Reaction Kinetics

Motivation

Thermodynamics answers whether a reaction can happen. A reaction with a negative Gibbs free energy is spontaneous in the Gibbs sense. That statement describes the direction of equilibrium. It does not describe how fast the system reaches equilibrium.

Diamond is thermodynamically unstable relative to graphite at ambient conditions. The Gibbs free energy change for the diamond to graphite transition is about minus 2.9 kJ per mol at 298 K and 1 atm. Diamond still sits on fingers for centuries. The carbon carbon bonds that would rearrange are kinetically locked. No thermal collision at 300 K carries the required activation energy. Kinetics answers the follow up question that thermodynamics does not ask. Will it happen on a timescale we care about?

The gap between a negative Gibbs free energy and an actually measurable rate is where the whole subject lives. Corrosion notes are about the kinetics of a thermodynamically inevitable process. Iron plus oxygen plus water forms rust. That reaction is downhill by about 1500 kJ per mol. Structural steel still lasts decades. The rate determining step is slow. Stainless steel lasts longer. Chromium doping changes the mechanism of film growth. It does not shift Gibbs free energy by much. Passivation is a kinetic phenomenon. The passive film is a kinetic barrier. Chloride pitting re exposes the thermodynamic driving force when the film breaks down.

This note is the non electrochemical parent of Butler Volmer and Tafel. Those topics appear in Corrosion and EIS. The existing Cyclic Voltammetry and chem_lpr notes lean on kinetics without a dedicated kinetics note. This is that note.

The rate law

For a reaction with reactants A and B and products, we define the instantaneous rate. The rate equals the negative change in concentration of A divided by its coefficient and by time. It also equals the negative change in concentration of B divided by its coefficient and by time. It equals the positive change in product concentration divided by its coefficient and by time.

The empirical rate law postulates a separable form that does not change with time. The rate equals k times A to the power m times B to the power n. The variable k is the rate constant. The exponents m and n are the reaction orders. These orders can be non integer values. Only an elementary step has orders equal to the stoichiometric coefficients by the law of mass action. The overall order is m plus n. The rate law is a fit from data. It is not a derivation.

Zeroth order

In zeroth order reactions the rate equals a constant k0. The concentration of A at time t equals the initial concentration minus k0 times t. The concentration declines linearly to zero. It then goes negative in an unphysical way. Zeroth order arises when the reactant saturates a catalyst surface. This happens in enzyme or surface catalyzed decomposition. The substrate fully occupies all sites. The rate is limited by site turnover. Adding more reactant does nothing. The half life is linear in the starting concentration. It equals the initial concentration divided by two times k0.

First order

In first order reactions the rate equals k1 times A. The concentration at time t equals the initial concentration times e to the minus k1 t. A plot of the natural log of concentration versus time is linear. The slope is minus k1. This linearity is the diagnostic test.

The half life is independent of concentration. It equals ln 2 divided by k1. First order kinetics dominate nuclear decay and unimolecular thermal decomposition. The half life does not depend on how much material you have. Most electrochemical electron transfer steps reduce to a pseudo first order rate law at small overpotential. This connects to Cyclic Voltammetry.

Second order

In second order reactions with one reactant the rate equals k2 times A squared. One over concentration at time t equals one over initial concentration plus k2 times t. A plot of one over concentration versus time is linear. The slope is k2.

The half life scales inversely with initial concentration. It equals one divided by k2 times the initial concentration. The more you start with, the faster the first half vanishes. This is the signature. Bimolecular gas phase reactions are canonically second order. A mixed second order reaction with two reactants needs excess B to use pseudo first order isolation.

Determining order from data

Given a single concentration versus time trace, the standard protocol uses integrated method linearisation. Try plotting concentration versus time for zeroth order. Try ln concentration versus time for first order. Try one over concentration versus time for second order. Pick the plot that is most linear by its R squared value. This approach is brittle when orders are fractional or when several reactants participate.

A more robust approach is the method of initial rates. Hold all but one reactant in large excess. Absorb its order into an effective rate constant. Measure the initial rate against the varied reactant initial concentration. A log log plot of rate versus initial concentration gives the order m directly as the slope. Both methods assume the rate constant stays constant over the run. It must be isothermal. Product inhibition can confound results. Product re binding the catalyst is the most common silent confound.

Worked example

A reaction A to P is monitored at two temperatures. The initial concentration is 1.00 M for both.

At 300 K the concentration drops from 1.00 at t=0 to 0.91 at t=100 s, to 0.83 at t=200 s, to 0.69 at t=400 s, and to 0.48 at t=800 s. At 320 K the concentration drops from 1.00 to 0.70 at t=100 s, to 0.49 at t=200 s, to 0.24 at t=400 s, and to 0.06 at t=800 s.

Order: Linearise each row. The 300 K data on ln A give a slope of about minus 9.17 times 10 to the minus 4 per s. The plot is linear. The plots for 1/A and A versus t are not linear. So the reaction is first order. The rate constant k1 at 300 K is 9.17 times 10 to the minus 4 per s. The 320 K data likewise linearise on ln A with slope minus 3.56 times 10 to the minus 3 per s. So k1 at 320 K is 3.56 times 10 to the minus 3 per s.

Activation energy: The two point Arrhenius relation is ln(k2/k1) = minus (Ea/R) times (1/T2 minus 1/T1). Plugging in the values gives ln of 3.56 times 10 to the minus 3 divided by 9.17 times 10 to the minus 4 which equals 1.359. This equals minus (Ea over 8.314) times (1/320 minus 1/300). That is minus (Ea/8.314) times minus 2.083 times 10 to the minus 4. Solving gives Ea of about 54 kJ per mol. A 20 K rise that quadruples the rate fits a barrier of about 50 kJ per mol. This is the regime where most thermal chemistry lives.

The Arrhenius equation

The empirical form is k of T equals A times e to the minus Ea over RT. A is the pre exponential factor. It is the hypothetical rate at infinite temperature. It combines an attempt frequency with a steric factor. Ea is the activation energy. Taking logs gives ln k equals ln A minus (Ea over R) times (1/T). An Arrhenius plot of ln k versus 1/T is linear. The slope is minus Ea/R. The intercept is ln A. This slope is the operational definition of Ea. It is whatever the slope gives you. It is not a fundamental quantity.

Derivation from the Boltzmann distribution

For a bimolecular gas phase reaction, the rate of productive collisions is the collision frequency Z times the fraction of collisions that carry energy above Ea along the reaction coordinate. The Boltzmann distribution gives the fraction of molecules with excess energy:

f(E >= Ea) = integral from Ea to infinity of (1/kBT) times e to the minus E/kBT dE = e to the minus Ea/kBT.

Multiplying by the per pair collision frequency ZAB folded with a sqrt T dependence into A, and a steric factor p, gives rate proportional to ZAB times p times e to the minus Ea over RT. Identifying A with ZAB times p times NA yields the Arrhenius form. The crucial move is that the exponential is the Boltzmann factor for crossing a barrier of height Ea. Kinetics is at root the thermal statistics of barrier crossing. The same exponential reappears with the electrochemical driving force folded into the effective barrier in Butler Volmer. See Corrosion.

The derivation assumes a single well defined barrier. It assumes an ideal gas collision picture. It assumes no tunneling. Condensed phase reactions inherit the form phenomenologically. The barrier is now a free energy Delta G‡ including an entropy cost. The Eyring Polanyi form is k = (kBT/h) times e to the minus Delta G‡ over RT. This reduces to Arrhenius with Ea = Delta H‡ + RT.

When it breaks

The Arrhenius equation is an empirical fit. It breaks in several well known ways:

• Non Arrhenius curvature: When the mechanism shifts with temperature, parallel pathways with different barriers exist. The ln k versus 1/T plot curves. A single Ea is a tangent value, not a constant. Combustion kinetics is the canonical case.
• Negative activation energies: Radical recombination and barrierless association reactions with pre equilibria show rate decreasing with temperature. The effective Ea is set by the T dependence of the pre equilibrium. This can flip the sign. It is impossible for an elementary step. It signals a composite mechanism.
• Tunneling: Below about 200 K especially for H atom transfer, the rate exceeds the classical Boltzmann prediction by orders of magnitude. Arrhenius plots bend upward at low T.
• Diffusion limit: In solution, once the rate nears the Smoluchowski limit of about 10 to the 10 M minus 1 s minus 1 in water, the rate is capped by how fast reactants find each other. Further heating does less than Stokes Einstein predicts.

Arrhenius is a model of a barrier, not a law of nature. Treat it as a linear regression that is informative within its regime.

Catalysis

A catalyst appears in the rate law. It changes the rate constant k. It is regenerated over the cycle. It does not appear in the overall stoichiometry. It does not shift the equilibrium composition. It lowers the effective activation energy by providing an alternative mechanism with a lower barrier transition state. It leaves the Gibbs free energy of the overall reaction untouched. This replaces one high barrier step with two or more lower barrier steps. The new path highest barrier lies below the old Ea. Rate rises. The thermodynamic endpoints stay unchanged because the catalyst returns to its original state at cycle end. A catalyst changes the mechanism, not just the barrier. The lower barrier is a symptom of a new mechanism, not an independent knob on the old one.

Homogeneous vs heterogeneous

• Homogeneous catalysts share a phase with the reactants. Examples are acid base catalysis in solution, organometallic catalysis, and enzymatic catalysis. They follow Michaelis Menten type rate laws with catalyst saturation and turnover number kcat. kcat is a first order rate constant at saturating substrate.
• Heterogeneous catalysts sit at a phase boundary. Examples are metal surfaces and zeolite pores. Their kinetics are dominated by adsorption steps. Langmuir Hinshelwood has both reactants adsorb then react on the surface. Eley Rideal has one adsorbed reactant and one from the gas. Saturation of surface sites gives the zeroth order regime above. Adding more reactant does nothing because every site is already occupied.

The same site saturation logic produces Michaelis saturation in enzymes. The mathematics is identical. The difference is whether the sites are in solution bound proteins or on a solid surface.

Electrocatalysis includes oxygen reduction at a Pt cathode and hydrogen evolution at a Ni cathode. It is heterogeneous catalysis with the electrode potential as an additional thermodynamic variable. The kinetic apparatus is Butler Volmer. See Corrosion and EIS. The electrode potential shifts the barrier symmetrically by alpha F eta. The variable eta is the overpotential. Alpha is a transfer coefficient. This electrochemical rate law is a direct descendant of the Arrhenius exponential.

Catalysis and thermodynamics, restated

A catalyst cannot make a thermodynamically uphill reaction spontaneous. It cannot fix CO2 to carbohydrates without free energy input. Photosynthesis runs on photons. Chlorophyll is a catalyst. The photon is the driving force. A common lay mistake confuses lowering the barrier with shifting the equilibrium. The correct statement is that the catalyst lowers the barrier in both directions equally. It accelerates the reverse reaction by the same factor as the forward. Keq = kf/kr stays untouched.

Reaction mechanisms and the rate determining step

A mechanism is a sequence of elementary steps. The steps sum to the overall stoichiometry. For a candidate mechanism with A plus B forming I rapidly with equilibrium constant K1, followed by I to P slowly with rate constant k2, the rate determining step approximation asserts that the slow step governs the rate. The preceding fast step is at quasi equilibrium. The rate equals k2 times I. The intermediate concentration I equals K1 times A times B. The observed law is second order overall and first order in each reactant. The mechanism is bimolecular then unimolecular. The observed rate constant kobs = k2 K1 is a product of a kinetic and a thermodynamic factor. Experimental rate laws invert to propose mechanisms. Any mechanism whose RDS approximation reproduces the observed order is a candidate.

Derivation from the steady state assumption

The RDS approximation is the limit of the steady state approximation when one rate constant is much smaller than the others. For the mechanism written irreversibly, the rate of change of I equals k1 times A times B minus k2 times I. The steady state approximation sets this to zero. I under steady state equals k1 over k2 times A times B. The rate equals k1 times A times B. Restoring k minus 1, the general steady state form is:

I_ss = k1 times A times B divided by k minus 1 plus k2. Rate = k1 k2 divided by k minus 1 plus k2 times A times B.

The RDS picture corresponds to k2 much less than k minus 1. This is the pre equilibrium limit with RDS as step 2. It gives rate = k2 K1 A B. If k minus 1 is much less than k2, the RDS is step 1 and rate = k1 A B. The full steady state expression interpolates between the two RDS limits. The RDS approximation is its singular limit.

When it breaks

The RDS approximation fails when multiple steps have comparable rates. No single bottleneck exists. The full steady state analysis or King Altman analysis is needed. Symptoms include:

• The rate law does not factorise as a single step mass action. It has sums in the denominator like Michaelis Menten or Langmuir Hinshelwood surface kinetics.
• Apparent orders are fractional under steady state radical chains.
• Isotopic substitution changes the rate inconsistently with any single RDS. Kinetic isotope effects distribute across the cycle.

Even the steady state assumption can fail. Highly reactive intermediates like transition metal catalysts with sub-millisecond turnover oscillate instead of settling. Transient microkinetic modeling is needed. The RDS picture is the zeroth order term in a hierarchy of approximations. It is useful for pedagogy and order of magnitude reasoning. Do not treat it as a law.

The electrochemical connection

The rate laws above apply to thermal reactions in bulk. Electrochemical reactions at an electrode are also rate limited by barrier crossing. The barrier is now the activation free energy of an electron transfer step. That barrier depends on the electrode potential. The Butler Volmer equation is:

i = i0 [ e^((1-alpha) F eta / RT) - e^(-alpha F eta / RT) ]

This is the Arrhenius equation with overpotential eta entering the exponential as an extra work term. i0 is the exchange current density. It is the electrochemical analogue of A times e to the minus Ea over RT at the equilibrium potential. Alpha is the symmetry factor splitting the barrier between anodic and cathodic directions.

The two Tafel limits exist for large overpotential values where one branch dominates. The equation is eta = a + b log i with b = 2.303 RT / (alpha F). These are first order rate laws in disguise. The value b encodes the effective activation energy per decade of current. An Arrhenius slope encodes the same thing per decade of temperature. Tafel slopes appear in mV per decade because they are Arrhenius slopes in the overpotential coordinate.

This unifies the chemistry wiki: Corrosion measures i_corr which is the exchange current of the coupled Fe to Fe2+ and O2 cell. EIS measures R_ct which is RT divided by F i0. Measuring R_ct equals measuring i0 which equals measuring the barrier. Cyclic Voltammetry measures i_p via the Randles Sekcik equation. It links peak current to diffusion and the heterogeneous rate constant k0. The Nernst equation in Electrochemical Thermodynamics gives the equilibrium potential. Butler Volmer gives the rate at which the system approaches it. This is exactly the thermodynamics versus kinetics split that opened this note, now specialised to the electrode interface.

See also

• Corrosion. Tafel kinetics and Butler Volmer as the electrochemical rate law.
• EIS. Charge transfer resistance R_ct is a kinetic parameter. It is the EIS analogue of an Arrhenius prefactor.
• Cyclic Voltammetry. Randles Sekcik is a kinetic equation relating peak current to diffusion and heterogeneous k0.
• Chemical Thermodynamics. Spontantety via Delta G < 0 versus rate. The diamond graphite and rust examples.
• General Chemistry Foundations. Rate law prerequisites.
• Electrochemical Thermodynamics. Nernst for equilibrium and Butler Volmer for kinetics. The same split specialised to the electrode.