Question

Ag crystallizes in fcc lattice. What is the total number of tetrahedral voids present in 540 g of Ag metal? (N$_A$ = Avagadro number, Ag atomic weight = 108 g mol$^{-1}$)

• A. 10 N$_A$
• B. 20 N$_A$
• C. 40 N$_A$
• D. 60 N$_A$

Hint:

For an fcc (or ccp) lattice:
Effective number of atoms per unit cell ($Z$) = 4.
Number of octahedral voids per unit cell = $Z = 4$.
Number of tetrahedral voids per unit cell = $2Z = 8$.
This means there are 1 octahedral void per atom and 2 tetrahedral voids per atom in an fcc structure.
Calculate the total number of atoms in the given mass of the substance.
Total tetrahedral voids = $2 \times (\text{Total number of atoms})$.

Answer 1

The Correct Option is A

Silver (Ag) crystallizes in a face-centered cubic (fcc) lattice. In an fcc lattice structure: The effective number of atoms per unit cell ($Z$) is 4. The number of octahedral voids per unit cell is equal to $Z$, so there are 4 octahedral voids. The number of tetrahedral voids per unit cell is equal to $2Z$, so there are $2 \times 4 = 8$ tetrahedral voids. First, calculate the number of moles of Ag in 540 g. Atomic weight of Ag = $108 \text{ g mol}^{-1}$. Number of moles ($n$) = $\frac{\text{Mass}}{\text{Atomic weight}} = \frac{540 \text{ g}}{108 \text{ g mol}^{-1}}$. $n = \frac{540}{108} = 5 \text{ moles}$. Next, calculate the total number of Ag atoms in 5 moles. Total number of atoms = Number of moles $\times$ Avogadro's number ($N_A$). Total Ag atoms = $5 \times N_A$. In an fcc lattice, there are $Z=4$ atoms per unit cell. The number of tetrahedral voids is $2Z = 2 \times 4 = 8$ per unit cell. This means for every 4 atoms, there are 8 tetrahedral voids. So, the ratio of tetrahedral voids to atoms is $8 \text{ voids} / 4 \text{ atoms} = 2 \text{ voids/atom}$. Therefore, the total number of tetrahedral voids is twice the total number of atoms. Total number of tetrahedral voids = $2 \times (\text{Total number of Ag atoms})$. Total number of tetrahedral voids = $2 \times (5 N_A)$. Total number of tetrahedral voids = $10 N_A$. This matches option (a). \[ \boxed{10 N_A} \]