Gold crystallizes in the face centred cubic lattice. Calculate the approximate number of unit cells in 2 mg of gold. (Atomic mass of gold= 197u).

Explanation

To solve the problem of calculating the approximate number of unit cells in 2 mg of gold, which crystallizes in a face-centered cubic (FCC) lattice, we can follow these steps:

### Step 1: Determine the number of atoms in one unit cell of gold.
In a face-centered cubic (FCC) lattice, the number of atoms per unit cell is calculated as follows:
- There are 8 corner atoms, each contributing \( \frac{1}{8} \) of an atom to the unit cell.
- There are 6 face-centered atoms, each contributing \( \frac{1}{2} \) of an atom to the unit cell.

Thus, the total number of atoms in one unit cell is:
\[ \text{Total atoms} = \left(8 \times \frac{1}{8}\right) + \left(6 \times \frac{1}{2}\right) = 1 + 3 = 4 \text{ atoms} \]

### Step 2: Convert the mass of gold from mg to grams.
Given that the mass of gold is 2 mg, we convert it to grams:
\[ 2 \text{ mg} = 2 \times 10^{-3} \text{ g} \]

### Step 3: Calculate the number of moles of gold in 2 mg.
The atomic mass of gold is given as 197 g/mol. To find the number of moles in 2 mg of gold, we use the formula:
\[ \text{Number of moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \]
Substituting the values:
\[ \text{Number of moles} = \frac{2 \times 10^{-3} \text{ g}}{197 \text{ g/mol}} \approx 1.014 \times 10^{-5} \text{ moles} \]

### Step 4: Calculate the number of atoms in 2 mg of gold.
Using Avogadro's number (\(6.022 \times 10^{23}\) atoms/mol), we can find the total number of atoms:
\[ \text{Number of atoms} = \text{Number of moles} \times \text{Avogadro's number} \]
\[ \text{Number of atoms} = 1.014 \times 10^{-5} \text{ moles} \times 6.022 \times 10^{23} \text{ atoms/mole} \approx 6.11 \times 10^{18} \text{ atoms} \]

### Step 5: Calculate the number of unit cells in 2 mg of gold.
Since we know that one unit cell contains 4 atoms, we can find the number of unit cells by dividing the total number of atoms by the number of atoms per unit cell:
\[ \text{Number of unit cells} = \frac{\text{Total atoms}}{\text{Atoms per unit cell}} = \frac{6.11 \times 10^{18} \text{ atoms}}{4 \text{ atoms/unit cell}} \approx 1.528 \times 10^{18} \text{ unit cells} \]

### Final Answer:
The approximate number of unit cells in 2 mg of gold is \(1.528 \times 10^{18}\).

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