Problem 1 - Chemistry and Chemical Reactivity

Question

Give the relative rates of disappearance of reactants and formation of products for each of the following reactions. (a) \(2 \mathrm{O}_{3}(\mathrm{g}) \longrightarrow 3 \mathrm{O}_{2}(\mathrm{g})\) (b) \(2 \mathrm{HOF}(\mathrm{g}) \longrightarrow 2 \mathrm{HF}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g})\)

Step-by-Step Solution

Answer

(a) \(R_{\mathrm{O}_{3}} = -\frac{1}{2} \frac{d[\mathrm{O}_{3}]}{dt}\), \(R_{\mathrm{O}_{2}} = \frac{1}{3} \frac{d[\mathrm{O}_{2}]}{dt}\); (b) \(R_{\mathrm{HOF}} = -\frac{1}{2} \frac{d[\mathrm{HOF}]}{dt}\), \(R_{\mathrm{HF}} = \frac{1}{2} \frac{d[\mathrm{HF}]}{dt}\), \(R_{\mathrm{O}_{2}} = \frac{d[\mathrm{O}_{2}]}{dt}\).

Step 1: Understanding Relative Rates

For a reaction, the relative rates of disappearance or formation can be understood in terms of stoichiometry from the balanced chemical equation. The general formula involves using the stoichiometric coefficients to express these rates. The rate of change in concentration for a component can be expressed as \(\frac{1}{\text{coefficient}} \times \text{rate of reaction}\).

Step 2: Calculate Relative Rates for Reaction (a)

(a) For the reaction \(2 \mathrm{O}_{3}(\mathrm{g}) \rightarrow 3 \mathrm{O}_{2}(\mathrm{g})\), the rate of disappearance of \(\mathrm{O}_{3}\) is \(-\frac{1}{2} \frac{d[\mathrm{O}_{3}]}{dt}\). The rate of formation of \(\mathrm{O}_{2}\) is \(+\frac{1}{3} \frac{d[\mathrm{O}_{2}]}{dt}\). These expressions reflect the stoichiometric coefficients of 2 for \(\mathrm{O}_{3}\) and 3 for \(\mathrm{O}_{2}\).

Step 3: Calculate Relative Rates for Reaction (b)

(b) For the reaction \(2 \mathrm{HOF}(\mathrm{g}) \rightarrow 2 \mathrm{HF}(\mathrm{g}) + \mathrm{O}_{2}(\mathrm{g})\), the rate of disappearance of \(\mathrm{HOF}\) is \(-\frac{1}{2} \frac{d[\mathrm{HOF}]}{dt}\). The rate of formation of \(\mathrm{HF}\) is \(+\frac{1}{2} \frac{d[\mathrm{HF}]}{dt}\), and for \(\mathrm{O}_{2}\), it is \(+\frac{1}{1} \frac{d[\mathrm{O}_{2}]}{dt}\). The rates are divided by the stoichiometric coefficients from the balanced equation.

Key Concepts

Stoichiometry

Stoichiometry involves the study of the quantitative relationships between reactants and products in a chemical reaction. It helps us understand how much of each substance is involved or produced, which is vital for calculating relative reaction rates.
The stoichiometric coefficients in a balanced chemical equation tell us the proportions of reactants and products. For example, in the reaction \(2 \mathrm{O}_{3}(\mathrm{g}) \rightarrow 3 \mathrm{O}_{2}(\mathrm{g})\), the coefficient 2 for \(\mathrm{O}_{3}\) and 3 for \(\mathrm{O}_{2}\) informs us that 2 moles of ozone yield 3 moles of oxygen gas.
This relationship helps determine how the rate of disappearance and formation are related. Stoichiometry ultimately provides the foundation to write expressions for reaction rates, allowing us to predict how quickly a reaction proceeds.

Rate of Disappearance

The rate of disappearance describes how quickly a reactant is consumed in a chemical reaction. It's a measure of the decrease in concentration of a reactant over time.
To find this rate, you use the stoichiometric coefficient from the equation. For example, with \(2 \mathrm{O}_{3}(\mathrm{g}) \rightarrow 3 \mathrm{O}_{2}(\mathrm{g})\), the rate at which \(\mathrm{O}_{3}\) disappears is:

• \(-\frac{1}{2} \frac{d[\mathrm{O}_{3}]}{dt}\)

This formula means the concentration of \(\mathrm{O}_{3}\) decreases by half of the rate of the overall reaction.
Negative signs indicate a reduction in reactant concentration. Understanding the rate of disappearance is important when determining how fast a reaction uses up reactants.

Balanced Chemical Equation

A balanced chemical equation is crucial as it represents the law of conservation of mass. It shows that atoms are neither created nor destroyed during a chemical reaction; they are simply rearranged.
Balancing equations involves ensuring the same number of each type of atom on both sides. This gives us the correct stoichiometric coefficients, which are essential for calculating reaction rates.
In reactions like \(2 \mathrm{HOF}(\mathrm{g}) \rightarrow 2 \mathrm{HF}(\mathrm{g}) + \mathrm{O}_{2}(\mathrm{g})\), each atom and molecule is accounted for:

• 2 molecules of \(\mathrm{HOF}\) yield 2 molecules of \(\mathrm{HF}\) and 1 molecule of \(\mathrm{O}_{2}\).

This balance assures accurate calculations of reaction rates and provides a clear understanding of the chemical process.

Rate of Formation

The rate of formation refers to the speed at which a product is generated in a chemical reaction. It is proportional to how rapidly the concentration of the product increases over time.
Using the stoichiometric coefficients, as with the rate of disappearance, we can determine this rate. For example, the \(\mathrm{O}_{2}\) formation in the reaction \(2 \mathrm{O}_{3}(\mathrm{g}) \rightarrow 3 \mathrm{O}_{2}(\mathrm{g})\) is:

• \(+\frac{1}{3} \frac{d[\mathrm{O}_{2}]}{dt}\)

Here, the positive sign shows an increase in concentration, indicating product generation.
Understanding the rate of formation helps in predicting how much product can be obtained in a given time, which is crucial for industrial and laboratory applications.