Question

The Rast method uses camphor \(\left(\mathrm{C}_{10} \mathrm{H}_{16} \mathrm{O}\right)\) as a solvent for determining the molar mass of a compound. When \(2.50 \mathrm{~g}\) of cortisone acetate are dissolved in \(50.00 \mathrm{~g}\) of camphor \(\left(k_{\mathrm{f}}=40.0^{\circ} \mathrm{C} / \mathrm{m}\right)\), the freezing point of the mixture is \(173.44^{\circ} \mathrm{C}\) that of pure camphor is \(178.40^{\circ} \mathrm{C}\). What is the molar mass of cortisone acetate?

Short answer

Answer: The molar mass of cortisone acetate in the given solution is approximately \(403.23\, \mathrm{g/mol}\).

Step-by-step solution

Calculate the change in freezing point

First, we need to find the change in freezing point, which is given by the difference between the freezing point of pure camphor and that of the mixture: \(\Delta T_f = T_{f, \text{pure}} - T_{f, \text{mixture}}\) Substitute the given values: \(\Delta T_f = 178.40^{\circ}\mathrm{C} - 173.44^{\circ}\mathrm{C} = 4.96^{\circ}\mathrm{C}\)

Find the molality using the freezing-point depression formula

Now that we have the change in freezing point, we can use the freezing-point depression formula to find the molality: \(\Delta T_f = K_f \cdot m\) Rearrange the formula to find the molality: \(m = \frac{\Delta T_f}{K_f}\) Substitute the given values: \(m = \frac{4.96^{\circ}\mathrm{C}}{40.0^{\circ}\mathrm{C}/\mathrm{m}} = 0.124\, \mathrm{mol/kg}\)

Calculate the molar mass of cortisone acetate

We can relate the molality and mass of cortisone acetate as follows: \(m = \frac{\text{moles of cortisone acetate}}{\text{mass of camphor (kg)}}\) Rearrange the formula to find moles of cortisone acetate: \(\text{moles of cortisone acetate} = m \cdot \text{mass of camphor (kg)}\) Substitute the given values: \(\text{moles of cortisone acetate} = 0.124\, \mathrm{mol/kg} \cdot 0.050\, \mathrm{kg} = 0.0062\, \mathrm{mol}\) Now we can find the molar mass of cortisone acetate using moles and mass: \(\text{molar mass} = \frac{\text{mass of cortisone acetate}}{\text{moles of cortisone acetate}}\) Substitute the given values: \(\text{molar mass} = \frac{2.50\, \mathrm{g}}{0.0062\, \mathrm{mol}} = 403.23\, \mathrm{g/mol}\) The molar mass of cortisone acetate is approximately \(403.23\, \mathrm{g/mol}\).

Key concepts

Molality

Understanding the concept of molality is fundamental when dealing with solutions and their properties. Molality is a measure of the concentration of a solute in a solution. In chemistry, it is defined as the number of moles of solute per kilogram of solvent. Unlike molarity, which is affected by changes in volume due to temperature variations, molality remains constant because it depends on the mass of the solvent, not its volume.

Molality is mathematically expressed as:
\[ m = \frac{\text{moles of solute}}{\text{mass of solvent in kilograms}} \]
This concept is particularly useful when studying colligative properties, like freezing point depression, because these properties rely on the number of particles in a solution rather than their identity. For students to grasp this topic, it's helpful to consider real-world examples, such as the addition of salt to ice on roads in winter, which lowers the freezing point of water due to the salt's solute particles interfering with the formation of ice.

Solvent Properties

Solvent properties are crucial when it comes to understanding the behavior of solutions and the calculation of colligative properties, such as freezing point depression. Every solvent has its unique physical properties that can affect the outcome of a chemical reaction or a solution's characteristics. In the context of the exercise, camphor is used as a solvent, and it has its own specific freezing point and a constant known as the freezing point depression constant (\(K_f\)).

When a non-volatile solute is dissolved in a solvent, it alters the solvent's freezing point, which is a key principle behind freezing point depression. The pure solvent has a higher freezing point compared to the solution because the presence of solute particles disrupts the formation of a solid structure, effectively requiring a lower temperature to achieve the same structure. This concept can be compared to adding an impurity to a perfect crystal lattice, where the impurity prevents the crystal from forming its regular structure at the usual temperature.

Molar Mass Calculation

Molar mass is the mass of one mole of a substance and is expressed in grams per mole (g/mol). It is a vital concept for stoichiometry, as it bridges the gap between the macroscopic world we observe and the microscopic world of atoms and molecules.

To calculate the molar mass of a compound, you must know the number of moles and the mass of the solute. The relationship is given by:
\[ \text{molar mass} = \frac{\text{mass of solute}}{\text{number of moles}} \]
In the exercise discussed, you would first determine the number of moles of cortisone acetate using the molality and the mass of the solvent (camphor). Once you have the number of moles, you can use the given mass of the cortisone acetate to calculate its molar mass. Being able to calculate molar mass is essential for students, not only for solving problems like these but also for being able to predict and understand the outcomes of reactions based on the amounts of substances involved.