10% (w/v) solution of sucrose (M.W = 342) is isotonic with 5% (w/v) solution of a solute X under identical conditions. The molecular weight of X is

Options

A. 171
B. 242
C. 684
D. 342

Answer

To solve the problem, we need to find the molecular weight of solute X, given that a 10% (w/v) solution of sucrose is isotonic with a 5% (w/v) solution of solute X.

### Step-by-Step Solution:

1. **Understand the Concept of Isotonic Solutions**:
- Isotonic solutions have the same osmotic pressure, which means they have the same concentration of solute particles. Therefore, the number of moles of sucrose must equal the number of moles of solute X.

2. **Calculate Moles of Sucrose**:
- A 10% (w/v) solution means there are 10 grams of sucrose in 100 mL of solution.
- The molecular weight (M.W) of sucrose is given as 342 g/mol.
- Calculate the number of moles of sucrose:
\[ \text{Moles of sucrose} = \frac{\text{mass}}{\text{molecular weight}} = \frac{10 \text{ g}}{342 \text{ g/mol}} = \frac{10}{342} \text{ mol} \]

3. **Calculate Moles of Solute X**:
- A 5% (w/v) solution means there are 5 grams of solute X in 100 mL of solution.
- Let the molecular weight of solute X be \( X \) g/mol.
- Calculate the number of moles of solute X:
\[ \text{Moles of solute X} = \frac{5 \text{ g}}{X \text{ g/mol}} = \frac{5}{X} \text{ mol} \]

4. **Set Up the Equation**:
- Since the solutions are isotonic, we can set the moles of sucrose equal to the moles of solute X:
\[ \frac{10}{342} = \frac{5}{X} \]

5. **Cross-Multiply to Solve for X**:
- Cross-multiplying gives:
\[ 10X = 5 \times 342 \]
- Calculate \( 5 \times 342 \):
\[ 5 \times 342 = 1710 \]
- Therefore, we have:
\[ 10X = 1710 \]
- Now, divide both sides by 10 to find \( X \):
\[ X = \frac{1710}{10} = 171 \text{ g/mol} \]

6. **Conclusion**:
- The molecular weight of solute X is 171 g/mol.

### Final Answer:
The molecular weight of solute X is **171 g/mol**.