Kinetics

Thermochemistry tells you whether a reaction can happen. Kinetics tells you how fast. Rate laws, the Arrhenius equation, and mechanistic analysis are the tools chemists and engineers use to design reactors, tune catalysts, model the atmosphere, and understand why a thermodynamically favorable reaction can still take geological ages.

1. Why kinetics matters

Diamond is thermodynamically unstable. At room temperature and pressure, graphite has a lower free energy, so every diamond in every ring on every hand is slowly turning into pencil lead. The catch is that "slowly" here means on the order of $10^{80}$ years. Thermochemistry said yes. Kinetics said "not in any useful sense." That gap — between "allowed" and "actually happens" — is the entire subject of kinetics.

The gap matters because almost every reaction of practical importance sits somewhere in the middle of the "in principle allowed but practically slow" spectrum, and the job of the chemist or engineer is to push it toward useful speeds without burning the reactor down. Every catalyst, every temperature setpoint, every solvent choice, and every enzyme in your body is a kinetic intervention.

• Industrial chemistry. The Haber-Bosch process runs nitrogen fixation ($\text{N}_2 + 3\text{H}_2 \to 2\text{NH}_3$) at 400–500°C and 200 bar over an iron catalyst. The thermodynamics actually prefer lower temperature; the kinetics force high temperature. The entire recipe is a compromise, and that compromise fixes half of the nitrogen in the global food supply.
• Pharma. Drug half-life in the body is first-order kinetics on a grand scale. Dosing schedules, therapeutic windows, and "take twice a day" instructions all come from kinetic modeling of absorption, distribution, metabolism, and excretion.
• Atmospheric and climate chemistry. Ozone destruction in the stratosphere is a catalytic radical chain — one chlorine atom can destroy $\sim 10^5$ ozone molecules before it's removed. That multiplier is why a small amount of CFC does so much damage, and it's pure chain kinetics.
• Batteries and fuel cells. Charge-transfer at the electrode is rate-limiting for most cells. The Butler-Volmer equation is Arrhenius with an extra knob (overpotential), and optimizing it is most of what electrochemical engineers do.
• Biology. Enzymes don't change the equilibrium of the reaction they catalyze — they only change the rate. The Michaelis-Menten equation is first-year biochemistry precisely because every metabolic step is describable by it.

2. Vocabulary cheat sheet

3. Rate laws and reaction order

For a generic reaction $a\text{A} + b\text{B} \to \text{products}$, experiment almost always finds a rate law of the form

How to measure reaction order

The cleanest experimental method is the method of initial rates. You run the reaction at several initial concentrations of one reactant while holding the others fixed, and watch how the initial rate changes. If doubling $[\text{A}]_0$ doubles the rate, the order in A is 1; if it quadruples, the order is 2; if it does nothing, the order is 0.

A worked example: for the reaction $2\text{NO}(g) + \text{O}_2(g) \to 2\text{NO}_2(g)$, experimental initial rates give

Runs 1→2 double [NO] and the rate goes up by $4\times$, so the order in NO is 2. Runs 1→3 double [O$_2$] and the rate doubles, so the order in O$_2$ is 1. Overall order: 3. The rate law is $r = k[\text{NO}]^2[\text{O}_2]$, and $k$ can be back-solved from any run: $k = 2.5\times 10^{-5}/(0.010^2 \cdot 0.010) = 25\ \text{M}^{-2}\text{s}^{-1}$.

4. Integrated rate laws and half-lives

Rate laws in their raw form are differential equations. Integrating them gives you the concentration-versus-time profile and lets you fit experimental data directly. Three cases cover almost everything you meet in practice.

Zero order

First order

Second order

How to tell them apart from data

A good first step with any kinetic data set is to plot the concentration three ways and see which one gives you a straight line:

• $[\text{A}]$ vs $t$ — linear means zero order.
• $\ln[\text{A}]$ vs $t$ — linear means first order.
• $1/[\text{A}]$ vs $t$ — linear means second order.

The slope of the linear plot gives $k$ directly (up to a sign). This "try all three" recipe is how undergraduates are supposed to do it, and it still works in research, because real reactions overwhelmingly fit one of these three to within experimental error.

5. Arrhenius and activation energy

The rate constant $k$ itself isn't a constant — it depends strongly on temperature. Arrhenius (1889) proposed the now-universal empirical form:

Taking logs gives the form you actually plot:

Worked example: extracting E_a from two measurements

A reaction has $k_1 = 1.5\times 10^{-3}$ s$^{-1}$ at 300 K and $k_2 = 1.2\times 10^{-2}$ s$^{-1}$ at 350 K. Find $E_a$.

$\ln(1.2\times 10^{-2}/1.5\times 10^{-3}) = \ln 8 \approx 2.08$. The temperature term is $(1/350 - 1/300) = -4.76\times 10^{-4}$ K$^{-1}$. So

A modest activation energy for a gas-phase reaction. The rate increased by a factor of 8 over 50 K, which Arrhenius correctly captures.

6. Collision and transition state theory

Arrhenius is an empirical fit. Two slightly more mechanistic theories try to derive the same form from what molecules are actually doing.

Collision theory

The simplest picture: two molecules react only if they collide, and only if the collision carries at least $E_a$ of energy along the line of approach, and only if they are oriented correctly. Putting these three factors together:

Transition state theory (Eyring)

A more sophisticated picture: reactants don't just collide, they climb a potential energy surface up to a saddle point — the transition state — and then slide down the other side into products. The rate is set by how fast molecules pass over the top of the barrier, and Eyring derived

7. Mechanisms and the steady-state approximation

Most real reactions are not elementary. They proceed through a sequence of steps, each of which is elementary, and the observed rate law is a function of the rate constants of all those steps. Deriving that overall rate law from a proposed mechanism is the central technical skill of kinetics.

Elementary steps and molecularity

An elementary step is one the reaction actually performs in a single collision or unimolecular event. Its rate law is given by its stoichiometry (unlike the overall reaction). A unimolecular step A $\to$ B has rate $k[\text{A}]$; a bimolecular step A + B $\to$ C has rate $k[\text{A}][\text{B}]$; termolecular steps exist but are rare because three-body collisions are rare.

A mechanism is a set of elementary steps whose sum is the overall reaction. Kinetics can disprove a mechanism (by showing the derived rate law doesn't match experiment) but it can't prove one — two different mechanisms can give the same overall rate law.

The rate-determining step

If one step is much slower than all the others, the reaction rate is set entirely by that slow step, and everything before it can be treated as a fast pre-equilibrium. This is the easiest case. Example: nucleophilic substitution at a tertiary carbon (SN1) goes via a slow ionization to a carbocation followed by fast attack of the nucleophile. The observed rate law is first-order in substrate and zeroth-order in nucleophile — which tells you the nucleophile isn't in the rate-determining step.

The steady-state approximation

When no single step is obviously rate-determining, the standard trick is to assume that the concentration of each reactive intermediate stays approximately constant — its production rate equals its consumption rate. For the Lindemann mechanism of unimolecular gas-phase reactions,

applying $d[\text{A}^*]/dt = 0$ gives $[\text{A}^*]_\text{ss} = k_1[\text{A}][\text{M}]/(k_{-1}[\text{M}] + k_2)$, and the overall rate is

The same machinery handles enzyme kinetics, radical chain reactions, and atmospheric photochemistry — you write down the elementary steps, apply the steady-state approximation to every intermediate, and out pops a closed-form rate law.

8. Catalysis and enzyme kinetics

A catalyst is a species that speeds up a reaction without being consumed. It works by providing a lower-energy path — a different transition state — from reactants to products. The thermodynamics are unchanged (a catalyst doesn't move equilibrium); only the activation barrier is.

Three flavors:

• Homogeneous catalysis. Catalyst and reactants in the same phase (usually solution). Transition-metal complexes doing cross-couplings, acid or base catalysis of ester hydrolysis, etc.
• Heterogeneous catalysis. Catalyst is a solid, reactants are gas or liquid. Industrially dominant — platinum catalytic converters, iron for Haber-Bosch, zeolites in petroleum cracking. Rate laws are typically derived from the Langmuir-Hinshelwood adsorption model.
• Enzymatic catalysis. Biology's version. Rate enhancements up to $10^{17}$ over the uncatalyzed reaction, with exquisite substrate specificity.

Michaelis-Menten

The workhorse model of enzyme kinetics. An enzyme E binds substrate S reversibly to form a complex ES, which breaks down to product P and free enzyme:

Applying the steady-state approximation to [ES] and using $[\text{E}]_0 = [\text{E}] + [\text{ES}]$ gives the Michaelis-Menten rate law:

Lineweaver-Burk and modern fitting

For decades, biochemists extracted $K_M$ and $V_\text{max}$ by plotting $1/v$ vs $1/[\text{S}]$ — a Lineweaver-Burk plot — which turns the hyperbola into a straight line with slope $K_M/V_\text{max}$ and intercept $1/V_\text{max}$. This is pedagogically nice but statistically terrible (it weights low-substrate points much more than it should). Modern practice is to fit the hyperbola directly with nonlinear least squares, which is one line of scipy.optimize.curve_fit.

9. Interactive: Arrhenius explorer

Drag $E_a$ and $A$ to see how $\ln k$ depends on $1/T$. Below is the standard Arrhenius plot ($\ln k$ vs $1000/T$). The slope is $-E_a/R$ — steeper slope, bigger $E_a$, more temperature-sensitive reaction.

10. Source code

Fitting a first-order decay and an Arrhenius plot in three-dozen lines of Python. Everything you need to analyze a real kinetic data set in a lab notebook.

import numpy as np
from scipy.optimize import curve_fit

# synthetic data: [A] vs t, first-order decay with k = 0.05 s^-1 and noise
t = np.linspace(0, 60, 13)
A0 = 1.0
k_true = 0.05
A = A0 * np.exp(-k_true * t) + 0.01 * np.random.randn(len(t))

# option 1: linearize and fit ln[A] vs t
mask = A > 0
slope, intercept = np.polyfit(t[mask], np.log(A[mask]), 1)
k_fit_linear = -slope
print(f"linear fit k = {k_fit_linear:.4f} s^-1")

# option 2: nonlinear fit of the full exponential (preferred)
def model(t, A0, k):
    return A0 * np.exp(-k * t)

(A0_fit, k_fit_nl), _ = curve_fit(model, t, A, p0=[1.0, 0.05])
print(f"nonlinear fit k = {k_fit_nl:.4f} s^-1, A0 = {A0_fit:.4f}")
print(f"half-life = {np.log(2)/k_fit_nl:.2f} s")

import numpy as np
from scipy.optimize import curve_fit

R = 8.314  # J/(mol K)

# measured rate constants at different temperatures
T = np.array([280, 300, 320, 340, 360, 380])         # K
k = np.array([1.2e-4, 1.5e-3, 1.3e-2, 8.5e-2, 0.42, 1.8])  # s^-1

# linearized Arrhenius: ln k = ln A - Ea / (R T)
slope, intercept = np.polyfit(1.0 / T, np.log(k), 1)
Ea_lin = -slope * R                 # J/mol
A_lin = np.exp(intercept)            # s^-1
print(f"linear Ea = {Ea_lin/1000:.1f} kJ/mol,  A = {A_lin:.2e} s^-1")

# nonlinear fit (better if you have many decades of k)
def arr(T, A, Ea):
    return A * np.exp(-Ea / (R * T))

(A_nl, Ea_nl), _ = curve_fit(arr, T, k, p0=[A_lin, Ea_lin])
print(f"nonlinear Ea = {Ea_nl/1000:.1f} kJ/mol,  A = {A_nl:.2e} s^-1")

# predict the rate constant at 310 K
k_310 = arr(310.0, A_nl, Ea_nl)
print(f"k(310 K) = {k_310:.3e} s^-1")

11. Cheat sheet

See also