Question

The normal boiling point of diethyl ether is \(34.5^{\circ} \mathrm{C} .\) A solution containing a nonvolatile solute dissolved in diethyl ether has a vapor pressure of 698 torr at \(34.5^{\circ} \mathrm{C}\). What is the mole fraction of diethyl ether in this solution?

Short answer

The mole fraction of diethyl ether in the solution is approximately 0.918, which can be calculated using Raoult's law: \(x_{solvent} = \frac{P_{solution}}{P^*_{} }\), where \(P_{solution}\) is the given vapor pressure of the solution (698 torr), and \(P^*_{}\) is the vapor pressure of pure diethyl ether at its normal boiling point (760 torr).

Step-by-step solution

Identify the given data

Boiling point of diethyl ether: \(34.5^{\circ} \mathrm{C}\) Vapor pressure of the solution: 698 torr

Note the property of normal boiling point

At the normal boiling point of diethyl ether, the vapor pressure of pure diethyl ether is equal to the external pressure, which is atmospheric pressure (normally around 760 torr).

Use Raoult's law

Raoult's law: \(P_{solution} = x_{solvent} \cdot P^*_{}\) where, \(P_{solution}\) is the vapor pressure of the solution, \(x_{solvent}\) is the mole fraction of the solvent (diethyl ether in this case), \(P^*_{}\) is the vapor pressure of the pure solvent (diethyl ether) at the given temperature.

Solve for mole fraction of the solvent

Rearranging the equation for Raoult's law to solve for \(x_{solvent}\): \(x_{solvent} = \frac{P_{solution}}{P^*_{} }\) Now, plug in the given values: \(x_{solvent} = \frac{698 \, torr}{760 \, torr}\)

Calculate the mole fraction of diethyl ether

Divide the vapor pressure of the solution by the vapor pressure of pure diethyl ether: \(x_{solvent} = \frac{698}{760} \approx 0.918 \) The mole fraction of diethyl ether in the solution is approximately 0.918.

Key concepts

Raoult's Law

Raoult's Law is a fundamental principle in chemistry that explains how the vapor pressure of a solution is affected by the presence of a solute. It states that the partial vapor pressure of each component in an ideal solution is directly proportional to its mole fraction. This relationship can be mathematically expressed as:
\[ P_{solution} = x_{solvent} \cdot P^{\ast}_{solvent} \]
where \(P_{solution}\) is the vapor pressure of the solution, \(x_{solvent}\) is the mole fraction of the solvent, and \(P^{\ast}_{solvent}\) is the vapor pressure of the pure solvent at the same temperature.

In simpler terms, if you have a pure solvent with a certain vapor pressure and you add a nonvolatile solute to it, the resulting solution will have a lower vapor pressure. This is because the solute particles occupy space at the surface of the liquid, preventing some of the solvent molecules from evaporating. This principle is crucial for calculating various physical properties of solutions, such as boiling points, freezing points, and vapor pressures.

Vapor Pressure

Vapor pressure is the pressure exerted by the vapor of a liquid (or solid) when it is in equilibrium with its condensed phase at a given temperature. It is a measure of a substance's tendency to evaporate, and it varies with temperature. For instance, when the temperature increases, the kinetic energy of the molecules also increases, allowing more molecules to escape into the vapor phase, thus increasing the vapor pressure.

Vapor pressure is an important property because it influences how substances transition between phases. A liquid with a high vapor pressure at room temperature is considered volatile and evaporates quickly. Understanding vapor pressure is essential for explaining phenomena like boiling and for solving problems related to Raoult's Law.

Nonvolatile Solute

A nonvolatile solute is a substance that has a negligible vapor pressure at a given temperature and does not readily evaporate. When a nonvolatile solute is dissolved in a solvent, it reduces the solution's overall vapor pressure compared to the pure solvent. This happens because the solute molecules interfere with the evaporation of the solvent molecules, taking up space at the surface and making it harder for the solvent to escape into the gas phase.

Nonvolatile solutes play a key role in various physical properties of solutions, such as boiling point elevation and freezing point depression. These properties are particularly important in real-world applications, for example, antifreeze in cars or the salting of roads in winter to prevent ice formation.

Boiling Point

The boiling point of a liquid is the temperature at which its vapor pressure equals the external pressure. Normally, for pure substances, this is atmospheric pressure, which is about 760 torr. When a solution contains a nonvolatile solute, its vapor pressure decreases, and thus it must be heated to a higher temperature to achieve the same external pressure and boil. This phenomenon is known as boiling point elevation.

When analyzing boiling points in chemistry, it's useful to understand how impurities, such as a nonvolatile solute, affect the boiling point of a solvent. The presence of the solute means more energy (higher temperature) will be required to turn the solvent into vapor, thus elevating the boiling point of the solution compared to the pure solvent.