Problem 23 - Chemistry The Central Science In Si Units

Question

For each of the following gas-phase reactions, indicate how the rate of disappearance of each reactant is related to the rate of appearance of each product: (a) \(\mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g) \longrightarrow \mathrm{CO}_{2}(\mathrm{~g})+\mathrm{H}_{2}(\mathrm{~g})\) (b) \(2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) \longrightarrow 2 \mathrm{NOCl}(g)\) (c) \(\mathrm{CH}_{4}(g)+2 \mathrm{O}_{2}(\mathrm{~g}) \longrightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)\) (d) \(\mathrm{N}_{2} \mathrm{O}_{4}(g) \longrightarrow 2 \mathrm{NO}_{2}(g)\)

Step-by-Step Solution

Answer

(a) Rates are equal; (b) Rate of NO is twice that of Cl2; (c) Rate of O2 is twice that of CH4; (d) Rate of N2O4 is half of NO2.

Step 1: Reaction (a) Analysis

For the reaction \(\mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g) \longrightarrow \mathrm{CO}_{2}(\mathrm{~g})+\mathrm{H}_{2}(\mathrm{~g})\), the stoichiometry is 1:1 for both reactants and products. Thus, the rate of disappearance of CO is equal to the rate of disappearance of H2O, and both are equal to the rates of appearance of CO2 and H2. Symbolically, \( \frac{-d[\mathrm{CO}]}{dt} = \frac{-d[\mathrm{H}_{2} \mathrm{O}]}{dt} = \frac{d[\mathrm{CO}_{2}]}{dt} = \frac{d[\mathrm{H}_{2}]}{dt} \).

Step 2: Reaction (b) Analysis

For the reaction \(2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) \longrightarrow 2 \mathrm{NOCl}(g)\), the stoichiometry indicates that 2 moles of NO react with 1 mole of Cl2 to produce 2 moles of NOCl. Therefore, the rate of disappearance of NO is twice that of Cl2, and equal to the rate of appearance of NOCl. Thus, \( \frac{-1}{2} \frac{d[\mathrm{NO}]}{dt} = \frac{-d[\mathrm{Cl}_{2}]}{dt} = \frac{1}{2} \frac{d[\mathrm{NOCl}]}{dt} \).

Step 3: Reaction (c) Analysis

In the reaction \(\mathrm{CH}_{4}(g)+2 \mathrm{O}_{2}(\mathrm{~g}) \longrightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)\), 1 mole of CH4 reacts with 2 moles of O2, producing 1 mole of CO2 and 2 moles of H2O. Hence, the rate of disappearance of CH4 is half the rate of disappearance of O2, equals the rate of appearance of CO2, and half the rate of appearance of H2O: \( \frac{-d[\mathrm{CH}_{4}]}{dt} = \frac{-1}{2} \frac{d[\mathrm{O}_{2}]}{dt} = \frac{d[\mathrm{CO}_{2}]}{dt} = \frac{1}{2} \frac{d[\mathrm{H}_{2} \mathrm{O}]}{dt} \).

Step 4: Reaction (d) Analysis

For the decomposition \(\mathrm{N}_{2} \mathrm{O}_{4}(g) \longrightarrow 2 \mathrm{NO}_{2}(g)\), 1 mole of N2O4 produces 2 moles of NO2. The rate of disappearance of N2O4 is half the rate of appearance of NO2. Thus, \( \frac{-d[\mathrm{N}_{2} \mathrm{O}_{4}]}{dt} = \frac{1}{2} \frac{d[\mathrm{NO}_{2}]}{dt} \).

Key Concepts

Stoichiometry

Stoichiometry is a fundamental concept in chemistry that helps us predict the quantities of reactants and products in a chemical reaction. It is based on the balanced chemical equation and the principle that matter is conserved in chemical processes. A balanced equation shows the exact ratio of molecules or moles of reactants and products. This ratio is crucial because it tells us how many particles, or moles, of each substance are involved in the reaction.

• For example, in the reaction \(\text{CO}(g) + \text{H}_2 \text{O}(g) \rightarrow \text{CO}_2(g) + \text{H}_2(g)\), the stoichiometry is 1:1:1:1. This means each mole of reactant produces one mole of each product.
• In more complex reactions, such as \(2\text{NO}(g) + \text{Cl}_2 (g) \rightarrow 2\text{NOCl}(g)\), stoichiometry indicates that two moles of NO and one mole of \text{Cl}_2 produce two moles of \text{NOCl}. This helps determine how much \text{Cl}_2 is needed for a certain amount of \text{NO} to react completely.

Stoichiometry is essential for understanding how reactants convert into products and ensuring that all reactants are used efficiently without leaving excess waste.

Gas-phase Reactions

Gas-phase reactions involve reactants that are gases, reacting in a gaseous environment to form products, also in the gas phase. These reactions are particularly significant in various natural and industrial processes, such as combustion and atmospheric reactions. Understanding gas-phase reactions helps us to handle and predict these processes better.

• Such reactions are represented by balanced equations, indicating how gases, like methane and oxygen in \(\text{CH}_4(g) + 2 \text{O}_2 (g) \rightarrow \text{CO}_2(g) + 2 \text{H}_2 \text{O}(g)\), interact to produce carbon dioxide and water vapor.
• The balanced equation provides stoichiometric information that allows us to calculate the amounts of reactants needed and products formed. This is crucial for industrial applications where precise mixing and reactions yield desirable outcomes.

Moreover, in gas-phase reactions, factors such as pressure and temperature can influence the rate and direction of the reaction, emphasizing the need for controlled environmental conditions to achieve optimal results.

Rate of Disappearance

The rate of disappearance refers to the speed at which a reactant is consumed in a chemical reaction. This is measured by observing how quickly the concentration of the reactant decreases over time. The rate of disappearance is often represented as \(-d[\text{Reactant}]/dt\), wherein a negative sign indicates a decrease in concentration.

• In the reaction \(\text{N}_2\text{O}_4(g) \rightarrow 2 \text{NO}_2(g)\), the rate of disappearance of \text{N}_2\text{O}_4 is half the rate of appearance of \text{NO}_2, showing how stoichiometry directly affects reaction rates.
• This aspect is crucial in processes such as combustion, where the speed at which \text{O}_2 is consumed can determine the efficiency of the reaction.

Understanding the rate of disappearance helps in controlling reactions, predicting how fast reactions will reach completion, and designing processes that require specific timing and sequences.

Rate of Appearance

The rate of appearance is a measure of how fast a product forms in a chemical reaction. It is calculated based on the increase in concentration of the product over time and often indicated as \(d[\text{Product}]/dt\). In chemical kinetics, this is an essential parameter for analyzing reaction efficiency.

• For instance, in the reaction \(2 \text{NO}(g) + \text{Cl}_2 (g) \rightarrow 2 \text{NOCl}(g)\), the rate of appearance of \text{NOCl} is directly related to the rates at which \text{NO} and\text{Cl}_2 disappear. The stoichiometric coefficients guide this relationship, where the appearance rate is directly correlated with the amount of reactants.
• In industrial processes, understanding the rate of appearance allows for optimizing the generation of desired products while minimizing waste.

By comprehending this rate, chemists can better design and predict outcomes in both laboratory and large-scale chemical manufacturing settings.