Question

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

Short answer

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

Step-by-step solution

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.

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.

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.

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.

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.

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.