Question

A \(2.00-\mathrm{g}\) sample of a large biomolecule was dissolved in \(15.0 \mathrm{~g}\) carbon tetrachloride. The boiling point of this solution was determined to be \(77.85^{\circ} \mathrm{C}\). Calculate the molar mass of the biomolecule. For carbon tetrachloride, the boiling-point constant is \(5.03^{\circ} \mathrm{C} \cdot \mathrm{kg} / \mathrm{mol}\), and the boiling point of pure carbon tetrachloride is \(76.50^{\circ} \mathrm{C}\).

Short answer

The molar mass of the biomolecule is approximately 492.95 g/mol.

Step-by-step solution

Calculate the change in boiling point

First, we need to calculate the change in boiling point of the solution compared to pure carbon tetrachloride. The difference can be obtained by subtracting the boiling point of pure carbon tetrachloride from the boiling point of the solution: ΔT = T_solution - T_pure ΔT = \(77.85^{\circ} \mathrm{C}\) - \(76.50^{\circ} \mathrm{C}\) ΔT = \(1.35^{\circ} \mathrm{C}\)

Calculate the molality

Next, we need to calculate the molality of the solution. We can do this using the formula: molality = ΔT / Kb Where molality is the molality of the solution, ΔT is the change in boiling point, and Kb is the boiling-point constant given for carbon tetrachloride. Now, substituting the values we have: molality = \(1.35^{\circ} \mathrm{C}\) / \(5.03^{\circ} \mathrm{C} \cdot \mathrm{kg} / \mathrm{mol}\) molality = 0.26839 mol/kg

Calculate the molar mass of the biomolecule

Finally, we can find the molar mass of the biomolecule by using the molality and the mass of the biomolecule in grams. The formula for molar mass (MM) is: MM = (mass of biomolecule) / (molality × mass of solvent in kg) First, let's convert the mass of carbon tetrachloride solvent from grams to kilograms: mass of solvent = 15.0 g = 0.015 kg Now, plug in the values to find the molar mass of the biomolecule: MM = (2.00 g) / (0.26839 mol/kg × 0.015 kg) MM = 492.95 g/mol So,the molar mass of the biomolecule is approximately 492.95 g/mol.

Key concepts

Molar Mass Calculation

Understanding the calculation of molar mass is key to this kind of exercise. Molar mass is the mass of a given substance (chemical element or compound) divided by the amount of substance in moles. It's typically expressed in units of grams per mole (g/mol).
The idea is to find out how much one mole of a substance weighs. This exercise involves finding the molar mass of a large biomolecule by using collative properties of solutions, specifically boiling point elevation.
First, we must find out the change in the boiling point of the solution, then use this to determine the molality. With the known mass of the solute (our biomolecule) and the solvent (carbon tetrachloride), the molar mass can be computed by rearranging and solving the formula for molality.

Chemical Solution

A chemical solution is a homogeneous mixture composed of two or more substances. The solvent is the substance in the larger amount that dissolves the other substances (solute).
In this exercise, carbon tetrachloride acts as the solvent, while the biomolecule is the solute. When dissolved, the solution displays its properties, such as its boiling point, which can differ from its pure components.
Understanding solutions is critical because it helps predict how different components in a mixture act together. This exercise illustrates how the addition of a solute can alter a solution's boiling point, which is foundational for determining the solute's molar mass.

Molality

Molality is a concentration unit of a solution expressed as the number of moles of solute per kilogram of solvent. Unlike molarity, which depends on the volume of solution, molality considers the mass of the solvent, which doesn't change with temperature.
In this problem, molality is used to find the change in boiling point when the biomolecule is dissolved in carbon tetrachloride. The formula used is \( \text{molality} = \frac{\Delta T}{K_b} \) where \( \Delta T \) is the change in boiling point and \( K_b \) is the boiling point elevation constant. Calculate molality as one of the steps toward determining the molecular mass of the solute.

Colligative Properties

Colligative properties are properties of solutions that depend on the ratio of the number of solute particles to solvent molecules, and not on the nature of the chemical species present. Examples include boiling point elevation, freezing point depression, and osmotic pressure.
The exercise utilizes the concept of boiling point elevation, a colligative property that indicates that the boiling point of a solvent increases when a solute is added.
This increase is proportional to the concentration of the solution as measured by molality. Boiling point elevation helps gather data needed to calculate the molar mass of obscure compounds like the biomolecule in this example, thus showing the practical application of colligative properties in determining molecular details.