An aqueous solution of `2` per cent `(wt.//wt)` non-volatile solute exerts a pressure of `1.004` bar at the boiling point of the solvent. What is the molecular mass of the solute?

Options

A. `"23.4 g mol"^(-1)`
B. `"41.35 g mol"^(-1)`
C. `"10 g mol"^(-1)`
D. `"20.8 g mol"^(-1)`

Explanation

To find the molecular mass of the non-volatile solute in the given aqueous solution, we can follow these steps:

### Step 1: Understand the given data
- We have a 2% (wt/wt) solution of a non-volatile solute.
- The pressure of the solution at boiling point is \( P_s = 1.004 \) bar.
- The vapor pressure of pure water (the solvent) is \( P_0 = 1.013 \) bar.

### Step 2: Calculate the mass of solute and solvent
From the definition of a 2% (wt/wt) solution:
- Mass of solute, \( w = 2 \) grams (since it is 2% of 100 grams of solution).
- Mass of solvent (water), \( W = 100 \, \text{grams} - 2 \, \text{grams} = 98 \) grams.

### Step 3: Apply Raoult's Law
According to Raoult's Law:
\[ \frac{P_0 - P_s}{P_s} = \frac{w}{M \cdot \frac{W}{M_w}} \]
Where:
- \( P_0 \) = vapor pressure of pure solvent
- \( P_s \) = vapor pressure of the solution
- \( w \) = mass of solute
- \( M \) = molar mass of the solute (what we need to find)
- \( M_w \) = molar mass of the solvent (water), which is \( 18 \, \text{g/mol} \)
- \( W \) = mass of solvent

### Step 4: Substitute the values into the equation
Substituting the known values into the equation:
\[ \frac{1.013 \, \text{bar} - 1.004 \, \text{bar}}{1.004 \, \text{bar}} = \frac{2 \, \text{grams}}{M \cdot \frac{98 \, \text{grams}}{18 \, \text{g/mol}}} \]

### Step 5: Calculate the left-hand side
Calculating the left-hand side:
\[ \frac{0.009 \, \text{bar}}{1.004 \, \text{bar}} \approx 0.00895 \]

### Step 6: Rearranging the equation
Rearranging the equation gives:
\[ 0.00895 = \frac{2 \cdot 18}{M \cdot 98} \]

### Step 7: Solve for \( M \)
Cross-multiplying to solve for \( M \):
\[ M = \frac{2 \cdot 18}{0.00895 \cdot 98} \]

Calculating the right-hand side:
\[ M = \frac{36}{0.8781} \approx 41.0 \, \text{g/mol} \]

### Step 8: Final answer
Thus, the molecular mass of the solute is approximately \( 41.0 \, \text{g/mol} \).