At 70°C the vapor pressure of pure water is 31 kPa. Which of the following is most likely the vapor pressure of a 2.0 molal aq. glucose solution at 70°C?

Options

A. 30.001 kPa
B. 29.915 kPa
C. 28.226 kPa
D. 32.392 kPa

Answer

To solve the problem of finding the vapor pressure of a 2.0 molal aqueous glucose solution at 70°C, we will use the concept of relative lowering of vapor pressure. Here’s a step-by-step solution:

### Step 1: Identify the given data
- Vapor pressure of pure water at 70°C, \( P^0 = 31 \, \text{kPa} \)
- Molality of glucose solution, \( m = 2.0 \, \text{molal} \)

### Step 2: Calculate the moles of water
Since molality is defined as moles of solute per kilogram of solvent, we need to first calculate the number of moles of water in 1 kg.
- Molecular weight of water = 18 g/mol
- Mass of water = 1000 g = 1 kg

\[ \text{Moles of water} = \frac{1000 \, \text{g}}{18 \, \text{g/mol}} \approx 55.56 \, \text{moles} \]

### Step 3: Calculate the total moles in the solution
The solution contains 2 moles of glucose (solute) and approximately 55.56 moles of water (solvent).
\[ \text{Total moles} = \text{Moles of glucose} + \text{Moles of water} = 2 + 55.56 = 57.56 \, \text{moles} \]

### Step 4: Calculate the mole fraction of glucose
The mole fraction of glucose (\( x_2 \)) can be calculated as follows:
\[ x_2 = \frac{\text{Moles of glucose}}{\text{Total moles}} = \frac{2}{57.56} \approx 0.0348 \]

### Step 5: Calculate the relative lowering of vapor pressure
Using the formula for relative lowering of vapor pressure:
\[ \Delta P = P^0 \cdot x_2 \]
Substituting the values:
\[ \Delta P = 31 \, \text{kPa} \cdot 0.0348 \approx 1.08 \, \text{kPa} \]

### Step 6: Calculate the vapor pressure of the glucose solution
The vapor pressure of the solution (\( P \)) can be calculated by subtracting the lowering from the vapor pressure of pure water:
\[ P = P^0 - \Delta P = 31 \, \text{kPa} - 1.08 \, \text{kPa} \approx 29.92 \, \text{kPa} \]

### Final Answer
The vapor pressure of the 2.0 molal aqueous glucose solution at 70°C is approximately **29.9 kPa**.

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