Question

The vapor pressure of benzene at \(75^{\circ} \mathrm{C}\) is 640 torr. \(\mathrm{A}\) solution of \(3.68 \mathrm{~g}\) of a solute in \(53.0 \mathrm{~g}\) benzene has a vapor pressure of 615 torr. Calculate the molecular weight of the solute. (MW of benzene \(=78.0 .)\)

Short answer

The molecular weight of the solute is approximately 137.3 g/mol.

Step-by-step solution

Calculate the moles of benzene

To do this, we first need to find the moles of benzene: moles of benzene = mass / molecular weight moles of benzene = \( \frac{53.0 g}{78.0 g/mol} \) moles of benzene = 0.679 moles

Apply Raoult's law for benzene in the solution

Raoult's law states that: Partial pressure of benzene in the solution = Mole fraction of benzene × Vapor pressure of pure benzene 615 torr = Mole fraction of benzene × 640 torr Mole fraction of benzene = \( \frac{615 torr}{640 torr} \) Mole fraction of benzene ≈ 0.9609

Find the mole fraction of the solute

Let's denote the mole fraction of solute as "x_solute". Since there are only two components in the solution, the sum of their mole fractions should be equal to 1. Mole fraction of solute = x_solute = 1 - Mole fraction of benzene x_solute = 1 - 0.9609 x_solute ≈ 0.0391

Calculate the moles of solute

Now that we have the mole fraction of the solute, it can be used to find the moles of solute: Mole fraction of solute = \( \frac{moles\: of\: solute}{moles\: of\: solute + moles\: of\: benzene} \) Rearrange the equation to find moles of solute: moles of solute = Mole fraction of solute × (moles of solute + moles of benzene) - moles of solute moles of solute ≈ \( 0.0391 × (moles\: of\: solute + 0.679) - moles\: of\: solute \) moles of solute ≈ 0.0268 moles

Calculate the molecular weight of the solute

Now we can use the moles of solute and its mass to find the molecular weight: molecular weight of solute = mass of solute / moles of solute molecular weight of solute = \( \frac{3.68g}{0.0268\: moles} \) molecular weight of solute ≈ 137.3 g/mol The molecular weight of the solute is approximately 137.3 g/mol.

Key concepts

Understanding Raoult's Law

Raoult's Law is a principle in chemistry that describes the behavior of a solvent in a solution. It states that the vapor pressure of a solvent above a solution is equal to the vapor pressure of the pure solvent multiplied by its mole fraction in the solution. This concept is particularly crucial when dealing with solutions of non-volatile solutes since it helps to understand how the presence of a solute affects the overall vapor pressure.

For instance, when a solute is dissolved in a solvent, the vapor pressure of the solvent is typically decreased. This property is often used to determine the molecular weight of an unknown solute by measuring the vapor pressure of the solution compared to that of the pure solvent, as seen in our example with benzene.

Deciphering Vapor Pressure

Vapor pressure is an essential concept in chemistry and is defined as the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases at a given temperature in a closed system. In simpler terms, it is the force per unit area that vapor molecules exert due to their motion above the liquid or solid form of a substance. Factors affecting vapor pressure include temperature and the nature of the liquid.

Vapor pressure plays a vital role in Raoult's law calculations. A higher vapor pressure indicates a tendency to evaporate more readily. Understanding this concept helps in explaining why the presence of a non-volatile solute reduces the vapor pressure of the solvent in a solution.

Calculating Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture or a solution. It is calculated as the ratio of the number of moles of one component to the total number of moles of all components. Mathematically, it is represented by:
\[ \text{Mole Fraction} = \frac{\text{moles of component}}{\text{total moles in solution}} \]

The mole fraction is unitless and has no dimensions, which makes it a convenient way of relating the proportions of components in a mixture. When working with solutions, the sum of mole fractions of all the components is always equal to one. This relationship makes it possible to use mole fraction to find the amount of a solute in a mixture, given the mole fraction of the other components, which we used in the example problem to find the moles of the solute.

Essentials of Solutions in Chemistry

Solutions in chemistry are homogenous mixtures composed of two or more substances. Typically, a solution is made up of a solute that gets dissolved in a solvent. The solute can be present in any phase—solid, liquid, or gas—but it must be distributed uniformly within the solvent. Solutions play an important role in many chemical processes and applications.

Understanding solutions is fundamental in calculating molecular weights, understanding solubility, and exploring properties like boiling point elevation or freezing point depression. Each component in a solution contributes to these properties collectively, which can be analyzed quantitatively using various laws, including Raoult's law, as demonstrated in our exercise.