Entropy

Entropy


In physics, entropy, symbolized by S,( from the Greek μετατροπή (metatropi) meaning "transformation"),[3][4] is a measure of the unavailability of a system’s energy to do work.[5] Entropy is central to the second law of thermodynamics and the combined law of thermodynamics, which deal with physical processes and whether they occur spontaneously. Spontaneous changes, in isolated systems, occur with an increase in entropy. Spontaneous changes tend to smooth out differences in temperature, pressure, density, and chemical potential that may exist in a system, and entropy is thus a measure of how far this smoothing-out process has progressed. In short Entropy is a function of a quantity of heat which shows the possibility of conversion of that heat into work. The increase in entropy is small when heat is added at high temperature and is greater when heat is added at lower temperature. Thus for maximum entropy there is minium availability for conversion into work and for minimum entropy there is maximum avilability for conversin into work.

The concept of entropy was developed in the 1850s by German physicist Rudolf Clausius who described it as the transformation-content, i.e. dissipative energy use, of a thermodynamic system or working body of chemical species during a change of state.[3] In contrast, the first law of thermodynamics, formalized through the heat-friction experiments of James Joule in 1843, deals with the concept of energy, which is conserved in all processes; the first law, however, lacks in its ability to quantify the effects of friction and dissipation. Entropy change has often been defined as a change to a more disordered state at a molecular level. In recent years, entropy has been interpreted in terms of the "dispersal" of energy. Entropy is an extensive state function that accounts for the effects of irreversibility in thermodynamic systems.



Ice melting - a classic example of entropy increasing described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice.


Quantitatively, entropy is defined by the differential quantity dS = δQ / T, where δQ is the amount of heat absorbed in an isothermal and reversible process in which the system goes from one state to another, and T is the absolute temperature at which the process is occurring.[6] Entropy is one of the factors that determines the free energy of the system. This thermodynamic definition of entropy is only valid for a system in equilibrium (because temperature is defined only for a system in equilibrium), while the statistical definition of entropy (see below) applies to any system. Thus the statistical definition is usually considered the fundamental definition of entropy.

When a system's energy is defined as the sum of its "useful" energy, (e.g. that used to push a piston), and its "useless energy", i.e. that energy which cannot be used for external work, then entropy may be (most concretely) visualized as the "scrap" or "useless" energy whose energetic prevalence over the total energy of a system is directly proportional to the absolute temperature of the considered system. (Note the product "TS" in the Gibbs free energy or Helmholtz free energy relations).

In terms of statistical mechanics, the entropy describes the number of the possible microscopic configurations of the system. The statistical definition of entropy is the more fundamental definition, from which all other definitions and all properties of entropy follow. Although the concept of entropy was originally a thermodynamic construct, it has been adapted in other fields of study, including information theory, psychodynamics, thermoeconomics, and evolution.

History

The short history of entropy begins with the work of French mathematician Lazare Carnot who in his 1803 work Fundamental Principles of Equilibrium and Movement postulated that in any machine the accelerations and shocks of the moving parts all represent losses of moment of activity. In other words, in any natural process there exists an inherent tendency towards the dissipation of useful energy. Building on this work, in 1824 Lazare's son Sadi Carnot published Reflections on the Motive Power of Fire in which he set forth the view that in all heat-engines whenever "caloric", or what is now known as heat, falls through a temperature difference, that work or motive power can be produced from the actions of the "fall of caloric" between a hot and cold body. This was an early insight into the second law of thermodynamics.

Carnot based his views of heat partially on the early 18th century "Newtonian hypothesis" that both heat and light were types of indestructible forms of matter, which are attracted and repelled by other matter, and partially on recent 1789 views of Count Rumford who showed that heat could be created by friction as when cannon bores are machined.Accordingly, Carnot reasoned that if the body of the working substance, such as a body of steam, is brought back to its original state (temperature and pressure) at the end of a complete engine cycle, that "no change occurs in the condition of the working body." This latter comment was amended in his foot notes, and it was this comment that led to the development of entropy.



Rudolf Clausius - originator of the concept of "entropy".


In the 1850s and 60s, German physicist Rudolf Clausius gravely objected to this latter supposition, i.e. that no change occurs in the working body, and gave this "change" a mathematical interpretation by questioning the nature of the inherent loss of usable heat when work is done, e.g., heat produced by friction.This was in contrast to earlier views, based on the theories of Isaac Newton, that heat was an indestructible particle that had mass. Later, scientists such as Ludwig Boltzmann, Willard Gibbs, and James Clerk Maxwell gave entropy a statistical basis. Carathéodory linked entropy with a mathematical definition of irreversibility, in terms of trajectories and integrability.

Definitions and descriptions

In science, the term "entropy" is generally interpreted in three distinct, but semi-related, ways, i.e. from macroscopic viewpoint (classical thermodynamics), a microscopic viewpoint (statistical thermodynamics), and an information viewpoint (information theory). Entropy in information theory is a fundamentally different concept from thermodynamic entropy. However, at a philosophical level, some argue that thermodynamic entropy can be interpreted as an application of the information entropy concept to a highly specific set of physical questions.

The statistical definition of entropy (see below) is the fundamental definition because the other two can be mathematically derived from it, but not vice versa. All properties of entropy (including second law of thermodynamics) follow from this definition.

Entropy in chemical thermodynamics

Thermodynamic entropy is central in chemical thermodynamics, enabling changes to be quantified and the outcome of reactions predicted. The second law of thermodynamics states that entropy in the combination of a system and its surroundings (or in an isolated system by itself) increases during all spontaneous chemical and physical processes. Spontaneity in chemistry means “by itself, or without any outside influence”, and has nothing to do with speed. The Clausius equation of δqrev/T = ΔS introduces the measurement of entropy change, ΔS. Entropy change describes the direction and quantitates the magnitude of simple changes such as heat transfer between systems – always from hotter to cooler spontaneously.Thus, when a mole of substance at 0 K is warmed by its surroundings to 298 K, the sum of the incremental values of qrev/T constitute each element's or compound's standard molar entropy, a fundamental physical property and an indicator of the amount of energy stored by a substance at 298 K.[14][15] Entropy change also measures the mixing of substances as a summation of their relative quantities in the final mixture.

Entropy is equally essential in predicting the extent of complex chemical reactions, i.e. whether a process will go as written or proceed in the opposite direction. For such applications, ΔS must be incorporated in an expression that includes both the system and its surroundings, Δ Suniverse = ΔSsurroundings + Δ S system. This expression becomes, via some steps, the Gibbs free energy equation for reactants and products in the system: Δ G [the Gibbs free energy change of the system] = Δ H [the enthalpy change] – T Δ S [the entropy change].

The second law

An important law of physics, the second law of thermodynamics, states that the total entropy of any isolated thermodynamic system tends to increase over time, approaching a maximum value; and so, by implication, the entropy of the universe (i.e. the system and its surroundings), assumed as an isolated system, tends to increase. Two important consequences are that heat cannot of itself pass from a colder to a hotter body: i.e., it is impossible to transfer heat from a cold to a hot reservoir without at the same time converting a certain amount of work to heat. It is also impossible for any device that can operate on a cycle to receive heat from a single reservoir and produce a net amount of work; it can only get useful work out of the heat if heat is at the same time transferred from a hot to a cold reservoir. This means that there is no possibility of a "perpetual motion" which is isolated. Also, from this it follows that a reduction in the increase of entropy in a specified process, such as a chemical reaction, means that it is energetically more efficient.

In general, according to the second law, the entropy of a system that is not isolated may decrease. An air conditioner, for example, cools the air in a room, thus reducing the entropy of the air. The heat, however, involved in operating the air conditioner always makes a bigger contribution to the entropy of the environment than the decrease of the entropy of the air. Thus the total entropy of the room and the environment increases, in agreement with the second law.



During steady-state continuous operation, an entropy balance applied to an open system accounts for system entropy changes related to heat flow and mass flow across the system boundary.


Entropy in quantum mechanics (von Neumann entropy)

In quantum statistical mechanics, the concept of entropy was developed by John von Neumann and is generally referred to as "von Neumann entropy". Von Neumann established the correct mathematical framework for quantum mechanics with his work Mathematische Grundlagen der Quantenmechanik. He provided in this work a theory of measurement, where the usual notion of wave collapse is described as an irreversible process (the so called von Neumann or projective measurement). Using this concept, in conjunction with the density matrix he extended the classical concept of entropy into the quantum domain.

It is well known that a Shannon based definition of information entropy leads in the classical case to the Boltzmann entropy. It is tempting to regard the Von Neumann entropy as the corresponding quantum mechanical definition. But the latter is problematic from quantum information point of view. Consequently Stotland, Pomeransky, Bachmat and Cohen have introduced a new definition of entropy that reflects the inherent uncertainty of quantum mechanical states. This definition allows to distinguish between the minimum uncertainty entropy of pure states, and the excess statistical entropy of mixtures.

Standard textbook definitions

Entropy – energy broken down in irretrievable heat.
Boltzmann's constant times the logarithm of a multiplicity; where the multiplicity of a macrostate is the number of microstates that correspond to the macrostate.
the number of ways of arranging things in a system (times the Boltzmann's constant).
a non-conserved thermodynamic state function, measured in terms of the number of microstates a system can assume, which corresponds to a degradation in usable energy.
a direct measure of the randomness of a system.
a measure of energy dispersal at a specific temperature.
a measure of the partial loss of the ability of a system to perform work due to the effects of irreversibility.
an index of the tendency of a system towards spontaneous change.
a measure of the unavailability of a system’s energy to do work; also a measure of disorder; the higher the entropy the greater the disorder.
a parameter representing the state of disorder of a system at the atomic, ionic, or molecular level.
a measure of disorder in the universe or of the availability of the energy in a system to do work.

Energy dispersal

The concept of entropy can be described qualitatively as a measure of energy dispersal at a specific temperature.Similar terms have been in use from early in the history of classical thermodynamics, and with the development of statistical thermodynamics and quantum theory, entropy changes have been described in terms of the mixing or "spreading" of the total energy of each constituent of a system over its particular quantized energy levels.

Ambiguities in the terms disorder and chaos, which usually have meanings directly opposed to equilibrium, contribute to widespread confusion and hamper comprehension of entropy for most students.As the second law of thermodynamics shows, in an isolated system internal portions at different temperatures will tend to adjust to a single uniform temperature and thus produce equilibrium. A recently developed educational approach avoids ambiguous terms and describes such spreading out of energy as dispersal, which leads to loss of the differentials required for work even though the total energy remains constant in accordance with the first law of thermodynamics.Physical chemist Peter Atkins, for example, who previously wrote of dispersal leading to a disordered state, now writes that "spontaneous changes are always accompanied by a dispersal of energy", and has discarded 'disorder' as a description.

Ice melting example

The illustration for this article is a classic example in which entropy increases in a small 'universe', a thermodynamic system consisting of the 'surroundings' (the warm room) and 'system' (glass, ice, cold water). In this universe, some heat energy δQ from the warmer room surroundings (at 298 K or 25 C) will spread out to the cooler system of ice and water at its constant temperature T of 273 K (0 C), the melting temperature of ice. The entropy of the system will change by the amount dS = δQ/T, in this example δQ/273 K. (The heat δQ for this process is the energy required to change water from the solid state to the liquid state, and is called the enthalpy of fusion, i.e. the ΔH for ice fusion.) The entropy of the surroundings will change by an amount dS = -δQ/298 K. So in this example, the entropy of the system increases, whereas the entropy of the surroundings decreases.

It is important to realize that the decrease in the entropy of the surrounding room is less than the increase in the entropy of the ice and water: the room temperature of 298 K is larger than 273 K and therefore the ratio, (entropy change), of δQ/298 K for the surroundings is smaller than the ratio (entropy change), of δQ/273 K for the ice+water system. To find the entropy change of our 'universe', we add up the entropy changes for its constituents: the surrounding room, and the ice+water. The total entropy change is positive; this is always true in spontaneous events in a thermodynamic system and it shows the predictive importance of entropy: the final net entropy after such an event is always greater than was the initial entropy.

As the temperature of the cool water rises to that of the room and the room further cools imperceptibly, the sum of the δQ/T over the continuous range, at many increments, in the initially cool to finally warm water can be found by calculus. The entire miniature "universe", i.e. this thermodynamic system, has increased in entropy. Energy has spontaneously become more dispersed and spread out in that "universe" than when the glass of ice water was introduced and became a "system" within it.

Topics in entropy

Entropy and life

For over a century and a half, beginning with Clausius' 1863 memoir "On the Concentration of Rays of Heat and Light, and on the Limits of its Action", much writing and research has been devoted to the relationship between thermodynamic entropy and the evolution of life. The argument that life feeds on negative entropy or negentropy as put forth in the 1944 book What is Life? by physicist Erwin Schrödinger served as a further stimulus to this research. Recent writings have utilized the concept of Gibbs free energy to elaborate on this issue. Tangentially, some creationists have argued that entropy rules out evolution.

In the popular 1982 textbook Principles of Biochemistry by noted American biochemist Albert Lehninger, for example, it is argued that the order produced within cells as they grow and divide is more than compensated for by the disorder they create in their surroundings in the course of growth and division. In short, according to Lehninger, "living organisms preserve their internal order by taking from their surroundings free energy, in the form of nutrients or sunlight, and returning to their surroundings an equal amount of energy as heat and entropy."

Evolution related definitions:
Negentropy - a shorthand colloquial phrase for negative entropy.
Ectropy - a measure of the tendency of a dynamical system to do useful work and grow more organized.
Syntropy - a tendency towards order and symmetrical combinations and designs of ever more advantageous and orderly patterns.
Extropy – a metaphorical term defining the extent of a living or organizational system's intelligence, functional order, vitality, energy, life, experience, and capacity and drive for improvement and growth.
Ecological entropy - a measure of biodiversity in the study of biological ecology.

The arrow of time

Entropy is the only quantity in the physical sciences that "picks" a particular direction for time, sometimes called an arrow of time. As we go "forward" in time, the Second Law of Thermodynamics tells us that the entropy of an isolated system can only increase or remain the same; it cannot decrease. Hence, from one perspective, entropy measurement is thought of as a kind of clock.

Entropy and cosmology

We have previously mentioned that a finite universe may be considered an isolated system. As such, it may be subject to the Second Law of Thermodynamics, so that its total entropy is constantly increasing. It has been speculated that the universe is fated to a heat death in which all the energy ends up as a homogeneous distribution of thermal energy, so that no more work can be extracted from any source.

If the universe can be considered to have generally increasing entropy, then - as Roger Penrose has pointed out - gravity plays an important role in the increase because gravity causes dispersed matter to accumulate into stars, which collapse eventually into black holes. Jacob Bekenstein and Stephen Hawking have shown that black holes have the maximum possible entropy of any object of equal size. This makes them likely end points of all entropy-increasing processes, if they are totally effective matter and energy traps. Hawking has, however, recently changed his stance on this aspect.

The role of entropy in cosmology remains a controversial subject. Recent work has cast extensive doubt on the heat death hypothesis and the applicability of any simple thermodynamic model to the universe in general. Although entropy does increase in the model of an expanding universe, the maximum possible entropy rises much more rapidly - thus entropy density is decreasing with time. This results in an "entropy gap" pushing the system further away from equilibrium. Other complicating factors, such as the energy density of the vacuum and macroscopic quantum effects, are difficult to reconcile with thermodynamical models, making any predictions of large-scale thermodynamics extremely difficult.

Miscellaneous definitions
Entropy unit - a non-S.I. unit of thermodynamic entropy, usually denoted "e.u." and equal to one calorie per kelvin
Gibbs entropy - the usual statistical mechanical entropy of a thermodynamic system.
Boltzmann entropy - a type of Gibbs entropy, which neglects internal statistical correlations in the overall particle distribution.
Tsallis entropy - a generalization of the standard Boltzmann-Gibbs entropy.
Standard molar entropy - is the entropy content of one mole of substance, under conditions of standard temperature and pressure.
Black hole entropy - is the entropy carried by a black hole, which is proportional to the surface area of the black hole's event horizon.
Residual entropy - the entropy present after a substance is cooled arbitrarily close to absolute zero.
Entropy of mixing - the change in the entropy when two different chemical substances or components are mixed.
Loop entropy - is the entropy lost upon bringing together two residues of a polymer within a prescribed distance.
Conformational entropy - is the entropy associated with the physical arrangement of a polymer chain that assumes a compact or globular state in solution.
Entropic force - a microscopic force or reaction tendency related to system organization changes, molecular frictional considerations, and statistical variations.
Free entropy - an entropic thermodynamic potential analogous to the free energy.
Entropic explosion – an explosion in which the reactants undergo a large change in volume without releasing a large amount of heat.
Entropy change – a change in entropy dS between two equilibrium states is given by the heat transferred dQrev divided by the absolute temperature T of the system in this interval.
Sackur-Tetrode entropy - the entropy of a monatomic classical ideal gas determined via quantum considerations.

Other relations

Other mathematical definitions
Kolmogorov-Sinai entropy - a mathematical type of entropy in dynamical systems related to measures of partitions.
Topological entropy - a way of defining entropy in an iterated function map in ergodic theory.
Relative entropy - is a natural distance measure from a "true" probability distribution P to an arbitrary probability distribution Q.
Rényi entropy - a generalized entropy measure for fractal systems.

Sociological definitions

The concept of entropy has also entered the domain of sociology, generally as a metaphor for chaos, disorder or dissipation of energy, rather than as a direct measure of thermodynamic or information entropy:
Entropology – the study or discussion of entropy or the name sometimes given to thermodynamics without differential equations.
Psychological entropy - the distribution of energy in the psyche, which tends to seek equilibrium or balance among all the structures of the psyche.
Economic entropy – a semi-quantitative measure of the irrevocable dissipation and degradation of natural materials and available energy with respect to economic activity.
Social entropy – a measure of social system structure, having both theoretical and statistical interpretations, i.e. society (macrosocietal variables) measured in terms of how the individual functions in society (microsocietal variables); also related to social equilibrium.
Corporate entropy - energy waste as red tape and business team inefficiency, i.e. energy lost to waste.


Quotes

“ Any method involving the notion of entropy, the very existence of which depends on the second law of thermodynamics, will doubtless seem to many far-fetched, and may repel beginners as obscure and difficult of comprehension. ”

--Willard Gibbs, Graphical Methods in the Thermodynamics of Fluids (1873)

“ My greatest concern was what to call it. I thought of calling it ‘information’, but the word was overly used, so I decided to call it ‘uncertainty’. When I discussed it with John von Neumann, he had a better idea. Von Neumann told me, ‘You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have the advantage. ”

--Conversation between Claude Shannon and John von Neumann regarding what name to give to the “measure of uncertainty” or attenuation in phone-line signals (1949)