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 $710.0$ 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 is approximately $0.9355$.

Step-by-step solution

Write down the given information

We know that the normal boiling point of methanol is 64.7°C, and the vapor pressure of the solution (containing methanol and a nonvolatile solute) is 710.0 torr at 64.7°C.

Recalling Raoult's law

Raoult's law states that the vapor pressure of a solution is directly proportional to the mole fraction of the solvent. Mathematically, it is represented as: $$P_{solution}=x_{methanol}\cdot P_{puremethanol}$$ Where $P_{solution}$ is the vapor pressure of the solution, $x_{methanol}$ is the mole fraction of methanol, and $P_{puremethanol}$ is the vapor pressure of pure methanol.

Find the vapor pressure of pure methanol

Since the normal boiling point of methanol is given as 64.7°C, we know that the vapor pressure of pure methanol at this temperature is equal to the standard atmospheric pressure (boiling occurs when vapor pressure equals atmospheric pressure). In torr, standard atmospheric pressure is approximately 760 torr. So, $P_{puremethanol}=760torr$.

Rearrange Raoult's law equation to solve for mole fraction

We will rearrange Raoult's law to isolate the mole fraction of methanol: $$x_{methanol}=\frac{P_{solution}}{P_{puremethanol}}$$ Now plug in the given values for $P_{solution}$ and $P_{puremethanol}$.

Calculate the mole fraction of methanol

Substitute the given values into the equation from step 4: $$x_{methanol}=\frac{710torr}{760torr}$$ Now, perform the calculation: $$x_{methanol}\approx 0.9355$$ The mole fraction of methanol in the solution is approximately $0.9355$.

Key concepts

Understanding Raoult's Law

Raoult's Law is a fundamental principle that dives into the relationship between the components of a mixture. It particularly focuses on how the vapor pressure of a solution is affected by the presence of a solute. According to Raoult's Law, the vapor pressure of a solvent in a solution is directly proportional to the mole fraction of the solvent. In other words, as the amount of solvent increases in a solution, so does its vapor pressure. The mathematical representation of Raoult's law is:
$$P_{solution}=x_{solvent}\times P_{puresolvent}$$.
Here, $P_{solution}$ is the vapor pressure of the solution, $x_{solvent}$ is the mole fraction of the solvent, and $P_{puresolvent}$ is the vapor pressure of the pure solvent. It's important to note that Raoult's Law applies ideally to solutions with nonvolatile solutes and volatile solvents.

The Concept of Vapor Pressure

Vapor pressure is a crucial term in understanding the physical properties of liquids. It is the pressure exerted by the vapor which is in equilibrium with its liquid or solid form, at a given temperature. Vapor pressure is influenced by the temperature of the system and the identity of the substance. Liquids with high vapor pressure at room temperature, such as ethanol, are known as volatile liquids. These liquids often have a strong tendency to evaporate. The vapor pressure of a pure solvent changes when a nonvolatile solute is added, normally resulting in a lower vapor pressure. The vapor pressure of a solution can be predicted by Raoult's Law, as seen in the exercise example.

Role of a Nonvolatile Solute in a Solution

A nonvolatile solute is a substance that has a negligible vapor pressure at a given temperature and, consequently, does not readily evaporate. When a nonvolatile solute is dissolved in a solvent, it disrupts the solute-solvent interactions, leading to a decrease in the overall vapor pressure of the solution. This phenomenon is excellently portrayed by Raoult's Law. Since the nonvolatile solute particles do not contribute to the vapor phase, the mole fraction of the solvent in the vapor phase is higher than that in the liquid phase, resulting in a lower vapor pressure of the solution compared to the pure solvent.

Defining the 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, which is 1 atmosphere or approximately 760 torr. Boiling occurs when the vapor pressure of the liquid overcomes atmospheric pressure, allowing the liquid to form bubbles and change into vapor throughout, not only from the surface. The presence of a nonvolatile solute elevates the boiling point of the solvent because it reduces the solvent’s vapor pressure, meaning that a higher temperature is needed to match the atmospheric pressure, leading to the phenomenon known as boiling point elevation.