Problem 65 - Chemistry: Atoms First

Question

The vapor pressure of methanol, \(\mathrm{CH}_{3} \mathrm{OH}\), is 94 torr at \(20^{\circ} \mathrm{C}\). The vapor pressure of ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\), is 44 torr at the same temperature. (a) Calculate the mole fraction of methanol and of ethanol in a solution of \(50.0 \mathrm{g}\) of methanol and \(50.0 \mathrm{g}\) of ethanol. (b) Ethanol and methanol form a solution that behaves like an ideal solution. Calculate the vapor pressure of methanol and of ethanol above the solution at \(20^{\circ} \mathrm{C}\). (c) Calculate the mole fraction of methanol and of ethanol in the vapor above the solution.

Step-by-Step Solution

Answer

Mole fraction methanol: 0.630, Mole fraction ethanol: 0.370; Vapor pressure of methanol: 59.22 torr, Vapor pressure of ethanol: 16.28 torr; Mole fraction of methanol in vapor: 0.784, Mole fraction of ethanol in vapor: 0.216.

Step 1: Calculate moles of methanol and ethanol

First, calculate the number of moles of methanol (CH3OH) and ethanol (C2H5OH) using their molecular weights. For methanol (CH3OH), the molar mass is approximately 32.04 g/mol, and for ethanol (C2H5OH), it is approximately 46.07 g/mol.

Step 2: Find the mole fraction of methanol and ethanol in the solution

The mole fraction for a component in a solution is calculated by dividing the number of moles of that component by the total number of moles in the solution. Use this formula to determine the mole fractions of methanol (CH3OH) and ethanol (C2H5OH).

Step 3: Calculate the vapor pressure of methanol in the solution

Using Raoult's Law, which states that the partial vapor pressure of each component in an ideal solution is equal to the vapor pressure of the pure component multiplied by its mole fraction in the solution.

Step 4: Calculate the vapor pressure of ethanol in the solution

Apply Raoult's Law again to find the partial vapor pressure of ethanol in the solution.

Step 5: Determine the total vapor pressure of the solution

The total vapor pressure of the solution is the sum of the partial vapor pressures of methanol and ethanol.

Step 6: Calculate the mole fraction of each component in the vapor

Use Dalton's Law of Partial Pressures to find the mole fractions of methanol and ethanol in the vapor above the solution.

Key Concepts

Raoult's Law

Raoult's Law is a principle of thermodynamics used for predicting the vapor pressures of ideal solutions. This law states that the partial vapor pressure of each component in a solution is directly proportional to its mole fraction. Raoult's Law is applicable to solutions where the intermolecular forces between different molecules are similar to those in the pure substances. In the context of our methanol and ethanol mixture, Raoult's Law allows us to determine the vapor pressure of each component, assuming the solution behaves ideally. The vapor pressure of each component is calculated by multiplying the mole fraction of the component by the vapor pressure of the pure substance at the same temperature.

Mole Fraction

In chemistry, the mole fraction is an expression of the concentration of a component in a mixture. It is defined as the ratio of moles of that component to the total moles of all the components in the mixture. This dimensionless quantity is used in various chemical calculations, including the application of Raoult's Law. In our problem, finding the mole fraction of methanol and ethanol is a necessary step before we can compute the vapor pressures of each substance in the solution. The mole fraction is calculated by dividing the moles of one component by the sum of moles of all components.

Ideal Solution Behavior

An ideal solution is a theoretical model in which the intermolecular forces between like and unlike molecules are equal. It allows for the simplification of many calculations because it implies that the solution follows Raoult's Law perfectly without deviations. In an ideal solution, the properties of the components mix additively. This means that the enthalpy change upon mixing is zero and the volume of mixing is also negligible. When dealing with actual substances, like methanol and ethanol in our exercise, we often assume ideal behavior to simplify the calculation of properties such as vapor pressures.

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures is an important concept in chemistry, particularly in the study of gases. It states that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of each individual gas in the mixture. The partial pressure of a gas is the pressure that gas would exert if it alone occupied the entire volume of the mixture. In the context of a solution's vapor phase, Dalton's Law helps us calculate the composition of the vapor by considering the contributions of each volatile component's partial pressure.

Molecular Weight Calculations

To determine the number of moles of a substance, we often use its molecular weight, which is the sum of the atomic weights of each atom in a molecule. It is expressed in units of grams per mole (g/mol). For our textbook exercise, understanding the molecular weights of methanol and ethanol is crucial for calculating their respective moles. Once we have the weight of each substance in the solution, we divide it by the molecular weight to find the number of moles. This enables us to proceed with calculating mole fractions and using Raoult's Law to find the vapor pressures.