Question

Calculate the number of unit cells in 0.79 g metal if product of density and volume of unit cell is 1.58 $\times$ 10$^{-22}$ g.

• A. 3.96 $\times$ 10$^{21}$
• B. 1.72 $\times$ 10$^{21}$
• C. 4.46 $\times$ 10$^{21}$
• D. 5.0 $\times$ 10$^{21}$

Hint:

The mass of a unit cell is the product of its density and its volume.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks to find the total number of unit cells in a given mass of a metal. We are provided with the total mass of the metal and the mass of a single unit cell (indirectly).
Step 2: Key Formula or Approach:
The total number of unit cells in a sample can be calculated by dividing the total mass of the sample by the mass of a single unit cell.
\[ \text{Number of unit cells} = \frac{\text{Total mass of metal}}{\text{Mass of one unit cell}} \] The mass of one unit cell can be found using its density (\(\rho\)) and volume (\(V_{\text{uc}}\)).
\[ \text{Mass of one unit cell} = \text{density} \times \text{volume of unit cell} = \rho \times V_{\text{uc}} \] Step 3: Detailed Explanation:
Given values:
- Total mass of metal = 0.79 g
- Product of density and volume of unit cell (\(\rho \times V_{\text{uc}}\)) = 1.58 \( \times \) 10\(^{-22}\) g.
This product directly gives us the mass of one unit cell.
Mass of one unit cell = 1.58 \( \times \) 10\(^{-22}\) g.
Now, we can calculate the number of unit cells:
\[ \text{Number of unit cells} = \frac{0.79 \text{ g}}{1.58 \times 10^{-22} \text{ g}} \] \[ \text{Number of unit cells} = \frac{0.79}{1.58} \times 10^{22} \] Since 0.79 is exactly half of 1.58 (1.58 / 2 = 0.79):
\[ \text{Number of unit cells} = 0.5 \times 10^{22} \] To express this in standard scientific notation:
\[ \text{Number of unit cells} = 5.0 \times 10^{21} \] Step 4: Final Answer:
The total number of unit cells in 0.79 g of the metal is 5.0 \( \times \) 10\(^{21}\), which corresponds to option (D).