the formula:
\[ V = \text{Area} \times C \]
Substituting the values we have:
\[ V = 7500\sqrt{3} \cdot \frac{200}{3} \]
Simplifying this:
\[ V = 7500 \cdot \frac{200\sqrt{3}}{3} = 500000\sqrt{3} \text{ pm}^3 \]

### Step 5: Calculate the numerical value
To find the numerical value of \( V \):
\[ \sqrt{3} \approx 1.732 \]
Thus,
\[ V \approx 500000 \cdot 1.732 \approx 866000 \text{ pm}^3 \]

### Step 6: Final answer
The volume of the hexagonal unit cell is approximately:
\[ V \approx 8.66 \times 10^5 \text{ pm}^3 \]

### Summary
The volume of the hexagonal unit cell is approximately \( 8.66 \times 10^5 \text{ pm}^3 \).

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