Question

A metallic element has a cubic lattice with edge length of unit cell 2Å. Calculate the number of unit cells in 200 g of the metal, if the density of metal is 2.5 g cm\(^{-3}\)?

• A. \( 6.25 \times 10^{25} \)
• B. \( 6.40 \times 10^{25} \)
• C. \( 1.0 \times 10^{25} \)
• D. \( 10.0 \times 10^{25} \)

Hint:

To calculate the number of unit cells, first calculate the total volume of the metal, then divide by the volume of a single unit cell.

Answer 1

The Correct Option is C

Step 1: Understanding the relationship.
To find the number of unit cells, we first need to calculate the volume of the metal in 200 g. Using the formula: \[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} = \frac{200}{2.5} = 80 \, \text{cm}^3 \] Next, we calculate the volume of one unit cell using the given edge length of 2Å (convert to cm: \(2 \times 10^{-8}\) cm): \[ \text{Volume of unit cell} = a^3 = (2 \times 10^{-8})^3 = 8 \times 10^{-24} \, \text{cm}^3 \] Step 2: Calculating the number of unit cells.
Now, we can calculate the number of unit cells by dividing the total volume by the volume of one unit cell: \[ \text{Number of unit cells} = \frac{\text{Volume of metal}}{\text{Volume of one unit cell}} = \frac{80}{8 \times 10^{-24}} = 1.0 \times 10^{25} \] Step 3: Conclusion.
The correct answer is (C) \( 1.0 \times 10^{25} \), the number of unit cells in 200 g of the metal.