# Steady state (chemistry) In chemistry, a **steady state** is a condition in which the concentration of a species stays constant in time even though the processes that make and consume it continue, because the two run at the same rate. The species is usually a [[Reaction_intermediate|reaction intermediate]], something formed in one step of a [[Reaction_mechanism|mechanism]] and destroyed in a later one, and the steady-state approximation of [[Chemical_kinetics|chemical kinetics]] is the working assumption that such an intermediate's net rate of change is negligible, `d[I]/dt ≈ 0`, once the reaction is under way.[^boyd-ssa] Setting that derivative to zero turns a system of differential equations into algebra and yields a single [[Rate_equation|rate law]] for the overall reaction, which is how most textbook rate laws for multistep mechanisms are derived. In the microsim below the reader slides the dimensionless rate ratio κ of a two-step mechanism across five decades, and the exact concentration of the intermediate is drawn over the steady-state value `i_ss = a/(λ·p + κ)`, with a ribbon showing where, and by how much, the approximation is wrong.[^boyd-mechanism] On the Chemistry flagship's spine this page is the main article for Part VI — Reaction, section *Mechanisms and the steady state*, the sibling that follows Chemical kinetics and its half-life test and stands beside [[Rate-determining_step|Rate-determining step]]; its sim is the one kinetics sim on the flagship that is baked offline rather than computed live, because the mechanism becomes stiff at the fast end of the slider. A steady state is not an [[Chemical_equilibrium|equilibrium]]. At equilibrium every step runs forward and backward at equal rates and nothing changes; in a steady state material flows through the intermediate from reactants to products, and only the intermediate's concentration is constant. The same idea appears wherever a small pool is fed and drained at matched rates: the [[Ozone|ozone]] of the stratosphere, the daughter nuclides of a [[Decay_chain|decay chain]], and the enzyme–substrate complex of [[Michaelis–Menten_kinetics|Michaelis–Menten kinetics]], whose saturating law is a steady-state result. ## Steady state approximation in chemical kinetics A mechanism is a list of [[Elementary_reaction|elementary steps]], and each step's [[Reaction_rate|rate]] follows its molecularity, which is the one case in which orders can be read from an equation rather than measured.[^af-mechanisms][^boyd-elementary] Writing an [[Ordinary_differential_equation|ordinary differential equation]] for every species gives an exact description, but the Portal Book notes that multistep mechanisms "more often than not" have no closed-form solution, so the exact route ends in numerical integration.[^boyd-noclosed] The steady-state approximation is the alternative: assume that each reactive intermediate is consumed as fast as it forms, solve the resulting algebraic equations for the intermediate concentrations, and substitute them into the rate of product formation. The approximation is presented in the book as an empirical device that works because intermediates are short-lived and never accumulate, and it must be checked against the exact solution rather than trusted on sight.[^boyd-ssa] The subsections below follow the pair's order: the rate equations, the exact solutions where they exist, the steady-state result, and its range of validity. ### Reaction rates The simplest mechanism with an intermediate is the consecutive first-order scheme A → B → C with rate constants k₁ and k₂. Its rate equations are `d[A]/dt = −k1·[A]`, `d[B]/dt = k1·[A] − k2·[B]` and `d[C]/dt = k2·[B]`, and their sum is zero, so [A] + [B] + [C] is conserved. B is the intermediate: fed by the first step, drained by the second. The Portal Book's own scheme, the one the microsim integrates, adds a reversible first step and a second reactant. Reconstructed from the book's rate terms, since the scheme figure was lost in the Wikitube extract, it is A ⇌ I + P with rate constants k₁ forward and k₋₁ reverse, followed by I + B → Q with rate constant k₂.[^boyd-mechanism] The intermediate I obeys `d[I]/dt = k1·[A] − k_-1·[I][P] − k2·[I][B]`, in which the reverse of the first step, proportional to the product P, competes with the second step for the intermediate. With B in large excess its concentration is constant, and the scheme scales to three dimensionless equations in the reduced time τ = k₁t: `da/dtau = −a + lambda·i·p`, `di/dtau = a − lambda·i·p − kappa·i` and `dq/dtau = kappa·i`, where a, i, p and q are the concentrations divided by [A]₀, λ = k₋₁[A]₀/k₁ measures the reverse step and κ = k₂[B]₀/k₁ measures the second step, with p = i + q and a = 1 − i − q by the material balances. ### Analytical solutions The consecutive scheme A → B → C can be solved exactly. The reactant follows the [[Exponential_decay|exponential decay]] `[A] = [A]0·exp(−k1·t)`, so the first step alone has the fixed [[Half-life|half-life]] of every first-order process.[^af-firstorder] Substituting into the equation for B and integrating gives `[B] = [A]0·(k1/(k2 − k1))·(exp(−k1·t) − exp(−k2·t))`, and [C] follows from the conservation of the total. The intermediate rises from zero, passes through a maximum at `t_max = ln(k2/k1)/(k2 − k1)`, and decays; with k₁ = 0.01 s⁻¹ and k₂ = 1 s⁻¹ the maximum comes at about 4.7 s and the peak [[Molar_concentration|concentration]] is under 1 % of [A]₀ (derived figures), which is the picture of a short-lived intermediate. The same double-exponential is the law of a parent and daughter nuclide in [[Radioactive_decay|radioactive decay]], where the case k₂ ≫ k₁ is called secular equilibrium in [[Nuclear_chemistry|nuclear chemistry]]. The two-step scheme of the book has no such solution. Its equations are nonlinear in the products i·p, and the Portal Book's remark that closed forms are usually unavailable applies directly.[^boyd-noclosed] The exact curves shown in the microsim are therefore not formulas but the output of a numerical integrator, the same kind of [[Simulation|simulation]] that [[Chemical_process_modeling|process modelling]] runs for whole reactors, run once offline for 51 values of κ spaced evenly in log₁₀ κ from −2 to 3, sampled at 501 points in τ from 0 to 10, and stored as a table that the page reads. A stiff solver is required because at κ = 1000 the intermediate relaxes a thousand times faster than the reactant decays; an explicit fourth-order [[Runge–Kutta_methods|Runge–Kutta]] step would have to be shorter than about 2.8/(κ + λp) in τ to remain stable, which is why this sim is never computed live in the browser. ### Steady state Setting `d[I]/dt = 0` in the two-step scheme gives `[I] = k1·[A]/(k_-1·[P] + k2·[B])`, and the rate of product formation, `r = k2·[I][B]`, becomes `r = k1·k2·[A][B]/(k_-1·[P] + k2·[B])`.[^boyd-ssa] This is the whole content of the approximation: one algebraic line replaces one differential equation. The result is not a simple power law. The orders with respect to B and P "cannot be specified", as the book puts it, because both appear in a sum in the denominator; the reaction is first order in A throughout, but its dependence on B and P changes with their concentrations.[^boyd-orders] In the scaled variables the steady-state intermediate is `i_ss = a/(lambda·p + kappa)` and the scaled rate is `r = kappa·i`. The microsim on this page is the exact solution set against that formula. The reader's one control is κ on a logarithmic slider from 0.01 to 1000, starting at 1, with λ fixed at 10. For each κ the page draws the exact i(τ) from the baked table and, over it, the steady-state value i_ss computed live from the exact a and p at every τ, so that the two curves differ only by the approximation itself. A ribbon between them shows the relative error, and a second panel plots the rate r = κ·i against the limit a. The HUD reads `r = k1*k2*[A][B]/(k_-1*[P] + k2*[B])`. Sliding κ upward, the intermediate's curve collapses onto the steady-state line after an ever-shorter start-up; sliding it downward, the ribbon widens, the intermediate builds up to a substantial fraction of [A]₀, and the rate falls away from a as P accumulates. The mechanism is labelled "reconstructed" in the sim header, since its figure did not survive the extract, and the trajectories are interpolated in log₁₀ κ rather than in κ. ### Validity The approximation is wrong twice: at the start, while the intermediate is still building up from zero and `d[I]/dt` is large and positive, and again just after, when the exact curve overshoots the steady-state line before settling onto it. After that induction period it is good whenever the intermediate is drained much faster than it is fed, which in the scaled scheme means κ + λp ≫ 1. The relative-error ribbon in the microsim makes the criterion visible: at κ = 1000 the exact rate κ·i matches the steady-state value κ·a/(λ·p + κ) to better than 1 % for all τ > 0.05, and both lie within 1 % of a itself, because λp/κ never exceeds 0.01; this is the acceptance test the sim is built against.[^boyd-limit] At κ = 1 the induction period stretches to a substantial fraction of the reaction, and at κ = 0.01 the intermediate accumulates to a level at which calling it a steady state is not honest. The two limits of the steady-state rate law are the two ends of the slider. When k₂[B] ≫ k₋₁[P] the denominator is dominated by the second step, the rate reduces to `r ≈ k1·[A]`, and the first step is the [[Rate-determining_step|rate-determining step]]: the reaction is first order in A and zero order in B.[^boyd-limit] When k₋₁[P] ≫ k₂[B] the reverse of the first step dominates, most intermediate returns to A before it can react, and the rate is `r ≈ (k1·k2/k_-1)·[A][B]/[P]`, inversely proportional to the product: this is product inhibition, the same effect that slows a [[Catalysis|catalyst]] whose product will not leave, and it is also the pre-equilibrium case, in which the first step is treated as a fast [[Dynamic_equilibrium|equilibrium]] with [[Equilibrium_constant|equilibrium constant]] K₁ = k₁/k₋₁ and the second step as slow. The steady-state law contains both limits and the transition between them, which neither limiting treatment can describe on its own. ## Example The Portal Book's pre-equilibrium example is the [[Chemical_reaction|reaction]] of nitric oxide with [[Chlorine|chlorine]], 2 NO + Cl₂ → 2 NOCl, whose measured rate law is second order in NO and first order in Cl₂, `rate = k·[NO]²[Cl2]`, with k = 3.0 L² mol⁻² s⁻¹ from three initial-rate trials (the rate constant is a derived value in the Wikitube extract).[^af-initial] The proposed mechanism is a fast equilibrium NO + Cl₂ ⇌ NOCl₂ with K₁ = k₁/k₋₁, followed by the slow step NOCl₂ + NO → 2 NOCl with rate constant k₂. Treating the intermediate NOCl₂ by the pre-equilibrium method gives `[NOCl2] = K1·[NO][Cl2]` and `rate = k2·K1·[NO]²[Cl2]`, which reproduces the measured orders with the observed k identified as k₂K₁; the same law follows from the steady-state treatment in the limit where the reverse of the first step is much faster than the second.[^af-preeq] A mechanism that reproduces the rate law is consistent with it, not proved by it, a distinction the book keeps. The scaled two-step scheme gives a worked number at the other end of the slider. At κ = 1000 the steady-state rate is `r_ss = kappa·a/(lambda·p + kappa)`; with λ = 10 and p ≤ 1 the correction term λp/κ is at most 0.01, so r_ss lies within 1 % of a for the whole reaction, and the exact rate from the baked trajectory lies within 1 % of r_ss for τ > 0.05. The overall reaction is then indistinguishable from a first-order decay of A with rate constant k₁, and a chemist measuring it would report first order in A and zero order in B without ever seeing the intermediate. At κ = 0.01 the same mechanism, unchanged in its steps, gives a rate that falls as the product accumulates and an intermediate that reaches a large fraction of [A]₀ (a limit test of the stated forms rather than a book number). The classic literature example of a steady-state rate law, the formation of hydrogen bromide from [[Hydrogen|hydrogen]] and [[Bromine|bromine]] by a chain of atom steps, has the same structure: a denominator containing the product concentration, and orders that cannot be stated as integers. ## See also - [[Rate-determining_step]] - [[Reaction_mechanism]] - [[Reaction_intermediate]] - [[Chemical_kinetics]] - [[Rate_equation]] - [[Michaelis–Menten_kinetics]] - [[Catalysis]] ## Notes and references The two-step mechanism on this page is reconstructed from the rate terms in the Portal Book's prose because the scheme figure was lost in text extraction; the scaling to τ, λ and κ, the value λ = 10, and the baked parameter grid are Wikitube choices for the microsim, not the book's. The footnotes are collected under References below. [^boyd-ssa]: Boyd, W. Christopher (2025). *Exploring Inorganic and Organometallic Chemistry*. Chapter 10 Kinetics, p. 281 (the steady-state approximation, d[I]/dt ≈ 0, presented as empirical; the intermediate concentration and the resulting rate law). https://open.umn.edu/opentextbooks/textbooks/exploring-inorganic-and-organometallic-chemistry [^boyd-mechanism]: Boyd, W. Christopher (2025). *Exploring Inorganic and Organometallic Chemistry*. Chapter 10 Kinetics, pp. 276, 281 (the two-step scheme A ⇌ I + P, I + B → Q, reconstructed from the rate terms; the scheme figure was lost in the Wikitube extract and is to be verified against the PDF page). [^boyd-noclosed]: Boyd, W. Christopher (2025). *Exploring Inorganic and Organometallic Chemistry*. Chapter 10 Kinetics, pp. 280–281 (multistep mechanisms "more often than not" lack closed-form solutions). [^boyd-orders]: Boyd, W. Christopher (2025). *Exploring Inorganic and Organometallic Chemistry*. Chapter 10 Kinetics, pp. 281–282 (the orders in B and P "cannot be specified"). [^boyd-limit]: Boyd, W. Christopher (2025). *Exploring Inorganic and Organometallic Chemistry*. Chapter 10 Kinetics, p. 282 (if k₂[B] ≫ k₋₁[P] the rate reduces to k₁[A]). The 1 % figures at κ = 1000 are Wikitube limit tests of the stated forms, not book numbers. [^af-initial]: Flowers, Paul; Neth, Edward; Robinson, William et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 17 Kinetics, §17.3 Rate Laws, Example 17.5, pp. 803–804 (NO + Cl₂ initial rates, three trials; second order in NO, first in Cl₂; k = 3.0 L² mol⁻² s⁻¹ is a derived value in the Wikitube extract). https://open.umn.edu/opentextbooks/textbooks/chemistry-atoms-first [^af-preeq]: Flowers, Paul; Neth, Edward; Robinson, William et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 17 Kinetics, §17.6 Reaction Mechanisms, Example 17.14, pp. 824–825 (fast pre-equilibrium NO + Cl₂ ⇌ NOCl₂, slow NOCl₂ + NO → 2 NOCl; the final rate law rate = k₂K₁[NO]²[Cl₂] is the standard form, to be verified against the PDF page). [^af-mechanisms]: Flowers, Paul; Neth, Edward; Robinson, William et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 17 Kinetics, §17.6 Reaction Mechanisms, pp. 822–827 (elementary steps, molecularity, intermediates and the rate-determining step; page to pin). [^boyd-elementary]: Boyd, W. Christopher (2025). *Exploring Inorganic and Organometallic Chemistry*. Chapter 10 Kinetics, p. 280 (orders may be taken from the stoichiometry only for an elementary step). [^af-firstorder]: Flowers, Paul; Neth, Edward; Robinson, William et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 17 Kinetics, §17.4 Integrated Rate Laws, pp. 806, 813–814 (first-order integrated law and half-life t½ = 0.693/k). ## External links - *Exploring Inorganic and Organometallic Chemistry* (2025), Portal Book 052: https://open.umn.edu/opentextbooks/textbooks/exploring-inorganic-and-organometallic-chemistry - *Chemistry: Atoms First 2e* (OpenStax, 2019), Portal Book 051: https://open.umn.edu/opentextbooks/textbooks/chemistry-atoms-first - The Wikipedia pair's External links section lists the pair's own links. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Steady_state_(chemistry).json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Steady state (chemistry)* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Steady_state_(chemistry).html" data-title="Steady state (chemistry)"></div> *Built from `MICROSIM_GUIDE/specs/sims/Steady_state_(chemistry).json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Steady_state_(chemistry)) : [Wikitube](https://en.wikitube.io/wiki/Steady_state_(chemistry)) · pinned revision [1361832957](https://en.wikipedia.org/w/index.php?oldid=1361832957) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. 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