`Al_(2)(SO_(4))_(3).XH_(2)O` has `8.1%` aluminium by mass. The value of `X` is:

Options

A. `4`
B. `10`
C. `16`
D. `18`

Explanation

To solve the problem, we need to determine the value of \( X \) in the compound \( Al_2(SO_4)_3 \cdot XH_2O \) given that it contains \( 8.1\% \) aluminum by mass. Here’s how we can approach the solution step by step:

### Step 1: Understand the Composition
The compound can be expressed as:
\[ \text{Molar mass of } Al_2(SO_4)_3 \cdot XH_2O = \text{Molar mass of } Al_2(SO_4)_3 + \text{Molar mass of } XH_2O \]

### Step 2: Calculate the Molar Mass of \( Al_2(SO_4)_3 \)
The molar mass of \( Al_2(SO_4)_3 \) can be calculated as follows:
- Aluminum (Al): \( 2 \times 27 \, \text{g/mol} = 54 \, \text{g/mol} \)
- Sulfur (S): \( 3 \times 32 \, \text{g/mol} = 96 \, \text{g/mol} \)
- Oxygen (O): \( 12 \times 16 \, \text{g/mol} = 192 \, \text{g/mol} \)

Adding these together:
\[ \text{Molar mass of } Al_2(SO_4)_3 = 54 + 96 + 192 = 342 \, \text{g/mol} \]

### Step 3: Calculate the Molar Mass of Water
The molar mass of water \( H_2O \) is:
\[ \text{Molar mass of } H_2O = 2 \times 1 + 16 = 18 \, \text{g/mol} \]
Thus, the molar mass of \( XH_2O \) is:
\[ \text{Molar mass of } XH_2O = 18X \, \text{g/mol} \]

### Step 4: Total Molar Mass of the Compound
The total molar mass of \( Al_2(SO_4)_3 \cdot XH_2O \) is:
\[ \text{Total molar mass} = 342 + 18X \, \text{g/mol} \]

### Step 5: Set Up the Equation for Aluminum Percentage
According to the problem, aluminum constitutes \( 8.1\% \) of the total mass. The mass of aluminum in the compound is:
\[ \text{Mass of Al} = 54 \, \text{g/mol} \]
Thus, we can set up the equation:
\[ \frac{54}{342 + 18X} \times 100 = 8.1 \]

### Step 6: Solve for \( X \)
Now, we can solve for \( X \):
1. Multiply both sides by \( (342 + 18X) \):
\[ 54 \times 100 = 8.1 \times (342 + 18X) \]
\[ 5400 = 8.1 \times 342 + 8.1 \times 18X \]
\[ 5400 = 2770.2 + 145.8X \]

2. Rearranging gives:
\[ 5400 - 2770.2 = 145.8X \]
\[ 2629.8 = 145.8X \]

3. Divide by \( 145.8 \):
\[ X = \frac{2629.8}{145.8} \approx 18 \]

### Final Answer
Thus, the value of \( X \) is \( 18 \).

---