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Step-by-Step Solution
Step 1: Identify the relevant colligative property
Since we are dealing with the freezing point of a solution, we use the depression in freezing point ( \Delta T_f ) formula for colligative properties.
The formula is:
\Delta T_f = K_f \times i \times m
where:
K_f is the cryoscopic constant (freezing point depression constant).
i is the van't Hoff factor.
m is the molality of the solution.
Step 2: Note the given values
Mass percentage of H_2SO_4 in solution = 38\% .
Van’t Hoff factor, i = 2.67 .
Freezing point depression constant, K_f = 1.8\, \text{K kg mol}^{-1} .
We assume 100\,\text{g} of the solution for simplicity:
Mass of H_2SO_4 = 38\,\text{g} .
Mass of water (solvent) = 62\,\text{g} = 0.062\,\text{kg}.
Step 3: Calculate the number of moles of H_2SO_4
The molar mass of H_2SO_4 is 98\,\text{g mol}^{-1} . Hence, the number of moles of H_2SO_4 in 38\,\text{g} is:
\text{moles of } H_2SO_4 = \frac{38}{98}.
Step 4: Determine molality ( m )
Molality is defined as the number of moles of solute per kilogram of solvent. So,
m = \frac{\frac{38}{98}}{0.062}\,\text{mol kg}^{-1}.
Step 5: Compute the freezing point depression ( \Delta T_f )
Using the relation:
\Delta T_f = K_f \times i \times m,
we substitute the values:
\Delta T_f = 1.8 \times 2.67 \times \frac{\frac{38}{98}}{0.062}.
Carrying out the multiplication:
\Delta T_f \approx 30\,\text{K}.
Step 6: Find the freezing point of the solution
The normal freezing point of water is 273\,\text{K} . Since the solution freezes at a temperature \Delta T_f below this:
T_{\text{freeze}} = 273 - 30 = 243\,\text{K}.
Final Answer
The freezing temperature of the battery acid solution is approximately 243\,\text{K} .
