Problem 23 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

Ethanol and methanol form nearly ideal solutions. At \(20^{\circ} \mathrm{C}\), the vapour pressure of pure ethanol is 5930 Pa and of methanol is \(11830 \mathrm{Pa}\). (Section 17.4 ) Calculate (a) the vapour pressure of each compound and (b) the total vapour pressure over a solution formed by mixing \(50 g\) of each. \\[\left(M_{r}\left(\mathrm{CH}_{3} \mathrm{OH}\right)=32.04 ; \mathrm{M}_{r}\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)=46.07 .\right)\\]

Step-by-Step Solution

Answer

(a) Partial vapour pressures are \( P_{\text{methanol}} \) and \( P_{\text{ethanol}} \); (b) Total vapour pressure is \( P_{\text{total}} \).

Step 1: Calculate Moles of Methanol and Ethanol

We first need to determine the number of moles of each component in the solution. The molar mass of methanol is given as \( M_r(\mathrm{CH}_3\mathrm{OH}) = 32.04 \) g/mol, and ethanol is \( M_r(\mathrm{C}_2\mathrm{H}_5\mathrm{OH}) = 46.07 \) g/mol. Thus, the moles of methanol and ethanol are calculated by dividing their masses by their respective molar masses: \[ n_{\text{methanol}} = \frac{50}{32.04}, \quad n_{\text{ethanol}} = \frac{50}{46.07} \].

Step 2: Calculate Mole Fraction of Each Component

The mole fraction is calculated by dividing the number of moles of each component by the total number of moles: \[ X_{\text{methanol}} = \frac{n_{\text{methanol}}}{n_{\text{methanol}} + n_{\text{ethanol}}}, \quad X_{\text{ethanol}} = \frac{n_{\text{ethanol}}}{n_{\text{methanol}} + n_{\text{ethanol}}} \].

Step 3: Calculate Partial Vapour Pressure of Each Compound

According to Raoult's Law, the partial vapour pressure of a component in a solution is the product of its mole fraction and its pure vapour pressure. Hence, the partial vapour pressures are given by: \[ P_{\text{methanol}} = X_{\text{methanol}} \times 11830 \text{ Pa}, \quad P_{\text{ethanol}} = X_{\text{ethanol}} \times 5930 \text{ Pa} \].

Step 4: Calculate Total Vapour Pressure Over the Solution

The total vapour pressure over the solution is simply the sum of the partial vapour pressures of the individual components: \[ P_{\text{total}} = P_{\text{methanol}} + P_{\text{ethanol}} \].

Key Concepts

Vapour Pressure

Vapour pressure is an important concept in chemistry that deals with the pressure exerted by a vapour in equilibrium with its liquid or solid form. At a given temperature, some molecules in a liquid gain enough energy to escape into the gas phase. These escaping molecules create the vapour pressure of a substance. For instance, both ethanol and methanol have specific vapour pressures at room temperature due to their tendency to evaporate.
The higher the temperature, the more molecules can enter the vapour phase, increasing the vapour pressure. It's important because it tells us about the volatility of the liquid—the ease with which it evaporates.

• High vapour pressure means the liquid evaporates easily.
• Low vapour pressure indicates less evaporation.

Understanding vapour pressure helps to predict how a liquid will behave when mixed with other substances.

Ideal Solutions

Ideal solutions are mixtures where the different components follow Raoult’s Law perfectly. This means that the interactions between unlike molecules are the same as those between like molecules.
In simpler terms, when you mix two substances, like ethanol and methanol, if they form an ideal solution, neither substance changes its behaviour drastically due to the presence of the other.

• No change in volume when mixed.
• Partial vapour pressures are directly proportional to their mole fractions and pure vapour pressures.

Ideal solutions provide a simplified model, helping chemists predict the properties and behaviour of real mixtures, even if they don't strictly follow ideality.

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture. It is the ratio of the number of moles of one component to the total number of moles of all components in the solution.
This dimensionless number is crucial in calculating many properties of solutions, like partial vapour pressure. For instance:

• If the mole fraction of methanol is 0.4, then methanol makes up 40% of the solution in terms of moles.
• The sum of the mole fractions of all components in a mixture equals 1.

This simple, yet powerful concept underpins the calculation of other properties like the partial pressures in a solution.

Partial Vapour Pressure

Partial vapour pressure refers to the contribution each component in a mixture makes to the total vapour pressure. According to Raoult's Law, this can be calculated by multiplying the mole fraction of each component by its pure vapour pressure.
For example, if the mole fraction of methanol is 0.4 and its pure vapour pressure is 11830 Pa, its partial vapour pressure would be

• \[P_{\text{methanol}} = 0.4 \times 11830 \, \text{Pa}\].

Similarly, each component in the mixture has its partial pressure, which adds up to form the total vapour pressure of the solution.
This helps predict how much each component contributes to the evaporation from a liquid solution, crucial in applications like distillation.