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 find the approximate number of unit cells in 2.0 g of gold, we will follow these steps:

### Step 1: Determine the number of moles of gold in 2.0 g.
The atomic mass of gold (Au) is given as 197 amu, which means 1 mole of gold weighs 197 g. We can calculate the number of moles of gold in 2.0 g using the formula:

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

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 2.0 g of gold.
Using Avogadro's number (\(N_A = 6.022 \times 10^{23} \, \text{atoms/mol}\)), we can find the total number of atoms in the gold sample:

\[ \text{Number of atoms} = \text{Number of moles} \times N_A \]

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 atoms per unit cell in FCC.
In a face-centered cubic (FCC) lattice, there are 4 atoms per unit cell. This is calculated as follows:

- Each face contributes \( \frac{1}{2} \) atom (6 faces) = 3 atoms
- Each corner contributes \( \frac{1}{8} \) atom (8 corners) = 1 atom

Thus, total atoms per unit cell = 3 + 1 = 4 atoms.

### Step 4: Calculate the number of unit cells.
To find the number of unit cells, we divide the total number of atoms by the number of atoms per unit cell:

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

Calculating this gives:

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

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

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