Question

Consider two solutions, one formed by adding $10\mathrm{g}$ of glucose $\left({\mathrm{C}}_6{\mathrm{H}}_{12}{\mathrm{O}}_6\right)$ to $1\mathrm{L}$ of water and the other formed by adding $10\mathrm{g}$ of sucrose $\left({\mathrm{C}}_{12}{\mathrm{H}}_{22}{\mathrm{O}}_{11}\right)$ to $1\mathrm{L}$ of water. Are the vapor pressures over the two solutions the same? Why or why not?

Short answer

The vapor pressures over the two solutions are not the same because the mole fractions of glucose and sucrose in the solutions are different. According to Raoult's law, the vapor pressure of a component in a solution is proportional to its mole fraction. Since the mole fractions of glucose and sucrose are different, this leads to different vapor pressures for the two solutions.

Step-by-step solution

Calculate the number of moles of solutes and solvent

Firstly, calculate the number of moles of solutes (glucose and sucrose) and solvent (water) in both solutions. The molecular weights of glucose, sucrose, and water are 180.16 g/mol, 342.3 g/mol, and 18 g/mol, respectively. Number of moles of glucose = $\frac{10g}{180.16g/mol}$ Number of moles of sucrose = $\frac{10g}{342.3g/mol}$ Number of moles of water = $\frac{1000g}{18g/mol}$

Calculate mole fractions of solutes and solvent

Next, calculate the mole fractions of solutes and solvent in both solutions. Mole fraction of glucose in solution 1 = $\frac{molesofglucose}{molesofglucose+molesofwater}$ Mole fraction of sucrose in solution 2 = $\frac{molesofsucrose}{molesofsucrose+molesofwater}$ Mole fraction of water in solution 1: $(1-molefractionofglucose)$ Mole fraction of water in solution 2: $(1-molefractionofsucrose)$

Apply Raoult's law

According to Raoult's law, the vapor pressure of a component in a solution is proportional to its mole fraction: Vapor pressure of water in solution 1 = $molefractionofwate{r}_1\times vaporpressureofpurewater$ Vapor pressure of water in solution 2 = $molefractionofwate{r}_2\times vaporpressureofpurewater$

Compare vapor pressures

Now, compare the vapor pressures calculated in step 3: - If the vapor pressure of water in solution 1 is equal to the vapor pressure of water in solution 2, then the vapor pressures over the two solutions are the same. - If the vapor pressure of water in solution 1 is not equal to the vapor pressure of water in solution 2, then the vapor pressures over the two solutions are different. After calculating the vapor pressures, you will find that the vapor pressures over the two solutions are not the same because the mole fractions of glucose and sucrose in the solutions are different, leading to different vapor pressures according to Raoult's law.

Key concepts

Raoult's law

Raoult's law is a fundamental principle in chemistry that deals with the vapor pressure of solutions. It states that the partial vapor pressure of each component in an ideal solution is directly proportional to the mole fraction of the component present in the solution. To put it simply, in a mixture of liquids, each substance contributes to the overall vapor pressure based on how much of it is present compared to the other substances.

The law can be expressed by the mathematical formula:

Vapor pressure of component A = Mole fraction of component A × Vapor pressure of pure component A

This relationship allows us to predict how the presence of a solute will affect the vapor pressure of the solvent. As a solute is added to a solvent, it generally decreases the vapor pressure of the solvent due to the solute particles occupying space at the surface, where evaporation occurs. However, for ideal solutions, each component affects the total vapor pressure independently of the other components.

In the context of the exercise given, the vapor pressures over two solutions containing different solutes (glucose and sucrose) in the same solvent (water) will differ, even if the masses of the solutes are the same. The molecular weights of glucose and sucrose are different, resulting in different numbers of moles and consequently different mole fractions when dissolved in water. According to Raoult's law, this leads to a variation in the vapor pressures of the two solutions.

Mole fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the number of moles of a particular component to the total number of moles of all components in the mixture. The beauty of mole fraction is that it's a dimensionless quantity—it doesn't depend on the volume of the mixture, which can change with temperature and pressure.

The formula to calculate the mole fraction (represented by χ) of a component is given as:

Mole fraction (χ) = Number of moles of the component / Total number of moles in the solution

For the glucose and sucrose solutions in the exercise, calculating the mole fractions involves determining the number of moles of solute and solvent and then applying this formula. Because the molecular weights of glucose and sucrose are different, 10 g of each will yield different amounts of moles, which will result in different mole fractions.

Mole fraction is particularly important because it is used in Raoult's law to determine the contribution of each component to the vapor pressure of the solution. The differences in mole fraction, due to the differences in molarity between glucose and sucrose, explain the variations in the vapor pressures of the two solutions.

Colligative properties

Colligative properties are unique in that they depend on the number of particles of solute in a solution rather than the identity of the solute. These properties are at the heart of many processes, both in industrial applications and in nature. For example, they are the reason automotive antifreeze lowers the freezing point of water and why saline solution has a higher boiling point than pure water.

The four main colligative properties are:

• Vapor pressure lowering
• Boiling point elevation
• Freezing point depression
• Osmotic pressure

When a non-volatile solute is dissolved in a solvent, it affects these physical properties of the solution. For instance, the introduction of a solute leads to a decrease in vapor pressure, which is directly tied to Raoult's law. Explaining this effect, the addition of a solute creates fewer solvent molecules at the surface to evaporate, thus lowering the vapor pressure.

The exercise involves comparing the vapor pressures of two solutions—a colligative property affected by the number of solute particles. The differing molecular weights of glucose and sucrose result in different mole fractions when the same mass is dissolved. This is why the vapor pressures differ—demonstrating the principle of vapor pressure lowering, which is directly related to the number of particles, and not their individual identity.