The temperature at a hill station is `-10^(@)C`. Will it be suitable to add ethylene glycol (mol mass = 62 )to water in the radiator so that the solution is `30%` by mass ? (`K_(f)` for water `=1.86"K m"^(-1)`)

Explanation

To determine whether it is suitable to add ethylene glycol to water in the radiator such that the solution is 30% by mass, we need to calculate the depression in freezing point caused by the addition of ethylene glycol. Here’s how to do it step by step:

### Step 1: Understand the given data
- Temperature at the hill station: \( -10^\circ C \)
- Mass percentage of ethylene glycol in the solution: \( 30\% \)
- Molar mass of ethylene glycol: \( 62 \, g/mol \)
- Freezing point depression constant for water, \( K_f = 1.86 \, K \cdot m^{-1} \)

### Step 2: Calculate the mass of solute and solvent
Since the solution is 30% ethylene glycol by mass, in 100 g of solution:
- Mass of ethylene glycol (solute) = \( 30 \, g \)
- Mass of water (solvent) = \( 100 \, g - 30 \, g = 70 \, g \)

### Step 3: Convert the mass of solvent to kilograms
To calculate molality, we need the mass of the solvent in kilograms:
- Mass of water = \( 70 \, g = \frac{70}{1000} \, kg = 0.07 \, kg \)

### Step 4: Calculate molality (m)
Molality (m) is defined as the number of moles of solute per kilogram of solvent:
- Moles of ethylene glycol = \( \frac{30 \, g}{62 \, g/mol} = 0.4839 \, mol \)
- Molality, \( m = \frac{0.4839 \, mol}{0.07 \, kg} = 6.913 \, mol/kg \)

### Step 5: Calculate the depression in freezing point (\( \Delta T_f \))
Using the formula for freezing point depression:
\[ \Delta T_f = m \cdot K_f \]
Substituting the values:
\[ \Delta T_f = 6.913 \, mol/kg \cdot 1.86 \, K \cdot m^{-1} = 12.87 \, K \]

### Step 6: Determine the new freezing point of the solution
The freezing point of pure water is \( 0^\circ C \). The new freezing point of the solution can be calculated as:
\[ T_f = T_{f0} - \Delta T_f \]
Where \( T_{f0} = 0^\circ C \):
\[ T_f = 0^\circ C - 12.87^\circ C = -12.87^\circ C \]

### Step 7: Conclusion
Since the freezing point of the solution (\( -12.87^\circ C \)) is lower than the temperature at the hill station (\( -10^\circ C \)), it is suitable to add ethylene glycol to water in the radiator.

### Final Answer
Yes, it is suitable to add ethylene glycol to water in the radiator so that the solution is 30% by mass.

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