Mole fraction fo solvent in an aqueous solution is 0.8. What is the molality of this solution in mol/kg?

Options

A. 55.55
B. 13.88
C. 27.77
D. 6.94

Answer

To find the molality of the solution given that the mole fraction of the solvent (water) is 0.8, we can follow these steps:

### Step 1: Understand the Definitions
- **Mole fraction of solvent (X_solvent)**: This is the ratio of the number of moles of the solvent to the total number of moles of the solution.
- **Molality (m)**: This is defined as the number of moles of solute per kilogram of solvent.

### Step 2: Set Up the Equation
Given that the mole fraction of the solvent (water) is 0.8, we can express the mole fraction of the solute (X_solute) as:
\[ X_{solute} = 1 - X_{solvent} = 1 - 0.8 = 0.2 \]

### Step 3: Relate Moles of Solvent and Solute
Let’s assume we have 1 kg of the solvent (water). The molar mass of water (H2O) is approximately 18 g/mol. Thus, the number of moles of solvent (water) in 1 kg (1000 g) is:
\[ n_{solvent} = \frac{1000 \, \text{g}}{18 \, \text{g/mol}} \approx 55.56 \, \text{mol} \]

### Step 4: Calculate Moles of Solute
Using the mole fraction of the solute:
\[ X_{solute} = \frac{n_{solute}}{n_{solute} + n_{solvent}} \]
Let \( n_{solute} \) be the number of moles of solute. We can set up the equation:
\[ 0.2 = \frac{n_{solute}}{n_{solute} + 55.56} \]

### Step 5: Solve for Moles of Solute
Cross-multiplying gives:
\[ 0.2(n_{solute} + 55.56) = n_{solute} \]
\[ 0.2n_{solute} + 11.112 = n_{solute} \]
Rearranging gives:
\[ 0.8n_{solute} = 11.112 \]
\[ n_{solute} = \frac{11.112}{0.8} \approx 13.89 \, \text{mol} \]

### Step 6: Calculate Molality
Now that we have the moles of solute, we can calculate molality:
\[ m = \frac{n_{solute}}{\text{mass of solvent in kg}} = \frac{13.89 \, \text{mol}}{1 \, \text{kg}} = 13.89 \, \text{mol/kg} \]

### Final Answer
The molality of the solution is approximately **13.89 mol/kg**.

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