Question

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

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

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

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

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

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.

Key concepts

Understanding Molality

Molality is a concept used to express the concentration of a solution. It measures the number of moles of solute per kilogram of solvent. Unlike molarity, which is affected by temperature changes because it depends on the volume of the solvent, molality remains constant because it is based on mass. This makes it particularly useful in scenarios involving temperature changes, like freezing point depression.
To calculate molality (\( m \)), you'll use the formula:

• \( m = \frac{\text{moles of solute}}{\text{kilograms of solvent}} \)

One common application is in determining how much a solute changes freezing or boiling points of a solvent. For instance, in freezing point depression, the greater the molality, the more the freezing point is lowered. To find this out, you'll often rearrange the freezing point depression formula, \( \Delta T_f = K_f \, \times \, m \), solving for molality.

Calculating Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It's calculated by adding up the atomic masses of all atoms in a chemical's formula. This is fundamental in chemistry for converting between moles and grams.
Let's break down the molar mass calculation for glycerin (\( \text{C}_3\text{H}_8\text{O}_3 \)):

• Carbon: 3 atoms \, \times \, 12.01 \, \text{g/mol} = 36.03 \, \text{g/mol}
• Hydrogen: 8 atoms \, \times \, 1.01 \, \text{g/mol} = 8.08 \, \text{g/mol}
• Oxygen: 3 atoms \, \times \, 16.00 \, \text{g/mol} = 48.00 \, \text{g/mol}
• Total: 36.03 \, + \, 8.08 \, + \, 48.00 = 92.11 \, \text{g/mol}

Knowing the molar mass helps in determining how much of a chemical is needed to reach a particular molality or concentration in solution.

Exploring Nonelectrolytes

A nonelectrolyte is a substance that, when dissolved in water, does not produce ions. This means it does not conduct electricity. Glycerin is an example of a nonelectrolyte.
In contrast to electrolytes, which disassociate into ions (like salt turning into sodium and chloride in water), nonelectrolytes dissolve without changing their molecular structure. This can be important when calculating the change in properties, such as boiling point or freezing point, since the number of particles in solution influences these changes more directly than the charge or nature of the solute.
Knowing whether a solute is a nonelectrolyte or an electrolyte can help you predict how its presence will affect the physical properties of a solution, which is vital in experiments involving freezing point depression.

Defining Solute

A solute is any substance that is dissolved in a solvent to make a solution. In the example of glycerin in water, glycerin is the solute, while water is the solvent.
Solutes can vary greatly in their characteristics, like being solids, liquids, or gases, or being acids, bases, or neutral molecules. The behavior and properties of a solute when dissolved—such as whether it forms ions (electrolytes) or stays as whole molecules (nonelectrolytes)—affect properties of the resulting solution.
Understanding the properties of a solute is crucial in areas such as chemical reactions, where the solute can change the concentration of solutions, influence the rate of reactions, or modify the physical properties of the solution like its freezing and boiling points. This understanding aids in predicting and controlling the outcome in various chemical processes.