Problem 97 - Chemistry: The Central Science

Question

Show that the vapor-pressure reduction, \(\Delta P_{\text {solvent }}\), associated with the addition of a nonvolatile solute to a volatile solvent is given by the equation \(\Delta P_{\text {solvent }}=\) \(X_{\text {solute }} \times P_{\text {solvent }}^{\circ}\).

Step-by-Step Solution

Answer

To show that the vapor-pressure reduction, \(\Delta P_{\text {solvent}}\), with the addition of a nonvolatile solute to a volatile solvent is given by the equation \(\Delta P_{\text {solvent }}= X_{\text {solute }} \times P_{\text {solvent }}^{\circ}\), we start by applying Raoult's law to the solvent: \(P_{\text {solvent}} = X_{\text {solvent}} \times P_{\text {solvent}}^{\circ}\). Since the solute is nonvolatile, its vapor pressure is zero, and we obtain \(\Delta P_{\text {solvent}} = P_{\text {solvent}}^{\circ} - P_{\text {solvent}}\). We then express \(X_{\text {solvent}}\) in terms of \(X_{\text {solute}}\): \(X_{\text {solvent}} = 1 - X_{\text {solute}}\). Substituting this into the Raoult's law equation and simplifying, we derive the desired equation: \(\Delta P_{\text {solvent}} = X_{\text {solute}} \times P_{\text {solvent}}^{\circ}\).

Step 1: Understand vapor pressure and its reduction

Vapor pressure is the pressure exerted by the vapor in equilibrium with its liquid phase at a given temperature. When a nonvolatile solute (a solute that does not contribute to vapor pressure) is added to a volatile solvent (a solvent that can evaporate), the vapor pressure of the solvent decreases. This decrease in vapor pressure is known as vapor-pressure reduction, denoted as \(\Delta P_{\text {solvent}}\). In this exercise, we want to find the relationship between \(\Delta P_{\text {solvent}}\) and the mole fraction of solute, \(X_{\text {solute }}\), as well as the vapor pressure of the pure solvent \(P_{\text {solvent }}^{\circ}\).

Step 2: Introduce Raoult's law

Raoult's law states that the vapor pressure of an ideal solution is dependent on the mole fraction of the components and their vapor pressures in the pure state. Mathematically, Raoult's law is given by: $$P_i = X_i \times P_i^{\circ}$$ where \(P_i\) is the vapor pressure of component \(i\) in the solution, \(X_i\) is the mole fraction of component \(i\), and \(P_i^{\circ}\) is the vapor pressure of pure component \(i\). In our case, we have a nonvolatile solute mixed with a volatile solvent, so component 1 will be the solvent, and component 2 will be the solute.

Step 3: Apply Raoult's law to the solvent

Let's apply Raoult's law to the solvent (component 1): $$P_{\text {solvent}} = X_{\text {solvent}} \times P_{\text {solvent}}^{\circ}$$ Since the solute is nonvolatile, there is no contribution to the vapor pressure from the solute: $$P_{\text {sol'n}} = P_{\text {solvent}}$$ Now, the vapor pressure reduction is the difference between the vapor pressure of the pure solvent and that of the solution: $$\Delta P_{\text {solvent}} = P_{\text {solvent}}^{\circ} - P_{\text {sol'n}} = P_{\text {solvent}}^{\circ} - P_{\text {solvent}}$$

Step 4: Express \(X_{\text {solvent}}\) in terms of \(X_{\text {solute}}\)#

The mole fraction of the solvent and solute in the solution is related by the following equation: $$X_{\text {solvent}} + X_{\text {solute}} = 1$$ So, $$X_{\text {solvent}} = 1 - X_{\text {solute}}$$

Step 5: Derive the equation for \(\Delta P_{\text {solvent}}\)#

Substitute the expression of \(X_{\text {solvent}}\) from Step 4 into the Raoult's law equation for the solvent from Step 3: $$P_{\text {solvent}} = (1 - X_{\text {solute}}) \times P_{\text {solvent}}^{\circ}$$ Finally, plug this \(P_{\text {solvent}}\) into the equation for \(\Delta P_{\text {solvent}}\) from Step 3: $$\Delta P_{\text {solvent}} = P_{\text {solvent}}^{\circ} - (1 - X_{\text {solute}}) \times P_{\text {solvent}}^{\circ}$$ Simplify the equation: $$\Delta P_{\text {solvent}} = X_{\text {solute}} \times P_{\text {solvent}}^{\circ}$$ Therefore, we have shown that the vapor-pressure reduction for a nonvolatile solute in a volatile solvent is given by the equation: $$\Delta P_{\text {solvent}} = X_{\text {solute}} \times P_{\text {solvent}}^{\circ}$$

Key Concepts

Raoult's Law

Raoult's Law is a fundamental principle in physical chemistry that explains how the vapor pressure of a solution is affected by its composition. The law states that the vapor pressure of a component in an ideal solution is directly proportional to the mole fraction of that component in the mixture.

The mathematical expression for Raoult's Law for any component 'i' in a mixture is: \[P_i = X_i \times P_i^{\circ}\]where \(P_i\) is the vapor pressure of component 'i' in the solution, \(X_i\) is its mole fraction, and \(P_i^{\circ}\) represents its vapor pressure when pure.

Raoult's Law is particularly useful because it enables us to understand the vapor pressure behavior of solutions without complex calculations. It implies that each solute's partial vapor pressure is independent of other components, yielding the total vapor pressure as the sum of individual pressures.

Vapor Pressure

Vapor pressure is a term used to describe the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases at a given temperature.

This equilibrium pressure reflects the rate at which molecules escape from the liquid (or solid) phase to the gas phase and vice versa.

• If the temperature increases, more molecules have enough energy to escape into the gas phase, which increases the vapor pressure.
• If a nonvolatile solute is added to a solvent, the surface area available for the solvent molecules to evaporate decreases, which leads to a reduction in vapor pressure.

The importance of understanding vapor pressure lies in its relevance to various everyday applications, such as formulating pharmaceuticals, designing distillation processes, and understanding weather patterns.

Nonvolatile Solute

A nonvolatile solute is a substance that does not easily vaporize; that is, it has low to negligible vapor pressure at the given temperature. When a nonvolatile solute is dissolved in a solvent, it affects the properties of the solution including its vapor pressure.

The addition of a nonvolatile solute to a solvent reduces the solvent's vapor pressure because it

• lowers the number of solvent molecules at the surface,
• changes the solvent's evaporation rate due to fewer solvent molecules being exposed,
• results in a phenomenon called vapor-pressure lowering, which is integral to understanding the colligative properties of solutions.

This aspect is critical not only in theoretical chemistry but also in practical applications like anti-freeze in car radiators and manufacturing of processed foods.

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 substance to the total number of moles of all substances present.

For a given component 'i', its mole fraction \(X_i\) is calculated by the formula: \[X_i = \frac{n_i}{n_{\text{total}}}\]where \(n_i\) is the number of moles of component 'i' and \(n_{\text{total}}\) is the sum of moles of all components in the solution.

Mole fractions are dimensionless quantities and always add up to 1 for a particular solution. Understanding mole fraction is critical for calculating many properties of solutions, including partial vapor pressures using Raoult's Law, which in turn helps predict how the solution will behave in various conditions.