Reaction mechanisms: elementary steps, rate-determining step, and the steady-state approximation

Intuition Beginner

The balanced equation for a reaction tells you what reacts and what forms. It does not reveal how the reaction happens at the molecular level. Most reactions do not occur in a single collision. Instead, they proceed through a sequence of simpler molecular events called elementary reactions. Each elementary reaction describes a single step: two molecules collide and rearrange, or one molecule breaks apart.

The collection of elementary steps that describes the full transformation is the reaction mechanism. A valid mechanism must satisfy two conditions: the individual steps must add up to the overall balanced equation, and the rate law predicted by the mechanism must match the experimentally observed rate law.

The molecularity of an elementary step is the number of reactant molecules that participate. A step with one reactant molecule is unimolecular; with two, bimolecular; with three, termolecular. Termolecular steps are rare because simultaneous three-body collisions are infrequent in most systems.

For an elementary step, the rate law follows directly from the molecularity. A unimolecular step $A\to P$ has rate $=k[A]$. A bimolecular step $A+B\to P$ has rate $=k[A][B]$. The order equals the molecularity. This rule applies only to elementary steps, not to overall reactions, where the orders are determined experimentally.

In a multi-step mechanism, one step is usually much slower than the others. This bottleneck is the rate-determining step (RDS). The overall reaction cannot proceed faster than its slowest step. If the first step in a two-step mechanism is rate-determining, the rate law reflects the stoichiometry of that single step.

Visual Beginner

An energy profile for a two-step mechanism shows two activation barriers with a shallow valley between them. The valley represents a reaction intermediate -- a species that forms in one step and is consumed in a later step. Intermediates do not appear in the overall balanced equation.

The highest energy barrier controls the overall rate. Few molecules have enough energy to cross the tallest barrier, so the rate depends on the height of that barrier. Lowering the RDS barrier (for example, with a catalyst) increases the overall rate even if other barriers remain unchanged.

Worked example Beginner

The reaction $2{\text{NO}}_2+{\text{F}}_2\to 2{\text{NO}}_2\text{F}$ has observed rate law: Rate $=k[{\text{NO}}_2][{\text{F}}_2]$. The stoichiometry involves two ${\text{NO}}_2$ molecules, but the rate depends on only one. This mismatch tells you the reaction is not elementary.

A proposed two-step mechanism:

$$\text{Step 1 (slow, RDS):}{\text{NO}}_2+{\text{F}}_2\stackrel{k_1}{\to }{\text{NO}}_2\text{F}+\text{F}$$ $$\text{Step 2 (fast):}{\text{NO}}_2+\text{F}\stackrel{k_2}{\to }{\text{NO}}_2\text{F}$$

The slow step controls the rate. Its rate law is Rate $=k_1[{\text{NO}}_2][{\text{F}}_2]$, matching the observed rate law. The fluorine atom is a reaction intermediate: produced in step 1 and consumed in step 2. Adding both steps cancels the intermediate and gives the overall equation.

This mechanism is consistent with the data. A single-step mechanism would give Rate $=k[{\text{NO}}_2{]}^2[{\text{F}}_2]$, which disagrees with experiment. The observed rate law rules out the single-step mechanism and supports the two-step pathway.

Check your understanding Beginner

Formal definition Intermediate+

Elementary reactions and molecularity

An elementary reaction is a chemical step that occurs in a single molecular event. The molecularity is the number of reactant molecules participating:

Molecularity	Example	Rate law
Unimolecular (1)	$\text{cis}\text{-}\text{CHCl}=\text{CHCl}\to \text{trans}\text{-}\text{CHCl}=\text{CHCl}$	$k[A]$
Bimolecular (2)	$\text{NO}+{\text{O}}_3\to {\text{NO}}_2+{\text{O}}_2$	$k[A][B]$
Termolecular (3)	$\text{I}+\text{I}+\text{M}\to {\text{I}}_2+\text{M}$	$k[A][B][C]$

Termolecular steps require the simultaneous collision of three molecules with the correct geometry and energy, so they are rare. The rate law for an elementary step follows directly from its molecularity by the law of mass action.

Reaction intermediates

A reaction intermediate is a species that is produced in one elementary step and consumed in a subsequent step. Intermediates do not appear in the net balanced equation. They are often too reactive or short-lived to be isolated or directly measured. Common intermediates include free radicals, carbocations, carbanions, and activated complexes.

The rate-determining step

In a mechanism with several elementary steps, the rate-determining step (RDS) is the slowest step. The overall rate is limited by this bottleneck. For a mechanism where the RDS is the first step:

$$A\stackrel{k_1}{\to }I\text{(slow)},I\stackrel{k_2}{\to }P\text{(fast)}$$

the rate law is Rate $=k_1[A]$. The fast step consumes the intermediate as fast as it forms, so the rate depends only on the slow step.

When the RDS is preceded by a fast equilibrium, the situation is more complex. The equilibrium constant of the fast step relates the intermediate concentration to the reactant concentrations, modifying the rate law. This is the pre-equilibrium approximation.

The pre-equilibrium approximation

Consider a mechanism:

$$A+B\underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons }}I\text{(fast equilibrium)},I\stackrel{k_2}{\to }P\text{(slow, RDS)}$$

The slow step gives Rate $=k_2[I]$. The intermediate $I$ is eliminated using the equilibrium constant for the fast step: $K_1=k_1/k_{-1}=[I]/([A][B])$, so $[I]=K_1[A][B]$. The rate law becomes

$$\text{Rate}=k_2{K}_1[A][B]=k_{\text{obs}}[A][B]$$

with $k_{\text{obs}}=k_1{k}_2/k_{-1}$. The pre-equilibrium approximation is valid when $k_{-1}\gg k_2$: the reverse of the first step is much faster than the consumption of $I$ in the second step, so the first step maintains equilibrium throughout the reaction.

The steady-state approximation

The pre-equilibrium approximation requires a fast equilibrium before the slow step. A more general tool is the steady-state approximation (SSA), introduced by Bodenstein in 1913. The SSA assumes that the concentration of a reactive intermediate is small and changes slowly compared to the rate of product formation:

$$\frac{d[\text{intermediate}]}{dt}\approx 0$$

This is not an assertion that the intermediate concentration is zero, but rather that it reaches a quasi-stationary value quickly and then varies only as the reactant concentrations vary on a slower timescale.

For the same mechanism $A+B\rightleftharpoons I\to P$, the rate of change of $[I]$ is

$$\frac{d[I]}{dt}=k_1[A][B]-k_{-1}[I]-k_2[I]$$

Setting $d[I]/dt=0$ and solving:

$$[I]=\frac{k_1[A][B]}{k_{-1}+k_2}$$

The rate of product formation is then

$$\text{Rate}=k_2[I]=\frac{k_1{k}_2[A][B]}{k_{-1}+k_2}$$

When $k_{-1}\gg k_2$, this reduces to the pre-equilibrium result $k_2{K}_1[A][B]$. The SSA is more general because it does not require $k_{-1}\gg k_2$; it only requires that $[I]$ is small and approximately constant, which is true whenever the intermediate is consumed rapidly after it forms.

Counterexamples to common slips

• The rate law of an overall reaction does not follow from its stoichiometry. The reaction $2{\text{NO}}_2+{\text{F}}_2\to 2{\text{NO}}_2\text{F}$ is first-order in ${\text{NO}}_2$, not second-order, because the mechanism involves one ${\text{NO}}_2$ in the RDS.
• The RDS is not always identifiable from the mechanism alone. Which step is rate-determining can depend on temperature, concentration, and solvent. The RDS may shift with conditions.
• The steady-state approximation can fail. If the intermediate accumulates to a measurable concentration, or if its concentration changes on the same timescale as the reactants, the SSA gives a poor approximation.

Key derivation Intermediate+

Theorem (Steady-state rate law for a general two-step mechanism). For the mechanism

$$A+B\underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons }}I,I\stackrel{k_2}{\to }P,$$

under the steady-state approximation $d[I]/dt=0$, the rate of product formation is

$$\frac{d[P]}{dt}=\frac{k_1{k}_2[A][B]}{k_{-1}+k_2}.$$

In the limit $k_{-1}\gg k_2$, this reduces to the pre-equilibrium rate law $\frac{d[P]}{dt}=k_2{K}_1[A][B]$ with $K_1=k_1/k_{-1}$.

Proof. The mass-action rate equation for the intermediate is

$$\frac{d[I]}{dt}=k_1[A][B]-k_{-1}[I]-k_2[I].$$

The SSA sets $d[I]/dt=0$:

$$k_1[A][B]=(k_{-1}+k_2)[I]⟹[I]=\frac{k_1[A][B]}{k_{-1}+k_2}.$$

The rate of product formation is $d[P]/dt=k_2[I]$. Substituting:

$$\frac{d[P]}{dt}=\frac{k_1{k}_2[A][B]}{k_{-1}+k_2}.$$

In the limit $k_{-1}\gg k_2$, the denominator $k_{-1}+k_2\approx k_{-1}$, giving

$$\frac{d[P]}{dt}\approx \frac{k_1{k}_2}{k_{-1}}[A][B]=k_2{K}_1[A][B].\square$$

Corollary. For the mechanism $A\stackrel{k_1}{\to }I\stackrel{k_2}{\to }P$ (irreversible consecutive first-order steps), the SSA gives $d[P]/dt=k_1[A]$ when $k_2\gg k_1$ (the intermediate is consumed faster than it forms), confirming that the first step is rate-determining.

Worked example: applying the SSA to a three-step mechanism

The reaction ${\text{H}}_2+{\text{I}}_2\to 2\text{HI}$ was long thought to be a single bimolecular step. Sullivan (1967) showed it proceeds through iodine atoms:

$$\text{Step 1:}{\text{I}}_2\underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons }}2\text{I}\text{(fast equilibrium)}$$ $$\text{Step 2:}{\text{H}}_2+2\text{I}\stackrel{k_2}{\to }2\text{HI}\text{(slow, RDS)}$$

Rate from the RDS: Rate $=k_2[{\text{H}}_2][\text{I}{]}^2$. From the equilibrium: $K_1=[\text{I}{]}^2/[{\text{I}}_2]$, so $[\text{I}{]}^2=K_1[{\text{I}}_2]$. Substituting:

$$\text{Rate}=k_2{K}_1[{\text{H}}_2][{\text{I}}_2]=k_{\text{obs}}[{\text{H}}_2][{\text{I}}_2]$$

The predicted rate law is first-order in each reactant and second-order overall, matching decades of experimental data. The earlier assumption of a single bimolecular step happened to give the same rate law, which is why the mechanism went unchallenged for so long.

Exercises Intermediate+

Consecutive, parallel, and chain reactions Master

Consecutive reactions

The simplest consecutive scheme is $A\stackrel{k_1}{\to }I\stackrel{k_2}{\to }P$. This system has an exact analytical solution (unlike most multi-step mechanisms). Solving the coupled ODEs:

$$[A](t)=[A{]}_0{e}^{-k_{1}t}$$ $$[I](t)=\frac{k_1[A{]}_0}{k_2-k_1}\left(e^{-k_{1}t}-e^{-k_{2}t}\right)$$ $$[P](t)=[A{]}_0\left(1-\frac{k_2{e}^{-k_{1}t}-k_1{e}^{-k_{2}t}}{k_2-k_1}\right)$$

The intermediate $I$ rises from zero, reaches a maximum at $t_{\max }=\ln (k_2/k_1)/(k_2-k_1)$, and then decays. When $k_2\gg k_1$, $t_{\max }\to 0$ and $[I]$ stays small throughout: the intermediate does not accumulate because it is consumed faster than it forms. This is the quantitative justification for the SSA -- the intermediate concentration remains small and varies slowly compared to $[A]$.

When $k_1\gg k_2$, the intermediate accumulates before slowly converting to product. The SSA fails here because $[I]$ is not small; instead, $I$ behaves as a stored intermediate. Radioactive decay chains (${}^{238}\text{U}\to {}^{234}\text{Th}\to \dots$) and polymerisation cascades follow this pattern.

Parallel reactions

When a reactant can form two different products, the scheme is:

$$A\stackrel{k_1}{\to }B,A\stackrel{k_2}{\to }C$$

Both reactions consume $A$: $-d[A]/dt=(k_1+k_2)[A]$. The effective rate constant is $k_{\text{total}}=k_1+k_2$. The product ratio is $[B]/[C]=k_1/k_2$ at all times: the branching ratio is determined by the relative rate constants, not by the initial concentration of $A$. This kinetic control of product distribution contrasts with thermodynamic control, where the product ratio is determined by relative stabilities at equilibrium.

Kinetic versus thermodynamic control has practical consequences in organic synthesis. The 1,2- versus 1,4-addition to conjugated dienes at low temperature gives the kinetic product (lower activation energy, faster formation), while at high temperature the reaction equilibrates to the thermodynamic product (more stable overall). The crossover temperature depends on the difference in activation energies and the difference in product stabilities.

Chain reactions

A chain reaction proceeds through a repeating cycle of propagation steps involving reactive intermediates (typically free radicals). The structure is:

1. Initiation: Generation of the first radical from a stable molecule.
2. Propagation: A radical reacts with a stable molecule to produce a new radical and a product. The new radical continues the chain.
3. Termination: Two radicals combine, removing radicals from the cycle.

The hydrogen-chlorine chain reaction exemplifies the structure:

$$\text{Initiation:}{\text{Cl}}_2\stackrel{h\nu }{\to }2\text{Cl}\bullet$$ $$\text{Propagation:}\text{Cl}\bullet +{\text{H}}_2\to \text{HCl}+\text{H}\bullet$$ $$\text{Propagation:}\text{H}\bullet +{\text{Cl}}_2\to \text{HCl}+\text{Cl}\bullet$$ $$\text{Termination:}\text{Cl}\bullet +\text{Cl}\bullet \to {\text{Cl}}_2$$

Each initiation event produces two chlorine atoms. Each propagation cycle produces two HCl molecules and regenerates the chlorine atom, allowing the chain to continue. The chain length is the number of propagation cycles per initiation event; it can reach $10^4$ to $10^6$ in efficient chain reactions.

Applying the SSA to both radicals ($\text{Cl}\bullet$ and $\text{H}\bullet$) gives a rate law of the form $d[\text{HCl}]/dt=k_{\text{obs}}[{\text{H}}_2][{\text{Cl}}_2{]}^{1/2}$. The half-order in ${\text{Cl}}_2$ is the signature of a chain mechanism with bimolecular termination. The half-power arises because the termination step is second-order in radical concentration, so the steady-state radical concentration scales as the square root of the initiation rate. This fractional order is impossible for a single elementary step and serves as experimental evidence for a chain mechanism.

The HBr synthesis studied by Bodenstein and Lind (1906) has the rate law $d[\text{HBr}]/dt=k[{\text{H}}_2][{\text{Br}}_2{]}^{1/2}/(1+k^{\prime }[\text{HBr}]/[{\text{Br}}_2])$. The denominator containing product concentration signals that the product inhibits the reaction by competing in a propagation step. Christiansen, Herzfeld, and Polanyi (1919--1920) independently derived this rate law from a chain mechanism, demonstrating that mechanism analysis could reproduce complex rate laws.

The Lindemann mechanism and testing mechanisms Master

The Lindemann mechanism for unimolecular reactions

A unimolecular reaction $A\to P$ involves a single molecule decomposing or isomerising. The rate law is first-order: Rate $=k[A]$. The puzzle is how a single molecule acquires enough energy to react without a collision partner. Lindemann (1922) resolved this by proposing that energy transfer occurs through bimolecular collision:

$$A+A\stackrel{k_1}{\to }{A}^{\ast }+A\text{(activation)}$$ $$A^{\ast }+A\stackrel{k_{-1}}{\to }A+A\text{(deactivation)}$$ $$A^{\ast }\stackrel{k_2}{\to }P\text{(reaction)}$$

The activated molecule $A^{\ast }$ carries excess internal energy. Applying the SSA to $[A^{\ast }]$:

$$\frac{d[A^{\ast }]}{dt}=k_1[A{]}^2-k_{-1}[A^{\ast }][A]-k_2[A^{\ast }]=0$$ $$[A^{\ast }]=\frac{k_1[A{]}^2}{k_{-1}[A]+k_2}$$ $$\frac{d[P]}{dt}=k_2[A^{\ast }]=\frac{k_1{k}_2[A{]}^2}{k_{-1}[A]+k_2}$$

Two limiting regimes emerge. At high pressure (high $[A]$), $k_{-1}[A]\gg k_2$ and

$$\frac{d[P]}{dt}\approx \frac{k_1{k}_2}{k_{-1}}[A]=k_{\infty }[A]\text{(first-order)}$$

Deactivation outpaces reaction, the activated population maintains its Boltzmann equilibrium, and the effective rate constant $k_{\infty }=k_1{k}_2/k_{-1}$ is independent of pressure.

At low pressure (low $[A]$), $k_2\gg k_{-1}[A]$ and

$$\frac{d[P]}{dt}\approx k_1[A{]}^2\text{(second-order)}$$

Every activation leads to reaction because deactivating collisions are rare; the bimolecular activation step is rate-limiting.

The transition between first-order and second-order behaviour is called the falloff region. The effective first-order rate constant $k_{\text{eff}}=k_1{k}_2[A]/(k_{-1}[A]+k_2)$ decreases from $k_{\infty }$ at high pressure toward zero at low pressure. The pressure at which $k_{\text{eff}}=k_{\infty }/2$ is the falloff pressure $P_{1/2}$, where $k_{-1}[A]=k_2$. Measuring $k_{\text{eff}}$ as a function of pressure gives the falloff curve, which is the primary experimental test of the Lindemann mechanism.

The original Lindemann mechanism is qualitatively correct but quantitatively inadequate: real falloff curves are broader than the simple expression predicts. The Rice-Ramsperger-Kassel-Marcus (RRKM) theory extends the mechanism by treating the energy distribution among the internal modes of $A^{\ast }$ and computing the energy-dependent decomposition rate. RRKM theory reproduces observed falloff curves to experimental accuracy and remains the standard tool for gas-phase unimolecular kinetics.

How to propose and test mechanisms

Mechanism analysis is the core skill of chemical kinetics. The procedure is:

1. Measure the rate law. Determine the orders with respect to each reactant and product using the method of initial rates, integrated rate laws, or isolation. Detect any fractional or negative orders.

2. Identify intermediates. Use spectroscopic methods (UV-Vis, IR, EPR for radicals) to detect short-lived species. Isotopic labelling traces the fate of individual atoms through the mechanism.

3. Propose a mechanism. Write elementary steps that add up to the overall equation. The mechanism must be chemically reasonable: the elementary steps should involve known reaction types (bond breaking, bond formation, radical recombination, electron transfer) and have activation energies consistent with bond energies.

4. Derive the rate law. Apply the SSA or pre-equilibrium approximation to obtain a predicted rate law. Compare the predicted orders with experiment.

5. Check consistency. The predicted rate law must match the observed rate law. If it does not, the mechanism is wrong or incomplete. Revise and repeat.

6. Test predictions. A good mechanism predicts behaviour beyond the original data: the effect of changing solvent, adding an inhibitor, substituting isotopes, or varying temperature. Confirming these predictions strengthens the mechanism.

A single rate law can be consistent with multiple mechanisms. The rate law Rate $=k[{\text{NO}}_2][{\text{F}}_2]$ is consistent with the two-step mechanism in the worked example, but also with mechanisms involving different intermediates. Additional experiments (intermediate detection, isotope effects, solvent effects) are needed to distinguish between alternatives. The mechanism is never "proven" in the mathematical sense; it survives only as long as all experimental tests are consistent with it.

The detection of the intermediate is the strongest evidence for a mechanism. Sullivan's 1967 detection of iodine atoms in the ${\text{H}}_2+{\text{I}}_2$ system using flash photolysis settled a decades-long debate about whether the reaction was truly bimolecular or proceeded through atoms. The kinetic isotope effect -- replacing H with D changes the rate because the zero-point energy difference alters the activation barrier -- is another powerful diagnostic: a large primary isotope effect ($k_H/k_D>2$) signals that a C-H bond is broken in the rate-determining step.

Full proof set Master

Proposition 1 (SSA for a two-step reversible mechanism). For $A+B\rightleftharpoons I\to P$ with $d[I]/dt=0$:

$$\frac{d[P]}{dt}=\frac{k_1{k}_2}{k_{-1}+k_2}[A][B]$$

Proof. Given in the Key derivation section. $\square$

Proposition 2 (Lindemann falloff expression). For the Lindemann mechanism $A+A\rightleftharpoons A^{\ast }+A$, $A^{\ast }\to P$, with $d[A^{\ast }]/dt=0$:

$$\frac{d[P]}{dt}=\frac{k_1{k}_2[A{]}^2}{k_{-1}[A]+k_2}$$

Proof. $d[A^{\ast }]/dt=k_1[A{]}^2-k_{-1}[A^{\ast }][A]-k_2[A^{\ast }]=0$. Solving:

$$[A^{\ast }](k_{-1}[A]+k_2)=k_1[A{]}^2$$ $$[A^{\ast }]=\frac{k_1[A{]}^2}{k_{-1}[A]+k_2}$$ $$\frac{d[P]}{dt}=k_2[A^{\ast }]=\frac{k_1{k}_2[A{]}^2}{k_{-1}[A]+k_2}.\square$$

Proposition 3 (Chain-reaction rate law with bimolecular termination). For the chain mechanism ${\text{Cl}}_2\to 2\text{Cl}\bullet$ (initiation, rate $r_i$), $\text{Cl}\bullet +{\text{H}}_2\to \text{HCl}+\text{H}\bullet$ and $\text{H}\bullet +{\text{Cl}}_2\to \text{HCl}+\text{Cl}\bullet$ (propagation), $2\text{Cl}\bullet \to {\text{Cl}}_2$ (termination, rate constant $k_t$), with SSA applied to both radicals:

$$\frac{d[\text{HCl}]}{dt}=2k_p{\left(\frac{r_i}{k_t}\right)}^{1/2}[{\text{H}}_2][{\text{Cl}}_2{]}^{1/2}$$

where $k_p$ is the propagation rate constant for $\text{Cl}\bullet +{\text{H}}_2$.

Proof. The steady-state radical concentrations satisfy $d[\text{Cl}\bullet ]/dt=0$ and $d[\text{H}\bullet ]/dt=0$. The initiation rate $r_i$ produces $\text{Cl}\bullet$. Termination consumes $2\text{Cl}\bullet$ at rate $k_t[\text{Cl}\bullet {]}^2$. At steady state, initiation balances termination: $r_i=k_t[\text{Cl}\bullet {]}^2$, giving $[\text{Cl}\bullet ]=(r_i/k_t{)}^{1/2}$. The rate of HCl formation from both propagation steps is

$$\frac{d[\text{HCl}]}{dt}=k_p[\text{Cl}\bullet ][{\text{H}}_2]+k_p^{\prime }[\text{H}\bullet ][{\text{Cl}}_2].$$

The SSA for $\text{H}\bullet$ requires $k_p[\text{Cl}\bullet ][{\text{H}}_2]=k_p^{\prime }[\text{H}\bullet ][{\text{Cl}}_2]$ (formation balances consumption), so both terms are equal. Therefore

$$\frac{d[\text{HCl}]}{dt}=2k_p[\text{Cl}\bullet ][{\text{H}}_2]=2k_p{\left(\frac{r_i}{k_t}\right)}^{1/2}[{\text{H}}_2].$$

If initiation is photolytic ($r_i\propto [{\text{Cl}}_2]$), the overall rate becomes proportional to $[{\text{H}}_2][{\text{Cl}}_2{]}^{1/2}$. $\square$

Proposition 4 (Exact solution for consecutive first-order reactions). For $A\stackrel{k_1}{\to }I\stackrel{k_2}{\to }P$ with $[I{]}_0=[P{]}_0=0$ and $k_1\ne k_2$:

$$[I](t)=\frac{k_1[A{]}_0}{k_2-k_1}\left(e^{-k_{1}t}-e^{-k_{2}t}\right)$$

Proof. The first step gives $[A](t)=[A{]}_0{e}^{-k_{1}t}$. The ODE for $[I]$ is $d[I]/dt=k_1[A{]}_0{e}^{-k_{1}t}-k_2[I]$. This is a linear first-order ODE with integrating factor $e^{k_{2}t}$:

$$\frac{d}{dt}\left([I]e^{k_{2}t}\right)=k_1[A{]}_0{e}^{(k_2-k_1)t}$$

Integrating from 0 to $t$ with $[I](0)=0$:

$$[I]e^{k_{2}t}=\frac{k_1[A{]}_0}{k_2-k_1}\left(e^{(k_2-k_1)t}-1\right)$$ $$[I](t)=\frac{k_1[A{]}_0}{k_2-k_1}\left(e^{-k_{1}t}-e^{-k_{2}t}\right).\square$$

Connections Master

• Chemical kinetics: rate laws and the Arrhenius equation 14.08.01. The rate law and temperature dependence of the rate constant developed in 14.08.01 are the inputs to mechanism analysis. The Arrhenius parameters $A$ and $E_a$ for each elementary step determine which step is rate-determining and how the mechanism responds to temperature changes.

• Chemical equilibrium 14.06.04. The pre-equilibrium approximation treats a fast step as maintaining equilibrium throughout the reaction. The equilibrium constant from 14.06.04 connects the forward and reverse rate constants of each elementary step: $K=k_f/k_r$.

• Electrochemistry 14.11.01. Electrochemical reactions proceed through elementary electron-transfer steps. The Butler-Volmer equation is a rate law derived from a mechanism in which the activation barrier depends on the electrode potential.

• Enzyme kinetics 17.04.01. The Michaelis-Menten mechanism $E+S\rightleftharpoons ES\to E+P$ is a direct application of the SSA to the enzyme-substrate complex. The derivation is identical in structure to the two-step mechanism treated here.

• Organic reaction mechanisms 15.14.01. Every organic reaction mechanism is a proposed sequence of elementary steps. The tools developed here -- proposing steps, applying the SSA or pre-equilibrium approximation, and testing the predicted rate law against experiment -- are the standard methods of physical organic chemistry.

• Statistical mechanics 14.07.01. The molecular-level interpretation of each elementary step's rate constant comes from transition-state theory (unit 14.08.01 and 14.07.04), which connects the macroscopic rate constant to partition functions and activation thermodynamics.

Historical context Master

The concept of a reaction mechanism emerged from the collision theory of reaction rates developed by Trautz (1916) and Lewis (1918). They recognised that reactions occur through molecular collisions and that the rate depends on the collision frequency and the fraction of collisions with sufficient energy. The limitation of simple collision theory -- it predicts rates that are often orders of magnitude off -- forced chemists to consider multi-step pathways.

Bodenstein's 1913 study of the hydrogen-bromine reaction was the first successful mechanism analysis. The complex rate law $d[\text{HBr}]/dt=k[{\text{H}}_2][{\text{Br}}_2{]}^{1/2}/(1+k^{\prime }[\text{HBr}]/[{\text{Br}}_2])$ could not be explained by any single-step model. Bodenstein introduced the steady-state approximation to handle the bromine-atom intermediate, and the resulting derivation reproduced the observed rate law. This was the founding moment of mechanism analysis as a quantitative discipline.

Lindemann's 1922 mechanism for unimolecular gas-phase decomposition resolved a standing paradox. Many gas-phase reactions appeared first-order (consistent with a unimolecular decomposition) yet required collisional activation (a bimolecular process). Lindemann's three-step mechanism with the SSA produced a rate law that was first-order at high pressure and second-order at low pressure, a prediction confirmed by experiments on the decomposition of ${\text{N}}_2{\text{O}}_5$, cyclopropane, and methylcyclobutane.

The hydrogen-iodine reaction ${\text{H}}_2+{\text{I}}_2\to 2\text{HI}$ was assumed for decades to be a single bimolecular step because its rate law was Rate $=k[{\text{H}}_2][{\text{I}}_2]$. Sullivan (1967, J. Chem. Phys. 46) used flash photolysis to detect iodine atoms and demonstrated that the mechanism proceeds through atomic intermediates. The single-step and atom-mediated mechanisms predict the same rate law, so kinetics alone could not distinguish them. Sullivan's spectroscopic detection of the intermediate was the decisive experiment.

The development of RRKM theory by Marcus (1952) extended the Lindemann mechanism to account for the energy dependence of the decomposition rate of the activated molecule. Marcus received the 1992 Nobel Prize for his theory of electron transfer, which applies the transition-state framework to redox reactions in solution.

Bibliography Master

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}

@book{Laidler1987,
  author = {Laidler, Keith J.},
  title = {Chemical Kinetics},
  edition = {3},
  publisher = {Harper and Row},
  year = {1987}
}

@book{ZumdahlDeCoste2017,
  author = {Zumdahl, Steven S. and DeCoste, Donald J.},
  title = {Chemical Principles},
  edition = {8},
  publisher = {Cengage},
  year = {2017}
}

@book{AtkinsPaula2023,
  author = {Atkins, Peter and de Paula, Julio},
  title = {Physical Chemistry},
  edition = {12},
  publisher = {Oxford University Press},
  year = {2023}
}