The boiling point of a glucose solution containing 12 g of glucose in 100 g of water is `100.34^@C`. Boiling point of water is `100^@C`. The molal elevation constant of water is

Options

A. `0.51^@C//"Molal"`
B. `51^@C//"Molal"`
C. `5.1^@C//"Molal"`
D. None of these

Answer

To find the molal elevation constant (K) of water given the boiling point elevation of a glucose solution, we can follow these steps:

### Step 1: Identify the given data
- Mass of glucose (solute), \( W_2 = 12 \, \text{g} \)
- Mass of water (solvent), \( W_1 = 100 \, \text{g} \)
- Boiling point of the solution, \( T_b = 100.34^\circ C \)
- Boiling point of pure water, \( T_{b0} = 100^\circ C \)

### Step 2: Calculate the change in boiling point (\( \Delta T_b \))
\[ \Delta T_b = T_b - T_{b0} = 100.34^\circ C - 100^\circ C = 0.34^\circ C \]

### Step 3: Find the molecular mass of glucose
The molecular mass of glucose (C₆H₁₂O₆) is given as:
\[ M_2 = 180 \, \text{g/mol} \]

### Step 4: Calculate the molality of the solution
Molality (m) is defined as the number of moles of solute per kilogram of solvent. First, we need to find the number of moles of glucose:
\[ \text{Number of moles of glucose} = \frac{W_2}{M_2} = \frac{12 \, \text{g}}{180 \, \text{g/mol}} = 0.06667 \, \text{mol} \]

Next, convert the mass of water from grams to kilograms:
\[ W_1 = 100 \, \text{g} = 0.1 \, \text{kg} \]

Now, calculate the molality:
\[ \text{Molality} (m) = \frac{\text{Number of moles of solute}}{\text{Mass of solvent in kg}} = \frac{0.06667 \, \text{mol}}{0.1 \, \text{kg}} = 0.6667 \, \text{mol/kg} \]

### Step 5: Use the formula for boiling point elevation
The formula for boiling point elevation is:
\[ \Delta T_b = K \cdot m \]
Where \( K \) is the molal elevation constant.

Rearranging the formula to solve for \( K \):
\[ K = \frac{\Delta T_b}{m} \]

### Step 6: Substitute the values into the equation
Substituting the values we calculated:
\[ K = \frac{0.34^\circ C}{0.6667 \, \text{mol/kg}} \approx 0.51 \, \text{°C kg/mol} \]

### Final Answer
The molal elevation constant of water is approximately:
\[ K \approx 0.51 \, \text{°C kg/mol} \]

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