Problem 27 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

A solution of iodoethane, \(1,\) and propanone, \(P\), with a mole fraction of \(I, x_{1}=0.55,\) had a partial vapour pressure of I of \(28.44 \mathrm{kPa}\) and a partial vapour pressure of \(\mathrm{P}\) of \(19.21 \mathrm{kPa}\) at \(50^{\circ} \mathrm{C} .\) At this temperature, the saturated vapour pressure of I is \(47.12 \mathrm{kPa}\) and of \(\mathrm{P}\) is \(37.38 \mathrm{kPa}\). Calculate the activity coefficients of both components in the solution. (Section 17.4 )

Step-by-Step Solution

Answer

The activity coefficients are \( \gamma_1 = 1.097 \) for iodoethane and \( \gamma_2 = 1.142 \) for propanone.

Step 1: Understand the Given Values

We are provided with the mole fraction of iodoethane (I), which is \( x_1 = 0.55 \). The partial vapor pressures of I and propanone (P) are \( P_1 = 28.44 \text{ kPa} \) and \( P_2 = 19.21 \text{ kPa} \), respectively. The saturated vapor pressures at \( 50^\circ \text{C} \) are \( P_1^0 = 47.12 \text{ kPa} \) for I and \( P_2^0 = 37.38 \text{kPa} \) for P.

Step 2: Activity Coefficient Formulation

The activity coefficient \( \gamma_i \) of a component is calculated using the formula:\[\gamma_i = \frac{P_i}{x_i P_i^0}\]where \( P_i \) is the partial vapor pressure of the component, \( x_i \) is the mole fraction, and \( P_i^0 \) is the saturated vapor pressure of the component.

Step 3: Calculate the Activity Coefficient of Iodoethane

Substitute the values for iodoethane into the formula:\[\gamma_1 = \frac{28.44}{0.55 \times 47.12} \]Calculating this gives:\[\gamma_1 = \frac{28.44}{25.916} = 1.097\]

Step 4: Calculate the Mole Fraction of Propanone

Since the mole fraction of iodoethane, \( x_1 = 0.55 \), the mole fraction of propanone, \( x_2 \), is \( 1 - x_1 = 0.45 \).

Step 5: Calculate the Activity Coefficient of Propanone

Substitute the values for propanone into the formula:\[\gamma_2 = \frac{19.21}{0.45 \times 37.38} \]Calculating this gives:\[\gamma_2 = \frac{19.21}{16.821} = 1.142\]

Key Concepts

Partial Vapor Pressure

Partial vapor pressure plays a crucial role in understanding mixtures of gases or solutions. It refers to the pressure that a single component of a mixture contributes to the total pressure, assuming it occupies the entire volume at the given temperature.
In our exercise, the partial vapor pressure of iodoethane is given as 28.44 kPa, while that for propanone is 19.21 kPa. These values tell us how much each component is contributing to the mixture's overall vapor pressure in the solution.
This concept is essential because it can help us understand how each component in a mixture behaves and interacts with each other, especially in terms of their tendency to escape into the gas phase.

Mole Fraction

Mole fraction is a way to express the concentration of a component in a mixture. It is defined as the ratio of moles of one component to the total moles of all components in the mixture.
For example, the mole fraction of iodoethane in the given solution is 0.55, meaning that iodoethane makes up 55% of the total number of moles in the solution.

• This is crucial for calculating other properties of mixtures, like partial pressures, because it gives a clear perspective of the mixture’s composition.
• In binary solutions, such as our example with iodoethane and propanone, knowing one component's mole fraction allows us to easily find the other as the sum of the mole fractions must always equal one.

Understanding mole fraction is fundamental when employing Raoult's Law and calculating activity coefficients to understand a solution's behavior better.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid (or solid) phase at a given temperature. It is a fundamental physical property of substances.
For pure substances, vapor pressure depends on the temperature; therefore, it's often given at a specific temperature.

• In this exercise, the saturated vapor pressure of iodoethane at 50°C is 47.12 kPa, and that for propanone is 37.38 kPa.
• A higher vapor pressure indicates a greater tendency of the substance to evaporate.

These values help us evaluate how close the actual solution’s behavior is to ideality, based on Raoult's Law.

Raoult's Law

Raoult's Law describes the relationship between the vapor pressure of an ideal solution and the properties of its individual components. It states that the partial vapor pressure of a component in a solution is directly proportional to its mole fraction and its vapor pressure when pure.
The formula can be expressed as: \[ P_i = x_i P_i^0 \]where:

• \( P_i \) is the partial vapor pressure of the component.
• \( x_i \) is the mole fraction of the component.
• \( P_i^0 \) is the vapor pressure of the pure component.

However, real solutions often deviate from this ideal behavior due to molecular interactions, which is where the concept of activity coefficients comes into play. These coefficients, calculated in this exercise, help adjust Raoult's Law to fit real-world observations by accounting for the non-ideality in mixtures.