Problem 76 - Chemistry The Central Science

Question

What is the freezing point of an aqueous solution that boils at \(105.0^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

The freezing point of the aqueous solution is -18.2 °C.

Step 1: Identify the formula for boiling point elevation and the given information

The formula for boiling point elevation is given by: ΔT_b = Kb * m Where: ΔT_b = boiling point elevation (increase in boiling point from the normal boiling point of the solvent) Kb = molal boiling point elevation constant (for water, Kb = 0.51 °C/mol) m = molality of the solution Here, we are given the boiling point of the aqueous solution (105.0 °C). We know the normal boiling point of water is 100.0 °C. So, we can find the boiling-point elevation.

Step 2: Calculate boiling point elevation

The boiling point elevation is the difference between the boiling point of the solution and the normal boiling point of water: ΔT_b = 105.0 °C - 100.0 °C = 5.0 °C Now that we have the boiling point elevation, we can find the molality of the solution.

Step 3: Calculate the molality of the solution

Using the boiling point elevation formula, we can rearrange it to solve for molality: m = ΔT_b / Kb Substitute the values into the formula: m = 5.0 °C / 0.51 °C/mol = 9.80 mol/kg Now, we have the molality of the solution.

Step 4: Identify the freezing point depression formula and given information

The formula for freezing point depression is given by: ΔT_f = Kf * m Where: ΔT_f = freezing point depression (decrease in freezing point from the normal freezing point of the solvent) Kf = molal freezing point depression constant (for water, Kf = 1.86 °C/mol) m = molality of the solution (which we already found in Step 3) The normal freezing point of water is 0.0 °C. We can use the molality and the freezing point depression formula to find the freezing point depression.

Step 5: Calculate freezing point depression

To find the freezing point depression, substitute the values into the formula: ΔT_f = Kf * m = (1.86 °C/mol) * (9.80 mol/kg) = 18.2 °C Now we have the freezing point depression.

Step 6: Calculate the freezing point of the solution

Finally, to find the freezing point of the solution, subtract the freezing point depression from the normal freezing point of water: Freezing point of the solution = Normal freezing point - ΔT_f = 0.0 °C - 18.2 °C = -18.2 °C So, the freezing point of the aqueous solution is -18.2 °C.