Question

A solution of methanol and water has a mole fraction of water of 0.312 and a total vapor pressure of 211 torr at \(39.9^{\circ} \mathrm{C}\) . The vapor pressures of pure methanol and pure water at this temperature are 256 torr and 55.3 torr, respectively. Is the solution ideal? If not, what can you say about the relative strengths of the solute-solvent interactions compared to the solute-solute and solvent-solvent interactions?

Short answer

The solution deviates from ideality; the relative interaction strengths can be determined by comparing the actual vapor pressure to that predicted by Raoult's Law.

Step-by-step solution

Understand and Express Mole Fraction

The mole fraction (\( X_i \)) of a component in a mixture is the ratio of the number of moles of that component to the total number of moles of all components in the mixture. Given that the mole fraction of water is 0.312, the mole fraction of methanol can be calculated as: \( X_{methanol} = 1 - X_{water} = 1 - 0.312 = 0.688 \) since the sum of mole fractions in a mixture is equal to 1.

Use Raoult's Law for Ideal Solutions

Raoult's law states that the partial pressure of each component of an ideal solution is equal to the vapor pressure of the pure component multiplied by its mole fraction in the solution. This means for water (\( P_{H2O} \frac{55.3 torr \times 0.312}\) and for methanol (\( P_{CH3OH} = 256 torr \times 0.688 \)

Calculate Expected Total Vapor Pressure of Ideal Solution

For an ideal solution, the total vapor pressure is the sum of the partial pressures of the components. Thus, \( P_{total}^{ideal} = P_{H2O} + P_{CH3OH} \) which can be calculated using the previously found partial pressures.

Compare with Actual Vapor Pressure

The given total vapor pressure of the solution is 211 torr. Now, compare this value with the calculated total vapor pressure for an ideal solution from the previous step. If the values match, the solution is ideal. If not, differences indicate deviations from ideality.

Determine the Type of Deviation from Ideality

If the actual vapor pressure is higher than the expected for an ideal solution, the solution exhibits positive deviation, indicating weaker intermolecular forces between unlike molecules than among like molecules. Conversely, if the actual vapor pressure is lower, the solution exhibits a negative deviation, indicating stronger intermolecular forces between unlike molecules than among like molecules.

Key concepts

Mole Fraction

The mole fraction is a way to express the composition of a mixture or solution by indicating what fraction of the whole mixture consists of a particular substance. It is written as \( X_i \) and is the number of moles of a substance divided by the total moles of all substances present.

For example, if you have a solution that contains water and methanol, and we know the mole fraction of water is 0.312, this tells us that water accounts for 31.2% of the total moles of the solution. Mathematically, if you have two components, A and B, and you know the mole fraction of A, you can easily calculate that of B as \( X_B = 1 - X_A \), because the sum of the mole fractions of all components in a solution always equals 1.

Vapor Pressure

Vapor pressure is the pressure exerted by the vapor that is in equilibrium with its liquid at a given temperature. It's a measure of a liquid's tendency to evaporate and become gas.

As the temperature increases, molecules gain energy and are more likely to escape into the vapor phase, hence the vapor pressure increases. The vapor pressure of a pure substance is constant at a fixed temperature. However, when a substance is mixed with others, as in a solution, its vapor pressure can change depending on the composition of the mixture.

Intermolecular Forces

Intermolecular forces (IMFs) are the forces of attraction or repulsion between particles (molecules, atoms, or ions) of a substance. They are weaker than the chemical bonds within molecules, yet they are crucial for determining the physical properties of substances.

Types of Intermolecular Forces

• Dipole-dipole interactions occur between polar molecules.
• London dispersion forces can exist between all molecules, especially non-polar ones.
• Hydrogen bonds are the strongest IMF, occurring between hydrogen and electronegative atoms like oxygen or nitrogen.

These forces influence the behavior of substances in a solution, including their vapor pressures.

Partial Pressure

Partial pressure refers to the contribution each gas in a mixture of gases makes to the total pressure. It's important when dealing with solutions because, in a mixture of volatile components, each contributes to the total vapor pressure.

According to Raoult's Law, the partial pressure of each component of an ideal solution is the product of the mole fraction of that component and the vapor pressure of the pure component. As an example, for a mixture of water and methanol, we would calculate the partial pressures of both and sum them to figure out the expected total vapor pressure if the solution were ideal.

Solution Ideality

Solution ideality is a condition wherein the solution behaves according to Raoult's Law. This means that the solution's physical properties can be directly calculated from the properties of the pure components and their proportions in the mixture.

For a solution to be ideal, the intermolecular forces between unlike molecules must be nearly the same as those between like molecules. Deviations from this behavior suggest that the actual interactions are stronger or weaker than those predicted by Raoult's Law, indicating that the solution is non-ideal. The exercise provided is an example of how to investigate whether a real solution is ideal or not based on comparing computed vapor pressures using Raoult's Law with actual measurements.