Problem 43 - Chemistry The Science in Context

Question

At \(20^{\circ} \mathrm{C},\) the vapor pressure of cthanol is 45 torr and the vapor pressure of methanol is 92 torr. What is the vapor pressure at \(20^{\circ} \mathrm{C}\) of a solution prepared by mixing \(25 \mathrm{g}\) of methanol and \(75 \mathrm{g}\) of ethanol?

Step-by-Step Solution

Answer

Answer: The vapor pressure of the solution at 20°C is 60.2 torr.

Step 1: Calculate the mole fractions of ethanol and methanol

We are given 25 grams of methanol and 75 grams of ethanol. We need to find the mole fraction of each component in the solution. To do this, we first need the molar mass of each component: - Molar mass of methanol (CH3OH): 12.01 (C) + 3x1.01 (H) + 16.00 (O) + 1.01 (H) = 32.04 g/mol - Molar mass of ethanol (C2H5OH): 12.01x2 (C) + 5x1.01 (H) + 16.00 (O) + 1.01 (H) = 46.07 g/mol Next, we find the moles of each component in the solution: - Moles of methanol: 25 g / 32.04 g/mol = 0.780 mol - Moles of ethanol: 75 g / 46.07 g/mol = 1.628 mol Now we can find the mole fraction of each component: - Mole fraction of methanol (X_methanol): 0.780 mol / (0.780 + 1.628) mol = 0.324 - Mole fraction of ethanol (X_ethanol): 1.628 mol / (0.780 + 1.628) mol = 0.676

Step 2: Apply Raoult's Law to find the partial vapor pressures

Raoult's Law states that the partial vapor pressure of a component in an ideal solution is equal to the mole fraction of that component multiplied by its vapor pressure in the pure state. We are given the pure vapor pressure of methanol (92 torr) and ethanol (45 torr). To calculate the partial vapor pressure of each component in the solution, use the following formula: Partial vapor pressure = mole fraction x pure vapor pressure - Partial vapor pressure of methanol: 0.324 x 92 torr = 29.8 torr - Partial vapor pressure of ethanol: 0.676 x 45 torr = 30.4 torr

Step 3: Calculate the total vapor pressure of the solution

The total vapor pressure of the solution is the sum of the partial vapor pressures of each component. So, we just need to add the partial vapor pressures of methanol and ethanol: Total vapor pressure = partial vapor pressure of methanol + partial vapor pressure of ethanol Total vapor pressure = 29.8 torr + 30.4 torr = 60.2 torr Therefore, the vapor pressure of the solution at 20°C is 60.2 torr.

Key Concepts

Vapor Pressure Concept

Vapor pressure is an important concept in chemistry, particularly when dealing with solutions and mixtures. It refers to the pressure exerted by a vapor when it is in equilibrium with its liquid form at a given temperature. This pressure is a measure of a liquid's tendency to evaporate and is determined by the intermolecular forces present.

Stronger forces within the liquid mean fewer molecules escape into the vapor phase, resulting in lower vapor pressure. Conversely, weaker intermolecular attractions allow for more evaporation, leading to higher vapor pressure.

• In pure substances, the vapor pressure is specific to a given temperature.
• For a mixture of substances, the overall vapor pressure is influenced by the individual vapor pressures of the components.

When a solution is formed, the presence of a solute can alter the vapor pressure of the solvent, often lowering it in comparison to its pure state. This behavior is described and quantified using Raoult's Law, which allows us to predict how the vapor pressures combine in a mixture.

Mole Fractions and Their Role

Mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the number of moles of one component divided by the total number of moles of all components in the mixture. This measure is unitless and gives us a simple ratio expressing how much of the total mixture is made up by a particular component.

The mole fraction is particularly useful because it directly relates to vapor pressure through Raoult's Law, providing a means to calculate how each component's potential to evaporate contributes to the overall vapor pressure.

• The mole fraction of a component can vary from 0 to 1, where 0 means it is absent, and 1 means it is the only substance present.
• In solutions, the sum of mole fractions of all components equals 1.

Calculating the mole fraction requires the knowledge of the amount (in moles) of each component, which can be found from the mass and molar mass for each substance, as seen in the exercise example for ethanol and methanol.

Understanding Partial Vapor Pressures

Partial vapor pressures refer to the individual pressures exerted by each component in a mixture. They are a key concept when using Raoult's Law to find the total vapor pressure of a solution. To determine the partial vapor pressure of each component, multiply its mole fraction by its vapor pressure in the pure state.

This gives insight into how each component contributes to the overall pressure observed.

• Partial pressures reflect the behavior and interaction of components within the mixture as they evaporate.
• The concept of partial pressure is crucial when understanding and predicting the behaviors of gases and volatile liquids.

In our example, the partial vapor pressures calculated for methanol and ethanol show how each contributes to the total vapor pressure, allowing us to understand and predict the behavior of the liquid mixture more accurately. This breakdown into partial components is essential for chemical engineers and scientists studying the thermodynamic properties of mixtures.