Problem 64 - Chemistry: The Central Science in SI Units, Global Edition

Question

The vapor pressure of pure water at \(70^{\circ} \mathrm{C}\) is \(31.2 \mathrm{kPa}\). The vapor pressure of water over a solution at \(70^{\circ} \mathrm{C}\) containing equal numbers of moles of water and glycerol \(\left(\mathrm{C}_{3} \mathrm{H}_{5}(\mathrm{OH})_{3}\right.\), a nonvolatile solute) is \(13.3 \mathrm{kPa}\). Is the solution ideal according to Raoult's law?

Step-by-Step Solution

Answer

The given solution does not follow Raoult's law and is not an ideal solution because the expected vapor pressure of an ideal solution (\(15.6 \mathrm{kPa}\)) is different from the given vapor pressure of the solution (\(13.3 \mathrm{kPa}\)).

Step 1: Calculate the mole fraction of water

Given that the solution contains equal numbers of moles of water and glycerol, this means the mole fraction of water denoted as \(x_{water}\) can be calculated as follows: \[x_{water} = \frac{\text{moles of water}}{\text{moles of water} + \text{moles of glycerol}}\] Since they both have equal moles, we can denote the moles of water as \(n\) and moles of glycerol as \(n\): \[x_{water} = \frac{n}{n + n} = \frac{n}{2n} = \frac{1}{2}\]

Step 2: Apply Raoult's law for an ideal solution

Now that we have the mole fraction of water, we'll apply Raoult's law for an ideal solution. Raoult's law states that: \[P_{solution}^{ideal} = x_{water} \times P_{water}\] Where \(P_{solution}^{ideal}\) is the expected vapor pressure of an ideal solution, \(x_{water}\) is the mole fraction of water, and \(P_{water}\) is the vapor pressure of pure water. Using the information given, we can calculate the expected vapor pressure of an ideal solution at \(70^{\circ} \mathrm{C}\): \[P_{solution}^{ideal} = \frac{1}{2} \times 31.2 \mathrm{kPa} = 15.6 \mathrm{kPa}\]

Step 3: Compare the expected vapor pressure to the given vapor pressure

To determine if the given solution follows Raoult's law and behaves ideally, we need to compare the expected vapor pressure of an ideal solution, \(P_{solution}^{ideal}\), to the given vapor pressure of the solution, \(P_{solution}\): Given vapor pressure, \(P_{solution} = 13.3 \mathrm{kPa}\) Expected vapor pressure of an ideal solution, \(P_{solution}^{ideal} = 15.6 \mathrm{kPa}\) Since the expected vapor pressure of an ideal solution is different from the given vapor pressure of the solution, we can infer that the solution does not follow Raoult's law and is not an ideal solution.

Key Concepts

Vapor Pressure

Vapor pressure is an essential concept in chemistry, especially when discussing solutions and their behavior. In simple terms, vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid or solid form in a closed system. It's all about molecules escaping a liquid and entering the vapor phase. The higher the vapor pressure, the more volatile the liquid.

Several factors can influence vapor pressure, such as temperature. Generally, as the temperature increases, the kinetic energy of the molecules also increases, leading to a higher vapor pressure. It means more molecules have the energy to escape into the gas phase.

In the context of Raoult's Law, we use the vapor pressure of a pure solvent to predict the vapor pressure of a solution. If you mix a non-volatile solute like glycerol with a volatile solvent such as water, the solute molecules occupy space at the surface, reducing the number of solvent molecules that can escape into the vapor phase. This leads to a lower vapor pressure compared to pure solvent. Understanding this concept is key when working with ideal solutions.

By comparing calculated and given vapor pressures, we can determine if a solution behaves ideally, following Raoult's Law, or if it shows deviations due to interactions between solute and solvent molecules.

Ideal Solution

An ideal solution is a key concept in the study of thermodynamics and physical chemistry. An ideal solution exhibits several characteristics:

• The interactions between solute and solvent molecules are similar to those present between molecules in pure substances.
• There are no significant energy changes or volume changes upon mixing.
• The solution obeys Raoult's Law.

Raoult's Law is crucial in understanding ideal solutions. It states that the partial vapor pressure of each component in a solution is equal to the mole fraction of that component multiplied by the vapor pressure of the pure component. That is, for a component "A" in a solution: \[ P_{A}^{ideal} = x_{A} \times P_{A}^{pure} \] Here, \(P_{A}^{ideal}\) is the partial pressure in the solution, \(x_{A}\) is the mole fraction of A, and \(P_{A}^{pure}\) is the vapor pressure of pure A.

For the given exercise, the solution with equal moles of water and glycerol showed a different vapor pressure from what Raoult's Law predicts for an ideal solution. This difference indicates non-ideal behavior, likely due to strong interactions between water and glycerol molecules that deviate from the simple interactions that define ideal solutions.

Mole Fraction

Mole fraction is a method to express the concentration of a component in a mixture and plays a critical role in Raoult's Law. It refers to the ratio of the number of moles of a particular substance to the total number of moles in the solution.

The formula for mole fraction of a component A, denoted as \(x_{A}\), is:\[ x_{A} = \frac{n_{A}}{n_{total}} \]Where \(n_{A}\) is the number of moles of A, and \(n_{total}\) is the total number of moles of all components in the solution. The sum of all mole fractions in a solution equals 1.

In the exercise, the solution had equal moles of water and glycerol, making the mole fraction of water \(0.5\). That's because:

• Number of moles of water = n
• Number of moles of glycerol = n
• Total moles = n + n = 2n

The mole fraction of water, \(x_{water}\), becomes \(\frac{n}{2n} = \frac{1}{2}\). This central idea allows calculations of expected vapor pressures in ideal solutions based on pure component vapor pressures and determines solution behavior in real-world scenarios.