What should be the freezing point of aqueous solution containing `17g` of `C_(2)H(5)OH` is `1000g` of water (`K_(f)` for water =`1.86 deg kg mol^(-1)`)?

Options

A. `-0.69^(@)C`
B. `-0.34^(@)C`
C. `0.0^(@)C`
D. `-0.34^(@)C`

Explanation

To find the freezing point of an aqueous solution containing 17 g of ethanol (C₂H₅OH) in 1000 g of water, we can follow these steps:

### Step 1: Calculate the molar mass of ethanol (C₂H₅OH)
The molecular formula of ethanol is C₂H₅OH. To find its molar mass, we sum the atomic masses of all the atoms in the formula:
- Carbon (C): 2 atoms × 12.01 g/mol = 24.02 g/mol
- Hydrogen (H): 6 atoms × 1.008 g/mol = 6.048 g/mol
- Oxygen (O): 1 atom × 16.00 g/mol = 16.00 g/mol

Adding these together:
\[ \text{Molar mass of C₂H₅OH} = 24.02 + 6.048 + 16.00 = 46.068 \text{ g/mol} \approx 46 \text{ g/mol} \]

### Step 2: Calculate the number of moles of ethanol
Using the mass of ethanol and its molar mass, we can calculate the number of moles:
\[ \text{Number of moles} = \frac{\text{mass}}{\text{molar mass}} = \frac{17 \text{ g}}{46 \text{ g/mol}} \approx 0.3696 \text{ moles} \]

### Step 3: Calculate the molality of the solution
Molality (m) is defined as the number of moles of solute per kilogram of solvent. Since we have 1000 g of water (which is 1 kg):
\[ \text{Molality} = \frac{\text{moles of solute}}{\text{kg of solvent}} = \frac{0.3696 \text{ moles}}{1 \text{ kg}} = 0.3696 \text{ mol/kg} \]

### Step 4: Calculate the freezing point depression (ΔTf)
The freezing point depression can be calculated using the formula:
\[ \Delta T_f = K_f \times m \]
Where \( K_f \) for water is given as 1.86 °C kg/mol. Thus:
\[ \Delta T_f = 1.86 \text{ °C kg/mol} \times 0.3696 \text{ mol/kg} \approx 0.687 \text{ °C} \]

### Step 5: Calculate the new freezing point of the solution
The normal freezing point of pure water is 0 °C. The new freezing point (Tf) of the solution is given by:
\[ T_f = 0 °C - \Delta T_f = 0 °C - 0.687 °C = -0.687 °C \]

### Final Answer
The freezing point of the aqueous solution containing 17 g of ethanol in 1000 g of water is approximately **-0.687 °C**.

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