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) \(2 \mathrm{O}_{3}(\mathrm{g}) \rightarrow 3 \mathrm{O}_{2}(\mathrm{g})\) (b) \(2 \mathrm{HOF}(\mathrm{g}) \rightarrow 2 \mathrm{HF}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g})\)

Step-by-Step Solution

Answer

(a) \(-\frac{1}{2}\frac{d[\mathrm{O}_3]}{dt} = \frac{1}{3}\frac{d[\mathrm{O}_2]}{dt}\); (b) \(-\frac{1}{2}\frac{d[\mathrm{HOF}]}{dt} = \frac{1}{2}\frac{d[\mathrm{HF}]}{dt} = \frac{d[\mathrm{O}_2]}{dt}\).

Step 1: Understanding Relative Rates

In a chemical reaction, the rate of disappearance of reactants is related to the rate of appearance of products by the stoichiometry of the reaction equation. For a reaction: \( aA + bB \rightarrow cC + dD \), the rates are written as: \( -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt} \).

Step 2: Applying to Reaction (a)

For the reaction \( 2 \mathrm{O}_{3}(\mathrm{g}) \rightarrow 3 \mathrm{O}_{2}(\mathrm{g}) \), compare the rate of disappearance of \( \mathrm{O}_{3} \) to the formation of \( \mathrm{O}_{2} \) using the stoichiometric coefficients: \( -\frac{1}{2}\frac{d[\mathrm{O}_3]}{dt} = \frac{1}{3}\frac{d[\mathrm{O}_2]}{dt} \). This equality reflects the stoichiometric balance indicating that for every 2 moles of \( \mathrm{O}_3 \) disappearing, 3 moles of \( \mathrm{O}_2 \) are formed.

Step 3: Applying to Reaction (b)

For the reaction \( 2 \mathrm{HOF}(\mathrm{g}) \rightarrow 2 \mathrm{HF}(\mathrm{g}) + \mathrm{O}_{2}(\mathrm{g}) \), set up the rates: \( -\frac{1}{2}\frac{d[\mathrm{HOF}]}{dt} = \frac{1}{2}\frac{d[\mathrm{HF}]}{dt} = \frac{d[\mathrm{O}_2]}{dt} \). Here, for every 2 moles of \( \mathrm{HOF} \) that disappear, 2 moles of \( \mathrm{HF} \) are formed, and 1 mole of \( \mathrm{O}_2 \) is formed.

Key Concepts

Stoichiometry

Stoichiometry is the heart of understanding chemical reactions. It is the relationship between the quantities of reactants and products in a chemical reaction. This concept helps us to predict how much of a reactant is needed to produce a desired amount of product. Or conversely, how much product we can produce from a given amount of reactant.
By using stoichiometric coefficients, which are the numbers placed in front of compounds in a chemical equation, we can determine the relative rates of disappearance and appearance in a reaction. For instance, if the reaction is \( aA + bB \rightarrow cC + dD \), the rates are connected as follows:
\[ -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt} \]
This shows that the rate at which the concentrations of reactants decrease is proportionate to the rate at which the products' concentrations increase.

Reactants and Products

Understanding the behavior of reactants and products in a chemical reaction is crucial. Reactants are substances that start a chemical reaction, while products are substances that are produced as a result of the reaction.
In the reactions given in the exercise, for example, \(2 \mathrm{O}_3(\mathrm{g}) \rightarrow 3 \mathrm{O}_2(\mathrm{g})\), \(\mathrm{O}_3\) is the reactant and \(\mathrm{O}_2\) is the product. The stoichiometric coefficients tell us that 2 moles of \(\mathrm{O}_3\) produce 3 moles of \(\mathrm{O}_2\).

• In reaction (a), every 2 moles of \(\mathrm{O}_3\) cut, cause 3 moles of \(\mathrm{O}_2\) to form.
• In reaction (b), \(2 \mathrm{HOF}\) decomposes to produce \(2 \mathrm{HF}\) and \(1 \mathrm{O}_2\).

This configuration shows the direct transformation from reactants to products, and highlights the fundamental balance of mass, as described by the law of conservation of mass.

Rate of Reaction

The rate of reaction is a measure of how quickly a reactant is consumed or a product is formed in a chemical reaction. This rate can change based on various factors including temperature, pressure, concentration, and the presence of catalysts.
For the given reactions, the rate at which \(\mathrm{O}_3\) disappears in reaction (a) is associated with the rate of formation of \(\mathrm{O}_2\), as given by their relationship:
\[ -\frac{1}{2}\frac{d[\mathrm{O}_3]}{dt} = \frac{1}{3}\frac{d[\mathrm{O}_2]}{dt} \]
This means for every 2 moles of \(\mathrm{O}_3\) disappearing, 3 moles of \(\mathrm{O}_2\) form, controlled by their reaction coefficients.
Similarly, in reaction (b), the disappearance of \(\mathrm{HOF}\) correlates with the formation of \(\mathrm{HF}\) and \(\mathrm{O}_2\), showing how each of these products form in relation to the reactant's consumption. Understanding rates helps in calculating the time required for reactions and in designing industrial processes for optimal product yield.