Question

The vapor pressure of pure water at \(60^{\circ} \mathrm{C}\) is 149 torr. The vapor pressure of water over a solution at \(60^{\circ} \mathrm{C}\) containing equal numbers of moles of water and ethylene glycol (a nonvolatile solute) is 67 torr. Is the solution ideal according to Raoult's law?

Short answer

The solution is not ideal according to Raoult's law. The mole fraction of water in the solution is 0.5. Applying Raoult's law, the theoretical vapor pressure of water over the solution should be \(74.5 \,\text{torr}\) (0.5 times the vapor pressure of pure water at \(60^{\circ}\mathrm{C}\), which is 149 torr). However, the given vapor pressure of water over the solution is \(67 \,\text{torr}\), which is different from the theoretical vapor pressure. Thus, the solution does not follow Raoult's law exactly.

Step-by-step solution

Calculate the mole fraction of water in the solution

Since there are equal moles of water and ethylene glycol in the solution, we can easily determine the mole fraction of each component. For water: Mole fraction = (moles of water) / (moles of water + moles of ethylene glycol) Mole fraction of water = 1 / (1 + 1) = 1 / 2 = 0.5

Apply Raoult's law to calculate the theoretical vapor pressure of water over the solution

According to Raoult's law, the partial vapor pressure of a component in a solution is equal to its mole fraction multiplied by the vapor pressure of the pure substance at the same temperature. Theoretical vapor pressure of water over the solution = mole fraction of water * vapor pressure of pure water = (0.5) * (149 torr) = 74.5 torr

Compare the theoretical and given vapor pressures of water over the solution

To determine if the solution is ideal, we need to compare the theoretical vapor pressure of water (74.5 torr) with the given vapor pressure of water over the solution (67 torr).

Determine if the solution is ideal

We'll compare the theoretical vapor pressure and the given vapor pressure: Theoretical vapor pressure of water over the solution = 74.5 torr Given vapor pressure of water over the solution = 67 torr Since these two values are different, the solution does not follow Raoult's law exactly. Therefore, the solution is not ideal according to Raoult's law.

Key concepts

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid or solid form in a closed system. This concept is essential for understanding how substances transition between phases, like liquid to vapor. At a given temperature, a substance reaches a specific vapor pressure, which aids in processes like evaporation and condensation.

In a pure liquid, the vapor pressure is determined by the temperature and the nature of the liquid itself. For instance, water has a specific vapor pressure at different temperatures. At 60°C, the vapor pressure of pure water is 149 torr, meaning if you have enough water at this temperature, it will exert this pressure naturally. However, when a nonvolatile solute such as ethylene glycol is added, the vapor pressure reduces. This occurrence is the crux of Raoult's law, which predicts such changes based on the composition of a solution.

Mole Fraction

Mole fraction is a measure of concentration and is defined as the ratio of moles of a component to the total moles of all components in a solution. This concept is critical for calculating how partial pressures relate to mixtures, especially in solutions. To determine the mole fraction of a solvent, like water in our example, you divide the moles of water by the total moles of all substances present, including both the solvent and the solute. If you have equal moles of water and ethylene glycol, the calculation results in a mole fraction of 0.5 for water. This tells us that half of the 'amount' is water, influencing how the vapor pressure of the system will change as per Raoult's law.

Nonvolatile Solute

A nonvolatile solute, such as ethylene glycol, does not readily evaporate or contribute directly to the vapor pressure in a solution. Its presence is significant because it disrupts the solvent’s ability to escape into the vapor phase, thereby lowering the vapor pressure of the solvent. The addition of any nonvolatile solute decreases the total vapor pressure from what it would be if the solvent were pure. This decrease is due to the fact that the solute molecules occupy space at the liquid's surface, lowering the number of solvent molecules reaching the surface to evaporate. The reduction in vapor pressure is measurable using Raoult's law. When you notice that the vapor pressure over a solution is less than expected for an ideal solution, it often implies the presence of nonideal interactions between solute and solvent, as is the case with water and ethylene glycol at 60°C.