Problem 22 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

The vapour pressure of pure \(\mathrm{CHCl}_{3}\) at \(318 \mathrm{K}\) is \(58 \mathrm{kPa}\). What would be the partial vapour pressure of \(\mathrm{CHCl}_{3}\) above a mixture of \(1 \mathrm{mol}\) of \(\mathrm{CHCl}_{3}\) with \(1 \mathrm{mol}\) of ethanol, assuming the mixture behaves as an ideal solution? Comment on the fact that the measured value of the partial vapour pressure of \(\mathrm{CHCl}_{3}\) above the mixture is \(42 \mathrm{kPa} \text { . (Section } 17.4)\).

Step-by-Step Solution

Answer

The ideal partial vapor pressure is 29 kPa, lower than the measured 42 kPa, indicating non-ideal behavior.

Step 1: Understand Raoult's Law for Ideal Solutions

Raoult's Law states that the partial vapor pressure of a component in a mixture is equal to the mole fraction of the component multiplied by its pure component vapor pressure. For a component \( A \), this is expressed as \( P_A = x_A \times P_A^0 \), where \( P_A \) is the partial vapor pressure of \( A \), \( x_A \) is the mole fraction of \( A \), and \( P_A^0 \) is the vapor pressure of pure \( A \).

Step 2: Calculate Mole Fraction of Chloroform

To find the mole fraction of \( \mathrm{CHCl_3} \), use its definition: \( x_{CHCl_3} = \frac{n_{CHCl_3}}{n_{CHCl_3} + n_{ethanol}} \). Given that there is 1 mol of \( \mathrm{CHCl_3} \) and 1 mol of ethanol, the total moles are 2. Thus, \( x_{CHCl_3} = \frac{1}{2} \).

Step 3: Apply Raoult's Law

Using Raoult's Law, calculate the partial vapor pressure of \( \mathrm{CHCl_3} \) in the solution: \( P_{CHCl_3} = x_{CHCl_3} \times P_{CHCl_3}^0 \). With \( P_{CHCl_3}^0 = 58 \text{ kPa} \) and \( x_{CHCl_3} = \frac{1}{2} \), we compute \( P_{CHCl_3} = \frac{1}{2} \times 58 = 29 \text{ kPa} \).

Step 4: Analyze the Discrepancy with the Measured Data

The calculated partial pressure of \( 29 \text{ kPa} \) is lower than the measured value of \( 42 \text{ kPa} \). This discrepancy suggests that the mixture does not behave ideally or that there are significant interactions between \( \mathrm{CHCl_3} \) and ethanol that cause an increase in the partial vapor pressure.

Key Concepts

Vapour Pressure

Vapour pressure is a critical concept in understanding how substances evaporate and subsequently exert pressure in a closed system. It refers to the pressure exerted by a vapor in thermodynamic equilibrium with its liquid at a given temperature. If you imagine a sealed container with a liquid inside, some of the liquid molecules will escape into the gas phase. These molecules exert a pressure on the walls of the container, which is known as vapour pressure.

Several factors influence vapour pressure:

• Temperature: As temperature increases, vapour pressure also increases. More molecules have enough kinetic energy to escape the liquid phase.
• Nature of the liquid: Substances with weaker intermolecular forces will have higher vapour pressures at a given temperature compared to those with stronger forces.

Understanding vapour pressure helps explain several phenomena in chemistry, including why certain liquids evaporate more readily than others. It's also a key element in studying how mixtures behave when combined, especially when considering their potential as ideal or non-ideal solutions.

Ideal Solutions

Ideal solutions are a fundamental concept when discussing mixtures in chemistry. An ideal solution is one where the components mix without any net energy change, meaning the intermolecular interactions between molecules of the mixed components are similar to those in the pure components. This assumption forms the basis of Raoult's Law, which predicts the behavior of the vapour pressures in such mixtures.

Raoult's Law can be mathematically expressed as: \[ P_A = x_A \times P_A^0 \]Here, \( P_A \) is the partial vapour pressure of component \( A \), \( x_A \) is the mole fraction of \( A \) in the solution, and \( P_A^0 \) is the vapour pressure of pure \( A \). Ideal solutions assume that the interactions are uniform and predictable, providing a simplified model to calculate the vapour pressures of the individual components in the mixture.

In real-world applications, many mixtures do not behave ideally due to differences in molecular size, polarity, or other interactions, as seen in the chloroform and ethanol mixture. Deviations from ideality can indicate strong molecular interactions which alter the expected vapor pressure.

Mole Fraction

Mole fraction is an essential concept in chemistry, particularly in solutions and mixtures. It represents the ratio of the number of moles of one component to the total number of moles of all components in the mixture. It is a dimensionless quantity that provides insight into the composition of a solution.

The formula to calculate mole fraction is:\[ x_i = \frac{n_i}{n_{total}} \]where \( x_i \) is the mole fraction of component \( i \), \( n_i \) is the number of moles of component \( i \), and \( n_{total} \) is the total number of moles in the solution.

Understanding mole fraction is crucial when applying Raoult's Law, particularly in predicting how each component's vapor pressure will contribute to the total pressure of a solution. For example, in a system with equal moles of chloroform and ethanol, each component would have a mole fraction of 0.5, reflecting an equal contribution of moles between the different elements in the mixture. This calculation is vital for determining the partial vapor pressures and understanding the solution’s behavior.