Question

How many total voids are present in 1 mole of a compound that forms an hcp structure?

• A. $1.806 \times 10^{24}$
• B. $1.204 \times 10^{24}$
• C. $3.011 \times 10^{23}$
• D. $4.204 \times 10^{23}$

Hint:

For any close-packed grid, the total number of voids is always three times the total number of atoms ($3N$). Multiplying the initial digit of Avogadro's number by 3 ($6 \times 3 = 18$) simplifies the choice down to something starting with $1.8$, which points safely to option (A).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to calculate the total combined number of structural voids (both tetrahedral and octahedral) present within exactly $1\ \text{mole}$ of a crystalline substance packed in a hexagonal close-packed (hcp) lattice.

Step 2: Key Formula or Approach:
In close-packed arrangements (like hcp or ccp) containing $N$ constituent atoms:
Number of Octahedral Voids $= N$
Number of Tetrahedral Voids $= 2N$
Total Number of Voids $= N + 2N = 3N$

Step 3: Detailed Explanation:
The question specifies that we have $1\ \text{mole}$ of the compound. Therefore, the total number of particles ($N$) matches Avogadro's number:
$$ N = 6.022 \times 10^{23} $$ Using our total structural void relationship:
$$ \text{Total Voids} = 3 \times N $$ $$ \text{Total Voids} = 3 \times (6.022 \times 10^{23}) $$ $$ \text{Total Voids} = 18.066 \times 10^{23} $$ Converting this value into proper scientific notation yields:
$$ \text{Total Voids} = 1.8066 \times 10^{24} \approx 1.806 \times 10^{24} $$

Step 4: Final Answer: The calculated value for the total number of voids perfectly matches option (A).