Book cover for Chemistry The Science in Context

Chemistry The Science in Context

Thomas R. Gilbert

ISBN #9780393615142

5th Edition

2,675 Questions

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191,124 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter provides a comprehensive overview of chemical kinetics, emphasizing the measurement and analysis of reaction rates. It covers developing rate laws from experimental data, determining reaction orders, calculating half-lives through integrated rate laws, and applying the Arrhenius equation. Important concepts such as reaction mechanisms, rate-determining steps, and the role of catalysts in modifying reaction pathways are also discussed. A strong grasp of these principles is critical for understanding how chemical reactions proceed in various environments, including the atmosphere.

Learning Objectives

1

Explain how reaction rates are measured and expressed through the relationship between reactant consumption and product formation.

2

Develop and interpret rate laws and determine reaction orders from experimental data.

3

Apply integrated rate laws to calculate half-lives and utilize the Arrhenius equation to relate temperature and activation energy to reaction rates.

4

Analyze reaction mechanisms including the concept of rate-determining steps and understand the role of catalysts in chemical reactions.

Key Concepts

CONCEPT

DEFINITION

Reaction Rate

The speed at which reactants are consumed or products are formed in a chemical reaction, usually expressed in concentration per unit time.

Rate Law

An equation that links the reaction rate with the concentrations of reactants, typically expressed in the form rate = k[A]^m[B]^n.

Reaction Order

The exponent of the concentration of a reactant in the rate law, reflecting its influence on the reaction rate.

Integrated Rate Law

An equation that relates the concentration of reactants to time, used to determine parameters like the half-life of a reaction.

Half-Life

The time required for the concentration of a reactant to decrease by half in a chemical reaction.

Arrhenius Equation

A formula that expresses the dependence of the reaction rate constant on temperature and activation energy: k = Ae^(-Ea/RT).

Activation Energy (Ea)

The minimum energy that colliding reactant molecules must possess for a reaction to occur.

Reaction Mechanism

A detailed sequence of elementary steps by which a chemical reaction proceeds to form products.

Rate-Determining Step

The slowest step in a reaction mechanism that controls the overall rate of the reaction.

Catalyst

A substance that increases the reaction rate by lowering the activation energy, without being consumed in the reaction.

Example Problems

Example 1

Nitrous oxide decomposes to nitrogen and oxygen in the following reaction: $$ 2 \mathrm{N}_{2} \mathrm{O}(g) \rightarrow 2 \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) $$ In Figure $P 13.1$, which curve represents $\left[\mathrm{N}_{2} \mathrm{O}\right]$ and which curve represents $\left[\mathrm{O}_{2}\right] ?$ (FIGURE CANNOT COPY)

Example 2

Sulfur trioxide is formed in the reaction $$ \mathrm{sO}_{2}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(g) \rightarrow \mathrm{SO}_{3}(\mathrm{g}) $$ In Figure $P 13.2,$ which curve represents $\left[\mathrm{SO}_{2}\right]$ and which curve represents $\left[\mathrm{O}_{2}\right]$ ? All three gases are present initially. (FIGURE CANNOT COPY)

Example 3

The rate law for the reaction $2 \mathrm{A} \rightarrow \mathrm{B}$ is second order in A. Figure $\mathrm{P} 13.3$ represents samples with different concentrations of $\mathrm{A}$; the red spheres represent molecules of A. In which sample will the reaction $A \rightarrow B$ proceed most rapidly? (FIGURE CANNOT COPY)

Example 4

The rate law for the reaction $A+B \rightarrow C$ is first order in both $\mathrm{A}$ and $\mathrm{B}$. Figure $\mathrm{P} 13.4$ represents samples with different concentrations of $\mathrm{A}$ (red spheres) and $\mathrm{B}$ (blue spheres). In which sample will the reaction $\mathrm{A}+\mathrm{B} \rightarrow \mathrm{C}$ proceed most rapidly? (FIGURE CANNOT COPY)

Example 5

Figure $\mathrm{P} 13.5$ shows plots of reactant concentrations versus time for four reactions. Which one has the greatest initial rate? (FIGURE CANNOT COPY)

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Step-by-Step Explanations

QUESTION

How can experimental data be used to develop a rate law for a given reaction?

STEP-BY-STEP ANSWER:

Step 1: Collect experimental data showing how the reaction rate changes with varying reactant concentrations.
Step 2: Assume a general rate law form, for example, rate = k[A]^m[B]^n.
Step 3: Use the experimental data to determine the values of exponents m and n by comparing how changes in concentration affect the rate.
Step 4: Calculate the rate constant k from the measured rates once the reaction orders are known.
Final Answer: The rate law is established with the appropriate reaction orders and rate constant derived from experimental observations.

Developing a Rate Law

QUESTION

How do you calculate the half-life of a reaction using its integrated rate law?

STEP-BY-STEP ANSWER:

Step 1: Write down the integrated rate law for the reaction. For a first-order reaction, it is ln[A] = -kt + ln[A]â‚€.
Step 2: Define the half-life (t½) where [A] = [A]₀/2.
Step 3: Substitute [A] = [A]₀/2 into the equation and simplify to find t½ in terms of the rate constant k.
Step 4: Solve for t½, leading to the relation t½ = ln2/k for a first-order reaction.
Final Answer: The half-life for a first-order reaction is given by t½ = ln2/k.

Calculating Half-Life Using Integrated Rate Law

QUESTION

How does the Arrhenius equation relate temperature to the reaction rate constant?

STEP-BY-STEP ANSWER:

Step 1: Write the Arrhenius equation: k = Ae^(-Ea/RT), where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the temperature.
Step 2: Identify how each parameter influences the rate constant k.
Step 3: Understand that as temperature increases, the exponent becomes less negative, thus increasing k.
Step 4: Use experimental data to determine Ea and A if needed.
Final Answer: The Arrhenius equation shows that the reaction rate constant increases with temperature due to a decrease in the effective energy barrier.

Applying the Arrhenius Equation

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Common Mistakes

  • Confusing stoichiometric coefficients with reaction orders in the rate law.
  • Neglecting the impact of temperature on reaction rates when not applying the Arrhenius equation correctly.
  • Assuming that a catalyst is consumed in a reaction, rather than understanding its role in lowering activation energy.
  • Overlooking the significance of the rate-determining step when analyzing overall reaction mechanisms.