Problem 113 - Chemistry An Atoms First Approach

Question

The normal boiling point of methanol is \(64.7^{\circ} \mathrm{C} .\) A solution containing a nonvolatile solute dissolved in methanol has a vapor pressure of 556.0 torr at \(64.7^{\circ} \mathrm{C}\). What is the mole fraction of methanol in this solution?

Step-by-Step Solution

Answer

The mole fraction of methanol in the solution can be found using Raoult's law: \(P_{solution} = P_{methanol} \cdot x_{methanol}\). Given the vapor pressure of the solution (556.0 torr) and the vapor pressure of pure methanol (760 torr) at 64.7°C, we can solve for the mole fraction of methanol: \(x_{methanol} = \frac{556.0}{760} \approx 0.7316\).

Step 1: Write down Raoult's law formula

Raoult's law states that the partial vapor pressure of a component in a mixture is equal to the product of its mole fraction and its vapor pressure in the pure state. The formula for Raoult's law is given by: \(P_{solution} = P_{methanol} \cdot x_{methanol}\) where: - \(P_{solution}\) is the vapor pressure of the solution - \(P_{methanol}\) is the vapor pressure of pure methanol - \(x_{methanol}\) is the mole fraction of methanol in the solution

Step 2: Convert the given pressure to the appropriate unit

The vapor pressure of the solution is given in torr. Raoult's law requires vapor pressures in the same units. We will keep the torr unit because it is commonly used for vapor pressures. The vapor pressure of the solution is given as 556.0 torr.

Step 3: Find the vapor pressure of pure methanol at the given temperature

We are given the normal boiling point of methanol, which is 64.7°C. At the normal boiling point, the vapor pressure of a substance is equal to the atmospheric pressure. The standard atmospheric pressure is 760 torr. Therefore, the vapor pressure of pure methanol at 64.7°C is 760 torr.

Step 4: Substitute the known values into Raoult's law formula and solve for the mole fraction of methanol

Now that we have the vapor pressure of the solution (556.0 torr) and the vapor pressure of pure methanol (760 torr), we can substitute these values into Raoult's law formula to solve for the mole fraction of methanol: \(556.0 = 760 \cdot x_{methanol}\) To find the mole fraction of methanol, \(x_{methanol}\), we will divide both sides of the equation by 760: \(x_{methanol} = \frac{556.0}{760}\)

Step 5: Calculate the mole fraction of methanol

Finally, calculate the mole fraction of methanol by dividing 556.0 by 760: \(x_{methanol} = \frac{556.0}{760} \approx 0.7316\) The mole fraction of methanol in the solution is approximately 0.7316.

Key Concepts

Mole Fraction

When studying solutions, the mole fraction is a vital concept to comprehend. It represents the proportion of moles of a particular component compared to the total moles of all components in the mixture. Mathematically, it is expressed as:

\( x_i = \frac{n_i}{_{total}} \)

where \( x_i \) is the mole fraction of component i, \( n_i \) is the number of moles of component i, and \( _{total} \) is the total number of moles of all components present. The sum of the mole fractions of all components in a mixture is always equal to 1. Understanding mole fractions is crucial when utilizing Raoult's law to determine vapor pressures in mixtures, as it can directly affect the resulting calculations and conclusions about the composition of a solution.

Normal Boiling Point

The normal boiling point of a liquid is the temperature at which its vapor pressure equals the atmospheric pressure at sea level, generally recognized as 760 torr or 1 atmosphere. This concept is especially important because it can be used as a reference point for determining the vapor pressures of solutions at this temperature.

At the normal boiling point, the liquid starts to boil and turns into a gas. If a nonvolatile solute is added to the liquid, this will typically raise the boiling point, a phenomenon known as boiling point elevation. Understanding the normal boiling point is essential for interpreting how solutes affect the physical properties of solvents and is a key piece of information when employing Raoult's law.

Nonvolatile Solute

A nonvolatile solute is a substance that has a negligible vapor pressure compared to the solvent at a given temperature. Due to its insignificant ability to vaporize, it doesn't appreciably contribute to the vapor pressure of the solution when dissolved.

Introducing a nonvolatile solute into a solvent disrupts the intermolecular forces at the surface, which reduces the number of solvent molecules able to escape into the vapor phase. This phenomenon leads to a decrease in the vapor pressure of the solution, a consequence central to Raoult's law. It also results in boiling point elevation and freezing point depression, thus altering the solvent’s normal boiling and freezing points. The addition of a nonvolatile solute is of fundamental importance in solutions chemistry and can significantly impact colligative properties.

Partial Vapor Pressure

Partial vapor pressure is the pressure exerted by a particular component of a mixture in the vapor phase when that component is at equilibrium with its liquid phase. In a solution, each substance contributes to the total vapor pressure based on its concentration and intrinsic vapor pressure.

According to Raoult's law, the partial vapor pressure of a component (A) in an ideal mixture is directly proportional to its mole fraction in the solution, represented mathematically:

\( P_A = x_A \cdot P^0_A \)

where \( P_A \) is the partial vapor pressure of component A, \( x_A \) is the mole fraction of A, and \( P^0_A \) is the vapor pressure of pure A. In a mixture with multiple components, each component contributes a 'part' of the total vapor pressure, and understanding this helps explain how changes in composition affect the overall vapor pressure and the properties of the mixture.