Mole fraction of solvent in an aqueous solution is 0.7. What is the molality of this solution in mol/kg?

Options

A. `23.88 xx 10^(-1)`
B. `23.88 xx 10^(-2)`
C. `23.8`
D. `23.88 xx 10^(-3)`

Answer

To find the molality of the solution given the mole fraction of the solvent, we can follow these steps:

### Step 1: Understand the relationship between mole fraction and molality
The mole fraction of the solute (X_solute) and the mole fraction of the solvent (X_solvent) are related by the equation:
\[ X_{\text{solute}} + X_{\text{solvent}} = 1 \]

### Step 2: Calculate the mole fraction of the solute
Given that the mole fraction of the solvent (X_solvent) is 0.7, we can find the mole fraction of the solute:
\[ X_{\text{solute}} = 1 - X_{\text{solvent}} = 1 - 0.7 = 0.3 \]

### Step 3: Use the formula for molality
Molality (m) is defined as the number of moles of solute per kilogram of solvent. The formula for molality in terms of mole fraction is:
\[ m = \frac{X_{\text{solute}}}{X_{\text{solvent}}} \times \frac{1000}{M_{\text{solvent}}} \]
where \( M_{\text{solvent}} \) is the molar mass of the solvent.

### Step 4: Substitute values into the formula
Here, the solvent is water, which has a molar mass of 18 g/mol (or 0.018 kg/mol). Now we can substitute the values:
\[ m = \frac{0.3}{0.7} \times \frac{1000}{0.018} \]

### Step 5: Calculate the molality
Calculating the above expression:
1. Calculate \( \frac{0.3}{0.7} \):
\[ \frac{0.3}{0.7} = 0.4286 \]
2. Calculate \( \frac{1000}{0.018} \):
\[ \frac{1000}{0.018} = 55555.56 \]
3. Now multiply these two results:
\[ m = 0.4286 \times 55555.56 \approx 238.1 \]

### Final Result
Thus, the molality of the solution is approximately:
\[ m \approx 23.8 \, \text{mol/kg} \]