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:

${\displaystyle {\ce {{A}<=> {B}}}}$

${\displaystyle {\frac {d\ln K}{dT}}={\frac {\Delta U}{RT^{2}}}}$

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:

${\displaystyle {\frac {d\ln k}{dT}}={\frac {\Delta E}{RT^{2}}}}$

Integration of this expression leads to the Arrhenius equation

${\displaystyle k=Ae^{-E_{a}/RT}}$

where k is the rate constant. A was referred to as the frequency factor (now called the pre-exponential coefficient), and E_a is regarded as the activation energy. By the early 20th century many had accepted the Arrhenius equation, but the physical interpretation of A and E_a 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 E_a 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

${\displaystyle k\propto \exp \left({\frac {-\Delta ^{\ddagger }G^{\ominus }}{RT}}\right)}$

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 following rate constant equation

${\displaystyle k\propto \exp \left({\frac {\Delta ^{\ddagger }S^{\ominus }}{R}}\right)\exp \left({\frac {-\Delta ^{\ddagger }H^{\ominus }}{RT}}\right)}$

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 collision between molecules are completely elastic.

Lewis applied his treatment to the following reaction and obtained good agreement with experimental result.

2HI → H₂ + I₂

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 equilibrium and rates in terms of molecular motions and the statistical distribution of molecular speeds.

It was not until 1912 when the French chemist A. Berthoud used the Maxwell–Boltzmann distribution law to obtain an expression for the rate constant.

${\displaystyle {\frac {d\ln k}{dT}}={\frac {a-bT}{RT^{2}}}}$

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 consideration.

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.^[7]

${\displaystyle {\ce {AB <=>[k_1][k_{-1}] {A}+ {B}}}}$

He obtained the following equation for the rate constant of the forward reaction^[8]

${\displaystyle k_{1}={\frac {k_{\mathrm {B} }T}{h}}\left(1-e^{-{\frac {h\nu }{k_{\text{B}}T}}}\right)\exp \left({\frac {-E^{\ominus }}{RT}}\right)}$

where ${\displaystyle \textstyle E^{\ominus }}$ is the dissociation energy at absolute zero, k_B is the Boltzmann constant, h is the Planck constant, T is thermodynamic temperature, ${\displaystyle \nu }$ is vibrational frequency of the bond. This expression is very important since it is the first time that the factor k_BT/h, which is a critical component of TST, has 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 below. This surface is a three-dimensional diagram based on quantum-mechanical principles as well as experimental data on vibrational frequencies and energies of dissociation.

H + H₂ → H₂ + H

A year after the Eyring and 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.

It has been typically assumed that the rate-limiting or lowest saddle point is located on the same energy surface as the initial ground state. However, it was recently found that this could be incorrect for processes occurring in semiconductors and insulators, where an initial excited state could go through a saddle point lower than the one on the surface of the initial ground state.^[9]

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

${\displaystyle k^{A\rightarrow B}={\frac {\omega _{a}\omega _{H}}{2\pi \gamma }}\exp \left(-{\frac {E_{H}-E_{A}}{k_{\text{B}}T}}\right)}$

where ${\displaystyle \omega _{a}}$ is the frequency of the well for state A, ${\displaystyle \omega _{H}}$ is the frequency of the barrier well, ${\displaystyle \gamma }$ is the viscous damping, ${\displaystyle E_{H}}$ is the energy of the top of the barrier, ${\displaystyle E_{A}}$ is the energy of bottom of the well for state A, and ${\displaystyle k_{\text{B}}T}$ is the temperature of the system times the Boltzmann constant.^[10]

For general damping (overdamped or underdamped), there is a similar formula.^[11]

Quasi-equilibrium assumption

Quasi-equilibrium is different from classical chemical equilibrium, but can be described using a similar thermodynamic treatment.^{[5] [13]} Consider the reaction below

${\displaystyle {\ce {{A}+{B}<=>{[AB]^{\ddagger }}->{P}}}}$

Figure 2: Potential energy diagram

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 [AB_l]^‡ (since they are moving from left to right). The remainder of them were product molecules in the immediate past ([AB_r]^‡).

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 [AB_r]^‡ stops, but there is still a flow from left to right. Hence, to be technically correct, the reactants are in equilibrium only with [AB_l]^‡, 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

${\displaystyle K^{\ddagger }={\frac {\ce {[AB]^{\ddagger }}}{\ce {[A][B]}}}}$.

So, the chemical activity of the transition state AB^‡ is

${\displaystyle [{\ce {AB}}]^{\ddagger }=K^{\ddagger }[{\ce {A}}][{\ce {B}}]}$.

Therefore, the rate equation for the production of product is

${\displaystyle {\frac {d[{\ce {P}}]}{dt}}=k^{\ddagger }[{\ce {AB}}]^{\ddagger }=k^{\ddagger }K^{\ddagger }[{\ce {A}}][{\ce {B}}]=k[{\ce {A}}][{\ce {B}}]}$,

where the rate constant k is given by

${\displaystyle k=k^{\ddagger }K^{\ddagger }}$.

Here, k^‡ is directly proportional to the frequency of the vibrational mode responsible for converting the activated complex to the product; the frequency of this vibrational mode is ${\displaystyle \nu }$. Every vibration does not necessarily lead to the formation of product, so a proportionality constant ${\displaystyle \kappa }$, referred to as the transmission coefficient, is introduced to account for this effect. So k^‡ can be rewritten as

${\displaystyle k^{\ddagger }=\kappa \nu }$.

For the equilibrium constant K^‡ , statistical mechanics leads to a temperature dependent expression given as

${\displaystyle K^{\ddagger }={\frac {k_{\text{B}}T}{h\nu }}K^{\ddagger '}}$ (${\displaystyle K^{\ddagger '}=:e^{\frac {-\Delta G^{\ddagger }}{RT}}}$).

Combining the new expressions for k^‡ and K^‡, a new rate constant expression can be written, which is given as

${\displaystyle k=k^{\ddagger }K^{\ddagger }=\kappa {\frac {k_{\text{B}}T}{h}}e^{\frac {-\Delta G^{\ddagger }}{RT}}=\kappa {\frac {k_{\text{B}}T}{h}}K^{\ddagger '}}$.

Since, by definition, ΔG^‡ = ΔH^‡ –TΔS^‡, the rate constant expression can be expanded, to give an alternative form of the Eyring equation:

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

For correct dimensionality, the equation needs to have an extra factor of (c^⊖)^1–m for reactions that are not unimolecular:

${\displaystyle k=\kappa {\frac {k_{\text{B}}T}{h}}e^{\frac {\Delta S^{\ddagger }}{R}}e^{\frac {-\Delta H^{\ddagger }}{RT}}(c^{\ominus })^{1-m}}$,

where c^⊖ is the standard concentration 1 mol⋅L^–1 and m is the molecularity.^[14]

Microcanonical variational TST

A fundamental flaw of transition state theory is that it counts any crossing of the transition state as a reaction from reactants to products or vice versa. In reality, a molecule may cross this "dividing surface" and turn around, or cross multiple times and only truly react once. As such, unadjusted TST is said to provide an upper bound for the rate coefficients. To correct for this, variational transition state theory varies the location of the dividing surface that defines a successful reaction in order to minimize the rate for each fixed energy. ^[25] The rate expressions obtained in this microcanonical treatment can be integrated over the energy, taking into account the statistical distribution over energy states, so as to give the canonical, or thermal rates.

Canonical variational TST

A development of transition state theory in which the position of the dividing surface is varied so as to minimize the rate constant at a given temperature.

Improved canonical variational TST

A modification of canonical variational transition state theory in which, for energies below the threshold energy, the position of the dividing surface is taken to be that of the microcanonical threshold energy. This forces the contributions to rate constants to be zero if they are below the threshold energy. A compromise dividing surface is then chosen so as to minimize the contributions to the rate constant made by reactants having higher energies.

Nonadiabatic TST

An expansion of TST to the reactions when two spin-states are involved simultaneously is called nonadiabatic transition state theory (NA-TST).

Semiclassical TST

Using vibrational perturbation theory, effects such as tunnelling and variational effects can be accounted for within the SCTST formalism.

Enzymatic reactions

Enzymes catalyze chemical reactions at rates that are astounding relative to uncatalyzed chemistry at the same reaction conditions. Each catalytic event requires a minimum of three or often more steps, all of which occur within the few milliseconds that characterize typical enzymatic reactions. According to transition state theory, the smallest fraction of the catalytic cycle is spent in the most important step, that of the transition state. The original proposals of absolute reaction rate theory for chemical reactions defined the transition state as a distinct species in the reaction coordinate that determined the absolute reaction rate. Soon thereafter, Linus Pauling proposed that the powerful catalytic action of enzymes could be explained by specific tight binding to the transition state species ^[26] Because reaction rate is proportional to the fraction of the reactant in the transition state complex, the enzyme was proposed to increase the concentration of the reactive species.

This proposal was formalized by Wolfenden and coworkers at University of North Carolina at Chapel Hill, who hypothesized that the rate increase imposed by enzymes is proportional to the affinity of the enzyme for the transition state structure relative to the Michaelis complex.^[27] Because enzymes typically increase the non-catalyzed reaction rate by factors of 10⁶-10²⁶, and Michaelis complexes often have dissociation constants in the range of 10⁻³-10⁻⁶ M, it is proposed that transition state complexes are bound with dissociation constants in the range of 10⁻¹⁴ -10⁻²³ M. As substrate progresses from the Michaelis complex to product, chemistry occurs by enzyme-induced changes in electron distribution in the substrate. Enzymes alter the electronic structure by protonation, proton abstraction, electron transfer, geometric distortion, hydrophobic partitioning, and interaction with Lewis acids and bases. Analogs that resemble the transition state structures should therefore provide the most powerful noncovalent inhibitors known.

All chemical transformations pass through an unstable structure called the transition state, which is poised between the chemical structures of the substrates and products. The transition states for chemical reactions are proposed to have lifetimes near 10⁻¹³ seconds, on the order of the time of a single bond vibration. No physical or spectroscopic method is available to directly observe the structure of the transition state for enzymatic reactions, yet transition state structure is central to understanding enzyme catalysis since enzymes work by lowering the activation energy of a chemical transformation.

It is now accepted that enzymes function to stabilize transition states lying between reactants and products, and that they would therefore be expected to bind strongly any inhibitor that closely resembles such a transition state. Substrates and products often participate in several enzyme catalyzed reactions, whereas the transition state tends to be characteristic of one particular enzyme, so that such an inhibitor tends to be specific for that particular enzyme. The identification of numerous transition state inhibitors supports the transition state stabilization hypothesis for enzymatic catalysis.

Currently there is a large number of enzymes known to interact with transition state analogs, most of which have been designed with the intention of inhibiting the target enzyme. Examples include HIV-1 protease, racemases, β-lactamases, metalloproteinases, cyclooxygenases and many others.

Adsorption on surfaces and reactions on surfaces

Desorption as well as reactions on surfaces are straightforward to describe with transition state theory. Analysis of adsorption to a surface from a liquid phase can present a challenge due to lack of ability to assess the concentration of the solute near the surface. When full details are not available, it has been proposed that reacting species' concentrations should be normalized to the concentration of active surface sites, an approximation called the surface reactant equi-density approximation (SREA).^[28]