Problem 97 - Chemistry The Science in Context

Question

Write the rate laws for the following elementary steps and identify them as uni-, bi-, or termolecular steps: a. \(\mathrm{SO}_{2} \mathrm{Cl}_{2}(g) \rightarrow \mathrm{SO}_{2}(g)+\mathrm{Cl}_{2}(g)\) b. \(\mathrm{NO}_{2}(g)+\mathrm{CO}(g) \rightarrow \mathrm{NO}(g)+\mathrm{CO}_{2}(g)\) c. \(2 \mathrm{NO}_{2}(g) \rightarrow \mathrm{NO}_{3}(g)+\mathrm{NO}(g)\)

Step-by-Step Solution

Answer

Question: Identify the molecularity of each elementary step in the following reactions and write the rate laws for each step: a. $$\mathrm{SO}_{2}\mathrm{Cl}_{2}(g) \rightarrow \mathrm{SO}_{2}(g) + \mathrm{Cl}_{2}(g)$$ b. $$\mathrm{NO}_{2}(g) + \mathrm{CO}(g) \rightarrow \mathrm{NO}(g) + \mathrm{CO}_{2}(g)$$ c. $$2 \mathrm{NO}_{2}(g) \rightarrow \mathrm{NO}_{3}(g) + \mathrm{NO}(g)$$ Answer: a. Unimolecular step; Rate law: $$\text{Rate} = k[\mathrm{SO}_{2}\mathrm{Cl}_{2}]$$ b. Bimolecular step; Rate law: $$\text{Rate} = k[\mathrm{NO}_{2}][\mathrm{CO}]$$ c. Bimolecular step; Rate law: $$\text{Rate} = k[\mathrm{NO}_{2}]^{2}$$

Step 1: Identify the molecularity of each reaction

First, we will analyze each reaction and determine the molecularity (number of molecules involved) for each one. a. In the first reaction, a single molecule of SO2Cl2 is breaking down into two molecules: $$\mathrm{SO}_{2}\mathrm{Cl}_{2}(g) \rightarrow \mathrm{SO}_{2}(g) + \mathrm{Cl}_{2}(g)$$ Since only one molecule is involved, it is a unimolecular step. b. In the second reaction, two molecules are involved: $$\mathrm{NO}_{2}(g) + \mathrm{CO}(g) \rightarrow \mathrm{NO}(g) + \mathrm{CO}_{2}(g)$$ Since we have two reactant molecules, it is a bimolecular step. c. In the third reaction, two molecules of NO2 are involved: $$2 \mathrm{NO}_{2}(g) \rightarrow \mathrm{NO}_{3}(g) + \mathrm{NO}(g)$$ Again, we have two reactant molecules, making it a bimolecular step.

Step 2: Write the rate laws for each elementary step

Now that we have identified the molecularity of each reaction, we can write the rate laws for these elementary steps. a. For the unimolecular reaction: $$\text{Rate} = k[\mathrm{SO}_{2}\mathrm{Cl}_{2}]$$ b. For the first bimolecular reaction: $$\text{Rate} = k[\mathrm{NO}_{2}][\mathrm{CO}]$$ c. For the second bimolecular reaction: $$\text{Rate} = k[\mathrm{NO}_{2}]^{2}$$ In these rate laws, k is the rate constant, and the brackets indicate the concentration of the reactants. The exponents represent the molecularity of the elementary step.

Key Concepts

Rate Laws

Rate laws describe the relationship between the rate of a chemical reaction and the concentration of the reactants. They are essential in understanding reaction kinetics. In an elementary reaction, the rate law can be directly written from the stoichiometry of the reaction. This is a unique feature of elementary reactions, making them simpler to analyze.

Consider the reaction \( ext{A} + ext{B} \rightarrow ext{C} \). If it is an elementary reaction, its rate law is \( ext{Rate} = k[ ext{A}][ ext{B}] \), reflecting that the rate depends on the concentration of A and B. Here, \( k \) is the rate constant, and the brackets indicate concentration. The exponents in the rate law are derived from the molecularity of the reaction, which we'll explore in the following sections.

Molecularity

Molecularity refers to the number of reactant molecules involved in an elementary step of a reaction. It provides an understanding of how molecules collide to cause the reaction. There are three types of molecularities: unimolecular, bimolecular, and termolecular.

• Unimolecular: Involves a single molecule.
• Bimolecular: Involves two molecules.
• Termolecular: Involves three molecules.

Although termolecular reactions are rare due to the low probability of three molecules colliding simultaneously with the correct orientation and energy, unimolecular and bimolecular reactions are quite common. Understanding molecularity helps in predicting the rate law for elementary reactions.

Elementary Reactions

Elementary reactions are the simplest type of chemical reactions that occur in a single step. Each involves a straightforward interaction between molecules, atoms, or radicals, without the formation of intermediates. They are vital for understanding complex reaction mechanisms, which often consist of multiple elementary steps.

An elementary reaction's key feature is that its rate law can be written directly from its stoichiometry. For instance, in the reaction \( ext{H}_2 + ext{I}_2 \rightarrow 2 ext{HI} \), if it were elementary, the rate law would be \( ext{Rate} = k[ ext{H}_2][ ext{I}_2] \). Since it happens in a single step, it simplifies the process of determining reaction rates.

Therefore, understanding the nature of elementary reactions provides a direct link between concentrations and reaction rates.

Unimolecular

Unimolecular reactions involve the decomposition or transformation of a single reactant molecule. They typically occur when a molecule gains enough energy to undergo a transformation or rearrangement. These processes often happen through internal molecular adjustments, like bond breaking or forming.

A simple example is the reaction \( ext{AB} \rightarrow ext{A} + ext{B} \), which is unimolecular because it involves only one molecule, AB. The rate law for such a reaction is typically \( ext{Rate} = k[ ext{AB}] \), reflecting its dependence solely on the concentration of the single reactant.

Unimolecular reactions are common in gas-phase reactions, where molecules can freely move and collide, gaining sufficient energy for internal transformations.

Bimolecular

Bimolecular reactions involve two reactant molecules colliding to form products. This type is the most common in chemical reactions due to the relatively higher chance of two molecules interacting with optimal energy and alignment compared to three molecules in termolecular steps.

In a typical bimolecular reaction, the rate law is derived based on the concentration of both reactant species. For example, for the reaction \( ext{A} + ext{B} \rightarrow ext{C} \), the rate law is \( ext{Rate} = k[ ext{A}][ ext{B}] \), indicating its dependency on the concentrations of both A and B.

• These reactions contribute to the overall rate of complex mechanisms, where several bimolecular steps might occur sequentially.
• Understanding bimolecular interactions provides insights into optimizing reaction conditions in both lab and industrial processes.