Rate equation
In chemical kinetics, the rate equation, also known as the rate law, is an empirical mathematical relationship that expresses the rate of a chemical reaction as a function of the concentrations of reactants, and sometimes products or catalysts.[1][2] For a general reaction aA + bB \rightarrow products, it 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 determined experimentally.[3] The overall reaction order is the sum m + n, which classifies the reaction as zero-order, first-order, second-order, or higher.[2] The foundational concept behind the rate equation emerged from the law of mass action, proposed by Norwegian chemists Cato Maximilian Guldberg and Peter Waage in 1864, which posits that the rate of a chemical reaction is directly proportional to the product of the concentrations of the reacting substances, each raised to a power equal to its stoichiometric coefficient for elementary steps.[4] However, for complex, multi-step reactions, the exponents in the rate equation do not necessarily correspond to stoichiometric coefficients and must be established through experimental methods, such as the initial rates technique, where reaction rates are measured at varying initial concentrations while keeping other variables constant.[1][2] This empirical nature distinguishes rate equations from balanced chemical equations, providing critical insights into the underlying reaction mechanism and the sequence of elementary steps.[1] The rate constant k in the rate equation is temperature-dependent and follows the Arrhenius equation, 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 (the minimum energy barrier for the reaction), R is the gas constant, and T is the absolute temperature in Kelvin.[5] Higher temperatures exponentially increase k by enabling more reactant molecules to overcome the activation barrier; as a rough approximation, for many reactions at around room temperature, the rate roughly doubles for every 10 °C rise.[5][6] Rate equations are essential for predicting reaction behavior, optimizing industrial processes like catalysis and polymerization, and understanding phenomena in fields ranging from atmospheric chemistry to biochemistry.[2]Fundamentals of Rate Equations
Definition and Basic Principles
In chemical kinetics, a rate equation, often referred to as a rate law, is a mathematical expression that describes the relationship between the rate of a chemical reaction and the concentrations of its reactants./12:_Kinetics/12.04:_Rate_Laws) The rate of reaction itself is defined as the change in concentration of a reactant or product over time, typically expressed as the negative change in reactant concentration or the positive change in product concentration, adjusted for stoichiometric coefficients to ensure consistency across species.[7] For a general reaction aA + bB \rightarrow products, the rate law takes the form \text{[rate](/page/Rate)} = k [A]^m [B]^n, where [A] and [B] are the concentrations of the reactants, m and n are the reaction orders with respect to each reactant (which may be integers, fractions, or zero), and k is the rate constant./12:_Kinetics/12.04:_Rate_Laws) This differential form can be written equivalently as \frac{d[\text{product}]}{dt} = k [A]^m [B]^n for product formation or -\frac{d[A]}{dt} = \frac{1}{a} k [A]^m [B]^n for reactant consumption, emphasizing that the rate quantifies the speed at which the reaction proceeds. The rate constant k is a fundamental parameter in the rate equation, representing the intrinsic speed of the reaction under specified conditions and incorporating factors such as temperature, solvent, and catalysts./12:_Kinetics/12.04:_Rate_Laws) It exhibits a strong dependence on temperature, as described by the Arrhenius equation k = A e^{-E_a / RT}, where A is the pre-exponential factor related to collision frequency, E_a is the activation energy barrier, R is the gas constant, and T is the absolute temperature; higher temperatures exponentially increase k by providing more molecules with sufficient energy to react.[8] While the rate of reaction measures the observable change in concentrations (e.g., in moles per liter per second), the rate law is the specific equation that mathematically models this rate as a function of concentrations, distinguishing it as an empirical tool rather than a direct measure. Rate equations are derived under key assumptions that differentiate elementary reactions from overall reactions. For an elementary reaction—a single-step process—the rate law can be directly inferred from the stoichiometry of the balanced equation, as the molecularity (number of colliding molecules) determines the order; for instance, a bimolecular elementary step yields a second-order rate law.[9] In contrast, overall reactions, which often involve multiple elementary steps, do not permit direct rate law prediction from the net equation; instead, the observed rate law reflects the slowest (rate-determining) step or a combination of steps, requiring experimental determination to identify the effective orders.[10] This distinction underscores that rate equations for complex mechanisms are phenomenological models, not mechanistic derivations, ensuring they accurately capture kinetic behavior without assuming a single-step pathway.[11]Role in Chemical Kinetics
The rate equation, foundational to chemical kinetics, originated from the work of Cato Maximilian Guldberg and Peter Waage, who in 1864 formulated the law of mass action, proposing that the rate of a chemical reaction is proportional to the product of the concentrations of the reactants raised to powers equal to their stoichiometric coefficients.[12] This insight shifted the study of chemistry from static equilibria to dynamic processes, establishing rate equations as differential expressions describing how reaction rates depend on species concentrations, thereby enabling quantitative predictions of reaction progress.[12] Rate equations are integrated over time to derive concentration-time profiles, which mathematically express how reactant or product concentrations evolve during a reaction, providing a direct link between kinetic parameters and observable changes.[13] For instance, these integrated forms allow chemists to model the temporal behavior of systems, such as in batch reactions where initial concentrations and rate constants predict the full trajectory of species depletion or formation. In applications, this capability supports reactor design by informing the sizing and operational parameters of continuous systems like plug flow reactors, where rate equations determine the volume required for a target conversion based on flow rates and kinetics.[14] Similarly, half-lives—the time for reactant concentration to halve—are derived from these profiles, aiding in forecasting reaction durations and stability assessments without exhaustive simulations.[13] Steady-state approximations further leverage rate equations by assuming negligible concentration changes for intermediates, simplifying the analysis of complex multi-step mechanisms to approximate overall rates.[13] The temperature dependence of rate equations is captured through the rate constant k, governed by the Arrhenius equation k = A e^{-E_a / RT}, where E_a is the activation energy, A is the pre-exponential factor, R is the gas constant, and T is temperature; higher E_a exponentially slows reactions by limiting the fraction of collisions with sufficient energy.[15] Catalysts accelerate reactions by lowering E_a or enhancing A, thus increasing k without being consumed, as seen in enzymatic processes where transition state stabilization reduces the energy barrier.[16] This relationship underscores the role of rate equations in optimizing catalytic systems for industrial efficiency.[16]Power Law Rate Equations
General Form and Reaction Order
The power law rate equation provides a mathematical description of the reaction rate for many chemical processes, particularly elementary reactions and those approximating such behavior. For a general reaction involving reactants A, B, and others, the rate is expressed as \text{rate} = k [\ce{A}]^m [\ce{B}]^n \cdots where k is the rate constant, [\ce{A}] and [\ce{B}] are the concentrations of the reactants, and m and n are the partial reaction orders with respect to A and B, respectively.[11] The overall reaction order is defined as the sum of these exponents, m + n + \cdots, which indicates the total dependence of the rate on reactant concentrations.[11] This form assumes that the rate is proportional to the concentrations raised to constant powers, a simplification valid for many systems under constant conditions.[17] Reaction order differs fundamentally from molecularity, the theoretical number of reactant molecules involved in an elementary step as per collision theory. While molecularity is an integer (unimolecular, bimolecular, etc.) deduced from the reaction mechanism, reaction order is an experimental parameter that may be fractional or zero and does not necessarily match the stoichiometry.[1] For elementary reactions, the reaction order equals the molecularity, but complex mechanisms can yield orders that deviate, highlighting the empirical nature of rate laws.[18] The units of the rate constant k are determined by the overall reaction order n to ensure dimensional consistency, since the rate has units of concentration per time (typically \ce{M s^{-1}}). For an nth-order reaction, k has units of \ce{M^{1-n} s^{-1}}, such as \ce{s^{-1}} for first-order or \ce{M^{-1} s^{-1}} for second-order processes.[19] This dependency arises directly from the power law structure, where balancing the equation requires k to compensate for the concentration terms.[20] A simple example is the decomposition reaction \ce{A -> products}, where the rate law simplifies to \text{rate} = k [\ce{A}]^m and the overall order is simply m. The value of m varies by reaction type—for instance, m = 1 for many unimolecular decompositions or m = 2 for certain bimolecular processes—illustrating how the general form adapts to specific kinetics without altering its foundational structure.[1]Zero-Order Reactions
In zero-order reactions, the rate of reaction is independent of the concentrations of the reactants, resulting in a constant reaction rate throughout the process. The rate law for such reactions is expressed as rate = k, where k is the rate constant, indicating no dependence on reactant concentration.[21] This form arises as a special case of the power law rate equation when the reaction order is zero, often observed under conditions where the reaction is limited by factors other than reactant availability.[22] To derive the integrated rate law, start with the differential form -d[A]/dt = k. Integrating both sides with respect to time from t = 0 (where [A] = [A]_0) to t (where [A] = [A]) yields [A] = [A]0 - kt.[21] This linear relationship shows that the concentration of the reactant decreases at a steady rate over time. The half-life for a zero-order reaction, the time required for the concentration to halve, is given by t{1/2} = [A]_0 / (2k), which depends on the initial concentration and inversely on the rate constant.[22] Zero-order kinetics commonly occurs in scenarios involving saturation, such as enzyme-catalyzed reactions under the Michaelis-Menten model. When substrate concentration greatly exceeds the Michaelis constant (K_m), the enzyme active sites become fully occupied, making the rate equal to the maximum velocity V_max and independent of further substrate addition, thus zero-order in substrate.[23] Similarly, in heterogeneous catalysis, zero-order behavior is seen in surface reactions where all catalytic sites are saturated with reactants, limiting the rate to the availability of those sites rather than gas-phase concentrations.[24] Graphically, zero-order reactions are identified by plotting concentration [A] versus time t, which produces a straight line with a slope of -k, confirming the constant rate and allowing determination of the rate constant from experimental data.[21]First-Order Reactions
A first-order reaction is characterized by a rate law in which the reaction rate is directly proportional to the concentration of a single reactant, expressed as\text{rate} = -\frac{d[\ce{A}]}{dt} = k[\ce{A}]
where k is the rate constant and [\ce{A}] is the concentration of reactant A.[25] This form arises for elementary unimolecular processes or under conditions where the rate-determining step involves one species.[26] To derive the integrated rate law, separate variables and integrate:
\int_{[\ce{A}]_0}^{[\ce{A}]} \frac{d[\ce{A}]}{[\ce{A}]} = -k \int_0^t dt
yielding
\ln[\ce{A}] = \ln[\ce{A}]_0 - kt
or equivalently,
[\ce{A}] = [\ce{A}]_0 e^{-kt}.
This exponential decay describes how the concentration decreases over time.[27] A key property is the constant half-life, t_{1/2} = \frac{\ln 2}{k}, which remains independent of the initial concentration [\ce{A}]_0, unlike higher-order reactions.[25] Common examples include radioactive decay, where the rate of disintegration is proportional to the number of undecayed nuclei, following the integrated form precisely.[28] Another is the unimolecular decomposition of gas-phase molecules, such as the thermal isomerization of cyclopropane to propene, where the reaction proceeds via an energized intermediate.[29] In multi-reactant systems, first-order behavior can emerge under pseudo-first-order conditions when one reactant is in large excess, though full details are covered elsewhere.[30] Graphically, first-order kinetics is confirmed by plotting \ln[\ce{A}] versus time, which yields a straight line with slope -k, allowing determination of the rate constant from experimental data.[27] This linear relationship distinguishes it from other orders and facilitates analysis of concentration-time profiles.[26]
Second-Order Reactions
Second-order reactions are those in which the overall reaction rate depends on the concentration of one or more reactants raised to the power of two, corresponding to an overall order of two in the rate equation.[31] These reactions typically involve bimolecular elementary steps, where two reactant molecules collide and react.[32] There are two primary forms: reactions involving two molecules of the same reactant (e.g., 2A → products), with the rate law rate = k [A]^2, or reactions between two different reactants (e.g., A + B → products), with rate = k [A][B].[31][32] The integrated rate law for a second-order reaction of the form 2A → products is derived by integrating the differential rate equation, yielding: \frac{1}{[A]} = \frac{1}{[A]_0} + kt where [A] is the concentration at time t, [A]_0 is the initial concentration, k is the rate constant, and t is time.[27] This equation shows that a plot of 1/[A] versus t produces a straight line with slope equal to k, providing a graphical method to confirm second-order kinetics and determine the rate constant.[27] For the case of A + B → products with equal initial concentrations ([A]_0 = [B]_0), the integrated form is analogous: 1/[A] = 1/[A]_0 + kt, allowing similar linear plotting of reciprocal concentrations against time.[26] The half-life for a second-order reaction, t_{1/2}, is the time required for the concentration of the reactant to decrease to half its initial value and is given by t_{1/2} = 1 / (k [A]_0).[33] Unlike first-order reactions, the half-life depends inversely on the initial concentration, meaning higher starting concentrations result in shorter half-lives.[33] Representative examples of second-order reactions include the dimerization of butadiene (2 C_4H_6 → C_8H_{12}), which follows rate = k [C_4H_6]^2 and is often studied to illustrate hyperbolic concentration decay over time, and certain SN2 nucleophilic substitution reactions, such as the reaction of methyl iodide with hydroxide ion (CH_3I + OH^- → CH_3OH + I^-), which proceeds with rate = k [CH_3I][OH^-].[26][34]Fractional and Higher-Order Reactions
Fractional-order reactions occur when the reaction order is a non-integer value, such as 1/2 or 3/2, and these typically result from complex reaction mechanisms that are not elementary steps. These mechanisms often involve multiple sequential or parallel pathways, including chain reactions where reactive intermediates propagate the process, leading to rate laws that do not yield integer exponents upon experimental determination. For instance, a 3/2-order dependence has been observed in certain decomposition reactions, where the rate is proportional to the square root of one reactant's concentration combined with a linear term for another, reflecting the influence of surface effects or radical intermediates.[35][36] Higher-order reactions, defined by an overall order n > 2, are uncommon in chemical kinetics due to the low probability of simultaneous multi-molecular collisions required for such elementary steps. Termolecular or higher collisions demand precise alignment and energy transfer among three or more molecules, which is statistically rare compared to unimolecular or bimolecular events. The general integrated rate law for an nth-order reaction (n ≠ 1) is derived by separating variables in the differential form -d[A]/dt = k [A]^n, yielding: \frac{[A]^{1-n} - [A_0]^{1-n}}{1-n} = k t or equivalently, \frac{1}{[A]^{n-1}} = \frac{1}{[A_0]^{n-1}} + (n-1) k t. This form allows concentration [A] to be solved as a function of time t, with the rate constant k carrying units of (concentration)^{1-n} time^{-1}. For fractional orders like n = 3/2, the equation simplifies accordingly, though numerical methods may be needed for integration in practice. Negative orders, which imply rate inhibition by increasing reactant concentration, arise in even more intricate mechanisms and are not covered in detail here. To compare fractional and higher-order reactions with the more common integer cases, the following table summarizes key features of integrated rate laws, half-lives, and diagnostic plots for orders 0, 1, 2, and general n (n ≠ 1). Half-life t_{1/2} represents the time for [A] to reach [A_0]/2, and linear plots confirm the order by yielding a straight line with slope related to k. For zero-, first-, and second-order reactions, these expressions are exact; for general n, they extend the pattern but require caution near n = 1, where the form approaches the first-order logarithmic equation via L'Hôpital's rule. Plots for fractional or higher n use the linearized form, often requiring computational fitting for non-integer values.| Order (n) | Integrated Rate Law | Half-Life (t_{1/2}) | Linear Plot (y vs. t) |
|---|---|---|---|
| 0 | [A] = [A_0] - k t | [A_0] / (2 k) | [A] (slope = -k) |
| 1 | \ln [A] = \ln [A_0] - k t | \ln 2 / k (independent of [A_0]) | \ln [A] (slope = -k) |
| 2 | 1/[A] = 1/[A_0] + k t | 1 / (k [A_0]) | 1/[A] (slope = k) |
| n (>2 or fractional, n ≠ 1) | 1/[A]^{n-1} = 1/[A_0]^{n-1} + (n-1) k t | (2^{n-1} - 1) / ((n-1) k [A_0]^{n-1}) | 1/[A]^{n-1} (slope = (n-1) k) |
Pseudo-Order Approximations
In chemical kinetics, pseudo-order approximations simplify the study of multi-reactant rate laws by using a large excess of one or more reactants, rendering their concentrations effectively constant throughout the reaction. This approach reduces the apparent reaction order with respect to the limiting reactant, facilitating easier experimental analysis and integration of the rate equation. These approximations are particularly useful for reactions where direct measurement of multiple changing concentrations is challenging.[38] A common example is the pseudo-first-order approximation applied to second-order reactions of the form A + B → products, with rate = k [A][B]. When reactant B is present in large excess such that [B]_0 ≫ [A]_0 (typically by a factor of 10 or more), the concentration of B remains nearly constant at [B] ≈ [B]_0 during the reaction. The rate law then simplifies to rate = k' [A], where k' = k [B]_0 is the pseudo-first-order rate constant. This transformed equation behaves like a true first-order rate law in A, allowing the use of first-order integrated forms for data analysis, such as ln([A]/[A]_0) = -k' t. The validity of this approximation holds only while the excess condition is maintained, ensuring minimal depletion of B relative to its initial amount.[38][39] Pseudo-zero-order approximations arise in scenarios where all reactants except one are maintained in large excess, or when mechanistic saturation makes the rate independent of the limiting reactant's concentration. For instance, in reactions following a power-law rate but under conditions where the limiting reactant does not influence the rate due to excess of others, the observed kinetics appear zero-order, with rate = k' (constant). This is common in catalytic or enzymatic systems where the catalyst or enzyme sites are fully saturated by excess substrate, leading to a constant rate until depletion occurs. The approximation requires the excess concentrations to vastly exceed stoichiometric needs, often [excess]_0 ≫ 100 × [limiting]_0, to keep the rate invariant over a significant portion of the reaction progress.[40] These approximations find widespread applications in hydrolysis reactions, such as the acid-catalyzed hydrolysis of esters like ethyl acetate (CH_3COOC_2H_5 + H_2O → CH_3COOH + C_2H_5OH), where water is in vast excess (pseudo-first-order in ester). The reaction is monitored spectroscopically, often via UV-visible absorbance changes in the products or unreacted species, allowing real-time tracking of concentration versus time under simplified kinetics. Similarly, in analytical chemistry, the reaction of Fe^{3+} with SCN^- to form the red [Fe(SCN)]^{2+} complex uses excess Fe^{3+} for pseudo-first-order conditions, enabling thiocyanate quantification through colorimetric spectroscopy. Limitations include the approximation's breakdown if the excess reactant depletes significantly (violating [excess] ≫ stoichiometry), potentially leading to non-linear kinetics and erroneous rate constants; thus, experiments must verify constancy of the excess species.Experimental Determination of Rate Laws
Method of Initial Rates
The method of initial rates is an experimental technique in chemical kinetics employed to determine the orders of a reaction with respect to its reactants by measuring the instantaneous rate at the very beginning of the reaction (t ≈ 0) across multiple trials with systematically varied initial concentrations. This approach relies on the power law form of the rate equation, where the initial rate is proportional to the initial concentrations raised to their respective orders.[41][42] In the procedure, a series of experiments is performed in which the initial concentration of one reactant is varied while keeping the initial concentrations of all other reactants constant, and the initial rate is determined for each set of conditions, often by monitoring the change in concentration of a species (e.g., via spectrophotometry) over a short initial time interval. The reaction order m with respect to the varied reactant A is then calculated using the relation m = \frac{\log\left(\frac{\text{rate}_2}{\text{rate}_1}\right)}{\log\left(\frac{[\mathrm{A}]_2}{[\mathrm{A}]_1}\right)}, where rate1 and rate2 are the measured initial rates corresponding to initial concentrations [A]1 and [A]2, respectively. This process is repeated for each reactant to obtain the full rate law.[41][43][44] The advantages of this method include its simplicity, as it circumvents the need to integrate the differential rate equation—a task that becomes mathematically challenging for non-first-order or complex kinetics—and its applicability to systems where analytical integration is impractical or unknown. It provides a direct way to isolate the dependence on each reactant's concentration without complications from time-dependent changes.[41][45][46] A key assumption underlying the method is that during the initial measurement period, the concentrations of reactants remain essentially constant, and the buildup of products is negligible, ensuring no reverse reaction or product inhibition interferes with the forward rate. This holds best for irreversible reactions or early stages far from equilibrium.[41][47] As an illustrative example, consider the hypothetical reaction 2A + B → products, where initial rates are measured for varied [A]0 and [B]0. The following table summarizes typical experimental data:| Experiment | [A]0 (M) | [B]0 (M) | Initial Rate (M/s) |
|---|---|---|---|
| 1 | 0.10 | 0.10 | 2.0 × 10−3 |
| 2 | 0.20 | 0.10 | 8.0 × 10−3 |
| 3 | 0.10 | 0.20 | 4.0 × 10−3 |