How much amount of `NaCl` should be added to 500 g of water `rho = 1.00g//mL)` to decrease the freezing point of water to `-0.3^@C` ? (The freezing point depression constant for water `= 2 Kkgmol^(-1))`

Options

A. `2.19 g`
B. `1.88 g`
C. `1.96 g`
D. `1.085 g`

Answer

To solve the problem of how much NaCl should be added to 500 g of water to decrease the freezing point to -0.3°C, we can follow these steps:

### Step 1: Understand the Freezing Point Depression Formula
The freezing point depression can be calculated using the formula:
\[ \Delta T_f = i \cdot K_f \cdot m \]
Where:
- \(\Delta T_f\) = change in freezing point (°C)
- \(i\) = van 't Hoff factor (number of particles the solute dissociates into)
- \(K_f\) = freezing point depression constant (K kg/mol)
- \(m\) = molality of the solution (mol/kg)

### Step 2: Calculate the Change in Freezing Point
Given that the normal freezing point of water is 0°C and we want to decrease it to -0.3°C:
\[ \Delta T_f = 0 - (-0.3) = 0.3°C \]

### Step 3: Identify the Van 't Hoff Factor for NaCl
Sodium chloride (NaCl) dissociates into two ions:
\[ \text{NaCl} \rightarrow \text{Na}^+ + \text{Cl}^- \]
Thus, \(i = 2\).

### Step 4: Use the Freezing Point Depression Constant
The freezing point depression constant for water is given as:
\[ K_f = 2 \, \text{K kg/mol} \]

### Step 5: Set Up the Equation for Molality
We can rearrange the freezing point depression formula to find molality:
\[ m = \frac{\Delta T_f}{i \cdot K_f} \]
Substituting the known values:
\[ m = \frac{0.3}{2 \cdot 2} = \frac{0.3}{4} = 0.075 \, \text{mol/kg} \]

### Step 6: Calculate the Mass of Solvent in kg
The mass of water is given as 500 g, which we convert to kg:
\[ \text{mass of water} = 500 \, \text{g} = 0.5 \, \text{kg} \]

### Step 7: Calculate the Number of Moles of NaCl Required
Using the definition of molality:
\[ m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} \]
Rearranging gives:
\[ \text{moles of solute} = m \cdot \text{mass of solvent in kg} = 0.075 \cdot 0.5 = 0.0375 \, \text{mol} \]

### Step 8: Calculate the Mass of NaCl Required
Now, we need to convert moles of NaCl to grams. The molar mass of NaCl is approximately 58.5 g/mol:
\[ \text{mass of NaCl} = \text{moles of NaCl} \cdot \text{molar mass of NaCl} = 0.0375 \cdot 58.5 \approx 2.19 \, \text{g} \]

### Final Answer
To decrease the freezing point of water to -0.3°C, you should add approximately **2.19 g of NaCl** to 500 g of water.

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