Hexane, \(\left(\mathrm{C}_{6} \mathrm{H}_{14}\right)\) and cyclohexane \(\left(\mathrm{C}_{6} \mathrm{H}_{12}\right)\) form an ideal solution. The vapor pressure of hexane is 151 Torr and that of cyclohexane is 98 Torr at \(25.0^{\circ} \mathrm{C}\). Calculate the vapor pressure of each of the following solutions and the mole fraction of each substance in the vapor phase above those solutions: (a) \(0.2 \mathrm{~S} \mathrm{~mol} \mathrm{C}_{6} \mathrm{H}_{14}\) mixed with \(0.65 \mathrm{~mol} \mathrm{C}_{6} \mathrm{H}_{12}\) (b) \(10.0 \mathrm{~g}\) of hexane mixed with \(10.0 \mathrm{~g}\) of cyclohexane.

An ideal solution is a mixture where the interactions between all types of molecules are similar to the interactions between molecules of the same kind. This concept is crucial because it lays the groundwork for understanding how substances mix without changing their physical properties other than their concentration.

In an ideal solution, the enthalpy of mixing is zero, meaning no heat is absorbed or released when the two substances are combined. The process is entropy-driven, making the solution's formation spontaneous due to the increase in disorder.

For students grappling with calculations involving ideal solutions, it's important to remember that the vapor pressure of each component in an ideal solution is directly proportional to its mole fraction. This principle allows us to use simple mathematical relationships, such as Raoult's Law, to predict the properties of the solution.

Raoult's Law provides a method to determine the vapor pressure of each component in a solution. The law states that the partial vapor pressure of each component is equal to the vapor pressure of the pure component multiplied by its mole fraction in the solution. Mathematically, we express this as: \( P_i = x_i * P^0_i \), where \( P_i \) is the partial vapor pressure of component \(i\), \( x_i \) is the mole fraction, and \( P^0_i \) is the vapor pressure of the pure component.

Understanding this relationship is vital for students because it explains how the vapor pressure above a solution changes as the composition of the solution changes. Raoult's Law is especially helpful in predicting the behavior of ideal solutions, where it applies most accurately.

The mole fraction, an essential concept in chemistry, is the ratio of the number of moles of a specific component to the total number of moles of all components in the mixture. The calculation of mole fraction is straightforward yet vital: \( x_i = n_i / \sum n_j \), where \( x_i \) is the mole fraction of component \(i\) and \( n_i \) and \( n_j \) are the number of moles of components \(i\) and \(j\), respectively.

When calculating vapor pressure using Raoult's Law, it's the mole fraction of the liquid phase we're interested in. In contrast, when dealing with gases, the mole fraction relates to Dalton's Law. For students, it's crucial to understand how mole fraction works because it links quantity (in moles) to the qualitative behavior of solutions and gases.

Dalton's Law focuses on the behavior of gas mixtures and states that the total pressure of a gas mixture is equal to the sum of the partial pressures of each individual gas component. This law can be represented mathematically as \( P_{total} = \sum P_i \), where \( P_{total} \) is the total pressure exerted by the gas mixture, and \( P_i \) are the partial pressures of each gas component.

For students looking at vapor pressures in solutions, Dalton's Law is integral to understanding how the partial pressures contribute to the total vapor pressure above a solution. In the context of an ideal solution, the mole fraction in the vapor phase can be calculated from the partial pressures. This means that understanding Dalton's Law is instrumental in calculating the composition of the vapor phase, which directly influences distillation and other separation processes.