Second law of thermodynamics

• 1. Introduction

  ›

  The second law of thermodynamics is easiest to grasp through a falling cup. Picture a cup tipping off a table and breaking on the floor. The first law of thermodynamics, which says energy is neither created nor destroyed, would also permit the reverse. The fragments would gather themselves and jump back onto the table. The second law allows the fall and denies the return.

  That one asymmetry comes from universal observation of heat and energy. Heat flows on its own from hotter regions to colder ones. And not all heat can be converted into work in a cyclic process.

  How did engineers and physicists turn those observations into a law? What is entropy, and why does it keep rising? Can anything, including living things, seem to run against the current?

  23205

• 2. Carnot's Engines

  ›

  In 1824, the French scientist Sadi Carnot studied how heat moves through steam engines, and that analysis became the historical seed of the second law. He worked when the caloric theory, which pictured heat as a fluid, dominated thinking. The first law had not yet been recognised, and entropy had no mathematical form.

  His central device was an ideal engine, run so slowly that heat and work passed only between parts that stayed in their own internal equilibrium. It is now called the Carnot engine. It marks the theoretical maximum efficiency for any engine working between two heat reservoirs at different temperatures.

  Carnot's own book framed the idea in terms of caloric. He wrote that "The motive power of heat is independent of the agents employed to realize it". In his picture, a steam engine did not use up caloric. It carried caloric from a warm body to a cold one, and power came from that transfer. Modern physics restates the point this way: the efficiency of a reversible Carnot cycle depends only on the two reservoir temperatures, whatever substance does the work.

  Carnot's theorem, also dated 1824, ranks real engines against that ideal. Any irreversible engine between two reservoirs is less efficient than a Carnot engine between the same pair. All reversible engines between them match the Carnot engine exactly. Carnot also suspected that some caloric is lost rather than turned into mechanical work, so no real engine could reach the reversibility of his cycle.

  A German scientist, Rudolf Clausius, would soon rebuild the idea around a new quantity.

  212223247

• 3. Clausius and Entropy

  ›

  In 1850, the German scientist Rudolf Clausius began laying the foundation of the second law by examining how heat transfer relates to work. He drew on James Prescott Joule's work on the conservation of energy. His conclusion was that heat does not flow spontaneously from cold bodies to hot ones. That ran against the caloric theory popular at the time. His formulation was published in German in 1854: "Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time."

  A refrigerator shows the rule at work. Heat does move from cold to hot inside it, but only because an outside agent, the refrigeration system, forces it. Clausius meant the net transfer of energy as heat, not the small contributing flows in each direction.

  From this starting point, Clausius could infer Carnot's principle and define entropy, which he did in 1865. That year he stated what he called the "second fundamental theorem in the mechanical theory of heat". It involved an "equivalence-value" of all the uncompensated transformations in a cyclical process, and he later defined that value as entropy.

  In a presentation to the Philosophical Society of Zurich on the 24th of April 1865, Clausius ended on a line that became the best-known phrasing of the law: "The entropy of the universe tends to a maximum."

  That loose wording has caused trouble. It says "universe" and names no conditions such as open, closed, or isolated. So many people take it to cover every subject imaginable. That is not true. It is only a simplified version of a longer and more precise description. For an isolated system, entropy cannot decrease with time, and the equal sign applies only once the system has settled into equilibrium.

  Lord Kelvin had already offered his own wording in 1851.

  3114173070

• 4. Rival Formulations

  ›

  Lord Kelvin gave the second law in several wordings. One says that no system operating in a cycle can deliver net work to its surroundings while taking in energy as heat from a single thermal reservoir. The ban applies only to cycles. A system that does not return to its starting state could still draw net work from one reservoir.

  Textbooks, such as the one by ter Haar and Wergeland, usually speak of a joint Kelvin-Planck statement, also called the heat engine statement. It rests on a proposition from Max Planck. Planck said it is impossible to build an engine that works in a complete cycle and produces no effect except work and the cooling of a heat reservoir. He regarded this as a starting point for deriving the second law, not as the law itself.

  A short argument shows that the Clausius and Kelvin statements amount to the same thing. Suppose an engine broke Kelvin's rule by turning drained heat completely into work in a cycle. Pair it with a reversed Carnot engine. The combined pair would move heat from a cooler reservoir to a hotter one with no other result. That breaks Clausius's rule. The first law guarantees the energy bookkeeping, and a similar argument runs the other way.

  Planck's own preferred statement used entropy. He held that every natural process runs in the direction that raises the sum of the entropies of all participating bodies. For reversible processes in the limit, that sum stays unchanged. George Uhlenbeck and G. W. Ford offered a similar line for irreversible change, using two bodies brought into contact and equalising their temperatures.

  In 1909, Constantin Carathéodory rebuilt thermodynamics on a purely mathematical, axiomatic base. His principle says: "In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S." With it he introduced adiabatic accessibility and opened a subfield often called geometrical thermodynamics. Textbooks often treat his principle as equal to the Clausius or Kelvin-Planck versions, but it is not. To carry the whole second law, it must be supplemented by Planck's principle that isochoric work always raises the internal energy of a closed system that began in its own internal equilibrium.

  Max Planck wrote an important paper on the foundations of thermodynamics in 1926. In it he proposed that the internal energy of a closed system rises in an adiabatic process during which the volume stays constant. The wording mentions no heat, temperature or entropy, yet it implies the second law. Planck wrote: "The production of heat by friction is irreversible." A close relative says: "Frictional pressure never does positive work."

  In 1965, G. N. Hatsopoulos and Joseph Keenan published the textbook Principles of General Thermodynamics. It restated the law in terms of stable equilibrium states. When an isolated system runs a process after a series of internal constraints is removed, it reaches a unique stable equilibrium, whatever the order of removal. Their statement entails the Clausius, Kelvin-Planck and Carathéodory versions. It also gave a basis for extending the definition of entropy to non-equilibrium situations.

  Hatsopoulos and Keenan argued that a satisfactory definition of heat is unlikely without a prior statement of the second law. Work by Hatsopoulos, E. P. Gyftopoulos, E. Zanchini and G. P. Beretta developed their statement into a rigorous formulation. It covers equilibrium and nonequilibrium, classical, statistical and quantum thermodynamics.

  833373841

• 5. Counting Microstates

  ›

  In 1860, James Clerk Maxwell made the first mechanical argument that molecular collisions equalise temperatures and push gases toward equilibrium. Ludwig Boltzmann followed in 1872 with his H-theorem. It argued that collisions make gases drift over time toward the Maxwell-Boltzmann distribution.

  Boltzmann's approach also leans on the ergodic hypothesis. Over long periods, a system spends time in a region of possible microstates in proportion to that region's volume. Every accessible microstate is therefore equally probable in the long run.

  Statistical mechanics treats matter as atoms and molecules in constant motion. One set of positions and velocities for every particle is a microstate, and the system keeps changing its microstate. At equilibrium, each microstate is equally likely. From that single assumption, the second law follows in a statistical sense. It holds on average, with variations on the order of one over the square root of N, where N is the number of particles.

  In everyday macroscopic situations, N is so large that the chance of a violation is practically zero. A system with only a small number of particles is another matter. Its entropy and other thermodynamic quantities may deviate significantly from what the law predicts. Classical thermodynamic theory does not deal with such fluctuations.

  Clausius started a traditional doctrine that entropy can be understood as molecular disorder within a macroscopic system. That doctrine is now considered obsolescent.

  Loschmidt's paradox puts a catch in every derivation. Any derivation of the second law must assume that the system was uncorrelated at some time in the past, which allows a simple probabilistic treatment. This assumption is usually taken as a boundary condition. Simple probability arguments are applied only to the future, while other sources of information tell us the past had low entropy.

  178798085

• 6. Work and Order

  ›

  Before the second law was established, many inventors of perpetual motion machines tried to slip past the first law. They hoped to draw on the enormous internal energy of the environment. Such a device is called a perpetual motion machine of the second kind, and the second law declared it impossible.

  Three examples show how the law decides which processes may occur. In an isolated system, heat goes from higher to lower temperature, mechanical energy becomes thermal energy, and a solute moves from higher to lower concentration. None of these runs backward on its own. Each can run backward in a system that exchanges energy with its surroundings, provided the combined entropy change of system and surroundings is not negative.

  Heat pumps and refrigerators do this for heat. They need enough mechanical work, converted to thermal energy inside the system. A heat engine can turn thermal energy into mechanical work at a single temperature, provided it passes enough heat to a lower-temperature region in the surroundings. In active transport, a solute travels from low to high concentration when enough work is supplied. The work can come from a concentration gradient of a chemical such as ATP, or from an electrochemical gradient.

  Robotic manufacturing of vehicles in a factory offers a larger example. The machines need electrical work and instructions. Yet the finished vehicles are more complex, and less uniform with their surroundings, than the raw materials were. The instructions might be pre-coded or sent by remote operation, humans might play part of the robots' role, and the electricity might come from on-site storage or the local grid.

  One conceptual statement sorts processes into four categories, by whether entropy and uniformity with the environment go up or down. Category IV, low disorder and low uniformity, counteracts the second law's pull toward uniformity. It takes a source of work, or exergy, plus some form of instruction or intelligence. Water freezing when its surroundings fall below freezing is category III. Its entropy falls, but it moves toward uniformity with its surroundings, whose entropy rises by more. Refrigerating water in a warm environment is category IV. It needs designed equipment and instructions, not only work.

  Engineers have a practical form of the law. Take the surroundings as an unlimited reservoir at fixed temperature and pressure. Then a quantity called exergy, or availability, limits the useful work a subsystem can deliver. A designer can use it without tracking entropy change in a whole isolated system. The approach is widely used in engineering practice, environmental accounting and systems ecology.

  Chemists have their own form. At constant temperature and pressure, a spontaneous process must lower the Gibbs free energy. At constant temperature and volume, the Helmholtz free energy must fall. These changes can be calculated from tabulated enthalpies of formation and standard molar entropies. At chemical equilibrium, at constant temperature and pressure with no electrical work, the change in Gibbs free energy is exactly zero.

  5732133653

• 7. Living Things

  ›

  Erwin Schrödinger, in his book What is Life?, and Léon Brillouin are among the many authors who have studied the thermodynamics of living organisms. To a fair approximation, an animal's physical state cycles by the day and leaves it nearly unchanged. It takes in food, water and oxygen. As a result of metabolism, it gives out breakdown products and heat.

  Plants take in radiative energy from the sun, along with carbon dioxide and water, and give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. That too is a cycle. The sun is a high-temperature source, and its energy ends up radiated into space, a lower-temperature sink. The entropy of the surroundings increases, so animals and plants obey the second law.

  Growth and rising complexity do not oppose the law. The same goes for the ability to form correlations with the environment through adaptation and memory. Such abilities are akin to general results that follow from it. Under some definitions, an increase in entropy also brings an increase in complexity. For a finite system interacting with finite reservoirs, an increase in entropy is equivalent to an increase in correlations between system and reservoirs.

  Living organisms are open systems, because matter passes into and out of them. Thermodynamics of open systems is often treated as passages between states of equilibrium, or as flows under local thermodynamic equilibrium. The problem can be simplified further by assuming a steady state with unchanging flows. General principles of entropy production under such approximations remain a subject of ongoing research.

  91929328100

• 8. Arrow of Time

  ›

  CPT symmetry is observed in the fundamental laws of physics, yet the second law is not symmetric under reversal of time. There is no conflict, because the second law applies statistically, on time-asymmetric boundary conditions. It has been related to the difference between moving forwards and backwards in time. It has also been related to the causal arrow of time, the principle that cause precedes effect.

  Irreversibility, on this account, comes from the asymmetric character of thermodynamic operations, not from any microscopic property of the bodies involved. Such operations are macroscopic outside interventions imposed on the bodies, not derived from their internal properties. Reputed paradoxes arise from failing to recognise this.

  The Big Bang is thought to have supplied the very low-entropy initial conditions behind the second law. Statistically, those conditions were very special. Yet they were also simple, since the part of the universe that grew into the observable universe seems to have been extremely uniform. That looks paradoxical, because uniform conditions, such as mixed gases, usually have high entropy. Gravity resolves it. Gravitational systems have negative heat capacity, so a uniform gas has lower entropy than non-uniform arrangements such as black holes in empty space.

  Another approach holds that the early universe had high, even maximal, entropy for its size. It grew so fast that it fell out of equilibrium, and its entropy rose only slightly compared with the rise in the maximum possible. So it ended up very low compared with that larger maximum. As for why the initial conditions were such, one suggestion is that cosmological inflation wiped out non-smoothness. Another is that the universe was created spontaneously, by a mechanism implying low-entropy beginnings.

  Stars and gas giants show the odd gravitational behaviour directly. Most systems have positive heat capacity, so when energy flows from a hot object to a cold one, the source cools and the sink warms. Systems bound by their own gravity, such as stars, can have negative heat capacities. As they contract, their total energy and entropy decrease, but their internal temperature may rise. That matters for protostars and for gas giant planets such as Jupiter. Once the entropy of the black-body radiation they emit is counted, the total entropy still increases.

  Classical thermodynamics is idealised. It assumes systems in their own internal states of thermodynamic equilibrium. Critical states are the exception, where critical opalescence shows to the naked eye, and studying them in the laboratory takes exceptionally long observation times. For non-equilibrium states there is no unqualified general definition of entropy. In looser terms, nothing in the entire universe is or has ever been in exact thermodynamic equilibrium.

  838485945