Question

Calculate the number of unit cells in 1 cm$^3$ volume of metal if unit cell edge length is $1.25 \times 10^{-8}$ cm.

• A. $1.40 \times 10^{23}$
• B. $3.35 \times 10^{23}$
• C. $5.12 \times 10^{23}$
• D. $2.25 \times 10^{23}$

Hint:

Converting cumbersome decimals (like 1.25) into fractions (like 5/4) before cubing them dramatically reduces the complexity of arithmetic in exams that do not permit calculators.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The total volume of a sample is equal to the number of unit cells multiplied by the volume of a single unit cell.

Step 2: Formula Application:
Volume of one unit cell $V_{cell} = a^3$. Number of unit cells $N = \frac{\text{Total Volume}}{V_{cell}}$.

Step 3: Explanation:
$a = 1.25 \times 10^{-8}$ cm. $V_{cell} = (1.25 \times 10^{-8})^3 = 1.953 \times 10^{-24}$ cm$^3$. $N = \frac{1}{1.953 \times 10^{-24}} \approx 0.512 \times 10^{24} = 5.12 \times 10^{23}$.

Step 4: Final Answer:
The number of unit cells is $5.12 \times 10^{23}$.