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Mole fraction is a crucial concept in chemistry when dealing with mixtures, particularly in solutions and gas phases. It represents the ratio of the number of moles of a component to the total number of moles in the mixture. The formula is simple:\[ x_i = \frac{n_i}{n_{\text{total}}} \]where \( n_i \) is the number of moles of the component, and \( n_{\text{total}} \) is the total number of moles of all components combined.
The mole fraction is dimensionless, meaning it has no units, and each component's mole fraction is a value between 0 and 1.
An important property is that the sum of all mole fractions in a mixture equals 1. This concept helps us understand how substances behave in mixtures, and it's widely used in calculating pressures and concentrations in solutions. In the case of a 1:1 molar ratio of hexane and cyclohexane in a liquid, their mole fractions would both initially be 0.5.

Equilibrium vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its liquid phase at a given temperature. It indicates how volatile a substance is—the higher the vapor pressure, the more readily the substance will evaporate.
For any liquid at a given temperature, there is a fixed equilibrium vapor pressure. This pressure is vital for understanding the vapor-liquid equilibrium in mixtures.

• For hexane, the equilibrium vapor pressure is given as 151.4 torr.
• For cyclohexane, it is 97.6 torr.

In a solution, the components' partial pressures are typically lower than their pure component pressures due to mixing, described by Raoult's Law.

Partial pressure refers to the pressure that each gas in a mixture would exert if it occupied the entire volume by itself at the same temperature. It is crucial in analyzing gas mixtures and solutions.Utilizing Raoult's Law, the partial pressure of a component is calculated as follows:\[ P_i = x_i P_i^0 \]where \( P_i \) is the partial pressure, \( x_i \) is the mole fraction, and \( P_i^0 \) is the equilibrium vapor pressure of the pure substance.For our exercise:

• The partial pressure of hexane is calculated as \( 0.5 \times 151.4 = 75.7 \) torr.
• The partial pressure of cyclohexane is \( 0.5 \times 97.6 = 48.8 \) torr.

The total pressure of the vapor mixture is the sum of these partial pressures, crucial for determining the mole fractions in the vapor phase.

Vapor-liquid equilibrium (VLE) describes the balance between a liquid's vapor phase and its liquid phase. It occurs when the rate of evaporation equals the rate of condensation, resulting in a stable system.In VLE, understanding the distribution of components between two phases is vital for various applications, like distillation and chemical engineering processes.Through the stoichiometric calculations of Raoult's Law, one derives the mole fractions in the vapor phase:

• The mole fraction of hexane, \( y_{C_6H_{14}} \), is calculated as \( \frac{75.7}{124.5} \approx 0.608 \).
• The mole fraction of cyclohexane, \( y_{C_6H_{12}} \), is \( \frac{48.8}{124.5} \approx 0.392 \).

These calculations illustrate the principle that the vapor phase contains a higher proportion of the more volatile component, hexane, as indicated by its higher mole fraction compared to cyclohexane.