Problem 67 - Chemistry: The Central Science in SI Units, Global Edition

Question

What is the molecularity of each of the following elementary reactions? Write the rate law for each. (a) \(\mathrm{H}_{2} \mathrm{O}(l)+\mathrm{CN}^{-}(a q) \longrightarrow \mathrm{HCN}(a q)\) (b) \(\mathrm{CH}_{3} \mathrm{Cl}(\) solv \()+\mathrm{OH}^{-}(\) solv \() \longrightarrow \mathrm{CH}_{3} \mathrm{OH}(\) solv \()+\mathrm{Cl}^{-}(\) solv \()\) (c) \(\mathrm{N}_{2} \mathrm{O}_{4}(g) \rightarrow 2 \mathrm{NO}_{2}\)

Step-by-Step Solution

Answer

(a) The molecularity of the reaction is 2 (bimolecular). The rate law is: \[Rate = k [\mathrm{H}_2 \mathrm{O}] [\mathrm{CN}^-]\] (b) The molecularity of the reaction is 2 (bimolecular). The rate law is: \[Rate = k [\mathrm{CH}_3 \mathrm{Cl}] [\mathrm{OH}^-]\] (c) The molecularity of the reaction is 1 (unimolecular). The rate law is: \[Rate = k [\mathrm{N}_2 \mathrm{O}_4]\]

Step 1: (a) Molecularity of Reaction 1

In this reaction, there are two reactants: H2O and CN. Hence, the molecularity of this elementary reaction is 2, making it a bimolecular reaction.

Step 2: (a) Rate Law for Reaction 1

The rate law will be the rate constant (k) multiplied by the concentrations of the reactants, each raised to the power of their respective stoichiometric coefficients. Since this is an elementary reaction, the coefficients are equal to the number of molecules of each reactant involved in the reaction. In this case, the rate law becomes: \[Rate = k [\mathrm{H}_2 \mathrm{O}] [\mathrm{CN}^-]\]

Step 3: (b) Molecularity of Reaction 2

In this reaction, there are two reactants: CH3Cl and OH-. Hence, the molecularity of this elementary reaction is 2, making it a bimolecular reaction.

Step 4: (b) Rate Law for Reaction 2

Following the same reasoning as in Reaction 1, the rate law for Reaction 2 will be the rate constant (k) multiplied by the concentrations of the reactants, each raised to the power of their respective stoichiometric coefficients. In this case, the rate law becomes: \[Rate = k [\mathrm{CH}_3 \mathrm{Cl}] [\mathrm{OH}^-]\]

Step 5: (c) Molecularity of Reaction 3

In this reaction, there is only one reactant: N2O4. Hence, the molecularity of this elementary reaction is 1, making it a unimolecular reaction.

Step 6: (c) Rate Law for Reaction 3

Following the same reasoning as in the previous reactions, the rate law for Reaction 3 will be the rate constant (k) multiplied by the concentration of the reactant, raised to the power of its stoichiometric coefficient. In this case, the rate law becomes: \[Rate = k [\mathrm{N}_2 \mathrm{O}_4]\]

Key Concepts

Rate Law

When studying chemical reactions, the concept of a rate law is essential to understanding how reactions proceed over time. A rate law expresses the relationship between the reaction rate and the concentration of reactants. For elementary reactions, which are reactions that occur in a single step with a single transition state, the rate law can be directly determined from the stoichiometry of the reactants involved.
This means that for an elementary reaction where

• one reactant transforms into products, the rate law will be: \[ \text{Rate} = k [\text{A}] \] where \( k \) is the rate constant and \([\text{A}]\) is the concentration of the reactant.
• two reactants are involved, the rate law takes the form: \[ \text{Rate} = k [\text{A}][\text{B}] \] with each concentration raised to the power of its stoichiometric coefficient.

These laws help chemists predict the speed of chemical reactions and are crucial for control in industrial processes.

Elementary Reaction

An elementary reaction is a single-step process with no intermediates. In this direct process, reactants proceed to form products without any recourse to more complicated mechanisms.
Elementary reactions are characterized by their simplicity and are used to understand more complex reaction mechanisms, as a complicated reaction can often be broken down into its elementary steps. It's important to remember that because they occur in a single step, the molecularity of an elementary reaction, which indicates the number of molecules colliding in the transition state, directly informs its rate law.
For instance, if two molecules collide and react, forming products immediately, it’s considered to be a bimolecular elementary reaction. This direct relationship allows chemists to form accurate rate equations based solely on stoichiometry, simplifying many calculations and interpretations of reaction kinetics.

Bimolecular Reaction

Bimolecular reactions involve two reactant molecules coming together to form products. In the context of elementary reactions, these occur in a single concerted step without any intermediate species.
The hallmark of a bimolecular reaction is that two species are involved in the crucial transition state – meaning both influence the rate of the reaction. The molecularity, in this case, is two, referring to the two entities colliding.

• The typical rate law for a bimolecular reaction is: \[ \text{Rate} = k [\text{Reactant1}][\text{Reactant2}] \]

Each reactant concentration is raised to the power of one, reflecting the actual number of molecules required to trigger the reaction. Bimolecular reactions are quite common and can describe many everyday chemical processes, from enzyme catalysis in biology to synthetic reactions in industrial chemistry.

Unimolecular Reaction

Unimolecular reactions are reactions where a single molecule undergoes change to form products. For these elementary reactions, the transformation is dependent on the concentration of just one reactant entity.
In a unimolecular process, the molecule might rearrange itself, break apart, or undergo some isomerization without the need for a second molecule to initiate the transformation. Such reactions are often preceded by some form of internal energy absorption, like heat or light, which prompts the molecule to surpass its energy barrier and proceed to products.
The rate law for a unimolecular reaction is straightforward:

• \[ \text{Rate} = k [\text{Reactant}] \]

This simplicity, reflecting direct dependence on a single reactant, allows for easier analysis and prediction of reaction kinetics, making unimolecular reactions essential in fields such as gas-phase chemistry and photochemistry.