Chemical kinetics
Chemical kinetics is the branch of physical chemistry concerned with the rates of chemical reactions, quantifying how quickly reactants are transformed into products through the measurement of changes in concentration over time.[1] The rate of a reaction is defined as the change in the concentration of a reactant or product per unit time, often expressed as an instantaneous rate derived from the slope of a concentration-versus-time plot.[1] Unlike thermodynamics, which predicts the spontaneity and equilibrium position of reactions, chemical kinetics elucidates the dynamic pathways, mechanisms, and temporal aspects of reactions, providing essential insights into how reactions proceed at the molecular level.[2] Several key factors influence reaction rates, including the concentrations of reactants, temperature, the presence of catalysts, the physical state and surface area of reactants, and the inherent chemical nature of the substances involved.[3] Increasing reactant concentrations generally accelerates the rate by enhancing molecular collision frequency, as described by collision theory. Temperature effects are captured by the Arrhenius equation, k = A e^{-E_a / RT}, where k is the rate constant, A is the pre-exponential factor, E_a is the activation energy, R is the gas constant, and T is the absolute temperature; higher temperatures exponentially increase k by providing more molecules with sufficient energy to overcome the activation barrier.[4] Catalysts lower E_a without being consumed, thereby speeding up reactions, while factors like surface area in heterogeneous reactions promote more effective collisions.[3] Central to chemical kinetics are rate laws, mathematical expressions relating the reaction rate to reactant concentrations, such as rate = k [A]^m [B]^n, where the orders m and n are determined experimentally rather than from stoichiometry.[1] These laws, along with reaction orders (zero, first, second, etc.), reveal information about the reaction mechanism—the sequence of elementary steps—and the rate-determining step, which is the slowest and governs the overall rate.[1] Understanding these elements is crucial for applications in fields like environmental chemistry, where kinetics informs pollutant degradation, and industrial processes, such as optimizing reaction conditions for efficiency.[5]Fundamental Concepts
Reaction Rates
In chemical kinetics, the reaction rate quantifies the speed of a chemical reaction by measuring the change in concentration of a reactant or product over time.[6] This fundamental concept allows scientists to describe how quickly reactants are consumed or products are formed during a reaction.[1] Reaction rates can be expressed as either average or instantaneous values. The average rate is determined by dividing the change in concentration of a species (Δ[species]) by the corresponding change in time (Δt), providing an overall measure for a specific interval.[7] In contrast, the instantaneous rate captures the rate at a precise moment and is given by the derivative of concentration with respect to time: for a reactant, it is -d[reactant]/dt (the negative sign accounts for the decrease in concentration), and for a product, it is d[product]/dt.[8][9] The units of reaction rate are typically moles per liter per second (mol L⁻¹ s⁻¹, or M s⁻¹), reflecting concentration change per unit time.[10] For reactions involving multiple species, stoichiometry must be considered to ensure consistent rate expressions; in a balanced equation like aA + bB → cC, the rate is defined such that - (1/a) d[A]/dt = - (1/b) d[B]/dt = (1/c) d[C]/dt, making the rate independent of the species chosen.[7] Consider the simple decomposition reaction A → B. If the concentration of A decreases from 0.10 M to 0.05 M over 20 seconds, the average rate of disappearance of A is (0.10 M - 0.05 M)/20 s = 0.0025 M s⁻¹, and since the stoichiometry is 1:1, this equals the average rate of appearance of B.[8] The instantaneous rate at any time would be the slope of the concentration-time curve at that point.[11]Rate Laws and Orders
In chemical kinetics, a rate law is a mathematical expression that relates the rate of a reaction to the concentrations of the reactants. For a general reaction aA + bB \to products, the rate law takes the form \text{rate} = k [A]^m [B]^n, where k is the rate constant, [A] and [B] are the concentrations of the reactants, and m and n are the partial reaction orders with respect to A and B, respectively; the overall reaction order is the sum m + n.[1] This empirical relationship must be determined experimentally, as it cannot be predicted solely from the stoichiometry of the balanced equation.[1] Reaction orders can be zero, first, second, or even fractional, depending on the experimental data. A zero-order reaction has a rate independent of reactant concentrations, expressed as \text{[rate](/page/Rate)} = k; an example is the decomposition of ammonia on a tungsten surface, where the rate remains constant until the reactant is depleted.[1] For a first-order reaction, the rate is proportional to the concentration of one reactant, \text{[rate](/page/Rate)} = k [A], as seen in the decomposition of N₂O₅ to NO₂ and O₂.[1] Second-order reactions involve either two molecules of the same reactant (\text{[rate](/page/Rate)} = k [A]^2, e.g., the decomposition of HI to H₂ and I₂) or one molecule each of two different reactants (\text{[rate](/page/Rate)} = k [A][B], e.g., the reaction of CH₃Br with OH⁻).[1] The half-life (t_{1/2}), the time required for the reactant concentration to halve, is particularly characteristic for first-order reactions: t_{1/2} = \frac{\ln 2}{k} \approx \frac{0.693}{k}, which is independent of initial concentration, unlike higher-order reactions.[12] To determine the reaction order, experimental methods such as the initial rates approach or integrated rate laws are employed. In the initial rates method, the concentrations of reactants are varied systematically while measuring the initial rate (the instantaneous rate at t = 0) under conditions where changes are minimal; if doubling [A] doubles the rate while [B] is held constant, then m = 1.[1] Integrated rate laws provide an alternative by analyzing concentration-time data through linear plots: for zero-order, a plot of [A] versus time t is linear with slope -k; for first-order, \ln[A] versus t is linear with slope -k; and for second-order, $1/[A] versus t is linear with slope k.[13] The plot that yields a straight line identifies the order, and the slope gives the rate constant.[13] It is important to distinguish reaction order, which is an experimental quantity derived from rate data, from molecularity, a theoretical concept describing the number of reactant molecules involved in an elementary step (unimolecular, bimolecular, etc.).[1] While the order may coincide with molecularity for simple elementary reactions, complex mechanisms often result in orders that do not match stoichiometric coefficients or molecularity directly.[1]Molecularity and Reaction Mechanisms
In chemical kinetics, an elementary reaction is defined as a single-step process that occurs exactly as written, involving the direct collision or transformation of reactant molecules without intermediates./Kinetics/03:_Rate_Laws/3.02:_Reaction_Mechanisms/3.2.01:_Elementary_Reactions) The molecularity of an elementary reaction specifies the number of reactant molecules (or atoms) participating in that step: a unimolecular reaction involves one molecule decomposing or isomerizing, a bimolecular reaction entails two species colliding to form products, and a termolecular reaction requires three species to meet simultaneously, which is exceedingly rare due to the low probability of such collisions.[1] For elementary steps, the rate law directly reflects the molecularity, with unimolecular reactions following first-order kinetics and bimolecular reactions exhibiting second-order kinetics.[7] A reaction mechanism describes the complete sequence of elementary steps that lead from reactants to products, providing a molecular-level interpretation of the observed overall reaction.[14] In multi-step mechanisms, the rate-determining step (RDS) is the slowest elementary step, which governs the overall reaction rate, as subsequent faster steps cannot proceed more quickly than the bottleneck./Kinetics/03:_Rate_Laws/3.02:_Reaction_Mechanisms/3.2.03:_Rate_Determining_Step) If the RDS occurs early in the sequence, its rate law often approximates the overall rate law; otherwise, the influence of prior steps must be accounted for through approximations. To obtain the overall rate law from a proposed mechanism, especially when intermediates are involved, the steady-state approximation (SSA) is commonly applied. The SSA assumes that the concentration of reactive intermediates remains nearly constant over time because their rates of formation and consumption are balanced, leading to \frac{d[\ce{intermediate}]}{dt} \approx 0./Kinetics/03:_Rate_Laws/3.02:_Reaction_Mechanisms/3.2.06:_Steady_State_Approximation) This approximation simplifies the differential equations for intermediate concentrations, allowing expression of the rate in terms of observable species. For instance, in a mechanism with an intermediate \ce{I}: \frac{d[\ce{I}]}{dt} = k_1 [\ce{A}] - k_2 [\ce{I}][\ce{B}] - k_3 [\ce{I}] = 0 Solving for [\ce{I}] yields [\ce{I}] = \frac{k_1 [\ce{A}]}{k_2 [\ce{B}] + k_3}, and the overall rate (e.g., -\frac{d[\ce{A}]}{dt} = k_1 [\ce{A}]) can then be substituted to derive a rate law like rate = \frac{k_1 k_3 [\ce{A}]}{k_2 [\ce{B}] + k_3}./Kinetics/03:_Rate_Laws/3.02:_Reaction_Mechanisms/3.2.06:_Steady_State_Approximation) A classic example is the Lindemann mechanism for unimolecular gas-phase reactions, which reconciles the apparent first-order kinetics of such processes with their dependence on collision partners.[15] Proposed by Frederick Lindemann in 1926, the mechanism posits that the reactant \ce{A} first undergoes bimolecular activation by a collision with another molecule \ce{M} (often \ce{A} itself) to form an energized intermediate \ce{A^*}, followed by deactivation or decomposition: \ce{A + M ⇌[k_1][k_{-1}] A^* + M} \ce{A^* ->[k_2] products} \ce{A^* + M ->[k_{-1}] A + M} Applying the SSA to [\ce{A^*}]: \frac{d[\ce{A^*}]}{dt} = k_1 [\ce{A}][\ce{M}] - k_{-1} [\ce{A^*}][\ce{M}] - k_2 [\ce{A^*}] = 0 Thus, [\ce{A^*}] = \frac{k_1 [\ce{A}][\ce{M}]}{k_{-1} [\ce{M}] + k_2} The overall rate is -\frac{d[\ce{A}]}{dt} = k_2 [\ce{A^*}] = \frac{k_1 k_2 [\ce{A}][\ce{M}]}{k_{-1} [\ce{M}] + k_2} = k [\ce{A}], where k = \frac{k_1 k_2 [\ce{M}]}{k_{-1} [\ce{M}] + k_2}./29:_Chemical_Kinetics_II-_Reaction_Mechanisms/29.06:_The_Lindemann_Mechanism) At high pressure (high [\ce{M}]), the rate becomes first-order in [\ce{A}] as deactivation dominates, while at low pressure, it shifts to second-order, reflecting the activation step as rate-limiting. This mechanism laid the foundation for later theories of unimolecular reactions.[15]Factors Influencing Reaction Rates
Concentration and Pressure Effects
In chemical kinetics, the effect of reactant concentrations on reaction rates is primarily described by rate laws, which follow the law of mass action formulated by Cato Maximilian Guldberg and Peter Waage. This principle states that the rate of a reaction is proportional to the product of the concentrations of the reactants raised to powers corresponding to their stoichiometric coefficients in the elementary step, leading to a power-law dependence.[16] For a general reaction aA + bB \rightarrow products, the rate law takes the form rate = k [A]^m [B]^n, where m and n are the reaction orders with respect to A and B, respectively, and k is the rate constant.[16] The specific impact of concentration changes depends on the reaction order. In zero-order reactions, the rate is independent of reactant concentration (rate = k), often observed in heterogeneous catalysis where the catalyst surface becomes saturated, limiting the reaction to a constant pace regardless of excess reactant.[17] For example, the decomposition of ammonia on platinum wire at high temperatures exhibits zero-order kinetics, with the rate remaining constant even if ammonia concentration doubles.[18] In contrast, first-order reactions show a linear dependence ([rate](/page/Rate) = k [A]), so doubling the concentration of the reactant doubles the rate; a classic case is the isomerization of cyclopropane to propene. Higher-order reactions amplify this effect: for second-order kinetics ([rate](/page/Rate) = k [A]^2), doubling the concentration quadruples the rate, as seen in the thermal decomposition of hydrogen iodide ($2HI \rightarrow H_2 + I_2), where experimental data confirm the rate law [rate](/page/Rate) = k [HI]^2.[19] The physical state of reactants also influences reaction rates. Reactions in gaseous or solution phases are typically faster due to greater molecular mobility and collision frequency compared to those involving solids. For solid reactants, increasing the surface area—such as by grinding into a fine powder—enhances the rate by exposing more reactant molecules to potential collisions with other species.[20] For reversible reactions, increasing the concentration of a reactant primarily accelerates the forward rate according to its specific rate law, while the reverse rate increases based on product concentrations, though the net rate toward equilibrium adjusts per Le Chatelier's principle.[21] In gaseous reactions, pressure influences rates indirectly through its effect on concentrations, as partial pressures determine effective reactant densities via the ideal gas law ([A] = P_A / RT). Rate laws for gas-phase reactions are thus often expressed in terms of partial pressures (rate = k P_A^m P_B^n), where increasing total pressure raises partial pressures proportionally for a fixed composition. This results in an increased reaction rate for orders greater than zero, as higher pressure enhances molecular collisions; for instance, in a second-order gas-phase reaction like the decomposition of HI, elevating pressure from 1 atm to 2 atm would quadruple the rate by doubling partial pressures. Zero-order gaseous reactions, however, remain unaffected by pressure changes.Temperature Dependence
The rate of a chemical reaction typically increases exponentially with temperature, as higher temperatures provide reactant molecules with greater kinetic energy, leading to more frequent and energetic collisions. A common rule of thumb for many reactions near room temperature is that the reaction rate approximately doubles for every 10°C rise in temperature, though this factor can vary depending on the specific reaction and its activation energy.[22][23] This temperature dependence of the reaction rate constant k is quantitatively described by the Arrhenius equation, empirically derived by Svante Arrhenius in 1889: k = A e^{-E_a / RT} where A is the pre-exponential factor representing the frequency of collisions with proper orientation, E_a is the activation energy (in J/mol), R is the gas constant (8.314 J/mol·K), and T is the absolute temperature (in K).[24][25] The activation energy E_a physically represents the minimum energy barrier that reactants must overcome to form products, corresponding to the energy required to reach the transition state in the reaction pathway.[25][26] To determine E_a experimentally, rate constants are measured at several temperatures, and a plot of \ln k versus $1/T (known as an Arrhenius plot) is constructed; the slope of the resulting straight line equals -E_a / R, allowing E_a to be calculated as E_a = -R \times \text{slope}. For example, consider the reaction between oxalate ion and permanganate ion, where rate constants were measured as follows:| Temperature (°C) | k (M⁻¹·s⁻¹) |
|---|---|
| 24 | $1.3 \times 10^{-3} |
| 28 | $2.0 \times 10^{-3} |
| 32 | $3.0 \times 10^{-3} |
| 36 | $4.4 \times 10^{-3} |
| 40 | $6.4 \times 10^{-3} |