A solution containing `10.2g` of glycerene per litre is found to be isotonic with a `2%` solution of glucose. Calculate the molecular weight of glycerene (mol.wt. of glucose `=180`)

Answer

To solve the problem, we need to use the concept of isotonic solutions, which states that the concentration of solute in two solutions must be equal for them to be isotonic.

### Step-by-Step Solution:

1. **Understanding the Isotonic Condition**:
Since the glycerin solution is isotonic with the glucose solution, we can set up the equation:
\[ \text{Concentration of glucose} = \text{Concentration of glycerin} \]

2. **Calculating the Concentration of Glucose**:
We are given a 2% glucose solution. This means there are 2 grams of glucose in 100 mL of solution. To find the concentration in grams per liter, we convert this to liters:
\[ \text{Mass of glucose in 1 L} = 2 \text{ g} \times 10 = 20 \text{ g} \]
Thus, the concentration of glucose in grams per liter is:
\[ \text{Concentration of glucose} = \frac{20 \text{ g}}{1 \text{ L}} = 20 \text{ g/L} \]

3. **Calculating the Moles of Glucose**:
The molar mass of glucose is given as 180 g/mol. Now, we can calculate the number of moles of glucose:
\[ \text{Moles of glucose} = \frac{\text{Mass}}{\text{Molar Mass}} = \frac{20 \text{ g}}{180 \text{ g/mol}} = \frac{1}{9} \text{ mol} \]

4. **Setting Up the Equation for Glycerin**:
We know the mass of glycerin in the solution is 10.2 g per liter. Let \( X \) be the molar mass of glycerin. The number of moles of glycerin can be calculated as:
\[ \text{Moles of glycerin} = \frac{10.2 \text{ g}}{X \text{ g/mol}} \]

5. **Equating the Moles of Glucose and Glycerin**:
Since the two solutions are isotonic, we can set the moles of glucose equal to the moles of glycerin:
\[ \frac{10.2}{X} = \frac{1}{9} \]

6. **Solving for \( X \)**:
Cross-multiplying gives:
\[ 10.2 \times 9 = X \]
\[ X = 91.8 \text{ g/mol} \]

Thus, the molecular weight of glycerin is **91.8 g/mol**.