If Boiling point of water is `100^(@)C`. How much gram of NaCl is added in 500 g of water to increase its boiling point of water by approx `1^(@)C.(K_(b))_(H_(2)O)="0.52 K kg/mole".`

Options

A. 2.812 g
B. 28.12 g
C. 14.06 g
D. 7.03 g

Explanation

To solve the problem of how much NaCl is needed to raise the boiling point of 500 g of water by approximately 1°C, we will use the formula for boiling point elevation:

\[ \Delta T_b = i \cdot K_b \cdot m \]

Where:
- \(\Delta T_b\) is the change in boiling point (in °C),
- \(i\) is the Van't Hoff factor (number of particles the solute dissociates into),
- \(K_b\) is the ebullioscopic constant of the solvent (water in this case),
- \(m\) is the molality of the solution.

### Step 1: Identify the known values
- \(\Delta T_b = 1 \, °C\)
- \(K_b = 0.52 \, \text{K kg/mole}\)
- For NaCl, \(i = 2\) (since NaCl dissociates into Na⁺ and Cl⁻ ions).

### Step 2: Rearrange the formula to find molality
We can rearrange the equation to solve for molality \(m\):

\[ m = \frac{\Delta T_b}{i \cdot K_b} \]

### Step 3: Substitute the known values into the equation
Substituting the known values into the equation gives:

\[ m = \frac{1}{2 \cdot 0.52} = \frac{1}{1.04} \approx 0.9615 \, \text{mol/kg} \]

### Step 4: Calculate the number of moles of NaCl needed
Molality \(m\) is defined as moles of solute per kg of solvent. Since we have 500 g of water, which is 0.5 kg, we can find the number of moles of NaCl needed:

\[ \text{moles of NaCl} = m \cdot \text{mass of solvent (in kg)} = 0.9615 \cdot 0.5 \approx 0.48075 \, \text{moles} \]

### Step 5: Calculate the mass of NaCl needed
To find the mass of NaCl, we need to multiply the number of moles by the molar mass of NaCl. The molar mass of NaCl is calculated as follows:

\[ \text{Molar mass of NaCl} = \text{mass of Na} + \text{mass of Cl} = 23 \, \text{g/mol} + 35.5 \, \text{g/mol} = 58.5 \, \text{g/mol} \]

Now, we can calculate the mass of NaCl:

\[ \text{mass of NaCl} = \text{moles of NaCl} \cdot \text{molar mass of NaCl} = 0.48075 \cdot 58.5 \approx 28.12 \, \text{g} \]

### Final Answer
The mass of NaCl required to raise the boiling point of 500 g of water by approximately 1°C is **28.12 g**.