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 torr at \(64.7^{\circ} \mathrm{C} .\) What is the mole fraction of methanol in this solution?

Short answer

The mole fraction of methanol in the solution can be found using Raoult's Law, which states that the vapor pressure of the solution equals the mole fraction of the solvent times the vapor pressure of pure solvent. At the normal boiling point (64.7°C), the vapor pressure of pure methanol is assumed to be 760 torr. Given the vapor pressure of the solution (556 torr) and the vapor pressure of pure methanol (760 torr), the mole fraction of methanol in the solution can be calculated as follows: \(X_{methanol} = \frac{556}{760} \approx 0.732\). Thus, the mole fraction of methanol in this solution is approximately 0.732.

Step-by-step solution

Find the vapor pressure of pure methanol at its normal boiling point

At the normal boiling point of a liquid, its vapor pressure is equal to the atmospheric pressure. As we are not given the actual atmospheric pressure, we will assume it to be 760 torr. Thus, at 64.7°C, the vapor pressure of pure methanol is 760 torr.

Apply Raoult's Law to relate the components' properties

Raoult's Law states that the vapor pressure of a solution is equal to the mole fraction of the solvent times the vapor pressure of pure solvent. In this case, solvent is methanol, and solute is a nonvolatile substance. Let \(P_{solution}\) be the vapor pressure of the solution, \(P_{methanol}\) be the vapor pressure of pure methanol, and \(X_{methanol}\) be the mole fraction of methanol in the solution. According to Raoult's Law, we have: \[P_{solution} = X_{methanol} \times P_{methanol}\]

Solve for the mole fraction of methanol in the solution

We are given the vapor pressure of the solution, \(P_{solution} = 556 \, \text{torr}\), and we found the vapor pressure of pure methanol, \(P_{methanol} = 760\, \text{torr}\). We can now solve for the mole fraction of methanol (\(X_{methanol}\)) using the Raoult's Law equation: \(556 = X_{methanol} \times 760\) Now, to find \(X_{methanol}\), we will divide both sides of the equation by 760: \[X_{methanol} = \frac{556}{760}\] By calculating the result, we find the mole fraction of methanol in the solution: \[X_{methanol} \approx 0.732 \] So, the mole fraction of methanol in this solution is approximately 0.732.

Key concepts

Vapor Pressure

Vapor pressure is a fundamental concept in understanding how substances behave when transitioning between liquid and gas phases. It is the pressure exerted by the vapor of a liquid in equilibrium with its liquid phase at a given temperature. This means the rate at which molecules escape from the liquid into the vapor phase is equal to the rate at which they return to the liquid. When considering a pure substance, the vapor pressure depends solely on the temperature and the specific compound’s properties. A higher temperature typically corresponds to higher vapor pressure because more molecules have enough energy to escape into the gas phase.
In the context of solutions, the vapor pressure is often affected by the presence of solutes, especially nonvolatile solutes. Nonvolatile solutes do not contribute to the vapor phase, therefore they reduce the vapor pressure of the solvent due to fewer available solvent molecules escaping into the vapor phase. This reduction is described by Raoult's Law, and it plays a crucial role in determining solution properties like boiling point elevation and freezing point depression.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture. It is calculated by taking the number of moles of that component and dividing it by the total number of moles of all components in the mixture. This dimensionless number lies between 0 and 1 and is vital in calculations involving Raoult's Law and vapor pressure. The mole fraction of a component signifies its proportion within the mixture.
In practical terms, if you have a solution with solvent and solute, the mole fraction of each gives insight into their relative amounts. For example, in solutions exhibiting partial vapor pressures, knowing the mole fraction of the solvent helps to predict how the presence of the solute influences those pressures. This becomes particularly important in determining the behavior of mixtures in chemical processes.

Boiling Point Elevation

Boiling point elevation is a colligative property, meaning it depends on the number of particles in a solution rather than the nature of the particles themselves. When a nonvolatile solute is dissolved in a solvent, the boiling point of the resultant solution is higher than that of the pure solvent. This is because the solute molecules interfere with the formation of vapor bubbles, requiring more heat energy (or higher temperature) to reach the boiling point.
This phenomenon can be mathematically described by the formula:\[\Delta T_b = i \cdot K_b \cdot m\]where \(\Delta T_b\) is the boiling point elevation, \(i\) is the van 't Hoff factor (which equals the number of particles the solute produces in solution), \(K_b\) is the ebullioscopic constant of the solvent, and \(m\) is the molality of the solution. Understanding this concept is essential for calculations involving boiling point adjustments in solutions and is directly related to alterations in vapor pressure due to solutes.