10/14 Chemical Kinetics

Scope and Core Ideas

Chemical kinetics focuses on two questions: how fast a reaction occurs and how the reaction proceeds at the molecular level. This differs from thermodynamics: thermodynamics addresses whether a process is energetically favorable, whereas kinetics describes its rate and pathway.

Important factors affecting reaction rate include:

• Reactant concentration

• Temperature

• Physical state

• Surface area

• Catalysts

A faster reaction has a greater change in concentration over a given interval, but speed alone does not identify the molecular pathway.

Takeaway: Kinetics concerns rate and mechanism, not simply whether a reaction is energetically favorable.

Measuring Reaction Rates

A reaction rate measures concentration change per unit time:

Because a reactant concentration decreases, its rate expression includes a negative sign. For the reaction

the stoichiometric coefficients make the rates consistent:

The usual units are $\mathrm{mol\,L^{-1}\,s^{-1}}$, or $\mathrm{M\,s^{-1}}$. An average rate is measured over a finite interval. An instantaneous rate applies at one moment and corresponds to the slope of a concentration-versus-time curve.

For

if $HI$ disappears at $0.040\ \mathrm{M\,s^{-1}}$, the reaction rate is

Thus, $H_2$ and $I_2$ are each produced at $0.020\ \mathrm{M\,s^{-1}}$.

Takeaway: Always divide each concentration-change rate by its stoichiometric coefficient before comparing substances.

Rate Laws and Reaction Order

A rate law connects reaction rate to reactant concentrations. For reactants $A$ and $B$, a common form is

Here, $k$ is the rate constant, $m$ and $n$ are the orders with respect to $A$ and $B$, and $m+n$ is the overall reaction order.

Reaction orders are usually determined experimentally. Compare experiments in which one concentration changes while the others remain constant:

• If doubling a concentration leaves the rate unchanged, the reaction is zero order in that reactant.

• If doubling it doubles the rate, the reaction is first order in that reactant.

• If doubling it quadruples the rate, the reaction is second order in that reactant.

For example, if doubling $[A]$ doubles the rate and doubling $[B]$ quadruples the rate, then

If an experiment gives $[A]=0.10\ \mathrm{M}$, $[B]=0.10\ \mathrm{M}$, and rate $=2.0\times10^{-3}\ \mathrm{M\,s^{-1}}$, then

For overall order $r$, the units of the rate constant are

The exponents in a rate law generally cannot be inferred from the coefficients of the overall balanced equation. A known elementary reaction is an important exception.

Takeaway: Determine reaction orders from concentration-and-rate data, then use one experiment to calculate $k$.

Concentration, Time, and Half-Life

An integrated rate law relates concentration to time. Use the form that matches the reaction order:

• Zero order:

  $$[A]_t=[A]_0-kt$$

  A plot of $[A]$ versus $t$ is linear, and $t_{1/2}=\dfrac{[A]_0}{2k}$.

• First order:

  $$\ln[A]_t=\ln[A]_0-kt$$

  A plot of $\ln[A]$ versus $t$ is linear, and $t_{1/2}=\dfrac{0.693}{k}$.

• Second order:

  $$\frac{1}{[A]_t}=\frac{1}{[A]_0}+kt$$

  A plot of $1/[A]$ versus $t$ is linear, and $t_{1/2}=\dfrac{1}{k[A]_0}$.

For a first-order reaction with $k=0.0250\ \mathrm{s^{-1}}$, $[A]_0=0.800\ \mathrm{M}$, and $t=40.0\ \mathrm{s}$,

which gives $[A]_t=0.294\ \mathrm{M}$. Its half-life is

The independence of first-order half-life from initial concentration is a useful diagnostic feature.

Takeaway: Match the concentration function that produces a straight-line plot to identify or test the reaction order.

Mechanisms and Elementary Steps

A reaction mechanism is a sequence of elementary reactions, each representing one molecular event. A species produced in one step and consumed in a later step is an intermediate.

For example:

Adding the steps gives

because intermediate $I$ cancels.

The molecularity of an elementary step is the number of reactant particles involved:

• Unimolecular: one particle, such as $A\rightarrow\text{products}$

• Bimolecular: two particles, such as $A+B\rightarrow\text{products}$ or $2A\rightarrow\text{products}$

• Termolecular: three particles colliding simultaneously; such steps are uncommon

For an elementary step, the rate law can be written directly from its reactants. For example,

The slowest elementary step often acts as the rate-determining step. However, a proposed mechanism is acceptable only if its steps add to the observed overall equation and its predicted rate law agrees with experimental data. If a fast equilibrium precedes the slow step, an intermediate concentration may need to be rewritten using reactant concentrations.

Takeaway: Do not write an overall rate law directly from a balanced equation unless the reaction is known to be elementary.

Collision Theory and Energy Barriers

Collision theory states that a reaction requires an effective collision. Reacting particles must collide, possess enough energy to overcome the barrier, and have a suitable orientation for bonds to break and form.

The minimum energy barrier is the activation energy $E_a$. The high-energy arrangement at the top of the barrier is the transition state, or activated complex. A larger $E_a$ generally means a slower reaction because fewer collisions have enough energy to react.

The Arrhenius equation is

where $A$ is the frequency factor, $R=8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}$, and $T$ is measured in kelvins. For two temperatures, use

A temperature increase can substantially increase the rate constant because of the exponential temperature dependence.

On an energy profile,

whereas

Therefore, $E_a$ measures a kinetic barrier, while $\Delta H$ compares initial and final energy states.

Takeaway: Temperature affects rate strongly through $k$, while activation energy and enthalpy describe different energy differences.

Catalysis and Quantitative Strategy

A catalyst increases reaction rate by providing an alternative mechanism with a lower activation energy. It participates in one or more elementary steps but is regenerated, so it is not consumed overall.

Catalysts may be:

• Homogeneous, when catalyst and reactants are in the same phase

• Heterogeneous, when they are in different phases

• Enzymatic, when the catalyst is a biological molecule with an active site

A catalyst does not change the overall stoichiometric equation, $\Delta H$, or the equilibrium constant. For a reversible reaction, it lowers the activation energy for both forward and reverse processes. Equilibrium is reached faster, but the final equilibrium composition is unchanged.

For an uncatalyzed pathway with $E_a=80\ \mathrm{kJ\,mol^{-1}}$ and a catalyzed pathway with $E_a=45\ \mathrm{kJ\,mol^{-1}}$, the catalyzed pathway has the larger rate constant at the same temperature because its Arrhenius factor is larger.

A catalytic mechanism can be represented by

Adding the steps gives $A+B\rightarrow AB$. Species $C$ is regenerated and therefore acts as the catalyst.

Problem-solving checklist:

1. Identify whether the task concerns a differential or integrated rate law.

2. Use experimental comparisons to determine reaction orders.

3. Check units, especially for $k$ and $E_a$.

4. Convert temperatures to kelvins before using the Arrhenius equation.

5. Check that the result is chemically reasonable, such as a positive rate constant and a concentration no greater than its initial value.

Takeaway: Catalysts change the pathway and speed of approach to equilibrium, not the reaction's overall energy change or equilibrium position.