Question

Write the rate laws for the following elementary reactions. a. \(\mathrm{CH}_{3} \mathrm{NC}(g) \rightarrow \mathrm{CH}_{3} \mathrm{CN}(g)\) b. \(\mathrm{O}_{3}(g)+\mathrm{NO}(g) \rightarrow \mathrm{O}_{2}(g)+\mathrm{NO}_{2}(g)\) c. \(\mathrm{O}_{3}(g) \rightarrow \mathrm{O}_{2}(g)+\mathrm{O}(g)\) d. \(\mathrm{O}_{3}(g)+\mathrm{O}(g) \rightarrow 2 \mathrm{O}_{2}(g)\)

Short answer

a. Rate = k[\(\mathrm{CH}_{3}\mathrm{NC}\)] b. Rate = k[\(\mathrm{O}_{3}\)][\(\mathrm{NO}\)] c. Rate = k[\(\mathrm{O}_{3}\)] d. Rate = k[\(\mathrm{O}_{3}\)][\(\mathrm{O}\)]

Step-by-step solution

a. Rate law for \(\mathrm{CH}_{3} \mathrm{NC}(g) \rightarrow \mathrm{CH}_{3} \mathrm{CN}(g)\) #

Since this reaction has only one reactant, \(\mathrm{CH}_{3} \mathrm{NC}(g)\), and its stoichiometric coefficient is 1, the rate law will be: Rate = k[\(\mathrm{CH}_{3}\mathrm{NC}\)]^1 Simplifying, we get: Rate = k[\(\mathrm{CH}_{3}\mathrm{NC}\)]

b. Rate law for \(\mathrm{O}_{3}(g)+\mathrm{NO}(g) \rightarrow \mathrm{O}_{2}(g)+\mathrm{NO}_{2}(g)\) #

This reaction has two reactants, \(\mathrm{O}_{3}(g)\) and \(\mathrm{NO}(g)\), both with stoichiometric coefficients of 1. The rate law will be: Rate = k[\(\mathrm{O}_{3}\)]^1[\(\mathrm{NO}\)]^1 Simplifying, we get: Rate = k[\(\mathrm{O}_{3}\)][\(\mathrm{NO}\)]

c. Rate law for \(\mathrm{O}_{3}(g) \rightarrow \mathrm{O}_{2}(g)+\mathrm{O}(g)\) #

Since this reaction has only one reactant, \(\mathrm{O}_{3}(g)\), and its stoichiometric coefficient is 1, the rate law will be: Rate = k[\(\mathrm{O}_{3}\)]^1 Simplifying, we get: Rate = k[\(\mathrm{O}_{3}\)]

d. Rate law for \(\mathrm{O}_{3}(g)+\mathrm{O}(g) \rightarrow 2 \mathrm{O}_{2}(g)\) #

This reaction has two reactants, \(\mathrm{O}_{3}(g)\) and \(\mathrm{O}(g)\), both with stoichiometric coefficients of 1. The rate law will be: Rate = k[\(\mathrm{O}_{3}\)]^1[\(\mathrm{O}\)]^1 Simplifying, we get: Rate = k[\(\mathrm{O}_{3}\)][\(\mathrm{O}\)]

Key concepts

Elementary Reactions

Elementary reactions are fundamental processes where reactants transform into products in a single step. Each elementary reaction has a direct relationship between its rate and the concentration of its reactants. This is different from more complex reactions that occur in multiple steps. For elementary reactions, the stoichiometric coefficients of reactants in a balanced equation directly determine the reaction order.

Consider the reaction \(\mathrm{CH}_{3} \mathrm{NC}(g) \rightarrow \mathrm{CH}_{3} \mathrm{CN}(g)\). Here, \(\mathrm{CH}_{3} \mathrm{NC}\) has a stoichiometric coefficient of 1, indicating a first-order reaction. This leads directly to the rate law: \[\text{Rate} = k[\mathrm{CH}_{3}\mathrm{NC}]\] where \(k\) is the rate constant. This straightforward approach to determining the rate law is unique to elementary reactions.

Chemical Kinetics

Chemical kinetics focuses on the rate at which chemical reactions occur and the factors that influence these rates. It helps us understand how reaction conditions like temperature and concentration affect the speed of reactions. For the elementary reaction \(\mathrm{O}_{3}(g)+\mathrm{NO}(g) \rightarrow \mathrm{O}_{2}(g)+\mathrm{NO}_{2}(g)\), the rate can be influenced by the concentration of reactant molecules.

The rate law for this reaction can be expressed as: \[\text{Rate} = k[\mathrm{O}_{3}]^1[\mathrm{NO}]^1\] Simplifying gives us \[\text{Rate} = k[\mathrm{O}_{3}][\mathrm{NO}]\] Understanding these rates helps chemists control reactions to optimize yields and efficiency.

Reaction Stoichiometry

Reaction stoichiometry involves the quantitative relationship between reactants and products in a chemical reaction. In elementary reactions, the stoichiometry is directly applied to formulating the rate laws. Each reactant's role is precisely quantified based on the balanced equation. For example, in the reaction: \(\mathrm{O}_{3}(g) \rightarrow \mathrm{O}_{2}(g)+\mathrm{O}(g)\), the stoichiometric coefficient of \(\mathrm{O}_{3}\) is 1. This correlates to a first-order reaction.

The reaction's rate law is: \[\text{Rate} = k[\mathrm{O}_{3}]\] This direct relationship between stoichiometry and rate laws is a hallmark of elementary reactions. For the reaction \(\mathrm{O}_{3}(g)+\mathrm{O}(g) \rightarrow 2 \mathrm{O}_{2}(g)\), the stoichiometric coefficients are confirmed by the balanced equation, leading to a rate law of: \[\text{Rate} = k[\mathrm{O}_{3}][\mathrm{O}]\]This demonstrates how stoichiometry and rate laws are tightly intertwined in chemical dynamics.