The temperature at which ice will begin to separate from a mixture of 20 mass percent of glycol `(C_(2)H_(6)O_(2))` in water, is : [`K_(f)` (water) = 1.86 K kg `mol^(-1)`]

Options

A. 280.5 K
B. 265. 5K
C. 276.5 K
D. 269.5 K

Answer

To determine the temperature at which ice will begin to separate from a mixture of 20 mass percent glycol (C₂H₆O₂) in water, we can follow these steps:

### Step 1: Understand the given data
- Mass percent of glycol = 20%
- Therefore, in 100 g of solution, the mass of glycol = 20 g and the mass of water (solvent) = 100 g - 20 g = 80 g.
- Kf (cryoscopic constant for water) = 1.86 K kg mol⁻¹.

### Step 2: Calculate the molar mass of glycol
The molar mass of glycol (C₂H₆O₂) can be calculated as follows:
- Carbon (C): 12 g/mol × 2 = 24 g/mol
- Hydrogen (H): 1 g/mol × 6 = 6 g/mol
- Oxygen (O): 16 g/mol × 2 = 32 g/mol
- Total molar mass of glycol = 24 + 6 + 32 = 62 g/mol.

### Step 3: Calculate the number of moles of glycol
Using the mass of glycol:
- Moles of glycol = mass of glycol / molar mass of glycol.
- Moles of glycol = 20 g / 62 g/mol = 0.3226 mol.

### Step 4: Calculate the molality of the solution
Molality (m) is defined as the number of moles of solute per kilogram of solvent:
- Mass of water = 80 g = 0.08 kg.
- Molality (m) = moles of glycol / mass of water in kg = 0.3226 mol / 0.08 kg = 4.0325 mol/kg.

### Step 5: Calculate the depression in freezing point (ΔTf)
Using the formula for depression in freezing point:
\[ \Delta T_f = i \cdot K_f \cdot m \]
Where:
- i (van 't Hoff factor) = 1 (since glycol is a non-electrolyte).
- Kf = 1.86 K kg mol⁻¹.
- m = 4.0325 mol/kg.

Now calculate ΔTf:
\[ \Delta T_f = 1 \cdot 1.86 \cdot 4.0325 = 7.5 K \]

### Step 6: Calculate the freezing point of the solution
The normal freezing point of water is 0 °C (273 K). The freezing point of the solution will be:
\[ \text{Freezing point} = 0 °C - \Delta T_f \]
\[ \text{Freezing point} = 0 °C - 7.5 °C = -7.5 °C \]
Convert this to Kelvin:
\[ \text{Freezing point in Kelvin} = 273 K - 7.5 K = 265.5 K \]

### Final Answer
The temperature at which ice will begin to separate from the mixture is **265.5 K**.

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