An alloy of `Cu, Ag` and `Au` is found to have copper constituting the `c.c.p.` lattice. If `Ag` atom occupy the edge centres and `Au` atom is present at body centre, the formula of this alloy is `:`

Options

A. `Cu_(4)Ag_(2)Au`
B. `Cu_(4)Ag_(4)Au`
C. `Cu_(4)Ag_(3)Au`
D. `CuAgAu`

Answer

To find the formula of the alloy consisting of copper (Cu), silver (Ag), and gold (Au) based on the given structure, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Lattice Structure**:
- The question states that copper constitutes a close-packed structure, specifically a cubic close-packed (CCP) lattice, which is equivalent to a face-centered cubic (FCC) lattice.
- In an FCC lattice, atoms are located at the corners and the centers of each face of the cube.

2. **Calculating Copper Atoms**:
- In a CCP (FCC) unit cell:
- 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.
- Total contribution from copper:
\[ \text{Total Cu} = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 1 + 3 = 4 \]
- Therefore, there are 4 copper atoms in the unit cell.

3. **Calculating Silver Atoms**:
- Silver atoms occupy the edge centers. In a cube, there are 12 edges.
- Each edge center atom contributes \( \frac{1}{4} \) of an atom to the unit cell.
- Total contribution from silver:
\[ \text{Total Ag} = 12 \times \frac{1}{4} = 3 \]
- Thus, there are 3 silver atoms in the unit cell.

4. **Calculating Gold Atoms**:
- Gold atoms are located at the body center of the cube.
- The body-centered atom contributes fully to the unit cell.
- Thus, there is 1 gold atom in the unit cell.

5. **Finding the Ratio of Atoms**:
- We now have the following contributions:
- Copper (Cu): 4 atoms
- Silver (Ag): 3 atoms
- Gold (Au): 1 atom
- The ratio of copper to silver to gold is:
\[ \text{Cu:Ag:Au} = 4:3:1 \]

6. **Writing the Formula**:
- Based on the ratio, the formula of the alloy can be written as:
\[ \text{Formula} = \text{Cu}_4\text{Ag}_3\text{Au} \]

### Final Answer:
The formula of the alloy is \( \text{Cu}_4\text{Ag}_3\text{Au} \).

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