Reaction Kinetics: Arrhenius, Rate Laws and Consecutive Reactions

Rate laws: how concentration sets speed

Chemical kinetics asks not whether a reaction happens (that's thermodynamics) but how fast. For an elementary step, the rate law says the reaction rate is proportional to reactant concentrations raised to powers called the reaction order:

rate = k · [A]^m · [B]^n        // k = rate constant, m+n = overall order
first-order:   d[A]/dt = -k[A]          → [A](t) = [A]0 · e^(-kt)
second-order:  d[A]/dt = -k[A]²         → 1/[A](t) = 1/[A]0 + kt

For a simple first-order decay, the concentration falls exponentially and the half-life t½ = ln(2)/k is constant regardless of starting concentration — a hallmark you can use to identify first-order kinetics experimentally. Second-order kinetics behaves very differently: its half-life depends on the starting concentration, getting longer as the reaction proceeds and concentration drops, because the rate itself depends on the square of a dwindling quantity.

The Arrhenius equation: why heat speeds everything up

Svante Arrhenius (1889) found that rate constants across an enormous range of reactions obey the same exponential temperature dependence:

k = A · exp( -Ea / (R·T) )
  A  = pre-exponential (frequency) factor — collision rate & orientation
  Ea = activation energy (J/mol)
  R  = gas constant, T = absolute temperature (K)

The physical picture is collision theory: reactant molecules must collide with enough kinetic energy to surmount an energy barrier — the activation energy Ea — and with roughly the right relative orientation for bonds to break and reform. The Boltzmann distribution says the fraction of molecules with kinetic energy above any threshold grows exponentially with temperature, which is exactly why a modest temperature rise produces such a large rate increase: the rule of thumb that reaction rate roughly doubles for every 10 °C rise is a direct, if approximate, consequence of typical Ea values (around 50 kJ/mol) plugged into the Arrhenius exponential near room temperature.

Consecutive reactions: A → B → C

When a reaction proceeds through an intermediate, A → B → C, the intermediate B's concentration first rises as it's produced from A, reaches a peak, and then falls as it's consumed to form C faster than it's replenished. Its governing equations are a small coupled linear system:

d[A]/dt = -k1[A]
d[B]/dt =  k1[A] - k2[B]
d[C]/dt =  k2[B]
[A](t) = [A]0·e^(-k1t)
[B](t) = [A]0 · k1/(k2-k1) · ( e^(-k1t) - e^(-k2t) )     (k1 ≠ k2)

Whichever step is slower dominates the observed overall rate — this is the rate-determining step, and when k1 ≪ k2 (A→B is much slower than B→C), [B] never builds up appreciably and the whole two-step sequence behaves, to a good approximation, just like simple first-order A→C decay with rate constant k1. When k1 ≫ k2, B accumulates almost completely before slowly leaking away to C, and the peak in [B] can persist for a long time — the shape of that concentration curve is itself a fingerprint you can use to diagnose which step is rate-limiting from experimental data alone.

Catalysts and reversibility

A catalyst speeds a reaction by providing an alternative reaction pathway with a lower activation energy — it does not change the reaction's thermodynamics (the equilibrium position and the overall free-energy change are identical with or without it), only the kinetic barrier standing between reactants and products, so a catalyst accelerates the approach to the same equilibrium rather than shifting where that equilibrium sits. Reversible reactions add a backward rate constant k₋₁ alongside the forward k₁; at equilibrium the forward and backward rates become equal, and the ratio of rate constants gives the equilibrium constant directly, K_eq = k₁/k₋₁ — the kinetic and thermodynamic descriptions of a reversible reaction are two views of the same underlying process, connected by that one relationship.

Frequently asked questions

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