The hydrated salt `Na_2 SO_4. nH_2 O` undergoes 56% loss in mass on heating and becomes anhydrous. The value of 'n' will be :

Options

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

Answer

To find the value of 'n' in the hydrated salt \( \text{Na}_2\text{SO}_4 \cdot n\text{H}_2\text{O} \) that undergoes a 56% loss in mass upon heating, we can follow these steps:

### Step 1: Determine the molar mass of the hydrated salt
The molar mass of \( \text{Na}_2\text{SO}_4 \) can be calculated as follows:
- Sodium (Na): \( 23 \, \text{g/mol} \times 2 = 46 \, \text{g/mol} \)
- Sulfur (S): \( 32 \, \text{g/mol} \)
- Oxygen (O): \( 16 \, \text{g/mol} \times 4 = 64 \, \text{g/mol} \)

Adding these together gives:
\[ \text{Molar mass of } \text{Na}_2\text{SO}_4 = 46 + 32 + 64 = 142 \, \text{g/mol} \]

Now, the molar mass of the water of crystallization \( n\text{H}_2\text{O} \):
- Water (H2O): \( 2 \times 1 + 16 = 18 \, \text{g/mol} \)

Thus, the total molar mass of the hydrated salt is:
\[ \text{Molar mass of } \text{Na}_2\text{SO}_4 \cdot n\text{H}_2\text{O} = 142 + 18n \, \text{g/mol} \]

### Step 2: Calculate the mass loss during heating
The problem states that there is a 56% loss in mass. If we assume the initial mass of the hydrated salt is \( m \), then the mass loss is:
\[ \text{Mass loss} = 0.56m \]

The remaining mass after heating (the anhydrous salt) is:
\[ \text{Remaining mass} = m - 0.56m = 0.44m \]

### Step 3: Relate mass loss to water of crystallization
The mass loss is due to the loss of water, which is \( n \) moles of \( \text{H}_2\text{O} \):
\[ \text{Mass loss} = n \times 18 \, \text{g/mol} \]

### Step 4: Set up the equation
From the above, we can set up the equation:
\[ 0.56m = n \times 18 \]

### Step 5: Relate the initial mass to the molar mass
The initial mass \( m \) can also be expressed in terms of the molar mass:
\[ m = \text{Molar mass of } \text{Na}_2\text{SO}_4 \cdot n\text{H}_2\text{O} = 142 + 18n \]

### Step 6: Substitute and solve for 'n'
Substituting \( m \) into the mass loss equation gives:
\[ 0.56(142 + 18n) = 18n \]

Expanding this:
\[ 79.52 + 10.08n = 18n \]

Rearranging the equation:
\[ 79.52 = 18n - 10.08n \]
\[ 79.52 = 7.92n \]

Now, solving for \( n \):
\[ n = \frac{79.52}{7.92} \approx 10.04 \]

### Step 7: Conclusion
Since \( n \) must be a whole number, we round \( 10.04 \) to \( 10 \). Thus, the value of \( n \) is:
\[ \boxed{10} \]