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Chapter 11: Problem 73

What mass of glycerin $\left(\mathrm{C}_{3} \mathrm{H}_{8} \mathrm{O}_{3}\right),$ a nonelectrolyte, must be dissolved in 200.0 \(\mathrm{g}\) water to give a solution with a freezing point of $-1.50^{\circ} \mathrm{C} ?$

Short Answer

Expert verified
To find the mass of glycerin that must be dissolved in 200.0 g of water to give a solution with a freezing point of -1.50°C, we first calculate the change in freezing point (ΔTf = 1.50°C), then find the molality using the freezing point depression formula (molality = 0.8065 mol/kg), determine the moles of solute (mol_solute = 0.1613 mol), and finally calculate the mass of glycerin (mass_solute = 14.86 g). Thus, 14.86 grams of glycerin are needed.

Step by step solution

01

Determine the change in freezing point

First, we need to determine the change in the freezing point due to the solute. The freezing point of pure water is 0°C and the question tells us that the freezing point of the solution is -1.50°C. We can find the change in freezing point by subtracting the solution's freezing point from the pure water freezing point. ΔTf = Tf_water - Tf_solution ΔTf = 0°C - (-1.50°C) ΔTf = 1.50°C
02

Calculate the molality of the solution

To calculate the molality of the solution, we need to apply the freezing point depression formula. For water, the freezing point depression constant, Kf, is 1.86 °C·kg/mol. The formula is: ΔTf = Kf · molality Rearranging the formula to solve for molality: molality = ΔTf / Kf molality = 1.50°C / 1.86 °C·kg/mol molality = 0.8065 mol/kg
03

Find the moles of solute in the solution

Now that we have the molality of the solution, we can find the amount of solute present in moles. Remember that molality is defined as moles of solute per kilogram of solvent. Here, we have 200.0 g of water as the solvent, which can be converted to kilogram: mass_water = 200.0 g = 0.200 kg Next, we can find the moles of solute: mol_solute = molality × mass_water mol_solute = 0.8065 mol/kg × 0.200 kg mol_solute = 0.1613 mol
04

Calculate the mass of solute

Finally, we can find the mass of glycerin (C3H8O3) by using its molar mass. The molar mass can be calculated as follows: molar_mass_glycerin = 3 × 12.01 g/mol (carbon) + 8 × 1.01 g/mol (hydrogen) + 3 × 16.00 g/mol (oxygen) molar_mass_glycerin = 36.03 g/mol + 8.08 g/mol + 48.00 g/mol = 92.11 g/mol The mass of glycerin (mass_solute) can then be calculated as follows: mass_solute = mol_solute × molar_mass_glycerin mass_solute = 0.1613 mol × 92.11 g/mol mass_solute = 14.86 g Therefore, 14.86 grams of glycerin must be dissolved in 200.0 grams of water to give a solution freezing point of -1.50°C.

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