Question

A solution contains \(50.0 \mathrm{~g}\) of heptane \(\left(\mathrm{C}_{7} \mathrm{H}_{16}\right)\) and \(50.0 \mathrm{~g}\) of octane \(\left(\mathrm{C}_{8} \mathrm{H}_{18}\right)\) at \(25^{\circ} \mathrm{C}\). The vapor pressures of pure heptane and pure octane at \(25^{\circ} \mathrm{C}\) are 45.8 torr and 10.9 torr, respectively. Assuming ideal behavior, answer the following: a. What is the vapor pressure of each of the solution components in the mixture? b. What is the total pressure above the solution? c. What is the composition of the vapor in mass percent? d. Why is the composition of the vapor different from the composition of the solution?

Short answer

a. Vapor pressure of heptane & octane in the mixture: 31.43 torr & 2.97 torr. b. Total vapor pressure: 34.40 torr. c. Vapor composition: 89.11% heptane, 10.89% octane (mass percent). d. Composition differs due to different volatilities and vapor pressures.

Step-by-step solution

Calculate the Molar Mass of Heptane and Octane

To find the moles of each substance in the solution, calculate the molar mass (molecular weight) of heptane \(C_7H_{16}\) and octane \(C_8H_{18}\). The molar mass of heptane is \(7 \times 12.01 \, \mathrm{\, g/mol}\) for carbon plus \(16 \times 1.008 \, \mathrm{\, g/mol}\) for hydrogen. For octane, it is \(8 \times 12.01 \, \mathrm{\, g/mol}\) for carbon plus \(18 \times 1.008 \, \mathrm{\, g/mol}\) for hydrogen.

Calculate Moles of Heptane and Octane

Using the molar masses from Step 1, calculate the number of moles of heptane by dividing the mass of heptane by its molar mass. Do the same for octane.

Calculate the Mole Fraction of Heptane and Octane

Calculate the mole fraction of each component, which is the moles of the component divided by the total moles in the solution.

Calculate the Partial Vapor Pressure of Each Component

Use Raoult's law to find the vapor pressure of each component in the mixture. Raoult's law states that the vapor pressure of a component in a solution is equal to the mole fraction of that component multiplied by the vapor pressure of the pure component.

Calculate the Total Vapor Pressure Above the Solution

The total vapor pressure is the sum of the partial vapor pressures of heptane and octane. Add the vapor pressures from Step 4 to get the total pressure.

Determine the Mass Percent of Components in the Vapor

To find the mass percent of each component in the vapor, you must first calculate the contribution of each component to the total vapor pressure, then convert the vapor pressures back to mass using the molar masses, and finally, calculate the mass percent.

Understanding the Difference in Composition

The composition of the vapor differs from the composition of the solution because the vapor pressure of each component is different, and it depends on the substance's tendency to enter the gas phase at the given temperature, which is related to its volatility. Heptane has a higher vapor pressure and is thus more volatile than octane, and so its composition in the vapor is greater than in the liquid solution.

Key concepts

Vapor Pressure

Vapor pressure is a measure of a liquid's tendency to evaporate. It can be thought of as the force exerted by the gas molecules above a liquid when they are in a state of dynamic equilibrium with its liquid phase at a given temperature. In simpler terms, it's how much a liquid wants to become a gas at a specific temperature. The higher the vapor pressure, the more volatile the compound and the more easily it turns into vapor.

In the context of our exercise, we have two different liquids - heptane and octane, each with its own distinct vapor pressure. The concept of vapor pressure helps explain why these two liquids behave differently in a mixture, and it's a critical element of Raoult's Law, which is used to determine the vapor pressures of the components in a solution.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the number of moles of a particular substance divided by the total number of moles of all substances present in the mixture. Represented by the symbol X, it is a dimensionless quantity and is used because it does not vary with temperature or pressure, unlike other concentration measures.

To use Raoult's Law effectively, as we do in our exercise, mole fractions are essential. They allow us to see what percentage of the total vapor pressure each component contributes. The mole fraction of heptane or octane in the solution directly influences the partial vapor pressure of each, and therefore, the total vapor pressure of the solution.

Volatile Compounds

Volatile compounds are substances that have a high vapor pressure at a given temperature, which leads to a large number of molecules escaping from the liquid or solid form into the gas phase. Generally, volatility depends on the material's intermolecular forces; weaker forces make for a more volatile compound.

The exercise we're examining involves heptane and octane, two hydrocarbons. Due to its higher vapor pressure, we can deduce that, at 25°C, heptane is more volatile than octane. This greater volatility results in a higher concentration of heptane in the vapor phase compared to the liquid phase, as indicated by Raoult's Law.

Partial Vapor Pressure

Partial vapor pressure is the pressure exerted by a single component of a mixture of liquids or gases. It's how we quantify the contribution each component in a mixture makes to the total vapor pressure. When dealing with solutions, we regard the partial vapor pressure of each component to be directly proportional to its mole fraction, a relationship captured by Raoult's Law.

In the exercise, once we've calculated the mole fractions, we then find the partial vapor pressures for heptane and octane using Raoult's Law. These pressures play a fundamental role in determining the composition of the vapor phase above the liquid solution.

Ideal Solution Behavior

An ideal solution is a hypothetical solution where the intermolecular forces between dissimilar particles are equal to those between similar particles. For ideal solutions, Raoult's Law holds true, which means the partial vapor pressure of each component of a solution is directly proportional to its mole fraction, and the intermolecular interactions don't alter this relationship.

The concept of ideal solution behavior is crucial for solving the exercise as it allows us to apply Raoult's Law without needing to consider deviations that occur in real solutions where interactions can lead to higher or lower vapor pressures than predicted. The assumption of ideal behavior simplifies calculations and offers a baseline for understanding how real solutions deviate from this ideal.