A 10% solution of sucrose (molar mass 342) is isotonic with 1.754% solution of urea. Calculate the molecular mass of urea.

Explanation

To solve the problem, we will use the concept of isotonic solutions and the formula for osmotic pressure. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the concept of isotonic solutions
Isotonic solutions are those that have the same osmotic pressure. Therefore, we can equate the osmotic pressures of the two solutions.

### Step 2: Write the formula for osmotic pressure
The osmotic pressure (π) can be expressed as:
\[ \pi = C \cdot R \cdot T \]
Where:
- \(C\) is the molarity of the solution,
- \(R\) is the universal gas constant,
- \(T\) is the temperature in Kelvin.

Since we are dealing with isotonic solutions, we can equate the concentrations of the two solutions:
\[ C_1 = C_2 \]

### Step 3: Define the variables for the solutions
Let:
- \(W_1\) = weight of sucrose = 10 g (for 100 mL solution)
- \(m_1\) = molar mass of sucrose = 342 g/mol
- \(W_2\) = weight of urea = 1.754 g (for 100 mL solution)
- \(m_2\) = molar mass of urea (unknown)

### Step 4: Calculate the number of moles of sucrose
To find the number of moles of sucrose, we use the formula:
\[ \text{Number of moles} = \frac{W}{m} \]
For sucrose:
\[ \text{Number of moles of sucrose} = \frac{10 \, \text{g}}{342 \, \text{g/mol}} = \frac{10}{342} \, \text{mol} \]

### Step 5: Calculate the concentration of sucrose
Since the solution volume is 1000 mL (1 L), the molarity \(C_1\) of the sucrose solution is:
\[ C_1 = \frac{10 \, \text{g}}{342 \, \text{g/mol} \cdot 1 \, \text{L}} = \frac{10}{342} \, \text{mol/L} \]

### Step 6: Calculate the number of moles of urea
For the urea solution:
\[ \text{Number of moles of urea} = \frac{W_2}{m_2} = \frac{1.754 \, \text{g}}{m_2} \]

### Step 7: Set the concentrations equal
Since the solutions are isotonic:
\[ \frac{10}{342} = \frac{1.754}{m_2} \]

### Step 8: Solve for \(m_2\)
Cross-multiplying gives:
\[ 10 \cdot m_2 = 1.754 \cdot 342 \]
\[ m_2 = \frac{1.754 \cdot 342}{10} \]
Calculating this:
\[ m_2 = \frac{600.828}{10} = 60.0828 \, \text{g/mol} \]

### Final Answer
The molecular mass of urea is approximately \(60 \, \text{g/mol}\).

---