Question

Imagine that the reaction \(2 \mathrm{CO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{CO}_{2}(g)\) occurs in a container that has a piston that moves to maintain a constant pressure when the reaction occurs at constant temperature. Which of the following statements describes how the volume of the container changes due to the reaction: (a) the volume increases by \(50 \%,\) (b) the volume increases by \(33 \%\), (c) the volume remains constant, (d) the volume decreases by \(33 \%,(\mathbf{e})\) the volume decreases by \(50 \%\). [Sections 10.3 and 10.4\(]\)

Short answer

The volume of the container decreases by approximately 33% due to the reaction, as the number of gas moles decreases from 3 to 2 moles. Therefore, the correct answer is (d).

Step-by-step solution

Calculate Moles on Both Sides of Reaction

Before the reaction, there are 2 moles of carbon monoxide (CO) and 1 mole of oxygen (O2). So, there are a total of 3 moles of gas before the reaction. After the reaction, there are 2 moles of carbon dioxide (CO2).

Compare Moles on Both Sides

Now, let's compare the moles of gas before and after the reaction: Moles before reaction = 3 moles Moles after reaction = 2 moles

Calculate Change in Volume

The number of moles of gas decreased from 3 to 2. At constant pressure and temperature, the volume of the gas is directly proportional to the number of moles. Since we observe that the moles are reduced by 1 mole, the percentage decrease in volume can be calculated by: Percentage decrease in volume = \(\frac{(initial \ moles - final \ moles)}{initial \ moles} \times 100\) Percentage decrease in volume = \(\frac{(3 - 2)}{3} \times 100\) Percentage decrease in volume = \(33.33 \% \) The volume decreases by approximately 33%. Therefore, the correct answer is (d).

Key concepts

Moles of Gas

Moles are a fundamental unit in chemistry that describe the amount of a substance. For gases, this concept becomes particularly vital because it connects to volume under the conditions of constant temperature and pressure. According to Avogadro's Law, equal volumes of gases at the same temperature and pressure contain the same number of moles, or particles. This principle is reflected in the Ideal Gas Law:\[ PV = nRT \]where:

• \( P \) is the pressure,
• \( V \) is the volume,
• \( n \) is the number of moles,
• \( R \) is the ideal gas constant,
• \( T \) is the temperature.

Therefore, understanding the moles of gas involved in a chemical reaction allows us to predict and calculate changes in the volume of gas produced or consumed. In the scenario given, before the reaction, there were 3 moles of gas (2 moles of CO and 1 mole of O2) involved, which reduced to 2 moles of CO2 after the reaction. This decrease directly impacts the volume due to Avogadro's Law, where fewer moles imply a smaller volume when pressure and temperature are constant.

Reaction Stoichiometry

Stoichiometry is the branch of chemistry that involves the calculation of reactants and products in chemical reactions. It uses the balanced equation to derive relationships between the quantities of reactants and products. In the balanced chemical equation provided:\[ 2 \ \text{CO} (g) + \text{ O}_2 (g) \rightarrow 2 \ \text{CO}_2 (g) \]The stoichiometric coefficients (the numbers in front of the molecules) indicate the proportions in which the reactants combine and the products form. Here, two moles of carbon monoxide react with one mole of oxygen to form two moles of carbon dioxide. Understanding stoichiometry is essential because it tells us how matter rearranges in a reaction:

• It provides conservation of mass, ensuring total atoms present before the reaction are equal to the total atoms after.
• Informs us that, for this reaction, three moles of gas react to produce two moles of gas, a change that's crucial for calculating volume changes.

This reduction in the number of moles under constant temperature and pressure conditions leads to a corresponding decrease in volume, as seen in our initial exercise.

Volume Change in Gases

The concept of volume change in gases is closely tied to the relationship established by the Ideal Gas Law. When pressure and temperature are held constant, the volume of a gas is directly proportional to its number of moles, as expressed in Avogadro's Law. This means that if the number of moles of gas decreases, the volume decreases as well. In the exercise scenario, we see a reaction where the initial 3 moles of CO and O2 become 2 moles of CO2 after the reaction:

• Moles before the reaction: 3
• Moles after the reaction: 2

The formula to calculate the percentage change in volume is:\[\text{Percentage decrease in volume} = \frac{(\text{initial moles} - \text{final moles})}{\text{initial moles}} \times 100\]Applying this formula:\[\frac{(3 - 2)}{3} \times 100 = 33.33\%\]This calculation shows that the volume decreases by approximately 33%, reflecting the direct relationship between gas volume and the number of moles at constant temperature and pressure. Therefore, understanding these principles allows us to quickly determine the nature of volume changes in similar scenarios.