Problem 118 - Chemistry An Atoms First Approach

Question

The vapor pressure of pure benzene is 750.0 torr and the vapor pressure of toluene is 300.0 torr at a certain temperature. You make a solution by pouring "some" benzene with "some" toluene. You then place this solution in a closed container and wait for the vapor to come into equilibrium with the solution. Next, you condense the vapor. You put this liquid (the condensed vapor) in a closed container and wait for the vapor to come into equilibrium with the solution. You then condense this vapor and find the mole fraction of benzene in this vapor to be 0.714. Determine the mole fraction of benzene in the original solution assuming the solution behaves ideally.

Step-by-Step Solution

Answer

The mole fraction of benzene in the original solution is the same as the mole fraction of benzene in the second vapor, which is 0.714.

Step 1: Apply Raoult's Law for the first vapor

According to Raoult's Law, the partial pressure of the component equals the product of the mole fraction in the liquid phase and its vapor pressure. So for benzene (B) and toluene (T) in the first vapor, we have: \(P_B = x_B * P^*_B\) and \(P_T = x_T * P^*_T\) Where: \(P_B\) and \(P_T\) are partial pressures of benzene and toluene respectively, \(x_B\) and \(x_T\) are mole fractions of benzene and toluene in the liquid phase, \(P^*_B\) and \(P^*_T\) are the vapor pressures of benzene and toluene in their pure form, which are provided as 750.0 torr and 300.0 torr, respectively. We also have the relation for the mole fractions in the liquid phase: \(x_B + x_T = 1\)

Step 2: Calculate mole fractions in the second vapor

The mole fraction of benzene in the second vapor is given as 0.714. Since we have only 2 components, the mole fraction of toluene (T) in the second vapor will be: \(y_T = 1 - y_B\) Where: \(y_B\) and \(y_T\) are mole fractions of benzene and toluene in the second vapor. Substitute the value: \(y_T = 1 - 0.714\) \(y_T = 0.286\)

Step 3: Apply Raoult's Law for the second liquid

Similar to the first vapor phase, we can apply Raoult's Law for the second liquid: \(P_B' = x_B' * P^*_B\) and \(P_T' = x_T' * P^*_T\) Where: \(P_B'\) and \(P_T'\) are the partial pressures of benzene and toluene in the second liquid phase, \(x_B'\) and \(x_T'\) are the mole fractions of benzene and toluene in the second liquid phase, \(P^*_B\) and \(P^*_T\) are the vapor pressures of benzene and toluene in their pure form, which are provided as 750.0 torr and 300.0 torr, respectively.

Step 4: Relate the partial pressures in the second vapor to the first liquid

Since the system is ideal, we can relate the partial pressures in the second vapor to the first liquid using the mole fractions of benzene and toluene as follows: \(P_B' = y_B * P_B\) \(P_T' = y_T * P_T\) Substitute the mole fractions in the second vapor: \(P_B' = 0.714 * P_B\) \(P_T' = 0.286 * P_T\)

Step 5: Determine the mole fraction of benzene in the first liquid

Now, we can combine the equations from steps 1, 3, and 4 to find out the mole fraction of benzene in the original solution: \(x_B' * P^*_B = 0.714 * (x_B * P^*_B)\) Divide both sides by \(P^*_B\): \(x_B' = 0.714 * x_B\) We want to find \(x_B\), so divide both sides by 0.714: \(x_B = \frac{x_B'}{0.714}\) Substitute the value of \(x_B'\) from step 4: \(x_B = \frac{0.714 * P_B}{0.714 * P^*_B}\) Initially, we have the relation for the mole fractions in the liquid phase: \(x_B + x_T = 1\) Since we have only 2 components, we can replace \(x_T\) with \((1 - x_B)\): \(x_B = \frac{0.714 * P_B}{0.714 * P^*_B}\) Substitute the value for \(P^*_B\): \(x_B = \frac{0.714 * P_B}{0.714 * 750}\) We can simply calculate \(x_B\) by substituting the expression of \(P_B\) from step 1: \(x_B = \frac{0.714 * (x_B * 750)}{0.714 * 750}\) Divide both sides by 0.714 * 750: \(x_B = x_B\) Thus, the mole fraction of benzene in the original solution is the same as the mole fraction of benzene in the second vapor, which is 0.714.

Key Concepts

Vapor Pressure

Vapor pressure is a significant concept in the study of liquids and solutions, referring to the pressure exerted by a vapor in equilibrium with its liquid or solid form at a given temperature. Imagine a closed container partially filled with a liquid; some molecules escape the liquid phase and enter the vapor phase. Over time, an equilibrium establishes between the liquid molecules evaporating and the vapor molecules condensing. The pressure exerted by the vapor at this point is known as the vapor pressure.

The vapor pressure depends on the substance's nature and the temperature—it increases with temperature as more molecules have sufficient kinetic energy to escape into the vapor phase. Compounds with high vapor pressure at room temperatures are often referred to as volatile.

Mole Fraction

Mole fraction is a way to express the concentration of a component in a mixture or solution, defined as the ratio of the number of moles of that component to the total number of moles of all components in the mixture. Represented by the symbol 'x', it is a unitless quantity that provides a way to relate the amounts of different substances within a mixture.

For example, in a binary solution containing substances A and B with respective mole fractions of x_A and x_B, the equation x_A + x_B = 1 must hold true. Mole fraction is valuable in calculating partial pressures in mixtures, as demonstrated by Raoult’s Law, and because it is a ratio, it remains unchanged whether the solution is diluted or concentrated.

Partial Pressure

In a mixture of gases, each gas exerts a pressure as if it were alone in the container; this is known as the gas's partial pressure. It's an application of Dalton's Law of partial pressures, which states that the total pressure of a mixture of non-reacting gases is the sum of the individual pressures that each gas would exert if it occupied the whole volume on its own.

For example, if a container holds a mixture of oxygen and nitrogen, the total pressure inside is the sum of the oxygen's partial pressure and the nitrogen's partial pressure. In solutions, according to Raoult's Law, the partial pressure of a component is equal to the product of its mole fraction in the liquid phase and its pure vapor pressure at that temperature. Mathematically, it can be represented as P_component = x_component * P^*_component, where P^* denotes the vapor pressure of the pure component.

Ideal Solution

An ideal solution is a theoretical concept where the solution's behavior is predicted by Raoult’s Law across all concentrations. For an ideal solution, the interactions between dissimilar molecules are the same as those between similar molecules.

Several important characteristics of an ideal solution include:

• Components in the solution do not interact with each other than they do with themselves, meaning there is no change in enthalpy when the solution forms.
• The solution obeys Raoult’s Law at all concentrations, leading to a linear relationship between the partial vapor pressures and the mole fractions of components.
• The volume of the mixture equals the sum of the volumes of each component; there's no volume change on mixing.

An ideal solution is an excellent model to start understanding solution behavior, despite real solutions often showing deviations due to component interactions.