Problem 77 - Chemical Principles

Question

Each of the following steps is an elementary reaction. Write its rate law and indicate its molecularity: (a) \(\mathrm{NO}+\mathrm{NO} \rightarrow \mathrm{N}_{2} \mathrm{O}_{2}\) (b) \(\mathrm{Cl}_{2} \rightarrow \mathrm{Cl}+\mathrm{Cl}\); (c) \(\mathrm{NO}_{2}+\mathrm{NO}_{2} \rightarrow \mathrm{NO}+\mathrm{NO}_{3}\); (d) Which o these reactions might be radical chain initiating?

Step-by-Step Solution

Answer

Rate laws: (a) Rate = k[NO]^2, bimolecular; (b) Rate = k[Cl2], unimolecular; (c) Rate = k[NO2]^2, bimolecular. Reaction (b) might be radical chain initiating.

Step 1 - Write the Rate Law for Reaction (a)

The rate law for an elementary reaction can be written directly from the reactants and their stoichiometric coefficients. For the reaction \(\mathrm{NO} + \mathrm{NO} \rightarrow \mathrm{N}_2\mathrm{O}_2\), since two molecules of NO are involved, the rate law is \(\text{Rate} = k[\mathrm{NO}]^2\), where \(k\) is the rate constant and \(\text{Rate}\) refers to the rate of the reaction.

Step 2 - Molecularity of Reaction (a)

Molecularity is the number of molecules participating in the elementary step. For the reaction in step 1, two molecules of NO are involved, so it is a bimolecular reaction.

Step 3 - Write the Rate Law for Reaction (b)

Reaction (b) \(\mathrm{Cl}_2 \rightarrow \mathrm{Cl} + \mathrm{Cl}\) involves the dissociation of a single molecule of \(\mathrm{Cl}_2\). Thus, the rate law is \(\text{Rate} = k[\mathrm{Cl}_2]\).

Step 4 - Molecularity of Reaction (b)

Only one molecule is involved in the elementary reaction (b), so it is a unimolecular reaction.

Step 5 - Write the Rate Law for Reaction (c)

For the reaction \(\mathrm{NO}_2 + \mathrm{NO}_2 \rightarrow \mathrm{NO} + \mathrm{NO}_3\), two molecules of \(\mathrm{NO}_2\) participate in the reaction. Thus, the rate law is \(\text{Rate} = k[\mathrm{NO}_2]^2\).

Step 6 - Molecularity of Reaction (c)

The reaction (c) involves two \(\mathrm{NO}_2\) molecules, which makes it a bimolecular reaction.

Step 7 - Identify the Radical Chain Initiating Reaction

A radical chain initiating reaction typically involves the homolytic cleavage of a bond, producing two radicals. Reaction (b) \(\mathrm{Cl}_2 \rightarrow \mathrm{Cl} + \mathrm{Cl}\) produces two chlorine radicals and is therefore a candidate for a radical chain initiation step.

Key Concepts

Molecularity

Molecularity is a term that refers to the number of reactant molecules that collide and interact in an elementary reaction step. This concept is particularly important because it directly influences the rate law for that reaction. An elementary reaction is a reaction which proceeds in a single step and its rate law can be determined from the reaction's stoichiometry.

For example, when we consider the reaction \( \mathrm{NO} + \mathrm{NO} \rightarrow \mathrm{N}_2\mathrm{O}_2 \), this is classified as bimolecular because two molecules of nitrogen monoxide (\mathrm{NO}) collide to form dinitrogen dioxide (\mathrm{N}_2\mathrm{O}_2). On the other hand, the dissociation of chlorine (\mathrm{Cl}_2) into two chlorine atoms is unimolecular as it involves the cleavage of one molecule into two distinct parts.

The determination of molecularity is straightforward for elementary reactions. It's an essential concept since it aids in the creation of the correct rate law, which is vital to predict how quickly a reaction will proceed under certain conditions.

Rate Constant

The rate constant, denoted as \(k\), is a crucial component in the rate law of a chemical reaction. It indicates the speed at which a reaction proceeds and depends both on the temperature and the specific reaction. Though its units can vary depending on the overall order of the reaction, it's important for students to understand that the rate constant is, indeed, constant for a given reaction at a constant temperature.

In the reactions provided, the rate constant appears in the form \(k[\mathrm{NO}]^2\) for the bimolecular reaction and \(k[\mathrm{Cl}_2]\) for the unimolecular reaction. These rate laws allow us to determine the rate at which the reactants are consumed and the products are formed. The rate constant could be thought of as the proportionality factor that connects the concentration of reactants with the rate of the reaction – a higher value of \(k\) means the reaction is faster under the same conditions.

Radical Chain Initiation

Radical chain initiation is the stage in a chain reaction where radicals—highly reactive species with unpaired electrons—are first produced. These radicals can go on to react in a series of propagation steps, potentially leading to a large number of reaction products.

In the given reactions, the dissociation of chlorine molecules (\(\mathrm{Cl}_2 \rightarrow \mathrm{Cl} + \mathrm{Cl}\)) represents a classic radical chain initiation. The homolytic cleavage of the chlorine molecule produces two chlorine radicals, which are highly reactive and capable of perpetuating the reaction sequence in subsequent propagation steps. This process often requires an initial input of energy, such as heat or light, to break the chemical bond and generate the radicals.

Understanding radical chain initiation is pivotal for grasping how certain reactions can lead to complex mechanisms and vast product mixtures, and it has immense implications in fields like polymer chemistry where radical initiators are routinely used to start the polymerization process.