Steady-State Approximation (SSA)
The steady-state approximation (SSA), also known as the Bodenstein approximation, is a powerful algebraic tool for deriving rate laws from mechanisms involving highly reactive, short-lived intermediates. It applies when the concentration of an intermediate I is small and changes slowly relative to the reactants.
Mathematical Principle
For intermediate I, we write:
d[I]/dt ≈ 0
This does not mean [I] = 0, but rather that the rates of formation and consumption of I are nearly equal:
Σ(rates forming I) − Σ(rates consuming I) = 0
Solve for [I] and substitute into the overall rate expression.
Application: H₂ + Br₂ Mechanism
The reaction H₂ + Br₂ → 2 HBr is one of the most-studied systems in chemical kinetics. Its radical chain mechanism is:
1. Br₂ → 2 Br· (initiation, k₁) 2. Br· + H₂ → HBr + H· (propagation, k₂) 3. H· + Br₂ → HBr + Br· (propagation, k₃) 4. H· + HBr → H₂ + Br· (inhibition, k₄) 5. 2 Br· → Br₂ (termination, k₅)
Applying the SSA to the radical intermediates Br· and H· yields the celebrated empirical rate law:
v = k[H₂][Br₂]^(1/2) / (1 + k'[HBr]/[Br₂])
This non-integer order law is completely inaccessible from the overall stoichiometry — demonstrating the power of the SSA.
| Radical species | d[·]/dt ≈ 0? | Justification |
|---|---|---|
| Br· | yes | highly reactive, [Br·] is stationary |
| H· | yes | even more reactive, [H·] very small |
Chain Mechanisms
In a radical chain mechanism:
- Initiation: generates radicals (homolytic bond cleavage or photochemical).
- Propagation: radicals regenerate; the chain is self-sustaining.
- Termination: two radicals combine → chain ends.
The chain length is the ratio of propagation rate to initiation rate. For H₂/Br₂ it can reach 10⁵ under certain conditions.
Validity of the SSA
The SSA is valid when:
- The intermediate concentration remains small compared to reactant concentrations.
- The initiation phase is short (quasi-stationary regime reached quickly).
- No explosive accumulation of the intermediate occurs.
The rate-determining step approximation is a limiting case of the SSA: it assumes one step is so slow that all subsequent steps are quasi-instantaneous.