Table of Contents
Introduction
Chemical kinetics deals with how fast chemical reactions occur and how their rates depend on conditions such as concentration, temperature, and the presence of catalysts. While thermodynamics answers the question “Will a reaction occur spontaneously?”, kinetics answers “How fast will it occur, and by what path (mechanism)?”
In this chapter, the focus is on:
- The concept and mathematical description of reaction rate
- How rates depend on reactant concentrations (rate laws)
- The idea of reaction mechanism and elementary steps
- The role of activated complexes and activation energy
- Distribution of molecular energies and collision ideas
Details of temperature dependence and catalysis are developed in later chapters within Chemical Kinetics.
Basic Concepts: Reaction Rate
Defining the Rate of a Reaction
The rate of a chemical reaction expresses how fast reactants are consumed or products are formed. It is defined as the change in concentration of a species per unit time.
For a general reaction
$$
a\,\mathrm{A} + b\,\mathrm{B} \rightarrow p\,\mathrm{P} + q\,\mathrm{Q},
$$
the rate $v$ can be expressed in terms of the time change of the concentrations $[\mathrm{A}], [\mathrm{B}], [\mathrm{P}], [\mathrm{Q}]$:
$$
v = -\frac{1}{a}\frac{d[\mathrm{A}]}{dt}
= -\frac{1}{b}\frac{d[\mathrm{B}]}{dt}
= \frac{1}{p}\frac{d[\mathrm{P}]}{dt}
= \frac{1}{q}\frac{d[\mathrm{Q}]}{dt}.
$$
Key points:
- The minus sign for reactants reflects that their concentrations decrease.
- The stoichiometric coefficients in the denominator ensure that all expressions for $v$ are equal for the same reaction.
- $[\;]$ stands for concentration, typically in $\mathrm{mol\,L^{-1}}$, and $t$ is time (often in seconds).
Average vs. Instantaneous Rate
The average rate over a finite time interval $\Delta t$ is
$$
v_{\text{avg}} = -\frac{1}{a}\frac{\Delta[\mathrm{A}]}{\Delta t}.
$$
As $\Delta t$ becomes very small, the average rate approaches the instantaneous rate, expressed by the derivative $d[\mathrm{A}]/dt$. Graphically, the instantaneous rate is given by the slope of the concentration–time curve at a particular time.
Because reactant concentrations usually decrease during the course of a reaction, rates often decrease with time.
Rate Laws and Reaction Order
The Empirical Rate Law
The rate law (or rate equation) relates the reaction rate to the concentrations of reactants (and sometimes other species, like catalysts) at a given temperature.
For a simple reaction
$$
\mathrm{A} + \mathrm{B} \rightarrow \text{products},
$$
the rate law might be
$$
v = k[\mathrm{A}]^{m}[\mathrm{B}]^{n},
$$
where:
- $k$ is the rate constant (depends on temperature and reaction conditions, but not on concentrations).
- $m$ and $n$ are the orders of the reaction with respect to A and B, respectively.
- The overall order of the reaction is $m + n$.
Important:
- The exponents $m$ and $n$ are determined experimentally, not simply from the stoichiometric coefficients.
- The rate law is valid for a specified reaction and set of conditions (e.g., temperature, solvent, pressure).
Reaction Orders
Common types of reactions (in terms of rate laws):
Zero-Order
Rate is independent of reactant concentration:
$$
v = k.
$$
Examples: Some surface-catalyzed reactions at high reactant concentration, where the surface is saturated.
First-Order
Rate is directly proportional to the concentration of one reactant:
$$
v = k[\mathrm{A}].
$$
Characteristic feature: The half-life (time needed for $[\mathrm{A}]$ to fall to half its initial value) is constant and independent of $[\mathrm{A}]_0$.
Second-Order
Typical form (single reactant):
$$
v = k[\mathrm{A}]^{2}.
$$
or (two reactants):
$$
v = k[\mathrm{A}][\mathrm{B}].
$$
Higher orders and fractional orders also occur but are less common in simple introductory examples.
Determining the Rate Law Experimentally
To find the rate law, one studies how the initial rate depends on the initial concentrations of reactants.
Qualitative approaches:
- If doubling $[\mathrm{A}]_0$ doubles the initial rate (with $[\mathrm{B}]_0$ constant), the reaction is first-order in A.
- If doubling $[\mathrm{A}]_0$ quadruples the initial rate, the reaction is second-order in A.
- If changing $[\mathrm{A}]_0$ does not change the rate, the reaction is zero-order in A.
Careful quantitative analysis uses:
- Initial-rate methods: Compare $v_0$ for different initial concentrations.
- Integrated rate laws: Analyze concentration–time data; see below.
Integrated Rate Laws and Concentration–Time Relationships
The differential rate law expresses rate as a function of concentration at a moment in time. Solving this differential equation yields an integrated rate law, relating concentration to time.
Zero-Order Reactions
For a zero-order reaction in A:
$$
v = -\frac{d[\mathrm{A}]}{dt} = k
$$
Integration gives:
$$
[\mathrm{A}] = [\mathrm{A}]_0 - kt.
$$
Graphical feature:
- Plot of $[\mathrm{A}]$ vs. $t$ is a straight line with slope $-k$.
Half-life:
$$
t_{1/2} = \frac{[\mathrm{A}]_0}{2k},
$$
so the half-life depends on the initial concentration.
First-Order Reactions
For a first-order reaction in A:
$$
v = -\frac{d[\mathrm{A}]}{dt} = k[\mathrm{A}].
$$
Integration yields:
$$
\ln[\mathrm{A}] = \ln[\mathrm{A}]_0 - kt
$$
or
$$
[\mathrm{A}] = [\mathrm{A}]_0 e^{-kt}.
$$
Graphical feature:
- Plot of $\ln[\mathrm{A}]$ vs. $t$ is linear with slope $-k$.
Half-life:
$$
t_{1/2} = \frac{\ln 2}{k}.
$$
Here, the half-life is independent of $[\mathrm{A}]_0$.
Second-Order Reactions (Single Reactant)
For a second-order reaction in A:
$$
v = -\frac{d[\mathrm{A}]}{dt} = k[\mathrm{A}]^2.
$$
Integration gives:
$$
\frac{1}{[\mathrm{A}]} = \frac{1}{[\mathrm{A}]_0} + kt.
$$
Graphical feature:
- Plot of $1/[\mathrm{A}]$ vs. $t$ is linear with slope $k$.
Half-life:
$$
t_{1/2} = \frac{1}{k[\mathrm{A}]_0},
$$
so the half-life decreases as the initial concentration increases.
Integrated rate laws for more complex rate expressions (e.g., $v = k[\mathrm{A}][\mathrm{B}]$ with changing both concentrations) are more involved but follow the same principle: integrate the differential rate equation.
Elementary Steps and Reaction Mechanisms
Overall Reaction vs. Elementary Step
The chemical equation you usually write is the overall reaction, which summarizes the net change. However, the reaction may proceed through several simpler steps called elementary steps.
Example:
Overall:
$$
2\,\mathrm{NO} + \mathrm{O_2} \rightarrow 2\,\mathrm{NO_2}.
$$
Possible mechanism:
- $\mathrm{NO} + \mathrm{NO} \rightleftharpoons \mathrm{N_2O_2}$ (fast equilibrium)
- $\mathrm{N_2O_2} + \mathrm{O_2} \rightarrow 2\,\mathrm{NO_2}$ (slow)
The mechanism is the sequence of such elementary steps that leads from reactants to products.
Molecularity of Elementary Steps
Each elementary step has a molecularity, the number of molecules (or ions) colliding in that step:
- Unimolecular: involves one reacting species (e.g., isomerization or decomposition).
- Bimolecular: involves two reacting species colliding.
- Termolecular: involves three species simultaneously colliding (rare).
For an elementary step, the rate law can be written directly from its molecularity:
- Unimolecular: $v = k[\mathrm{A}]$.
- Bimolecular: $v = k[\mathrm{A}][\mathrm{B}]$ or $v = k[\mathrm{A}]^2$.
This direct connection applies only to elementary steps, not to the overall reaction.
Rate-Determining Step
In many mechanisms, one step is significantly slower than the others. This rate-determining step (RDS) limits the overall reaction rate.
Characteristics:
- The rate law of the overall reaction is often closely related to the rate law of the RDS.
- Fast steps before or after the RDS can influence the observed rate law via equilibria or intermediate concentrations.
Example (qualitative):
If the slow step is
$$
\mathrm{A} + \mathrm{B} \rightarrow \text{intermediate} \quad (\text{slow}),
$$
then the rate may be
$$
v \approx k[\mathrm{A}][\mathrm{B}],
$$
even if the overall reaction shows a more complex stoichiometry.
Reaction Intermediates
An intermediate is a species that is formed in one step of the mechanism and consumed in a subsequent step. It does not appear in the overall balanced equation.
Features:
- Usually present in low concentration.
- Often highly reactive and short-lived.
- Important for understanding the reaction path, but not directly observable in many simple experiments.
Mechanisms are constructed to be consistent with:
- The overall stoichiometry
- The experimentally determined rate law
- Any evidence for intermediates (e.g., spectroscopic detection)
Energy Profile, Activated Complex, and Activation Energy
Reaction Coordinate and Energy Profile
A reaction can be visualized along a reaction coordinate, which describes the progress from reactants to products. An energy profile diagram plots potential energy vs. reaction coordinate.
Typical features for a simple one-step reaction:
- Reactants at some energy level.
- A maximum (the transition state or activated complex).
- Products at another energy level.
Activated Complex (Transition State)
At the top of the energy barrier between reactants and products lies the activated complex (or transition state):
- It is a highly unstable arrangement of atoms.
- It exists only momentarily, at the point of highest potential energy.
- Its structure is intermediate between reactants and products.
The transition state is not the same as an intermediate: the transition state is a maximum in energy along the path, while an intermediate corresponds to a local minimum between two maxima.
Activation Energy
The activation energy $E_\mathrm{a}$ is the minimum energy that reacting molecules must possess (in the direction of the reaction) to reach the activated complex.
For a simple reaction:
- Forward activation energy: energy difference between reactants and transition state.
- Reverse activation energy: energy difference between products and transition state.
The magnitude of $E_\mathrm{a}$ strongly influences the rate:
- Large $E_\mathrm{a}$ → few molecules have enough energy → slow reaction.
- Small $E_\mathrm{a}$ → more molecules can react → faster reaction.
The mathematical link between $k$, $E_\mathrm{a}$, and temperature is treated in detail in the chapter on the temperature dependence of reaction rates.
Molecular Kinetic Concepts (Qualitative)
Distribution of Molecular Energies
In a gas or solution, molecules move with a distribution of kinetic energies. At a given temperature:
- Most molecules have moderate energies.
- Some have low energy.
- Some have high energy.
For a reaction requiring a certain minimum energy $E_\mathrm{a}$, only molecules in the high-energy tail of the distribution can react on collision. Increasing temperature:
- Shifts the distribution so more molecules exceed $E_\mathrm{a}$.
- Thus increases the reaction rate.
Collision and Orientation
For many reactions, the rate depends not only on energy but also on how molecules collide:
- Collision frequency: The number of collisions per unit time increases with concentration and temperature.
- Orientation factor: Colliding molecules must be oriented properly for bonds to break and form.
Even if two molecules collide with energy exceeding $E_\mathrm{a}$, a reaction might not occur if the orientation is unfavorable.
Summary
- Chemical kinetics focuses on reaction rates and how they depend on conditions.
- Reaction rate is defined in terms of concentration change per unit time and related to stoichiometry.
- Rate laws express the dependence of rate on concentration and involve the rate constant and reaction orders, which are determined experimentally.
- Integrated rate laws describe how concentrations change over time and give useful characteristics like half-lives.
- Real reactions often proceed via multistep mechanisms including elementary steps, intermediates, and a rate-determining step.
- The concepts of activated complex, activation energy, and molecular energy distributions explain why reactions have finite rates and why rates change with conditions.
- The detailed influence of temperature and catalysts on rate is addressed in the subsequent chapters of Chemical Kinetics.