What weight of glycerol would have to be added to 1000g of water in order to lower its freezing point by `10^(@)C`? `K_(f)=1.86`

Answer

To solve the problem of determining the weight of glycerol needed to lower the freezing point of 1000 g of water by 10°C, we can follow these steps:

### Step 1: Understand the Formula
The freezing point depression (\( \Delta T_f \)) can be calculated using the formula:
\[ \Delta T_f = K_f \cdot m \]
where:
- \( \Delta T_f \) = change in freezing point (in °C)
- \( K_f \) = freezing point depression constant of the solvent (for water, \( K_f = 1.86 \, °C \cdot kg/mol \))
- \( m \) = molality of the solution (in mol/kg)

### Step 2: Calculate Molality
We know:
- \( \Delta T_f = 10 \, °C \)
- \( K_f = 1.86 \, °C \cdot kg/mol \)

Rearranging the formula to find molality (\( m \)):
\[ m = \frac{\Delta T_f}{K_f} = \frac{10}{1.86} \approx 5.376 \, mol/kg \]

### Step 3: Calculate Moles of Glycerol Required
Since we are using 1000 g of water (which is 1 kg), the molality is also equal to the number of moles of solute (glycerol) per kg of solvent. Therefore:
\[ \text{Moles of glycerol} = m \cdot \text{kg of solvent} = 5.376 \, mol/kg \cdot 1 \, kg = 5.376 \, mol \]

### Step 4: Find the Molar Mass of Glycerol
The molar mass of glycerol (\( C_3H_8O_3 \)) is calculated as follows:
- Carbon (C): 12.01 g/mol × 3 = 36.03 g/mol
- Hydrogen (H): 1.008 g/mol × 8 = 8.064 g/mol
- Oxygen (O): 16.00 g/mol × 3 = 48.00 g/mol

Adding these together:
\[ \text{Molar mass of glycerol} = 36.03 + 8.064 + 48.00 = 92.094 \, g/mol \approx 92 \, g/mol \]

### Step 5: Calculate the Weight of Glycerol Required
Now, we can calculate the weight of glycerol needed using the number of moles and the molar mass:
\[ \text{Weight of glycerol} = \text{Moles} \times \text{Molar mass} = 5.376 \, mol \times 92 \, g/mol \approx 494.592 \, g \]

### Final Answer
The weight of glycerol that must be added to 1000 g of water to lower its freezing point by 10°C is approximately:
\[ \text{Weight of glycerol} \approx 494.6 \, g \]

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