Bromomethane, \(\mathrm{CH}_{3} \mathrm{Br}\), and iodomethane, \(\mathrm{CH}_{3} \mathrm{I}\), form an ideal solution. The vapor pressure of bromomethane is 661 Torr and that of iodomethane is 140 Torr at \(0.0^{\circ} \mathrm{C}\). Calculate the vapor pressure of each of the following solutions and the mole fraction of each substance in the vapor phase above those solutions at \(0.0^{\circ} \mathrm{C}:\) (a) \(0.33 \mathrm{~mol}\) of bromomethane mixed with \(0.67 \mathrm{~mol}\) of iodomethane; (b) \(35.0 \mathrm{~g}\) of bromomethane mixed with \(35.0 \mathrm{~g}\) of iodomethane.

When a liquid is at equilibrium with its vapor in a closed container, the pressure exerted by the vapor is known as vapor pressure. This pressure is a result of the molecules escaping from the liquid phase into the gas phase. Vapor pressure depends on the temperature and the nature of the liquid; it increases with temperature and varies depending on the substance's volatility. For instance, the vapor pressure of bromomethane at a certain temperature might differ notably from that of iodomethane, due to differences in their molecular structures and intermolecular forces. Understanding vapor pressure is crucial when studying mixtures, as it influences the behavior of solutions and the distribution of components between the liquid and vapor phases.

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 substance to the total number of moles of all the substances present in the mixture. Represented by the symbol \( X \), it is dimensionless and always ranges between 0 and 1. For example, if you have a container with bromomethane and iodomethane, the mole fraction of bromomethane \( X_{\text{Br}} \) would be calculated by dividing the moles of bromomethane by the total moles of both bromomethane and iodomethane. It's an essential concept in applying Raoult's Law since vapor pressures of individual components in a solution are proportional to their respective mole fractions.

An ideal solution is one where the components mix perfectly at the molecular level, behaving ideally in accordance to Raoult's Law. In such a solution, the interactions between different molecules (A-B) are equal to the interactions among the same kind of molecules (A-A or B-B). There is no change in enthalpy when an ideal solution is formed, implying that no heat is absorbed or released. These solutions also follow a linear mixture rule concerning properties like vapor pressure. In an ideal solution, each component contributes to the total vapor pressure in proportion to its mole fraction, reflecting the essence of Raoult's Law. Real solutions often deviate from ideal behavior due to various intermolecular forces at play, but substances with similar molecular sizes and intermolecular forces, like bromomethane and iodomethane, may closely approximate ideal solution behavior.

Partial vapor pressure refers to the pressure exerted by a single component of a mixture of liquids in the vapor phase when that component is in equilibrium with the liquid phase. According to Raoult's Law, the partial vapor pressure \( P_i \) of component \( i \) in a mixture is equal to the product of the mole fraction \( X_i \) of that component in the liquid phase and the vapor pressure \( P^0_i \) of the pure component at the same temperature: \( P_i = X_i \times P^0_i \). When regarding an ideal solution such as a bromomethane and iodomethane mixture, understanding the concept of partial vapor pressure is critical to determining the contribution each substance makes to the total vapor pressure of the solution. It indicates the tendency of each component to escape into the vapor phase and is pivotal in the calculation of total vapor pressures and mole fractions in the vapor phase above a solution.