Liquids A (molar mass \(100 \mathrm{~g} / \mathrm{mol}\) ) and \(\mathrm{B}\) (molar mass \(110 \mathrm{~g} / \mathrm{mol}\) ) form an ideal solution. At \(55^{\circ} \mathrm{C}, \mathrm{A}\) has a vapor pressure of \(95 \mathrm{mmHg}\) and \(\mathrm{B}\) has a vapor pressure of \(42 \mathrm{mmHg}\). A solution is prepared by mixing equal masses of \(\mathrm{A}\) and \(\mathrm{B}\). (a) Calculate the mole fraction of each component in the solution. (b) Calculate the partial pressures of A and B over the solution at \(55^{\circ} \mathrm{C}\). (c) Suppose that some of the vapor described in (b) is condensed to a liquid in a separate container. Calculate the mole fraction of each component in this liquid and the vapor pressure of each component above this liquid at \(55^{\circ} \mathrm{C}\).

An ideal solution can be conceptualized as a mixture of two or more substances that obeys Raoult's law at all concentrations for both the liquid and vapor phases. In simple terms, in an ideal solution, the interactions between the unlike molecules are similar to those between the like molecules. This means that the molecules of different substances interact with each other in the same way they would with their own kind.

In the context of our exercise, liquids A and B forming an ideal solution signifies that the blending of A and B doesn't involve any significant heat release or absorption. Moreover, the volumes of the liquids simply add up. Another key feature is that the vapor pressure of each component in the ideal solution is directly proportional to its mole fraction. In essence, Raoult's Law, which governs the behavior of ideal solutions, allows us to predict the vapor pressure and composition of the mixture when the pure components' vapor pressures are known at the same temperature.

The concept of partial pressures is pivotal when dealing with mixtures of gases, or in our case, vapors above a liquid in a solution. The partial pressure of a gas is the pressure that the gas would exert if it were alone in a container at the same temperature. It is a measure of the impact of that particular gas on the total pressure of the mixture.

In terms of the vapor over a liquid mixture, the partial pressure of each component is obtained by multiplying its vapor pressure in pure form (as a pure substance at the same temperature) by its mole fraction in the mixture. This approach complies with Raoult's law for ideal solutions. Hence, the total vapor pressure over the mixture is simply the sum of the partial pressures of its components. Our exercise demonstrates the calculation of partial pressures using the vapor pressures of pure liquids A and B and their mole fractions in the mixture.

The mole fraction is a way of expressing the composition of a mixture by comparing the amount of one constituent to the total amount of substances present. It is defined as the number of moles of a particular component divided by the total number of moles of all components in the mixture.

To calculate the mole fraction, we need the number of moles of each substance, which we can find using their respective molar masses and the mass of each one in the solution. The mole fraction is dimensionless, ranging from 0 to 1, and is invaluable when working with the colligative properties of solutions, like boiling point elevation, freezing point depression, and the vapor pressures as we've focused on in our exercise. It's worth noting that the mole fraction will remain consistent for the substance in both the vapor and liquid phase, assuming that the system is at equilibrium and no chemical reactions occur that would change the mole amounts.