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 0.9184.

Step-by-step solution

List the given information

We are given: - Normal boiling point of diethyl ether: \(34.5^{\circ} \mathrm{C}\) - Vapor pressure of the solution: \(698\ torr\) - Atmospheric pressure at normal boiling point: \(760\ torr\)

Calculate the mole fraction of diethyl ether

Using the Raoult's Law formula, we can write: \(P_{solution} = (mole\ fraction\ of\ diethyl\ ether) * P^{0}_{diethyl\ ether}\) Now, plug in the given values: \(698\ torr = (mole\ fraction\ of\ diethyl\ ether) * 760\ torr\) Next, solve for the mole fraction of diethyl ether: \(mole\ fraction\ of\ diethyl\ ether = \frac{698\ torr}{760\ torr}\)

Calculate the mole fraction

Now, calculate the mole fraction: \(mole\ fraction\ of\ diethyl\ ether = \frac{698}{760} = 0.9184\) The mole fraction of diethyl ether in this solution is approximately 0.9184.

Key concepts

Boiling Point

The boiling point of a liquid is the temperature at which its vapor pressure equals the atmospheric pressure. When a liquid reaches this temperature, it starts to transition into a gas. In the example you provided, diethyl ether has a normal boiling point of \(34.5^{\circ} \mathrm{C}\). This is because its vapor pressure at that temperature matches the atmospheric pressure of \(760\ torr\).
Boiling occurs because the molecules within the liquid gain sufficient energy to overcome intermolecular forces and enter the gas phase.
Adding a nonvolatile solute can actually raise the boiling point of the solution. This is because the solute particles take up space at the surface of the liquid, which decreases the number of solvent molecules that can escape into the vapor phase.

Vapor Pressure

Vapor pressure is the pressure exerted by the vapor of a liquid in a closed system when the liquid and its vapor are in dynamic equilibrium. In simple terms, it reflects how easily molecules can escape from the liquid phase to the vapor phase.
Each liquid has its own vapor pressure, which increases with temperature. Diethyl ether, for instance, has a vapor pressure of \(760\ torr\) at its normal boiling point of \(34.5^{\circ} \mathrm{C}\).
A nonvolatile solute lowers the vapor pressure of a solvent, like diethyl ether, by occupying surface space and reducing the escape rate of solvent molecules. This phenomenon is described by Raoult's Law: \( P_{solution} = (mole\ fraction\ of\ solute) \times P^{0}_{solvent} \). In your problem, the solution's vapor pressure is \(698\ torr\), indicating the presence of a solute.

Mole Fraction

The mole fraction is the ratio of the number of moles of a component to the total number of moles in the solution. It is a way to express the composition of a mixture.
Mole fraction is dimensionless and is used in Raoult's Law to calculate the vapor pressure of the solution. For diethyl ether, the mole fraction can help determine how the addition of a solute affects its behavior in solution.
Based on Raoult's Law, the vapor pressure of a solution is proportionate to the mole fraction of the solvent. In the given exercise, the formula used was \( \frac{698}{760} = 0.9184 \), demonstrating that \(0.9184\) is the mole fraction of diethyl ether in the solution. This means \(91.84\%\) of the composition is diethyl ether, with the remaining \(8.16\%\) being the solute.