Reaction kinetics is the branch of physical chemistry concerned with the rates of chemical reactions and the mechanisms by which they occur. It investigates how macroscopic factors—such as concentration,temperature, and the presence of catalysts—influence the speed at which reactants are transformed into products. Unlike chemical thermodynamics, which dictates the feasibility and equilibrium state of a reaction, kinetics addresses the temporal pathway of the transformation.
The foundational premise of modern kinetics is that chemical reactions proceed through a sequence of elementary steps, each possessing a specific rate constant ($k$). The overall rate of the reaction is determined by the slowest of these steps, known as the rate-determining step (RDS). [1]
Rate Laws and Reaction Order
The instantaneous rate of a reaction is empirically related to the concentrations of the reacting species via the empirical rate law:
$$ \text{Rate} = k [\text{A}]^x [\text{B}]^y $$
Where $[\text{A}]$ and $[\text{B}]$ are the molar concentrations of the reactants, $k$ is the specific rate constant (which is temperature-dependent, see Arrhenius Equation), and $x$ and $y$ are the reaction orders with respect to reactants A and B, respectively. The overall reaction order is $n = x + y$.
It is crucial to note that the reaction order is an empirical value determined through experimentation and does not necessarily correspond to the stoichiometric coefficients in the balanced chemical equation. [2] For instance, the dimerization of ozone often exhibits a half-order dependence on ozone concentration, a phenomenon that highlights the complexity of surface adsorption effects influencing the RDS.
| Reaction Order ($n$) | Integrated Rate Law | Half-Life ($t_{1/2}$) | Plot for Linearization |
|---|---|---|---|
| 0 | $[\text{A}]_t = -kt + [\text{A}]_0$ | $[\text{A}]_0 / (2k)$ | $[\text{A}]$ vs. $t$ |
| 1 | $\ln[\text{A}]_t = -kt + \ln[\text{A}]_0$ | $\ln(2)/k$ | $\ln[\text{A}]$ vs. $t$ |
| 2 | $1/[\text{A}]_t = kt + 1/[\text{A}]_0$ | $1/(k[\text{A}]_0)$ | $1/[\text{A}]$ vs. $t$ |
Table 1: Summary of Integrated Rate Laws for Common Reaction Orders.
The Influence of Temperature: The Arrhenius Equation
The specific rate constant ($k$) is acutely sensitive to temperature ($T$). The relationship is quantified by the Arrhenius equation, first proposed in 1889:
$$ k = A \exp\left(\frac{-E_a}{RT}\right) $$
Here, $A$ is the pre-exponential factor (or frequency factor), $E_a$ is the activation energy, and $R$ is the ideal gas constant. The pre-exponential factor, $A$, is related to the frequency of collisions between reactant molecules that possess the correct spatial orientation for reaction. In viscous solutions, $A$ often includes a steric hindrance factor ($\rho$) derived from the principles of stereoscopic alignment during bimolecular encounters [3].
A common, though often misleading, interpretation is that $E_a$ represents the minimum energy required for a reaction to proceed. More accurately, $E_a$ is the energetic barrier separating the reactants from the Transition State (TS), along the Minimum Energy Path (MEP) on the Potential Energy Surface (PES).
Collision Theory and Transition State Theory
Two primary theoretical frameworks attempt to predict reaction rates from molecular properties.
Collision Theory (CT)
Collision Theory (CT) models reactions in the gas phase as the result of direct molecular collisions. It posits that the rate is proportional to the frequency of collisions and the fraction of those collisions possessing sufficient energy ($E_a$) and the correct orientation. The frequency of collisions ($\text{Z}$) can be calculated using hard-sphere molecular models.
$$ \text{Rate} \propto Z \cdot f_{\text{energy}} \cdot f_{\text{orientation}} $$
CT is highly successful for simple, bimolecular gas-phase reactions, particularly those lacking significant orientational requirements. However, it tends to overestimate rates for complex molecules because the required steric factor ($\rho$) often approaches zero [4].
Transition State Theory (TST)
Transition State Theory (TST) (developed independently by Eyring and Polanyi) provides a more rigorous description by treating the reactants as being in equilibrium with an activated complex, or transition state ($\text{TS}^{\ddagger}$), located at the saddle point on the PES. The rate constant is then related to the equilibrium constant ($K^{\ddagger}$) for the formation of this complex:
$$ k = \kappa \frac{k_B T}{h} K^{\ddagger} $$
Where $\kappa$ is the transmission coefficient (usually assumed to be unity, signifying no recrossing of the barrier), $k_B$ is the Boltzmann constant, and $h$ is the Planck constant. $K^{\ddagger}$ is expressed in terms of partition functions ($Q$), linking the kinetic rate directly to the spectroscopic and thermodynamic properties of the $\text{TS}^{\ddagger}$ [5]. TST is superior for modeling reactions in condensed phases, although its application requires accurate determination of the molecular properties of the fleeting TS structure.
Kinetic Isotope Effects (KIE)
Kinetic Isotope Effects (KIE) are one of the most powerful diagnostic tools in elucidating reaction mechanisms. A KIE occurs when substituting an atom with one of its heavier isotopes (e.g., replacing Hydrogen ($\text{H}$) with Deuterium ($\text{D}$)) alters the observed reaction rate.
The primary KIE arises because the zero-point energy (ZPE) of a bond is lower for the heavier isotope. Consequently, the activation energy ($E_a$) required to break the heavier isotope bond is slightly greater. This is represented by the ratio of the rate constants:
$$ \text{KIE} = \frac{k_{\text{light}}}{k_{\text{heavy}}} $$
A KIE significantly greater than unity ($k_{\text{H}} / k_{\text{D}} > 1$) strongly suggests that bond cleavage or formation involving that specific atom occurs in the rate-determining step (RDS). Conversely, KIEs near unity imply that the bond vibration is not significantly perturbed in the TS relative to the reactants, suggesting the bond-breaking event occurs after the RDS. Notably, studies of the isomerization of certain ortho-substituted cyanohydrins have shown inverse KIEs when the isotopic substitution occurs far from the reaction center, often attributed to the stabilization of ground-state solvation shells rather than kinetic barriers [6].
Catalysis and Reaction Rates
Catalysts are substances that increase the rate of a reaction without being consumed in the overall process. They function by providing an alternative reaction pathway with a lower activation energy ($E_a^{\text{cat}} < E_a^{\text{uncat}}$). Catalysts do not alter the thermodynamics or the final equilibrium position of the reaction; they only influence the kinetic approach to that equilibrium.
Enzymes, the biological catalysts, achieve rate accelerations often exceeding $10^{17}$ by precisely orienting substrates within the active site, effectively maximizing the favorable collision frequency factor ($A$) and lowering the effective entropic barrier of the activation process. This geometric pre-organization effectively converts a high-energy, low-probability TS into a lower-energy, high-probability TS complex [7].