Question

A solution contains \(3.75 \mathrm{~g}\) of a nonvolatile pure hydrocarbon in \(95 \mathrm{~g}\) acetone. The boiling points of pure acetone and the solution are \(55.95^{\circ} \mathrm{C}\) and \(56.50^{\circ} \mathrm{C}\), respectively. The molal boilingpoint constant of acetone is \(1.71^{\circ} \mathrm{C} \cdot \mathrm{kg} / \mathrm{mol}\). What is the molar mass of the hydrocarbon?

Short answer

The molar mass of the hydrocarbon is approximately 122.56 g/mol.

Step-by-step solution

To calculate the change in boiling point (ΔTb), subtract the boiling point of the pure solvent from the boiling point of the solution: ΔTb = Boiling Point of Solution - Boiling Point of Pure Acetone ΔTb = \(56.50^{\circ} \mathrm{C}\) - \(55.95^{\circ} \mathrm{C}\) ΔTb = \(0.55^{\circ} \mathrm{C}\) #Step 2: Calculate Molality (m) Using Boiling Point Constant (Kb)#

We know that ΔTb = Kb * molality. Therefore, we can find the molality (m) by dividing the change in boiling point by the boiling point constant: molality = ΔTb / Kb molality = \(0.55^{\circ} \mathrm{C}\) / \(1.71^{\circ} \mathrm{C} \cdot \mathrm{kg} / \mathrm{mol}\) molality = 0.322 mol/kg #Step 3: Calculate Moles of Hydrocarbon (n)#

Now that we have the molality, we can calculate the number of moles (n) of hydrocarbon in 1 kg of acetone using the formula: n = molality * mass of solvent (in kg) n = 0.322 mol/kg * (95 g / 1000 g/kg) # Convert mass of acetone to kg n = 0.03061 mol #Step 4: Calculate Molar Mass of Hydrocarbon (M)#

Finally, we can find the molar mass (M) of the hydrocarbon by dividing the mass of hydrocarbon (given) by the number of moles (calculated in Step 3): M = mass of hydrocarbon / moles of hydrocarbon M = 3.75 g / 0.03061 mol M ≈ 122.56 g/mol The molar mass of the hydrocarbon is approximately 122.56 g/mol.

Key concepts

Molality

Molality is a way to express the concentration of a solution and is especially useful when dealing with colligative properties such as boiling point elevation. Unlike molarity, which depends on volume, molality (m) is based on the mass of the solvent, making it temperature-independent. This is key for problems involving temperature changes.
To define it precisely, molality is the number of moles of solute per kilogram of solvent. The formula for calculating molality is:

• \( m = \frac{moles \ of \ solute}{mass \ of \ solvent \ (in \ kg)} \)

Molality is especially useful in boiling point elevation problems because it helps account for how solute particles affect the boiling temperature. By maintaining a constant mass of the solvent, calculations involving temperature alterations like boiling and freezing points are more straightforward. In this exercise, we used molality to relate the change in the boiling point to the amount of solute present in acetone.

Molar Mass Calculation

Calculating the molar mass of a substance is crucial in identifying unknown compounds and understanding their properties. To find the molar mass, you must have the mass of the substance in question and be able to calculate the number of moles present in that mass.
The formula for molar mass (M) is:

• \( M = \frac{mass \ of \ substance \ (in \ g)}{moles \ of \ substance} \)

In this exercise, we determined the molar mass of a hydrocarbon by dividing the given mass of the hydrocarbon (3.75 g) by the number of moles calculated using molality. This process involves determining how much of the solute (in this case, the hydrocarbon) is needed to achieve the measured change in boiling point, allowing for precise calculations of molar mass in solutions.

Colligative Properties

Colligative properties are characteristics of solutions that depend on the number of solute particles rather than the type of particles. These properties include boiling point elevation, freezing point depression, vapor pressure lowering, and osmotic pressure.
Boiling point elevation, one of the central concepts here, occurs because solute particles disrupt the solvent's ability to vaporize. This effect is proportional to the number of solute particles in the solution.
The change in boiling point (ΔTb) is calculated using the equation:

• \( \Delta Tb = Kb \times molality \)

Where \( Kb \) is the ebullioscopic constant of the solvent, which is unique for each solvent. This relationship underscores the idea that the greater the number of solute particles, the higher the boiling point of the solvent. By understanding colligative properties, we recognize how different solute concentrations impact the thermodynamic properties of a solvent, as seen in this acetone solution example.