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Chapter 12: Problem 36

A student dissolves \(45.0 \mathrm{~g}\) of an unknown solid in \(225.0 \mathrm{~g}\) of cyclohexane. It is known to dissolve without dissociating. She cools the solution and finds that the temperature remains at \(2.70^{\circ} \mathrm{C}\) while the solution changes phase from liquid to solid. What is the molar mass of the solid?

Short Answer

Expert verified
The molar mass of the unknown solid is approximately 1061 g/mol.

Step by step solution

01

Find the change in freezing point

The freezing point of pure cyclohexane is \(6.47^{\circ} \mathrm{C}\), and the freezing point of the solution is given as \(2.70^{\circ} \mathrm{C}\). We can find the change in freezing point (∆T) by subtracting the freezing point of the solution from the freezing point of pure cyclohexane: ∆T = T°(pure) - T°(solution) = \(6.47^{\circ} \mathrm{C}\) - \(2.70^{\circ} \mathrm{C}\) = \(3.77^{\circ} \mathrm{C}\)
02

Calculate the molality of the solution

We can now use the formula for freezing point depression: ∆T = Kf × molality × i where Kf is the freezing point depression constant for cyclohexane (20.0 °C·kg/mol), i is the van't Hoff factor, and we know that the solid does not dissociate (i = 1). Therefore, the formula simplifies to: ∆T = Kf × molality Now we can solve for molality: molality = ∆T / Kf = \(3.77^{\circ} \mathrm{C}\) / \(20.0 °C \cdot kg / mol\) = 0.1885 mol/kg
03

Determine the moles of the unknown solid

Molality is defined as moles of solute per kilogram of solvent. We can rearrange the formula for molality to find the moles of solute, which is the unknown solid: moles of solid = molality × mass of solvent (in kg) = 0.1885 mol/kg × 0.225 kg = 0.04242 mol
04

Calculate the molar mass of the unknown solid

We can now calculate the molar mass of the unknown solid using its mass and the moles calculated in the previous step: Molar mass = mass / moles = 45.0 g / 0.04242 mol = 1061 g/mol Therefore, the molar mass of the unknown solid is approximately 1061 g/mol.

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