Question

Liquid A has vapor pressure \(x\), and liquid B has vapor pressure \(y\). What is the mole fraction of the liquid mixture if the vapor above the solution is \(30 . \%\) A by moles? \(50 . \%\) A? \(80 . \%\) A? (Calculate in terms of \(x\) and \(y .)\) Liquid A has vapor pressure \(x\), liquid B has vapor pressure \(y\). What is the mole fraction of the vapor above the solution if the liquid mixture is \(30 . \%\) A by moles? \(50 . \%\) A? \(80 . \% \mathrm{~A}\) ? (Calculate in terms of \(x\) and \(y .\) )

Short answer

In conclusion, the mole fractions of A in the vapor phase for each case are: 1. 30% A by moles: \( YA=\frac{0.30x}{0.30x + 0.70y} \) 2. 50% A by moles: \( YA=\frac{0.50x}{0.50x + 0.50y} \) 3. 80% A by moles: \( YA=\frac{0.80x}{0.80x + 0.20y} \)

Step-by-step solution

Determining the mole fractions of A and B in the liquid mixture for each case

First, we need to find the mole fractions of A (XA) and B (XB) in the liquid mixture for each case. For 30% A by moles, XA = 0.30 and XB = 1 - XA = 0.70. For 50% A by moles, XA = 0.50 and XB = 1 - XA = 0.50. For 80% A by moles, XA = 0.80 and XB = 1 - XA = 0.20.

Applying Raoult's Law

Now, we apply Raoult's law, which states the partial pressure of a component in a solution is equal to the mole fraction of that component in the liquid phase multiplied by its vapor pressure in the pure state. For liquid A, PA = XA * x For liquid B, PB = XB * y

Calculating the total vapor pressure of the solution

Next, to find the total vapor pressure of the solution (P_total), we sum the partial pressures of A and B. P_total = PA + PB

Calculating the mole fractions of A and B in the vapor phase

Finally, we can find the mole fractions of A and B in the vapor phase using the partial pressures and the total vapor pressure. For vapor A, YA = PA / P_total For vapor B, YB = PB / P_total Now we will calculate the mole fraction of vapor A (YA) for each case.

Case 1: 30% A by moles

XA = 0.30 XB = 0.70 PA = 0.30 * x PB = 0.70 * y P_total = PA + PB = 0.30 * x + 0.70 * y YA = (0.30 * x) / (0.30 * x + 0.70 * y)

Case 2: 50% A by moles

XA = 0.50 XB = 0.50 PA = 0.50 * x PB = 0.50 * y P_total = PA + PB = 0.50 * x + 0.50 * y YA = (0.50 * x) / (0.50 * x + 0.50 * y)

Case 3: 80% A by moles

XA = 0.80 XB = 0.20 PA = 0.80 * x PB = 0.20 * y P_total = PA + PB = 0.80 * x + 0.20 * y YA = (0.80 * x) / (0.80 * x + 0.20 * y) In conclusion, the mole fractions of A in the vapor phase for each case are: 1. 30% A by moles: \( \displaystyle YA=\frac{0.30x}{0.30x + 0.70y} \) 2. 50% A by moles: \( \displaystyle YA=\frac{0.50x}{0.50x + 0.50y} \) 3. 80% A by moles: \( \displaystyle YA=\frac{0.80x}{0.80x + 0.20y} \)

Key concepts

Vapor Pressure

Vapor pressure is a measure of a liquid's tendency to evaporate into a gas. It is defined as the pressure exerted by the vapor in equilibrium with its liquid at a given temperature. The term ‘equilibrium’ here means that the rate of evaporation of the liquid equals the rate of condensation from the vapor phase.

Each liquid has its unique vapor pressure, which depends on the substance's intrinsic properties and temperature. In a mixture of liquids, each component contributes to the total vapor pressure according to its mole fraction presence in the mixture, which leads us to Raoult's Law. This law allows us to calculate the partial vapor pressures and understand how a mixture will behave when exposed to different conditions.

Mole Fraction

Mole fraction is an important concept in chemistry, used extensively when studying mixtures and solutions. It is a dimensionless quantity that represents the ratio of the number of moles of a particular component to the total number of moles of all components in the mixture.

In our exercise, to calculate the mole fraction, we simply take the percentage given for liquid A and convert it to a decimal. For instance, 30% A by moles translates to a mole fraction (XA) of 0.30 for A, meaning that for every 100 moles of the mixture, 30 are of A and the remaining 70 are of another component, typically referred to as B, leading to its mole fraction (XB) being 0.70.

Partial Pressure

Partial pressure arises from Dalton’s Law, which states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases. This concept can be applied to vapors over a liquid mixture.

According to Raoult's Law, the partial pressure (PA) of a component A is defined as the product of its mole fraction (XA) in the liquid phase and its pure component vapor pressure (x). Therefore, for a liquid mixture with a known composition, we can use the mole fractions and pure vapor pressures to determine the contribution of each liquid to the total vapor pressure, aiding in the prediction of the behavior of mixtures under various conditions.

Liquid-Vapor Equilibrium

Liquid-vapor equilibrium is a state where a liquid and its vapor are in balance, with the rate of evaporation equal to the rate of condensation. At this point, the system's vapor pressure is the highest possible at a given temperature without added heat, which is representative of the liquid's volatility.

In the context of our exercise, when mixing two volatile liquids like A and B, the final vapor pressure is influenced by the vapor pressures of the pure components and their amounts in the mixture. Raoult’s Law is instrumental here, as it allows us to relate the composition of the liquid phase to the composition of the vapor phase at equilibrium, thus understanding the behavior of each component within the mixture.