In chemistry, transition state theory (TST) explains the reaction rates of elementary chemical reactions. The theory assumes a special type of chemical equilibrium (quasi-equilibrium) between reactants and activated transition state complexes.
TST is used primarily to understand qualitatively how chemical reactions take place. TST has been less successful in its original goal of calculating absolute reaction-rate constants because the calculation of absolute reaction rates requires precise knowledge of potential-energy surfaces, but it has been successful in calculating the standard enthalpy of activation (ΔH‡, also written Δ‡Hɵ), the standard entropy of activation (ΔS‡ or Δ‡Sɵ), and the standard Gibbs energy of activation (ΔG‡ or Δ‡Gɵ) for a particular reaction if its rate constant has been experimentally determined (the ‡ notation refers to the value of interest at the transition state; ΔH‡ is the difference between the enthalpy of the transition state and that of the reactants).
This theory was developed simultaneously in 1935 by Henry Eyring, then at Princeton University, and by Meredith Gwynne Evans and Michael Polanyi of the University of Manchester. TST is also referred to as "activated-complex theory", "absolute-rate theory", and "theory of absolute reaction rates".
Before the development of TST, the Arrhenius rate law was widely used to determine energies for the reaction barrier. The Arrhenius equation derives from empirical observations and ignores any mechanistic considerations, such as whether one or more reactive intermediates are involved in the conversion of a reactant to a product. Therefore, further development was necessary to understand the two parameters associated with this law, the pre-exponential factor (A) and the activation energy (Ea). TST, which led to the Eyring equation, successfully addresses these two issues; however, 46 years elapsed between the publication of the Arrhenius rate law, in 1889, and the Eyring equation derived from TST, in 1935. During that period, many scientists and researchers contributed significantly to the development of the theory.
Theory The basic ideas behind transition state theory are as follows:
Rates of reaction can be studied by examining activated complexes near the saddle point of a potential-energy surface. The details of how these complexes are formed are not important. The saddle point itself is called the transition state. The activated complexes are in a special equilibrium (quasi-equilibrium) with the reactant molecules. The activated complexes can convert into products, and kinetic theory can be used to calculate the rate of this conversion.
Development In the development of TST, three approaches were taken as summarized below.
Thermodynamic treatment In 1884, Jacobus van 't Hoff proposed the Van 't Hoff equation describing the temperature dependence of the equilibrium constant for a reversible reaction: {A} <=> {B}
where ΔU is the change in internal energy, K is the equilibrium constant of the reaction, R is the universal gas constant, and T is thermodynamic temperature. Based on experimental work, in 1889, Svante Arrhenius proposed a similar expression for the rate constant of a reaction, given as follows:
Integration of this expression leads to the Arrhenius equation
where k is the rate constant. A was referred to as the frequency factor (now called the pre-exponential coefficient), and Ea is regarded as the activation energy. By the early 20th century, many had accepted the Arrhenius equation, but the physical interpretation of A and Ea remained vague. This led many researchers in chemical kinetics to offer different theories of how chemical reactions occurred in an attempt to relate A and Ea to the molecular dynamics directly responsible for chemical reactions.
In 1910, French chemist René Marcelin introduced the concept of standard Gibbs energy of activation. His relation can be written as
At about the same time as Marcelin was working on his formulation, Dutch chemists Philip Abraham Kohnstamm, Frans Eppo Cornelis Scheffer, and Wiedold Frans Brandsma introduced standard entropy of activation and the standard enthalpy of activation. They proposed the rate-constant equation
However, the nature of the constant was still unclear.
Kinetic-theory treatment In early 1900, Max Trautz and William Lewis studied the rate of the reaction using collision theory, based on the kinetic theory of gases. Collision theory treats reacting molecules as hard spheres colliding with one another; this theory neglects entropy changes, since it assumes that the collisions between molecules are completely elastic. Lewis applied his treatment to the reaction and obtained good agreement with experimental result. However, later when the same treatment was applied to other reactions, there were large discrepancies between theoretical and experimental results.
Statistical-mechanical treatment Statistical mechanics played a significant role in the development of TST. However, the application of statistical mechanics to TST was developed very slowly given the fact that in mid-19th century, James Clerk Maxwell, Ludwig Boltzmann, and Leopold Pfaundler published several papers discussing reaction equilibria and rates in terms of molecular motions and the statistical distributions of molecular speeds.
It was not until 1912 when the French chemist Alfred Berthoud used the Maxwell–Boltzmann distribution law to obtain an expression for the rate constant:
where a and b are constants related to energy terms.
Two years later, René Marcelin made an essential contribution by treating the progress of a chemical reaction as a motion of a point in phase space. He then applied Gibbs' statistical-mechanical procedures and obtained an expression similar to the one he had obtained earlier from thermodynamic considerations.
In 1915, another important contribution came from British physicist James Rice. Based on his statistical analysis, he concluded that the rate constant is proportional to the "critical increment". His ideas were further developed by Richard Chace Tolman. In 1919, Austrian physicist Karl Ferdinand Herzfeld applied statistical mechanics to the equilibrium constant and kinetic theory to the rate constant of the reverse reaction, k−1, for the reversible dissociation of a diatomic molecule:
AB <=>[k_1][k_{-1}] {A} + {B}.
He obtained the following equation for the rate constant of the forward reactionKeith J. Laidler, Chemical Kinetics (3rd ed., Harper & Row 1987), p.88
where is the dissociation energy at absolute zero, kB is the Boltzmann constant, h is the Planck constant, T is thermodynamic temperature, and is vibrational frequency of the bond. This expression is very important since it is the first time that the factor kBT/h, which is a critical component of TST, appeared in a rate equation.
In 1920, the American chemist Richard Chace Tolman further developed Rice's idea of the critical increment. He concluded that critical increment (now referred to as activation energy) of a reaction is equal to the average energy of all molecules undergoing reaction minus the average energy of all reactant molecules.
Potential-energy surfaces The concept of potential-energy surface was very important in the development of TST. The foundation of this concept was laid by René Marcelin in 1913. He theorized that the progress of a chemical reaction could be described as a point in a potential-energy surface with coordinates in atomic momenta and distances.
In 1931, Henry Eyring and Michael Polanyi constructed a potential-energy surface for the reaction . This surface is a three-dimensional diagram based on quantum-mechanical principles as well as experimental data on vibrational frequencies and energies of dissociation. A year after the Eyring–Polanyi construction, Hans Pelzer and Eugene Wigner made an important contribution by following the progress of a reaction on a potential energy surface. The importance of this work was that it was the first time that the concept of col or saddle point in the potential-energy surface was discussed. They concluded that the rate of a reaction is determined by the motion of the system through that col.
Kramers theory of reaction rates By modeling reactions as Langevin motion along a one dimensional reaction coordinate, Hendrik Kramers was able to derive a relationship between the shape of the potential-energy surface along the reaction coordinate and the transition rates of the system. The formulation relies on approximating the potential-energy landscape as a series of harmonic wells. In a two-state system, there will be three wells: a well for state A, an upside-down well representing the potential energy barrier, and a well for state B.
In the overdamped (or "diffusive") regime, the transition rate from state A to B is related to the resonant frequency of the wells via
where is the frequency of the well for state A, is the frequency of the barrier well, is the viscous damping, is the energy of the top of the barrier, is the energy of bottom of the well for state A, and is the temperature of the system times the Boltzmann constant.
For general damping (overdamped or underdamped), there is a similar formula.
Justification for the Eyring equation One of the most important features introduced by Eyring, Polanyi, and Evans was the notion that activated complexes are in quasi-equilibrium with the reactants. The rate is then directly proportional to the concentration of these complexes multiplied by the frequency (kBT/h) with which they are converted into products. Below, a non-rigorous plausibility argument is given for the functional form of the Eyring equation. However, the key statistical-mechanical factor kBT/h will not be justified, and the argument presented below does not constitute a true "derivation" of the Eyring equation.For an introductory treatment of the statistical mechanics and an elementary derivation of the Eyring equation, see: Lowry and Richardson, Mechanism and Theory in Organic Chemistry, 3rd ed. (Harper & Row, 1987), pp. 248–253.
Quasi-equilibrium assumption Quasi-equilibrium is different from classical chemical equilibrium, but can be described using a similar thermodynamic treatment. Consider the reaction
{A} + {B} <=>[][] [AB]^\ddagger -> {P},
where complete equilibrium is achieved between all the species in the system including activated complexes, [AB]‡ . Using statistical mechanics, concentration of [AB]‡ can be calculated in terms of the concentration of A and B.
TST assumes that even when the reactants and products are not in equilibrium with each other, the activated complexes are in quasi-equilibrium with the reactants. As illustrated in Figure 2, at any instant of time, there are a few activated complexes, and some were reactant molecules in the immediate past, which are designated [ABl]‡ (since they are moving from left to right). The remainder of them were product molecules in the immediate past ([ABr]‡).
In TST, it is assumed that the flux of activated complexes in the two directions are independent of each other. That is, if all the product molecules were suddenly removed from the reaction system, the flow of [ABr]‡ stops, but there is still a flow from left to right. Hence, to be technically correct, the reactants are in equilibrium only with [ABl]‡, the activated complexes that were reactants in the immediate past.
Plausibility argument The activated complexes do not follow a Boltzmann distribution of energies, but an "equilibrium constant" can still be derived from the distribution they do follow. The equilibrium constant K‡ for the quasi-equilibrium can be written as
So, the chemical activity of the transition state AB‡ is
Therefore, the rate equation for the production of product is
where the rate constant k is given by