Lecture 1 Introduction to Chemical Kinetics

1.1 Why Study Reaction Kinetics?

Chemistry is fundamentally a study of change—the transformation of substances into new forms with different properties. When we mix reactants together, we often want to know not just whether a reaction will occur, but how rapidly it will proceed. While thermodynamics provides powerful tools for determining whether a particular chemical change is spontaneous and what the final equilibrium composition will be, it cannot tell us how quickly this equilibrium will be reached.

A reaction may be thermodynamically favourable, yet proceed so slowly that it appears not to happen at all. Consider the conversion of diamond to graphite—thermodynamically spontaneous at room temperature and pressure, yet the process occurs at such a negligible rate that diamonds remain stable for millions of years. Conversely, some reactions may reach equilibrium in fractions of a second.

Chemical kinetics addresses these questions of reaction rates and mechanisms. Where thermodynamics concerns itself with the energy relationships that govern chemical equilibria, kinetics focuses on the rates of chemical processes and the molecular pathways by which they occur. Through kinetic analysis, we can determine how quickly reactants transform into products, what factors influence reaction speed, and what sequence of molecular events constitutes the reaction mechanism.

The complementary nature of thermodynamics and kinetics provides a more complete picture of chemical change. Thermodynamics allows us to predict whether a reaction is possible and what the equilibrium state will be; kinetics reveals the rate at which this equilibrium is approached and the molecular steps involved in the transformation. Both perspectives are essential for understanding chemical processes in laboratory settings, industrial applications, and natural systems.

1.2 Chemical Kinetics → Describing [A] as a function of time

Let us start by examining a simple reaction, the conversion of a reactant A into a product B:

\[{\mathrm{A}} \rightarrow {\mathrm{B}}\]

As the reaction proceeds, the concentrations of both species change. We can track the progress of the reaction by monitoring the concentrations of A and B as functions of time. Using square brackets to denote concentration (in mol dm\(^{-3}\)), we write these as [\(\mathrm{A}\)] and [\(\mathrm{B}\)].

Starting from initial values at \(t=0\), the system evolves toward its final state. Throughout this process, the total concentration remains constant:

\[[\mathrm{A}]_t + [\mathrm{B}]_t = [\mathrm{A}]_{t=0} + [\mathrm{B}]_{t=0}\]

This equation reflects the stoichiometric conversion between reactant and product—each molecule of A that disappears gives rise to exactly one molecule of B.

We might ask questions such as:

  • How much product B is there after time \(t\)?
  • How long will it take for a certain percentage of A to convert to B?
  • If we change pressure, temperature, or initial concentrations, how will the reaction speed be affected?

1.3 Reaction Rates → Describing d[A]/dt

We can quantify how quickly a reaction occurs in terms of the rate at which the concentrations of reactants and products change with time. This is what we call the reaction rate.

For our simple A → B reaction, the rate can be expressed as either the rate of disappearance of A or the rate of appearance of B:

\[\text{rate} = -\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{d}t} = +\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{d}t}\]

The negative sign in the first term ensures that the rate is a positive quantity, since [\(\mathrm{A}\)] decreases with time (giving a negative derivative).

For a constant reaction rate, we can calculate the change in [A] per unit time as:

\[\text{change in [A] per unit time} = \frac{[\mathrm{A}]_{t_2} - [\mathrm{A}]_{t_1}}{t_2 - t_1} = \frac{\Delta[\mathrm{A}]}{\Delta t}\]

In most cases, however, reaction rates aren’t constant but change as the reaction progresses and reactant concentrations decrease.

For reactions with more complex stoichiometry, the relationship between concentration changes requires appropriate stoichiometric factors.

Consider a reaction where one molecule of A combines with two molecules of B to form one molecule of C:

\(\mathrm{A} + 2\mathrm{B} \rightarrow \mathrm{C}\)

Here, B is consumed twice as rapidly as A, and the rates are related by:

\[\text{rate} = -\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{d}t} = -\frac{1}{2}\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{d}t} = +\frac{\mathrm{d}[\mathrm{C}]}{\mathrm{d}t}\]

In general, for a reaction with stoichiometry \(\nu_\mathrm{A}\mathrm{A} + \nu_\mathrm{B}\mathrm{B} + \ldots \rightarrow \nu_\mathrm{P}\mathrm{P} + \nu_\mathrm{Q}\mathrm{Q} + \ldots\)

The rate can be written as:

\[\text{rate} = -\frac{1}{\nu_\mathrm{A}}\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{d}t} = -\frac{1}{\nu_\mathrm{B}}\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{d}t} = \ldots = \frac{1}{\nu_\mathrm{P}}\frac{\mathrm{d}[\mathrm{P}]}{\mathrm{d}t} = \frac{1}{\nu_\mathrm{Q}}\frac{\mathrm{d}[\mathrm{Q}]}{\mathrm{d}t} = \ldots\]

These relationships ensure that the stoichiometric ratios are properly accounted for when defining the reaction rate.

1.4 Calculus for Kinetics

Chemical kinetics relies heavily on the mathematical tools of calculus to connect concentration-time profiles with reaction rates:

These two aspects of kinetic data are related through differentiation and integration. If we know how concentration varies with time, we can determine the instantaneous reaction rate by taking the derivative:

\[\text{rate} = \nu = -\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{d}t}\]

Conversely, if we know the rate law for a reaction, we can predict the concentration-time profile by integration:

\[[\mathrm{A}]_t = [\mathrm{A}]_0 - \int_0^t \nu\,\mathrm{d}t\]

where \(\nu\) represents the reaction rate as a function of time. This mathematical relationship forms the foundation for analysing kinetic data and determining rate laws experimentally.

1.5 Rate Laws

The rates of chemical reactions typically depend on the concentrations of the reactants. The mathematical relationship between the concentrations of reactants and the rate of a reaction is called a “rate law”.

Many reactions follow simple rate laws, where the rate of the reaction is proportional to the concentration of each reactant, raised to some power:

\[\text{rate} = k[\mathrm{A}]^a[\mathrm{B}]^b[\mathrm{C}]^c\cdots\]

This equation contains several important components:

  • \(k\) is the rate constant, a proportionality factor that depends on temperature but not on concentration
  • The exponents \(a\), \(b\), \(c\), etc. are called the order of reaction with respect to reactants A, B, C, etc.
  • The sum \((a + b + c + \cdots)\) is called the overall reaction order

It’s important to note that these orders need not reflect the stoichiometry of the overall reaction equation. For example:

\(\mathrm{H}_2 + \mathrm{I}_2 \rightarrow 2\mathrm{HI}\) has rate law \(\nu = k[\mathrm{H}_2][\mathrm{I}_2]\)

\(3\mathrm{ClO}^- \rightarrow \mathrm{ClO}_3^- + 2\mathrm{Cl}^-\) has rate law \(\nu = k[\mathrm{ClO}^-]^2\)

Rate laws can also be complex, as in the case of hydrogen bromide formation:

\(\mathrm{H}_2 + \mathrm{Br}_2 \rightarrow 2\mathrm{HBr}\) with rate law \(\nu = \frac{k[\mathrm{H}_2][\mathrm{Br}_2]^{1/2}}{1+k'[\mathrm{HBr}]/[\mathrm{Br}_2]}\)

A complex rate law always implies a multistep reaction mechanism.

1.6 Elementary Processes

Any chemical reaction can be broken down into a sequence of simple molecular-level steps called elementary processes. An elementary process represents a single molecular event—a simple transformation that cannot be broken down further.

The molecularity of an elementary process tells us how many reactant molecules must come together for the reaction to occur. This concept allows us to classify elementary steps into three categories:

  1. Unimolecular processes involve the transformation of a single molecule: \(\mathrm{A} \rightarrow \text{products} \qquad \text{rate} = k[\mathrm{A}]\)

    Typical examples include isomerisation reactions (where a molecule rearranges its bonds without changing composition) and decomposition reactions (where a molecule breaks into smaller fragments).

  2. Bimolecular processes require the collision of two molecules: \(\mathrm{A} + \mathrm{B} \rightarrow \text{products} \qquad \text{rate} = k[\mathrm{A}][\mathrm{B}]\)

    These represent the most common elementary reactions, as they involve the relatively probable event of two molecules encountering each other.

  3. Termolecular processes involve the simultaneous collision of three molecules: \(\mathrm{A} + \mathrm{B} + \mathrm{C} \rightarrow \text{products} \qquad \text{rate} = k[\mathrm{A}][\mathrm{B}][\mathrm{C}]\)

    Such processes are rare in solution chemistry due to the low probability of three molecules colliding simultaneously with the correct orientation. When observed, they often involve a third body that absorbs excess energy from an energetic collision between two reactive species.

For elementary processes, there is a direct correspondence between molecularity and reaction order—the rate law exponents exactly match the number of molecules participating in the reaction. This correspondence provides a powerful tool for deducing reaction mechanisms from experimental rate laws.

1.7 Complex Reactions

For complex reactions involving multiple elementary steps, we cannot deduce the rate law from the reaction stoichiometry alone. Consider the hydrogen bromide formation again:

\[\mathrm{H}_2 + \mathrm{Br}_2 \rightarrow 2\mathrm{HBr}\]

This overall reaction actually proceeds through a mechanism involving four elementary steps:

  1. \(\mathrm{Br}_2 \rightarrow \mathrm{Br} + \mathrm{Br}\) (bimolecular)
  2. \(\mathrm{Br} + \mathrm{H}_2 \rightarrow \mathrm{H} + \mathrm{HBr}\) (bimolecular)
  3. \(\mathrm{H} + \mathrm{Br}_2 \rightarrow \mathrm{Br} + \mathrm{HBr}\) (bimolecular)
  4. \(\mathrm{Br} + \mathrm{Br} \rightarrow \mathrm{Br}_2\) (unimolecular)

Rate laws for complex reactions must be determined empirically through analysis of experimental data. However, as we shall see later, if we know the mechanism of a reaction (the sequence of elementary steps), we can predict its rate law. Conversely, experimental rate laws can help us deduce possible reaction mechanisms.

1.8 Units of the Rate Constant

The units of a rate constant depend on the form of the rate law in which it appears. This dependence arises from the requirement of dimensional consistency—both sides of a rate equation must have the same units.

For reactions following first-order kinetics (rate = \(k[\mathrm{A}]\)), we have:

\[\text{mol}\,\text{dm}^{-3}\text{s}^{-1} = k \times \text{mol}\,\text{dm}^{-3}\]

Solving for \(k\) shows that its units must be s\(^{-1}\):

\(k \text{ has units } \text{s}^{-1}\)

For second-order reactions (rate = \(k[\mathrm{A}]^2\)):

\[\text{mol}\,\text{dm}^{-3}\text{s}^{-1} = k \times (\text{mol}\,\text{dm}^{-3})^2\]

Here, the rate constant has units of mol\(^{-1}\)dm\(^3\)s\(^{-1}\):

\[k \text{ has units } \text{mol}^{-1}\text{dm}^3\text{s}^{-1}\]

In general, for a reaction of order \(n\), the rate constant has units of mol\(^{1-n}\) dm\(^{3(n-1)}\)s \(^{-1}\). This pattern reflects the fundamental relationship between reaction order and the dimensions of the rate constant.

Understanding these units is essential for interpreting kinetic data correctly and for comparing rate constants across different reaction systems.