Collision Theory, Transition-State Theory, and the Origin of Rate Constants

Chemical kinetics describes how fast chemical reactions occur and how reaction rates depend on variables such as concentration and temperature. While an experimentally determined rate law describes the dependence of rate on concentration, molecular theories of kinetics attempt to explain the magnitude and temperature dependence of the rate constant.

A reaction may be thermodynamically favorable and still proceed very slowly. Thermodynamics determines the relative stability of reactants and products, whereas kinetics describes the pathway between them and the rate at which that pathway is followed.

Collision Theory

Consider a gas-phase bimolecular reaction

$$A+B\rightarrow P.$$

Before reaction can occur, molecules of $A$ and $B$ must encounter each other.

For a simple hard-sphere model, the collision rate depends on the collision cross-section and the relative velocity of the molecules. The collision contribution is proportional to

$$\sigma_{AB}\langle v_{\mathrm{rel}}\rangle,$$

where $\sigma_{AB}$ is the collision cross-section and $\langle v_{\mathrm{rel}}\rangle$ is the mean relative speed.

For molecules with reduced mass $\mu$,

$$\langle v_{\mathrm{rel}}\rangle = \sqrt{\frac{8k_{\mathrm B}T}{\pi\mu}},$$

where

$$\mu=\frac{m_A m_B}{m_A+m_B}.$$

The mean relative velocity therefore increases approximately as

$$T^{1/2}.$$

An increase in temperature increases the collision frequency, but collision frequency alone cannot explain the strong temperature dependence observed for many chemical reactions.

Not every collision leads to reaction. The molecules must have sufficient energy and must approach each other in a suitable orientation. A simple collision-theory expression for a bimolecular rate constant can be written as

$$k \approx pZ_{AB}\exp\left(-\frac{E_a}{RT}\right),$$

where $Z_{AB}$ represents the collision contribution and $p$ is a steric factor accounting for the fraction of collisions having a suitable orientation.

The exponential term represents the fraction of collisions with sufficient energy to overcome the activation barrier. This term is usually much more sensitive to temperature than the collision frequency itself.

Therefore, the large increase in reaction rate with temperature is mainly caused by an increase in the fraction of molecules capable of overcoming the activation barrier rather than simply by an increase in the number of collisions.

The Arrhenius Equation

For many reactions, the temperature dependence of the rate constant can be described by the Arrhenius equation:

$$\boxed{ k=A\exp\left(-\frac{E_a}{RT}\right) }$$

where $A$ is the pre-exponential factor, $E_a$ is the activation energy, $R$ is the gas constant, and $T$ is the absolute temperature.

Taking the natural logarithm gives

$$\ln k = \ln A-\frac{E_a}{RT}.$$

If $A$ and $E_a$ are approximately constant over the temperature range considered, a plot of $\ln k$ against $1/T$ is linear:

$$\text{slope}=-\frac{E_a}{R}.$$

This relation can be used to determine the activation energy experimentally.

For measurements at two temperatures,

$$\ln k_1 = \ln A-\frac{E_a}{RT_1}$$

and

$$\ln k_2 = \ln A-\frac{E_a}{RT_2}.$$

Subtracting the two equations eliminates $A$:

$$\boxed{ \ln\frac{k_2}{k_1} = \frac{E_a}{R} \left( \frac{1}{T_1}-\frac{1}{T_2} \right) }$$

This expression is useful when the rate constant is known at one temperature and its value at another temperature is required.

Temperature in the Arrhenius equation must always be expressed in kelvin.

Activation Energy and the Reaction Coordinate

A chemical reaction can be represented using a potential-energy diagram along a reaction coordinate.

In a simple one-step reaction,

$$\text{reactants} \rightarrow \text{transition state} \rightarrow \text{products}.$$

The transition state corresponds to the high-energy region separating reactants from products.

The forward activation energy can be represented approximately as

$$E_{a,\mathrm f} \approx E_{\mathrm{TS}}-E_R,$$

while the reverse activation energy is

$$E_{a,\mathrm r} \approx E_{\mathrm{TS}}-E_P.$$

Therefore,

$$E_{a,\mathrm f}-E_{a,\mathrm r} \approx E_P-E_R.$$

For an exothermic reaction, the products lie lower in energy than the reactants. The reverse activation barrier is therefore larger than the forward activation barrier.

For an endothermic reaction, the products lie higher in energy, and the forward activation barrier is larger.

These relationships describe the energy profile of a reaction pathway. They do not imply that thermodynamic favorability determines how rapidly a reaction occurs.

Transition-State Theory

Collision theory provides a useful physical picture for simple gas-phase reactions, but it becomes less satisfactory for more complicated molecular systems. Molecular orientation, vibrations, rotations, solvent interactions, and molecular organization can all influence reaction rates.

Transition-state theory describes a reaction in terms of passage through an activated configuration separating reactants from products.

The rate constant is described by the Eyring equation:

$$\boxed{ k = \kappa \frac{k_{\mathrm B}T}{h} \exp\left( -\frac{\Delta G^\ddagger}{RT} \right) }$$

where $\Delta G^\ddagger$ is the Gibbs energy of activation, $k_{\mathrm B}$ is the Boltzmann constant, $h$ is Planck’s constant, and $\kappa$ is the transmission coefficient.

The Gibbs energy of activation can be written as

$$\Delta G^\ddagger = \Delta H^\ddagger - T\Delta S^\ddagger.$$

Substitution into the Eyring equation gives

$$\boxed{ k = \kappa \frac{k_{\mathrm B}T}{h} \exp\left( \frac{\Delta S^\ddagger}{R} \right) \exp\left( -\frac{\Delta H^\ddagger}{RT} \right) }$$

The rate constant therefore depends on both the enthalpy and entropy of activation.

A large positive $\Delta H^\ddagger$ corresponds to a large energetic barrier and generally decreases the reaction rate.

A negative $\Delta S^\ddagger$ indicates that formation of the transition state requires greater molecular organization. This is common when two independently moving molecules must adopt a specific relative orientation before reaction can occur.

Thus two reactions with similar energetic barriers may still have substantially different rate constants because their activation entropies are different.

Relation Between the Arrhenius and Eyring Equations

The Arrhenius equation is

$$k=Ae^{-E_a/RT},$$

whereas transition-state theory gives

$$k = \kappa\frac{k_{\mathrm B}T}{h} e^{\Delta S^\ddagger/R} e^{-\Delta H^\ddagger/RT}.$$

The Arrhenius pre-exponential factor should therefore not always be interpreted simply as a collision frequency. It can contain contributions associated with molecular organization and entropy.

For an elementary process under common approximations, the Arrhenius activation energy and the enthalpy of activation are related approximately by

$$E_a \approx \Delta H^\ddagger+RT.$$

The two descriptions are therefore closely related, although they arise from different approaches to reaction kinetics.

Catalysis and Activation Energy

A catalyst increases the rate of a reaction by providing an alternative reaction pathway.

The catalyzed pathway may contain different intermediates and transition states from the uncatalyzed pathway. Instead of crossing one large energy barrier, the system may pass through several steps with smaller barriers.

A catalyst does not change the equilibrium constant at a fixed temperature. It changes the rates at which equilibrium is approached.

If a catalyst accelerates the forward reaction, the corresponding reverse pathway is also accelerated. The equilibrium composition therefore remains unchanged.

Reaction Order and Molecularity

Reaction order and molecularity describe different properties of a reaction.

Consider an experimentally determined rate law

$$v=k[A]^2[B].$$

The reaction is second order with respect to $A$, first order with respect to $B$, and third order overall.

This does not imply that two molecules of $A$ and one molecule of $B$ collide simultaneously.

Reaction order is defined from the experimentally observed rate law.

Molecularity refers to the number of reacting species involved in a single elementary step.

For example, if

$$A+B\rightarrow I$$

is an elementary bimolecular step, the law of mass action gives

$$v=k[A][B].$$

For an overall reaction consisting of several elementary steps, the rate law generally cannot be obtained directly from the stoichiometric equation. It must instead be determined experimentally or derived from an appropriate reaction mechanism.

Competing Reaction Pathways

A reactant may undergo more than one reaction at the same time. Consider two parallel first-order pathways:

$$A\xrightarrow{k_1}P_1$$

and

$$A\xrightarrow{k_2}P_2.$$

Their rates are

$$v_1=k_1[A]$$

and

$$v_2=k_2[A].$$

The ratio of the two rates is therefore

$$\frac{v_1}{v_2} = \frac{k_1}{k_2}.$$

If both rate constants follow Arrhenius behavior,

$$k_1=A_1e^{-E_{a,1}/RT}$$

and

$$k_2=A_2e^{-E_{a,2}/RT}.$$

Therefore,

$$\boxed{ \frac{k_1}{k_2} = \frac{A_1}{A_2} \exp\left[ -\frac{E_{a,1}-E_{a,2}}{RT} \right] }$$

The relative importance of competing pathways can therefore change with temperature.

If one pathway has a larger activation energy, its rate constant generally increases more strongly with temperature. However, the final ratio of the two rate constants also depends on their pre-exponential factors.

References

• IUPAC Compendium of Chemical Terminology (Gold Book), entries on activation energy, reaction order, molecularity, and transition-state theory.
• OpenStax, Chemistry 2e, section “Collision Theory.”
• Atkins, P.; de Paula, J.; Keeler, J. Atkins’ Physical Chemistry.
• Laidler, K. J. Chemical Kinetics.