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 Question:
We are asked to find the total quantity of tiny unit cells that can fit inside a macroscopic 1 cm$^3$ block of metal, given the edge length ($a$) of a single cubic unit cell.

Step 2: Key Formula or Approach:
The total number of unit cells is simply the total volume of the metal block divided by the volume of a single unit cell.
$$\text{Number of unit cells} = \frac{\text{Total Volume ($V_{total}$)}}{\text{Volume of one unit cell ($a^3$)}}$$

Step 3: Detailed Explanation:
First, we must calculate the volume of one single unit cell.
The unit cell edge length is $a = 1.25 \times 10^{-8} \text{ cm}$.
To make manual calculation easier, convert the decimal to a fraction: $1.25 = \frac{5}{4}$.
$$V_{cell} = a^3 = \left(\frac{5}{4} \times 10^{-8}\right)^3$$
$$V_{cell} = \frac{125}{64} \times 10^{-24} \text{ cm}^3$$
Evaluating the fraction $\frac{125}{64} \approx 1.953$.
$$V_{cell} \approx 1.953 \times 10^{-24} \text{ cm}^3$$
Now, calculate the total number of unit cells in $1 \text{ cm}^3$:
$$\text{Number} = \frac{1}{1.953 \times 10^{-24}}$$
$$\text{Number} = \frac{64}{125} \times 10^{24}$$
$$\text{Number} = 0.512 \times 10^{24}$$
$$\text{Number} = 5.12 \times 10^{23}$$

Step 4: Final Answer:
There are $5.12 \times 10^{23}$ unit cells, which corresponds to option (c).