Question

A 2.00 -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 497 g/mol.

Step-by-step solution

Understand the Boiling-Point Elevation formula

Boiling-point elevation is the phenomenon that occurs when a non-volatile solute is dissolved in a solvent, resulting in an increase in the boiling point of the solvent. The relationship is described by the following formula: ΔT = Kₘ ⋅ molality where ΔT is the boiling point elevation, Kₘ is the boiling-point constant for the solvent, and the molality is the concentration of the solute in the solution.

Calculate the boiling point elevation

First, let's calculate the boiling point elevation (ΔT). We are given the boiling point of the pure solvent (carbon tetrachloride) and the boiling point of the solution: ΔT = Boiling point of the solution - Boiling point of pure solvent ΔT = ${77.85}^{\circ }\mathrm{C}-{76.50}^{\circ }\mathrm{C}$ ΔT = 1.35 °C

Rearrange the formula to calculate molality

Now we can rearrange the boiling-point elevation formula to solve for the molality of the biomolecule in the solution: molality = ΔT / Kₘ

Calculate the molality of the biomolecule

The boiling-point constant for carbon tetrachloride $Kₘ={5.03}^{\circ }\mathrm{C}\cdot \mathrm{kg}/\mathrm{mol}$ and ΔT = 1.35 °C. So, let's plug the values: molality = (1.35 °C) / ${5.03}^{\circ }\mathrm{C}\cdot \mathrm{kg}/\mathrm{mol}$ molality = 0.2684 mol / kg

Calculate the amount of moles of the biomolecule

Molality is defined as the number of moles of solute per kilogram of solvent. We know the mass of the solvent and the molality of the biomolecule. Using this information, we can calculate the number of moles of the biomolecule present in the solution: moles = molality × mass of solvent in kg moles = (0.2684 mol / kg) × (15.0 g CCl₄ × (1 kg / 1000 g)) moles = 0.004026 mol

Calculate the molar mass of the biomolecule

We are given the mass of the biomolecule (2.00 g) and we have calculated the number of moles present in the solution. We can now calculate the molar mass of the biomolecule: molar mass = mass of biomolecule / moles of biomolecule molar mass = (2.00 g) / (0.004026 mol) molar mass ≈ 497 g/mol The molar mass of the biomolecule is approximately 497 g/mol.

Key concepts

Molar Mass Calculation

Calculating the molar mass of a substance is an essential skill in chemistry. Molar mass allows chemists to relate mass in grams to the amount of substance in moles. To find molar mass, remember this simple formula:

• Molar Mass (g/mol) = Mass of the Substance (g) / Moles of the Substance (mol)

Steps to Calculate:
- First, you need the mass of the substance, which in our example is the biomolecule weighing 2.00 g.
- Next, determine the amount in moles, which requires understanding the solute’s concentration in the solvent, often done through colligative properties.
By dividing, you’ll arrive at the molar mass, which here is approximately 497 g/mol.
Molar mass is a fundamental concept because it bridges the gap between atomic-level interactions and measurable quantities, allowing scientists to predict and balance chemical reactions.

Colligative Properties

Colligative properties depend on the number of solute particles in a solvent rather than the identity of the solute. These properties are crucial for understanding how solutions behave. Among them are boiling-point elevation, freezing-point depression, vapor pressure lowering, and osmotic pressure.

In the context of boiling-point elevation:

• When a non-volatile solute, like our biomolecule, is added to a solvent such as carbon tetrachloride, it causes the boiling point to increase.
• This increase is quantified by the formula: ΔT = Kₘ ⋅ molality, where ΔT represents the boiling point elevation and Kₘ is the boiling-point elevation constant.
• Molality, a measure of solute concentration, changes the physical properties of the solvent.

Colligative properties are powerful tools to determine molar mass and understand solution compositions. When dealing with solutions, these properties illustrate how solute particles, regardless of type, influence overall behavior.

Solution Chemistry

Solution chemistry is the study of how substances dissolve and interact within a medium, usually a liquid solvent. A solution is composed of a solute — the substance being dissolved — and a solvent — the substance doing the dissolving.

Key principles include:

• Solubility: The amount of solute that can be dissolved in a solvent at a given temperature.
• Concentration: Represents the quantity of solute present in a certain amount of solution, expressed commonly in molarity or molality.
• Interactions: When a solute dissolves, it may dissociate into ions or stay as molecules, affecting how properties manifest.

Understanding these components is vital to predicting the properties of mixtures. Solution chemistry is foundational to the study of thermodynamics and kinetics in reactions. It helps chemists control conditions to optimize reaction outcomes and duplicate results reliably.