Problem 67 - Chemistry The Molecular Science

Question

Assuming that each reaction is elementary, predict the rate law. (a) \(\mathrm{NO}(\mathrm{g})+\mathrm{NO}_{3}(\mathrm{~g}) \longrightarrow 2 \mathrm{NO}_{2}(\mathrm{~g})\) (b) \(\mathrm{O}(\mathrm{g})+\mathrm{O}_{3}(\mathrm{~g}) \longrightarrow 2 \mathrm{O}_{2}(\mathrm{~g})\) (c) \(\left(\mathrm{CH}_{3}\right)_{3} \mathrm{CBr}(\mathrm{aq}) \longrightarrow\left(\mathrm{CH}_{3}\right)_{3} \mathrm{C}^{+}(\mathrm{aq})+\mathrm{Br}^{-}(\mathrm{aq})\) (d) \(2 \mathrm{HI}(\mathrm{g}) \longrightarrow \mathrm{H}_{2}(\mathrm{~g})+\mathrm{I}_{2}(\mathrm{~g})\)

Step-by-Step Solution

Answer

(a) Rate = k[NO][NO₃]; (b) Rate = k[O][O₃]; (c) Rate = k[(CH₃)₃CBr]; (d) Rate = k[HI]².

Step 1: Understanding Rate Laws and Elementary Reactions

In chemical kinetics, the rate law of a reaction describes how the rate depends on the concentration of the reactants. For an elementary reaction, which occurs in a single step, the rate law can be directly determined from the stoichiometry of the balanced equation.

Step 2: Predict Rate Law for Reaction (a)

The reaction is \(NO\,(g) + NO_3\,(g) \rightarrow 2NO_2\,(g)\). Since this is an elementary reaction, the rate law is based on the stoichiometric coefficients of the reactants: \[\text{Rate} = k[NO][NO_3]\] where \(k\) is the rate constant.

Step 3: Predict Rate Law for Reaction (b)

The reaction is \(O\,(g) + O_3\,(g) \rightarrow 2O_2\,(g)\). As an elementary reaction, the rate law is based on the coefficients: \[\text{Rate} = k[O][O_3]\] where \(k\) is the rate constant.

Step 4: Predict Rate Law for Reaction (c)

The reaction is \((CH_3)_3 CBr\,(aq) \rightarrow (CH_3)_3 C^+\,(aq) + Br^-\,(aq)\). This is a unimolecular reaction, meaning the rate depends on the concentration of a single reactant: \[\text{Rate} = k[(CH_3)_3 CBr]\] where \(k\) is the rate constant.

Step 5: Predict Rate Law for Reaction (d)

The reaction is \(2 HI\,(g) \rightarrow H_2\,(g) + I_2\,(g)\). For this bimolecular reaction, the rate law is given by: \[\text{Rate} = k[HI]^2\] where \(k\) is the rate constant, and the exponent reflects the coefficients of the reactants.

Key Concepts

Chemical Kinetics

Chemical kinetics is the branch of chemistry that deals with the rates of chemical reactions and the factors that affect them. It helps us understand how quickly a reaction occurs and what influences this speed.
Chemical reactions vary significantly in how fast they happen. Some might be instantaneous, while others may take years to complete.
The rate of a reaction is often dependent on:

• The concentration of the reactants: Often, higher concentrations lead to faster reactions.
• The temperature: Generally, increasing the temperature increases the rate of reaction.
• The presence of a catalyst: Catalysts can speed up reactions without being consumed.
• The physical state and surface area of the reactants: Finer particles react faster than larger chunks.
• The light exposure, in some cases.

Understanding rate laws, which mathematically describe reaction rates, is crucial in chemical kinetics. The general form is: \[\text{Rate} = k[A]^m[B]^n\]where:- \(k\) is the rate constant,- \([A]\) and \([B]\) are the concentrations of the reactants, and- \(m\) and \(n\) represent the reaction order with respect to each reactant.

Elementary Reactions

Elementary reactions are single-step processes in reaction mechanisms. Unlike complex reactions, which consist of multiple steps, elementary reactions occur in one go and directly reflect the exact molecular interaction involved.
Such reactions provide clear insights into the reaction kinetics, as the rate law can be directly deduced from their stoichiometry (i.e., the coefficients in a balanced chemical equation).
For instance, in an elementary reaction like \( NO_3 + NO \rightarrow 2NO_2 \), the rate law is determined directly from the components of the reaction:

• The reaction is first-order with respect to each reactant.
• Thus, the overall reaction is second-order, given by \( Rate = k[NO][NO_3] \).

Elementary reactions are crucial for understanding catalysis and particle collisions at a molecular level and are a foundation for developing more complex reaction mechanisms.

Stoichiometry

Stoichiometry is the study of the quantitative relationships, or ratios, that are derived from the balanced chemical equations of reactions.
It connects the amounts of reactants and products in a chemical reaction.
In the context of rate laws for elementary reactions, stoichiometry directly influences the exponents in the rate law expression:

• Each stoichiometric coefficient in a balanced equation becomes the exponent of its respective reactant concentration in the rate law.
• For example, in the reaction \( 2 HI ightarrow H_2 + I_2 \), the rate law is \( Rate = k[HI]^2 \) due to the stoichiometric coefficient "2" in front of \( HI \).

Stoichiometry ensures that chemical equations represent the mass, energy, and charge balance correctly, serving as a critical check for chemists predicting reaction outcomes.

Reaction Mechanisms

Reaction mechanisms provide a detailed step-by-step description of how reactions occur at the molecular level.
They help in identifying elementary reactions and the intermediates formed in complex reactions.
Understanding the reaction mechanism is crucial because it sheds light on the pathway taken during a chemical transformation, beyond what is apparent from the overall reaction equation.
Here's why knowing reaction mechanisms is beneficial:

• They explain why certain products are formed preferentially over others, based on the pathway's energetic favorability.
• Mechanisms allow chemists to design experiments to test specific pathways, supporting or refuting proposed mechanisms.
• They help in understanding catalytic effects which lower activation energy and increase the reaction rate.

Reactions are often influenced at multiple levels, including the molecular encounters, energy changes, and the presence of catalysts, all of which are intricately described by the reaction mechanism.