# Irreversible process In [[Thermodynamics|thermodynamics]], an **irreversible process** is a change of state that cannot be undone by any sequence of steps that returns both the system *and* its surroundings exactly to where they began. The idealised alternative, a [[Reversible_process_(thermodynamics)|reversible process]], runs through a continuous chain of equilibrium states and can be reversed at any instant by an infinitesimal push. No real process is reversible, because every real process involves at least one of a short list of dissipative effects: [[Friction|friction]], unrestrained expansion, mixing, heat crossing a finite [[Temperature|temperature]] difference, electrical resistance, inelastic deformation and [[Chemical_reaction|chemical reaction]].[^yan-269] The [[Second_law_of_thermodynamics|second law]] converts that qualitative statement into a number: an irreversible process generates [[Entropy|entropy]], S_gen > 0, and the entropy it generates is the exact measure of the [[Work_(thermodynamics)|work]] the universe can no longer get out of the change. In the microsim below the reader sends a fixed quantity of [[Heat|heat]] Q through a wall separating a hot reservoir at T_H from a cold one at T_L, and controls the temperature drop across the wall, ΔT = T_H − T_L, from 0 to 300 K, together with T_H itself. The wall stores nothing, so the same Q leaves as arrives, but the entropy that leaves the hot side, Q/T_H, is smaller than the entropy that enters the cold side, Q/T_L. The difference is the entropy generated, S_gen = Q·(1/T_L − 1/T_H) ≥ 0, and at a dead-state temperature T₀ the work destroyed is W_lost = T₀·S_gen. The readouts give S_gen, W_lost, the [[Exergy|exergy]] of the heat itself, Ex = Q·(1 − T₀/T), and the second-law efficiency η_II = W/W_rev; a temperature–entropy rectangle shows the wedge of available work that the wall has eaten. Push ΔT to zero and the wedge closes: heat transfer becomes reversible only in the limit where it takes forever. On the [[Energy]] flagship this article is the child of Part V — Transformation, section *Irreversibility and lost work* (row E53), the framework root for the shared C08 embed that the [[Heat_engine|heat-engine]] and [[Thermodynamic_cycle|cycle]] sections reuse. ## Absolute versus statistical reversibility The equations that govern the microscopic constituents of matter are almost all symmetric under reversal of the time coordinate. Run a film of two colliding molecules backwards and you see another legal collision; the same is true of [[Maxwell's_equations|Maxwell's equations]] and, for the systems of interest here, of the [[Schrödinger_equation|Schrödinger equation]]. Irreversibility is therefore not written into the microscopic laws. It is a property of the *description*: a macroscopic state specified by a handful of numbers — [[Pressure|pressure]], temperature, composition — corresponds to an enormous number of microscopic arrangements, and the states we call "later" are simply the ones that can be realised in overwhelmingly more ways. Ludwig Boltzmann made that counting argument the definition of entropy in 1877, writing the entropy of a macrostate as proportional to the logarithm of the number of microstates compatible with it — the relation S = k_B·ln W that [[Max_Planck|Planck]] later cast in its familiar form.[^boltzmann1877] The paper was a direct answer to the objection, raised by Josef Loschmidt, that time-symmetric mechanics cannot produce a time-asymmetric law: Boltzmann's reply was that the reverse trajectory is not forbidden, only absurdly improbable. The numbers are not close. Let one mole of an ideal gas expand freely into twice its volume; the reverse, every molecule wandering back into the original half at the same instant, has probability 2^(−6.022×10²³), a number with more zeros than the observable universe has particles (derived). On that scale "never" and "almost never" are the same word. The distinction the section title draws is therefore between *absolute* reversibility, which the microscopic laws permit and which no macroscopic process exhibits, and *statistical* reversibility, which is a question of system size and observation time. Small systems observed briefly do run backwards: a colloidal bead in a laser trap, or a single [[Kinetic_theory_of_gases|gas molecule]] near a wall, spends measurable time in states of lower entropy than the one before. For the systems an engineer meets — a [[Steam_turbine|turbine]] stage, a wall, a [[Electric_battery|battery]] — the fluctuation terms are smaller than any instrument, and the second law is as firm as any law in physics. The practical content of "irreversible" is then not that reversal is impossible in principle but that reversal costs more work than the forward process delivered, and the excess is exactly T₀·S_gen.[^yan-ch6] ## History The subject begins with a question about machines. [[Nicolas_Léonard_Sadi_Carnot|Sadi Carnot]]'s *Réflexions sur la puissance motrice du feu* of 1824 asked what limits the work obtainable from a given flow of heat, and answered that the limit depends only on the two temperatures between which the engine works, not on the working substance — and that it is reached only by an engine in which every step can be run backwards.[^carnot1824] Carnot's argument is the first appearance of reversibility as a *criterion*: the best machine is the one whose every process is reversible, and every departure from that ideal is a loss. [[Rudolf_Clausius|Rudolf Clausius]] turned the criterion into an inequality. Writing the heat exchanged by a system around a closed cycle, he showed that ∮ δQ/T ≤ 0, with equality only if the cycle is reversible, and in 1865 named the state function whose differential is δQ_rev/T the entropy.[^clausius1865] The [[First_law_of_thermodynamics|first law]] says that the energy of the universe is constant; the second, in Clausius's formulation, that its entropy tends to a maximum. [[Lord_Kelvin|William Thomson]] had already stated the same content in the language of engineering in 1852, as a universal tendency in nature to the dissipation of mechanical energy: energy is never destroyed, but it is continually degraded into forms from which no work can be extracted.[^thomson1852] The modern engineering form of the subject splits the Clausius inequality into an equation. For any process, the entropy change of a closed system equals the entropy carried in by heat plus the entropy made inside it — ΔS = ∫ δQ/T + S_gen, with S_gen ≥ 0 and S_gen = 0 only for a reversible process. That decomposition is what makes irreversibility measurable rather than merely deplorable: S_gen is computed from the same property tables that give enthalpy and specific volume, and every component of a plant can be given its own entropy-generation budget.[^yan-ch6] The companion idea, that the work lost to irreversibility is T₀·S_gen, where T₀ is the temperature of the environment the plant actually rejects heat to, gives the loss in the units the owner cares about. [[Ilya_Prigogine|Ilya Prigogine]] received the 1977 Nobel Prize in Chemistry for carrying the analysis into systems held far from equilibrium, where entropy production is not a defect to be minimised but the price of maintaining structure.[^prigogine-nobel] ## Examples of irreversible processes Engineering thermodynamics works from a short, closed list. Claire Yu Yan's text enumerates the irreversibilities as friction, unrestrained expansion, mixing of two fluids, heat transfer across a finite temperature difference, electrical resistance, inelastic deformation of solids and chemical reaction.[^yan-269] Each is a mechanism by which organised energy becomes disorganised, and each can be priced in the same currency. ### Heat across a finite temperature difference This is the case the microsim runs, and the one that dominates real plant losses, because every [[Heat_exchanger|heat exchanger]], boiler tube and condenser works by holding a temperature difference across a wall. Send Q = 1.00 MJ from a reservoir at T_H = 800 K to one at T_L = 500 K. The hot side loses 1.00×10⁶/800 = 1,250 J/K; the cold side gains 1.00×10⁶/500 = 2,000 J/K; the wall keeps nothing, so S_gen = 750 J/K (derived). With the environment at T₀ = 298 K the destroyed work is W_lost = 298 × 750 = 223.5 kJ — more than a fifth of the heat, lost to nothing but a temperature drop. The same number appears from the other side of the ledger: the exergy of 1 MJ delivered at 800 K is 1.00×10⁶ × (1 − 298/800) = 627.5 kJ, and at 500 K it is 404.0 kJ, a fall of 223.5 kJ (derived). The two routes agree because they are the same statement. The sliders make the scaling visible. S_gen ∝ ΔT/(T_H·T_L) for small drops, so halving the temperature difference halves the entropy made, and the loss vanishes only as ΔT → 0 — which is why a reversible heat exchanger would need infinite area and infinite time. Real designs buy back exergy with surface: the manual's worked plant delivers 500 MW net from a cycle whose Carnot ceiling at T_H = 250 °C is 44.3 %, and rejects 627.5 MW of waste heat into a lake at 18 °C flowing at 20 m³/s, warming it by 7.50 °C; run the same duty on a real 35 % cycle and the rejected heat rises to 928.6 MW and the lake warms by 11.10 °C.[^yan-lake] The extra 4 °C in the lake *is* the irreversibility, made visible as [[Waste_heat|waste heat]]. ### Friction, mixing and unrestrained expansion Friction is the simplest case to price. Work W done against a frictional resistance at ambient temperature T₀ is wholly dissipated, so S_gen = W/T₀ and the lost work is W itself: none of it survives. [[Joule_heating|Resistive heating]] in a conductor is the electrical version of the same sentence, and inelastic deformation the mechanical one, which is why a hysteresis loop in a [[Stress–strain_curve|stress–strain curve]] has an area with units of energy per volume. Unrestrained expansion — the [[Joule_expansion|Joule expansion]], in which a gas is allowed into an evacuated vessel — takes no heat and does no work, so the [[Internal_energy|internal energy]] of an [[Ideal_gas_law|ideal gas]] is unchanged and its temperature does not move. Yet entropy is made: doubling the volume of one mole gives ΔS = R·ln 2 = 5.76 J/(mol·K) (derived), and at T₀ = 298 K that is 1.72 kJ of work thrown away per mole. Mixing is the same arithmetic applied to composition rather than volume; the [[Entropy_of_mixing|entropy of mixing]] two ideal gases is −R·Σ xᵢ·ln xᵢ per mole, 5.76 J/(mol·K) for an equimolar pair, and separating them again costs at least T₀ times that. The [[Diffusion|diffusion]] of a dye through water and the cooling of coffee are the domestic versions. ### Lost work and second-law efficiency The ratio that closes the accounting is the second-law efficiency, η_II = W/W_rev, the work actually delivered divided by the work a reversible device between the same states would deliver. It differs from the ordinary thermal efficiency in asking not "how much of the heat became work?" but "how much of what was *available* became work?" A steam plant taking heat at 250 °C has a Carnot ceiling of 44.3 % and, at a realistic 35 %, a second-law efficiency of 0.35/0.443 = 0.79 (derived): a fifth of the available work is destroyed inside the machine, and the rest of the gap to 100 % is not a defect but the [[Carnot_cycle|Carnot]] limit itself.[^yan-272] [[Heat_pump|Heat pumps]] show the same split. Yan's worked case delivers 20 kW into a house at 24 °C from outdoor air at 0 °C with a rated [[Coefficient_of_performance|coefficient of performance]] of 5.5, saving 16.354 kW of electricity against direct heating; the reversible COP between the same reservoirs is 12.38, so η_II = 5.5/12.38 = 0.44.[^yan-heatpump] Likharev's survey makes the same point about domestic equipment: a Carnot refrigerator working across 10 K near room temperature would have a COP near 30, while real air-conditioning manages 3 to 4.[^likharev-carnot] Whole economies can be audited this way. Thermal power stations reject two and a half to three times their electrical output as heat, which corresponds to a thermal efficiency of about 25 to 29 %;[^kerlin-185] a modern [[Nuclear_reactor|fission plant]] converting 2.5 GW of thermal power into 1 GW of electricity runs at 40 %.[^murphy-276] The missing 60 % is not an engineering failure to be scolded but a mixture of the Carnot ceiling and a measurable, itemisable pile of S_gen. ## Complex systems Classical thermodynamics compares equilibrium states and is silent about the path between them. Non-equilibrium thermodynamics keeps the entropy generation as its central object and asks how it is distributed. Lars Onsager showed in 1931 that near equilibrium each flux — a reaction rate, a [[Fick's_laws_of_diffusion|diffusion]] flux, a heat flow — is a linear combination of the thermodynamic forces, and that the matrix of coefficients is symmetric, so that a temperature gradient can drive a flow of matter exactly as strongly as a concentration gradient drives a flow of heat.[^onsager1931] Entropy production is then a sum of force–flux products, each term non-negative. Far from equilibrium the linear laws fail and something less obvious happens: a system held open by a steady throughput of energy can organise itself. Prigogine named the resulting patterns [[Dissipative_system|dissipative structures]] — [[Rayleigh–Bénard_convection|convection rolls]] in a heated fluid layer, oscillating reactions, flames — and their defining property is that they exist *because* of entropy production, not in spite of it.[^prigogine-nobel] A living cell is the same kind of object: it maintains a low-entropy internal state by exporting entropy to its surroundings faster than it makes it internally, and the export is paid for by [[Photosynthesis|photosynthesis]] or by [[Food_energy|food]]. Nothing in any of this repeals the second law. The entropy of the cell plus its surroundings still rises; irreversibility is the engine, and [[Self-organization|self-organisation]] is what it drives. The largest-scale version of the argument runs the other way. If the [[Expansion_of_the_universe|universe]] as a whole is treated as a closed system, its entropy rises monotonically toward a state in which no temperature difference anywhere is large enough to drive a machine — the [[Heat_death_of_the_universe|heat death]] that Thomson's 1852 paper already anticipated in outline.[^thomson1852] Between the cell and the cosmos sit the objects this flagship is about: [[Energy_transformation|energy conversions]] that are never free, whose price is always T₀·S_gen, and whose design is the art of keeping that product small. ## See also - [[Exergy]] - [[Reversible_process_(thermodynamics)]] - [[Thermodynamic_free_energy]] - [[Entropy_production]] - [[Second_law_of_thermodynamics]] - [[Heat_engine]] - [[Carnot_cycle]] - [[Waste_heat]] ## References [^yan-269]: Yan, Claire Yu (2022). *Introduction to Engineering Thermodynamics*. Chapter 6, "Entropy and the Second Law of Thermodynamics", p. 269 (the list of irreversibilities: friction, unrestrained expansion, mixing, heat transfer across a finite temperature difference, electrical resistance, inelastic deformation, chemical reaction). Portal Book 115, https://open.umn.edu/opentextbooks/textbooks/introduction-to-engineering-thermodynamics [^yan-ch6]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 6, "Entropy and the Second Law of Thermodynamics", pp. 239–348 (the Clausius inequality, entropy generation and isentropic efficiency; page to pin). The exergy forms W_lost = T₀·S_gen and Ex = Q·(1 − T₀/T) used on this page are the standard statements of that accounting; the book's equation displays were lost in extraction and standard forms were supplied. Portal Book 115. [^yan-272]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 6, p. 272 (η = 1 − Q_L/Q_H = 1 − T_L/T_H; an actual engine falls below the Carnot value and a device above it is "impossible") and p. 273 (large steam plants raise T_H with 300–600 °C steam). Portal Book 115. [^yan-lake]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 6, pp. 273–276 (the 500 MW plant rejecting into an 18 °C lake at 20 m³/s, with c_p = 4.181 kJ/(kg·K) and ΔT = Q_L/(ṁ·c_p) at p. 275). The printed results were lost in extraction; η = 44.3 %, Q_L = 627.5 MW and ΔT = 7.50 °C at T_H = 250 °C, and Q_L = 928.6 MW with ΔT = 11.10 °C for the actual 35 % cycle, are computed from the book's stated inputs. Portal Book 115. [^yan-heatpump]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 6, pp. 277–280 (COP_HP = T_H/(T_H − T_L); the 20 kW heat pump at 24 °C indoors and 0 °C outdoors with COP 5.5 "will save 16.354 kW"; heat pumps are "preferably used in mild winter conditions"). The reversible COP of 12.38 is computed. Portal Book 115. [^likharev-carnot]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics*. Chapter 1, "Review of Thermodynamics", pp. 22–26 (Q_H = Q_L + W; the Carnot cycle as a rectangle in the T–S plane; η = 1 − T_L/T_H as a bound on any engine; COP_c = T_L/(T_H − T_L) near 30 across a 10 K span against 3–4 for real equipment; the Nernst argument that absolute zero is unreachable in finitely many cycles). Portal Book 075, https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics [^kerlin-185]: Kerlin, Thomas (2013). *Future Energy: Opportunities & Challenges*, p. 185 (a thermal plant rejects 2.5–3 times its electrical output, corresponding to a thermal efficiency of about 25–29 %, computed). Portal Book 048, https://open.umn.edu/opentextbooks/textbooks/future-energy-opportunities-challenges [^murphy-276]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*, Chapter 6 "Alternative Energy", p. 276 (a 2.5 GW-thermal, 1 GW-electric fission plant runs at 40 %; fission plants generally at about one-third). The entropy and heat-engine treatment used here is Chapter 5 "Energy and Fossil Fuels", pp. 87–182 (page to pin). Portal Book 097, https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet [^carnot1824]: Carnot, Sadi (1824). *Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance*. Paris: Bachelier. [^clausius1865]: Clausius, R. (1865). "Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie." *Annalen der Physik und Chemie* 125 (7): 353–400. [^thomson1852]: Thomson, William (1852). "On a Universal Tendency in Nature to the Dissipation of Mechanical Energy." *Proceedings of the Royal Society of Edinburgh* 3: 139–142. [^boltzmann1877]: Boltzmann, Ludwig (1877). "Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respective den Sätzen über das Wärmegleichgewicht." *Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, Mathematisch-Naturwissenschaftliche Classe* 76: 373–435. The paper answers Loschmidt's reversibility objection by defining entropy through the number of microstates. [^onsager1931]: Onsager, L. (1931). "Reciprocal Relations in Irreversible Processes. I." *Physical Review* 37 (4): 405–426. https://doi.org/10.1103/PhysRev.37.405 [^prigogine-nobel]: Nobel Prize Outreach. "The Nobel Prize in Chemistry 1977 — Ilya Prigogine." https://www.nobelprize.org/prizes/chemistry/1977/summary/ <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Irreversible_process.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Irreversible process* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Irreversible_process.html" data-title="Irreversible process"></div> *Built from `MICROSIM_GUIDE/specs/sims/Irreversible_process.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/Irreversible_process) : [Wikitube](https://en.wikitube.io/wiki/Irreversible_process) · pinned revision [1373728133](https://en.wikipedia.org/w/index.php?oldid=1373728133) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. 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