What mass of non-volatile solute, sucrose, need to be dissolved in 100g of water to decrease the vapour pressure of water by 30%?

Explanation

To solve the problem of finding the mass of non-volatile solute (sucrose) needed to decrease the vapor pressure of water by 30%, we can follow these steps:

### Step 1: Write down the given data
- Mass of water (solvent), \( W_1 = 100 \, \text{g} \)
- Decrease in vapor pressure = 30%

### Step 2: Calculate the molar mass of sucrose
The molecular formula of sucrose is \( C_{12}H_{22}O_{11} \).
- Molar mass calculation:
- Carbon (C): \( 12 \times 12 = 144 \, \text{g/mol} \)
- Hydrogen (H): \( 22 \times 1 = 22 \, \text{g/mol} \)
- Oxygen (O): \( 11 \times 16 = 176 \, \text{g/mol} \)

Total molar mass of sucrose:
\[ M = 144 + 22 + 176 = 342 \, \text{g/mol} \]

### Step 3: Calculate the number of moles of water
Using the formula for moles:
\[ n_1 = \frac{W_1}{M_{water}} = \frac{100 \, \text{g}}{18 \, \text{g/mol}} \approx 5.56 \, \text{mol} \]

### Step 4: Set up the equation for vapor pressure lowering
According to Raoult's law, the relative lowering of vapor pressure is equal to the mole fraction of the solute:
\[ \Delta P = \frac{n_2}{n_1 + n_2} \]
Where:
- \( \Delta P = 0.30 \) (30% decrease)
- \( n_2 \) is the number of moles of sucrose.

### Step 5: Express mole fraction in terms of \( n_2 \)
We can express the mole fraction of sucrose as:
\[ 0.30 = \frac{n_2}{n_1 + n_2} \]
Rearranging gives:
\[ 0.30(n_1 + n_2) = n_2 \]
\[ 0.30n_1 + 0.30n_2 = n_2 \]
\[ 0.30n_1 = n_2 - 0.30n_2 \]
\[ 0.30n_1 = 0.70n_2 \]
\[ n_2 = \frac{0.30n_1}{0.70} = \frac{3}{7}n_1 \]

### Step 6: Substitute \( n_1 \) to find \( n_2 \)
Substituting \( n_1 \):
\[ n_2 = \frac{3}{7} \times 5.56 \approx 2.38 \, \text{mol} \]

### Step 7: Calculate the mass of sucrose
Using the number of moles to find the mass of sucrose:
\[ W_2 = n_2 \times M_{sucrose} = 2.38 \, \text{mol} \times 342 \, \text{g/mol} \approx 813.96 \, \text{g} \]

### Final Answer
The mass of sucrose needed to decrease the vapor pressure of water by 30% is approximately \( 814 \, \text{g} \).

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