The hydrated salt `Na_(2)SO_(4).xH_(2)O` undergoes `55%` loss in weight on heating and becomes anhydrous. The value of x will be

Options

A. 5
B. 3
C. 7
D. 10

Answer

To solve the problem of determining the value of \( x \) in the hydrated salt \( Na_2SO_4 \cdot xH_2O \) that undergoes a 55% loss in weight upon heating, we can follow these steps:

### Step 1: Understand the Composition of the Hydrated Salt
The hydrated salt is represented as \( Na_2SO_4 \cdot xH_2O \). This means that for every mole of sodium sulfate, there are \( x \) moles of water.

### Step 2: Calculate the Molar Mass of the Components
1. **Molar mass of \( Na_2SO_4 \)**:
- Sodium (Na): \( 23 \, \text{g/mol} \times 2 = 46 \, \text{g/mol} \)
- Sulfur (S): \( 32 \, \text{g/mol} \)
- Oxygen (O): \( 16 \, \text{g/mol} \times 4 = 64 \, \text{g/mol} \)
- Total: \( 46 + 32 + 64 = 142 \, \text{g/mol} \)

2. **Molar mass of water (\( H_2O \))**:
- Hydrogen (H): \( 1 \, \text{g/mol} \times 2 = 2 \, \text{g/mol} \)
- Oxygen (O): \( 16 \, \text{g/mol} \)
- Total: \( 2 + 16 = 18 \, \text{g/mol} \)

### Step 3: Write the Total Molar Mass of the Hydrated Salt
The total molar mass of the hydrated salt \( Na_2SO_4 \cdot xH_2O \) is:
\[ \text{Total Molar Mass} = 142 + 18x \]

### Step 4: Calculate the Mass of Water Lost
When heated, the hydrated salt loses \( x \) moles of water, which corresponds to a mass of:
\[ \text{Mass of water lost} = 18x \, \text{g} \]

### Step 5: Set Up the Equation for Percent Loss
We know that the percent loss in weight is given as 55%. Therefore, we can set up the equation:
\[ \frac{\text{Mass of water lost}}{\text{Total mass of hydrated salt}} \times 100 = 55 \]
Substituting the known values:
\[ \frac{18x}{142 + 18x} \times 100 = 55 \]

### Step 6: Solve the Equation
1. Rearranging gives:
\[ \frac{18x}{142 + 18x} = 0.55 \]
2. Cross-multiplying:
\[ 18x = 0.55(142 + 18x) \]
3. Expanding:
\[ 18x = 77.1 + 9.9x \]
4. Rearranging:
\[ 18x - 9.9x = 77.1 \]
\[ 8.1x = 77.1 \]
5. Solving for \( x \):
\[ x = \frac{77.1}{8.1} \approx 9.56 \]

### Step 7: Round to the Nearest Whole Number
Since \( x \) must be a whole number, we round \( 9.56 \) to \( 10 \).

### Final Answer
The value of \( x \) is approximately \( 10 \).

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