A face-centered cubic unit cell contains 8 ‘X’ atoms at the corners of the cell and 6 ‘Y’ atoms at the faces. If the empirical formula of the solid is `X_(a)Y_(b)` then, what is `(b)/(a)`?

Explanation

To solve the problem, we need to determine the number of 'X' and 'Y' atoms in a face-centered cubic (FCC) unit cell and then find the empirical formula in the form of \(X_aY_b\) to calculate the ratio \(\frac{b}{a}\).

### Step 1: Calculate the contribution of 'X' atoms
In a face-centered cubic unit cell:
- There are 8 corner atoms.
- Each corner atom contributes \(\frac{1}{8}\) of an atom to the unit cell.

So, the total contribution of 'X' atoms from the corners is:
\[ \text{Total contribution of 'X'} = 8 \times \frac{1}{8} = 1 \text{ atom} \]

### Step 2: Calculate the contribution of 'Y' atoms
In a face-centered cubic unit cell:
- There are 6 face atoms.
- Each face atom contributes \(\frac{1}{2}\) of an atom to the unit cell.

So, the total contribution of 'Y' atoms from the faces is:
\[ \text{Total contribution of 'Y'} = 6 \times \frac{1}{2} = 3 \text{ atoms} \]

### Step 3: Write the empirical formula
From the contributions calculated:
- We have 1 atom of 'X' and 3 atoms of 'Y'.
- Therefore, the empirical formula can be written as:
\[ X_1Y_3 \quad \text{or simply} \quad XY_3 \]

### Step 4: Determine the ratio \(\frac{b}{a}\)
In the empirical formula \(X_aY_b\):
- Here, \(a = 1\) and \(b = 3\).
- Thus, the ratio \(\frac{b}{a}\) is:
\[ \frac{b}{a} = \frac{3}{1} = 3 \]

### Final Answer
The ratio \(\frac{b}{a}\) is \(3\).

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