A mixture of ethanol and 1-propanol behaves ideally at \(36^{\circ} \mathrm{C}\) and is in equilibrium with its vapor. If the mole fraction of ethanol in the solution is \(0.62,\) calculate its mole fraction in the vapor phase at this temperature. (The vapor pressures of pure ethanol and 1 -propanol at \(36^{\circ} \mathrm{C}\) are \(108 \mathrm{mmHg}\) and \(40.0 \mathrm{mmHg}\), respectively.

Understanding the partial pressure of a component in a mixture is crucial for many fields, including chemistry and environmental science. Partial pressure represents the pressure that a component of a mixture would exert if it occupied the entire volume of the mixture alone at the same temperature.

To calculate the partial pressure, we apply Raoult's Law, which is expressed in the equation: \( P_i = x_i \times P_i^0 \), where \( P_i \) is the partial pressure of component \(i\), \( x_i \) is the mole fraction of \(i\) in the liquid phase, and \( P_i^0 \) is the vapor pressure of the pure component \(i\). In our textbook exercise, the mole fraction of ethanol in the solution was given as 0.62, and the vapor pressures for pure ethanol and 1-propanol were provided, allowing us to compute their respective partial pressures.

For example, the partial pressure of ethanol was calculated using its mole fraction (0.62) and its vapor pressure as a pure substance (108 mmHg), leading to \( P_{ethanol} = 0.62 \times 108 \mathrm{mmHg} = 66.96 \mathrm{mmHg} \). Similarly, we calculated the partial pressure for 1-propanol. Partial pressure calculations are essential to identify how each component contributes to the total pressure of the system and to understand the distribution of molecules in vapor and liquid phases.

The mole fraction in the vapor phase gives us an understanding of the composition of the gaseous mixture above a liquid solution. It is crucial for predicting how a liquid will evaporate, which is important in fields like distillation and meteorology. In the context of our textbook problem, the mole fraction in the vapor phase is symbolized as \( y_i \) for component \(i\).

To determine the mole fraction in the vapor phase, one can use the calculated partial pressures. The mole fraction is calculated by the formula: \( y_i = \frac{P_i}{P_{total}} \), with \( P_i \) being the partial pressure of component \(i\) and \( P_{total} \) representing the total pressure of the solution. In our exercise, after finding the total pressure to be 82.16 mmHg, we could find the mole fraction of ethanol in the vapor phase by dividing its partial pressure (66.96 mmHg) by the total pressure. This resulted in a mole fraction in the vapor phase for ethanol of approximately 0.815.

This means that ethanol, due to its higher volatility, is more prevalent in the vapor phase compared to its presence in the liquid solution. This indicates that ethanol will evaporate more quickly than 1-propanol when a mixture of both is exposed to the same conditions.

Vapor-liquid equilibrium (VLE) describes a state in which a liquid and its vapor are in balance. At this point, the rate of evaporation equals the rate of condensation, meaning no net change in the amounts of liquid and vapor occurs over time. VLE is vital across various industries, such as chemical engineering and environmental studies, where the design and analysis of separation processes are essential.

In the exercise, a mixture of ethanol and 1-propanol at a certain temperature was considered to behave ideally, adhering to Raoult’s law. Ideal behavior implies that the interactions between like molecules are similar to those between unlike molecules, which is a simplification often used for mixtures of similar substances.

By reaching vapor-liquid equilibrium, the system achieves a stable state characterized by specific mole fractions in both the vapor and liquid phases. These fractions are dependent on the vapor pressures of the individual components and their interactions in the mixture. The mole fractions are used to calculate partial pressures and eventually get to a point where the mole fraction of a component in both phases can be determined, such as done for ethanol in our problem. Understanding VLE enables the prediction of how mixtures will behave when heated or cooled, a fundamental concept for numerous applications including distillation, extraction, and chromatography.