Metallic gold crystallizes in the face centered cubic lattice. What is the approximate number of unit cells in `2.0 g` of gold ? (Atomic mass of gold is 197 amu)

Options

A. `3.06xx10^(11)`
B. `6.023xx10^(23)`
C. `1.53xx10^(21)`
D. `1.25xx10^(22)`

Answer

To determine the approximate number of unit cells in 2.0 g of gold, we can follow these steps:

### Step 1: Calculate the number of moles of gold in 2.0 g
To find the number of moles, we use the formula:

\[ \text{Number of moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \]

Given:
- Mass of gold = 2.0 g
- Molar mass of gold = 197 g/mol

Substituting the values:

\[ \text{Number of moles} = \frac{2.0 \, \text{g}}{197 \, \text{g/mol}} \approx 0.01015 \, \text{mol} \]

### Step 2: Calculate the number of atoms in the given moles of gold
Using Avogadro's number (\(6.022 \times 10^{23} \, \text{atoms/mol}\)), we can find the total number of atoms:

\[ \text{Number of atoms} = \text{Number of moles} \times \text{Avogadro's number} \]

Substituting the values:

\[ \text{Number of atoms} = 0.01015 \, \text{mol} \times 6.022 \times 10^{23} \, \text{atoms/mol} \approx 6.11 \times 10^{21} \, \text{atoms} \]

### Step 3: Determine the number of unit cells
In a face-centered cubic (FCC) lattice, there are 4 atoms per unit cell. Therefore, we can calculate the number of unit cells using the formula:

\[ \text{Number of unit cells} = \frac{\text{Number of atoms}}{\text{Number of atoms per unit cell}} \]

Substituting the values:

\[ \text{Number of unit cells} = \frac{6.11 \times 10^{21} \, \text{atoms}}{4 \, \text{atoms/unit cell}} \approx 1.53 \times 10^{21} \, \text{unit cells} \]

### Final Answer
The approximate number of unit cells in 2.0 g of gold is \( \approx 1.53 \times 10^{21} \, \text{unit cells} \).

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