entropy

Entropy 1

Entropy

Ice melting in a warm room is a commonexample of increasing entropy,[1] described in1862 by Rudolf Clausius as an increase in the

disgregation of the molecules of ice.[2]

Entropy articles

Introduction

History

Classical

Statistical

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Sculpture dedicated to entropy at the Universidad de Monterrey, México

Entropy is a thermodynamic property that is ameasure of the energy not available for usefulwork in a thermodynamic process, such as inenergy conversion devices, engines, or machines.Such devices can only be driven by convertibleenergy, and have a theoretical maximumefficiency when converting energy to work.During this work entropy accumulates in thesystem, but has to be removed by dissipation inthe form of waste heat.

The concept of entropy is defined by the secondlaw of thermodynamics, which states that theentropy of a closed system always increases orremains constant. Thus, entropy is also measureof the tendency of a process, such as a chemicalreaction, to be entropically favored, or to proceedin a particular direction. It determines thatthermal energy always flows spontaneously fromregions of higher temperature to regions of lowertemperature, in the form of heat. These processesreduce the state of order of the initial systems,and therefore entropy is an expression of disorderor randomness. This model is the basis of themicroscopic interpretation of entropy in statisticalmechanics describing the probability of the constituents of a thermodynamic system to be occupying accessiblequantum mechanical states, a model directly related to the information entropy.

Thermodynamic entropy has the dimension of energy divided by temperature, and a unit of joules per kelvin (J/K) inthe International System of Units.The term entropy was coined in 1865 by Rudolf Clausius based on the Greek εντροπία [entropía], a turning toward,from εν- [en-] (in) and τροπή [tropē] (turn, conversion).[3] [4]

Thermodynamical and statistical descriptions

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

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

There are two related definitions of entropy: the thermodynamic definition and the statistical mechanics definition. The thermodynamic definition was developed in the early 1850s by Rudolf Clausius and essentially describes how to measure the entropy of an isolated system in thermodynamic equilibrium. Importantly, it makes no reference to the microscopic nature of matter. The statistical definition was developed by Ludwig Boltzmann in the 1870s by analyzing the statistical behavior of the microscopic components of the system. Boltzmann went on to show that this definition of entropy was equivalent to the thermodynamic entropy to within a constant number which has since been known as Boltzmann's constant. In summary, the thermodynamic definition of entropy provides the experimental definition of entropy, while the statistical definition of entropy extends the concept, providing an explanation and a

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deeper understanding of its nature.Thermodynamic entropy is a non-conserved state function that is of great importance in the sciences of physics andchemistry.[5] [6] Historically, the concept of entropy evolved in order to explain why some processes are spontaneousand others are not; systems tend to progress in the direction of increasing entropy.[7] Entropy is as such a function ofa system's tendency towards spontaneous change.[7] [8] For isolated systems, entropy never decreases.[6] This fact hasseveral important consequences in science: first, it prohibits "perpetual motion" machines; and second, it suggests anarrow of time. Increases in entropy correspond to irreversible changes in a system, because some energy must beexpended as waste heat, limiting the amount of work a system can do.[5] [9] [9] [10] [11] [12]

In statistical mechanics, entropy is essentially a measure of the number of ways in which a system may be arranged,often taken to be a measure of "disorder" (the higher the entropy, the higher the disorder).[5] [10] [11] [13] [14] [15] [16]

Specifically, this definition describes the entropy as being proportional to the logarithm of the number of possiblemicroscopic configurations of the individual atoms and molecules of the system (microstates) which could give riseto the observed macroscopic state (macrostate) of the system. The constant of proportionality is the Boltzmannconstant.

The second law of thermodynamicsThe second law of thermodynamics states that in general the total entropy of any system will not decrease other thanby increasing the entropy of some other system. Hence, in a system isolated from its environment, the entropy of thatsystem will tend not to decrease. It follows that heat will not flow from a colder body to a hotter body without theapplication of work (the imposition of order) to the colder body. Secondly, it is impossible for any device operatingon a cycle to produce net work from a single temperature reservoir; the production of net work requires flow of heatfrom a hotter reservoir to a colder reservoir. As a result, there is no possibility of a perpetual motion system. Finally,it follows that a reduction in the increase of entropy in a specified process, such as a chemical reaction, means that itis energetically more efficient.It follows from the second law of thermodynamics that the entropy of a system that is not isolated may decrease. Anair conditioner, for example, may cool the air in a room, thus reducing the entropy of the air of that system. The heatexpelled from the room (the system), involved in the operation of the air conditioner, will always make a biggercontribution to the entropy of the environment than will the decrease of the entropy of the air of that system. Thus,the total of entropy of the room plus the entropy of the environment increases, in agreement with the second law ofthermodynamics.In mechanics, the second law in conjunction with the fundamental thermodynamic relation places limits on asystem's ability to do useful work.[17] The entropy change of a system at temperature T absorbing an infinitesimalamount of heat in a reversible way, is given by . More explicitly, an energy TRS is not available to do useful

work, where TR is the temperature of the coldest accessible reservoir or heat sink external to the system. For furtherdiscussion, see Exergy.Statistical mechanics demonstrates that entropy is governed by probability, thus allowing for a decrease in disordereven in a closed system. Although this is possible, such an event has a small probability of occurring, making itunlikely. Even if such event were to occur, it would result in a transient decrease that would affect only a limitednumber of particles in the system.[18]

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Definitions and descriptionsThermodynamic entropy is more generally defined from a statistical thermodynamics viewpoint, in which themolecular nature of matter is explicitly considered. Alternatively entropy can be defined from a classicalthermodynamics viewpoint, in which the molecular interactions are not considered and instead the system is viewedfrom perspective of the gross motion of very large masses of molecules and the behavior of individual molecules isaveraged and obscured. Historically, the classical thermodynamics definition developed first, and it has morerecently been extended in the area of non-equilibrium thermodynamics.

Statistical thermodynamicsThe interpretation of entropy in statistical mechanics is the measure of uncertainty, or mixedupness in the phrase ofGibbs, which remains about a system after its observable macroscopic properties, such as temperature, pressure andvolume, have been taken into account. For a given set of macroscopic variables, the entropy measures the degree towhich the probability of the system is spread out over different possible microstates. In contrast to the macrostate,which characterizes plainly observable average quantities, a microstate specifies all molecular details about thesystem including the position and velocity of every molecule. The more such states available to the system withappreciable probability, the greater the entropy.More specifically, entropy is a logarithmic measure of the density of states:

where kB is the Boltzmann constant, equal to 1.38065 × 10−23 J K−1. The summation is over all the microstates thesystem can be in, and Pi is the probability that the system is in the ith microstate. [19] For almost all practicalpurposes, this can be taken as the fundamental definition of entropy since all other formulas for S can bemathematically derived from it, but not vice versa. (In some rare and recondite situations, a generalization of thisformula may be needed to account for quantum coherence effects, but in any situation where a classical notion ofprobability makes sense, the above is the entropy.)In what has been called the most famous equation of statistical thermodynamics, the entropy of a system in which allstates, of number Ω, are equally likely, is given by

In thermodynamics, such a system is one in which the volume, number of molecules, and internal energy are fixed(the microcanonical ensemble).In essence, the most general interpretation of entropy is as a measure of our uncertainty about a system. Theequilibrium state of a system maximizes the entropy because we have lost all information about the initial conditionsexcept for the conserved variables; maximizing the entropy maximizes our ignorance about the details of thesystem.[20] This uncertainty is not of the everyday subjective kind, but rather the uncertainty inherent to theexperimental method and interpretative model.The interpretative model has a central role in determining entropy. The qualifier "for a given set of macroscopicvariables" above has very deep implications: if two observers use different sets of macroscopic variables, then theywill observe different entropies. For example, if observer A uses the variables U, V and W, and observer B uses U,V, W, X, then, by changing X, observer B can cause an effect that looks like a violation of the second law ofthermodynamics to observer A. In other words: the set of macroscopic variables one chooses must includeeverything that may change in the experiment, otherwise one might see decreasing entropy![21]

In general, entropy can be defined for any Markov processes with reversible dynamics and the detailed balanceproperty.In Boltzmann's 1896 Lectures on Gas Theory, he showed that this expression gives a measure of entropy for systemsof atoms and molecules in the gas phase, thus providing a measure for the entropy of classical thermodynamics.

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Classical thermodynamicsAccording to the Clausius equality, for a reversible process

That means the line integral is path independent.

So we can define a state function S called entropy, which satisfied

With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolutevalue, we need Third Law of Thermodynamics, which states that S=0 at absolute zero for perfect crystals.From a macroscopic perspective, in classical thermodynamics the entropy is interpreted as a state function of athermodynamic system: that is, a property depending only on the current state of the system, independent of how thatstate came to be achieved. The state function has the important property that, when multiplied by a referencetemperature, it can be understood as a measure of the amount of energy in a physical system that cannot be used todo thermodynamic work; i.e., work mediated by thermal energy. More precisely, in any process where the systemgives up energy ΔE, and its entropy falls by ΔS, a quantity at least TR ΔS of that energy must be given up to thesystem's surroundings as unusable heat (TR is the temperature of the system's external surroundings). Otherwise theprocess will not go forward. In classical thermodynamics, the entropy of a system is defined only if it is inthermodynamic equilibrium.In a thermodynamic system, pressure, density, and temperature tend to become uniform over time because thisequilibrium state has higher probability (more possible combinations of microstates) than any other; see statisticalmechanics. In the ice melting example, the difference in temperature between a warm room (the surroundings) andcold glass of ice and water (the system and not part of the room), begins to be equalized as portions of the thermalenergy from the warm surroundings spread to the cooler system of ice and water.

A thermodynamic system

Over time the temperature of the glass and its contents and thetemperature of the room become equal. The entropy of the room hasdecreased as some of its energy has been dispersed to the ice andwater. However, as calculated in the example, the entropy of thesystem of ice and water has increased more than the entropy of thesurrounding room has decreased. In an isolated system such as theroom and ice water taken together, the dispersal of energy fromwarmer to cooler always results in a net increase in entropy. Thus,when the "universe" of the room and ice water system has reached atemperature equilibrium, the entropy change from the initial state is ata maximum. The entropy of the thermodynamic system is a measure ofhow far the equalization has progressed.

A special case of entropy increase, the entropy of mixing, occurs when two or more different substances are mixed.If the substances are at the same temperature and pressure, there will be no net exchange of heat or work - theentropy change will be entirely due to the mixing of the different substances. At a statistical mechanical level, thisresults due to the change in available volume per particle with mixing.[22]

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History

Rudolf Clausius, originator of the concept ofentropy

The first law of thermodynamics, formalized based on the heat-frictionexperiments of James Joule in 1843, deals with the concept of energy,which is conserved in all processes; the first law, however, lacks in itsability to quantify the effects of friction and dissipation.

Entropy began with the work of French mathematician Lazare Carnotwho in his 1803 paper Fundamental Principles of Equilibrium andMovement proposed that in any machine the accelerations and shocksof the moving parts all represent losses of moment of activity. In otherwords, in any natural process there exists an inherent tendency towardsthe dissipation of useful energy. Building on this work, in 1824Lazare's son Sadi Carnot published Reflections on the Motive Power ofFire in which he set forth the view that in all heat-engines whenever"caloric", or what is now known as heat, falls through a temperaturedifference, that work or motive power can be produced from theactions of the "fall of caloric" between a hot and cold body. This wasan early insight into the second law of thermodynamics.[23]

Carnot based his views of heat partially on the early 18th century "Newtonian hypothesis" that both heat and lightwere types of indestructible forms of matter, which are attracted and repelled by other matter, and partially on thecontemporary views of Count Rumford who showed in 1789 that heat could be created by friction as when cannonbores are machined.[24] Accordingly, Carnot reasoned that if the body of the working substance, such as a body ofsteam, is brought back to its original state (temperature and pressure) at the end of a complete engine cycle, that "nochange occurs in the condition of the working body". This latter comment was amended in his foot notes, and it wasthis comment that led to the development of entropy.

In the 1850s and 1860s, German physicist Rudolf Clausius gravely objected to this latter supposition, i.e. that nochange occurs in the working body, and gave this "change" a mathematical interpretation by questioning the natureof the inherent loss of usable heat when work is done, e.g. heat produced by friction.[25] Clausius described entropyas the transformation-content, i.e. dissipative energy use, of a thermodynamic system or working body of chemicalspecies during a change of state.[25] This was in contrast to earlier views, based on the theories of Isaac Newton, thatheat was an indestructible particle that had mass.Later, scientists such as Ludwig Boltzmann, Josiah Willard Gibbs, and James Clerk Maxwell gave entropy astatistical basis. In 1877, Boltzmann visualized a probabilistic way to measure the entropy of an ensemble of idealgas particles, in which he defined entropy to be proportional to the logarithm of the number of microstates such a gascould occupy. Henceforth, the essential problem in statistical thermodynamics, i.e. according to Erwin Schrödinger,has been to determine the distribution of a given amount of energy E over N identical systems. Carathéodory linkedentropy with a mathematical definition of irreversibility, in terms of trajectories and integrability.

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Consequences and applications

The arrow of timeEntropy is the only quantity in the physical sciences that seems to imply a particular direction of progress, sometimescalled an arrow of time. As time progresses, the second law of thermodynamics states that the entropy of an isolatedsystem never decreases. Hence, from this perspective, entropy measurement is thought of as a kind of clock.

The fundamental thermodynamic relationThe entropy of a system depends on its internal energy and the external parameters, such as the volume. In thethermodynamic limit this fact leads to an equation relating the change in the internal energy to changes in theentropy and the external parameters. This relation is known as the fundamental thermodynamic relation. If thevolume is the only external parameter, this relation is:

Since the internal energy is fixed when one specifies the entropy and the volume, this relation is valid even if thechange from one state of thermal equilibrium to another with infinitesimally larger entropy and volume happens in anon-quasistatic way (so during this change the system may be very far out of thermal equilibrium and then theentropy, pressure and temperature may not exist).The fundamental thermodynamic relation implies many thermodynamic identities that are valid in general,independent of the microscopic details of the system. Important examples are the Maxwell relations and the relationsbetween heat capacities.

Entropy in chemical thermodynamicsThermodynamic entropy is central in chemical thermodynamics, enabling changes to be quantified and the outcomeof reactions predicted. The second law of thermodynamics states that entropy in an isolated system, the combinationof a subsystem under study and its surroundings, increases during all spontaneous chemical and physical processes.The Clausius equation of δqrev/T = ΔS introduces the measurement of entropy change, ΔS. Entropy change describesthe direction and quantifies the magnitude of simple changes such as heat transfer between systems – always fromhotter to cooler spontaneously.[26]

The thermodynamic entropy therefore has the dimension of energy divided by temperature, and the unit joule perkelvin (J/K) in the International System of Units (SI).Thermodynamic entropy is an extensive property, meaning that it scales with the size or extent of a system. In manyprocesses it is useful to specify the entropy as an intensive property independent of the size, as a specific entropycharacteristic of the type of system studied. Specific entropy may be expressed relative to a unit of mass, typicallythe kilogram (unit: Jkg-1K-1). Alternatively, in chemistry, it is also referred to one mole of substance, in which case itis called the molar entropy with a unit of Jmol-1K-1.Thus, when one mole of substance at 0K is warmed by its surroundings to 298K, the sum of the incremental valuesof qrev/T constitute each element's or compound's standard molar entropy, a fundamental physical property and anindicator of the amount of energy stored by a substance at 298K.[27] [28] Entropy change also measures the mixing ofsubstances as a summation of their relative quantities in the final mixture.[29]

Entropy is equally essential in predicting the extent and direction of complex chemical reactions. For suchapplications, ΔS must be incorporated in an expression that includes both the system and its surroundings, ΔSuniverse= ΔSsurroundings + ΔS system. This expression becomes, via some steps, the Gibbs free energy equation for reactantsand products in the system: ΔG [the Gibbs free energy change of the system] = ΔH [the enthalpy change] −T ΔS [theentropy change].[27]

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Entropy changeWhen an ideal gas undergoes a change, its entropy may also change. For cases where the specific heat doesn't changeand either volume, pressure or temperature is also constant, the change in entropy can be easily calculated.[30]

When specific heat and volume are constant, the change in entropy is given by:

.

When specific heat and pressure are constant, the change in entropy is given by:

.

When specific heat and temperature are constant, the change in entropy is given by:

.

In these equations is the specific heat at constant volume, is the specific heat at constant pressure, is theideal gas constant, and is the number of moles of gas.For some other transformations, not all of these properties (specific heat, volume, pressure or temperature) areconstant. In these cases, for only 1 mole of an ideal gas, the change in entropy can be given by[31] either:

or

.

Entropy balance equation for open systemsIn chemical engineering, the principles of thermodynamics are commonly applied to "open systems", i.e. those inwhich heat, work, and mass flow across the system boundary. In a system in which there are flows of both heat ( ) and work, i.e. (shaft work) and P(dV/dt) (pressure-volume work), across the system boundaries, the heat flow,but not the work flow, causes a change in the entropy of the system. This rate of entropy change is where Tis the absolute thermodynamic temperature of the system at the point of the heat flow. If, in addition, there are massflows across the system boundaries, the total entropy of the system will also change due to this convected flow.

During steady-state continuous operation, an entropy balance applied to an opensystem accounts for system entropy changes related to heat flow and mass flow

across the system boundary.

To derive a generalized entropy balancedequation, we start with the general balanceequation for the change in any extensivequantity Θ in a thermodynamic system, aquantity that may be either conserved, suchas energy, or non-conserved, such asentropy. The basic generic balanceexpression states that dΘ/dt, i.e. the rate ofchange of Θ in the system, equals the rate atwhich Θ enters the system at the boundaries,minus the rate at which Θ leaves the systemacross the system boundaries, plus the rateat which Θ is generated within the system.Using this generic balance equation, withrespect to the rate of change with time of the

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extensive quantity entropy S, the entropy balance equation for an open thermodynamic system is:[32]

where

= the net rate of entropy flow due to the flows of mass into and out of the system (where =

entropy per unit mass).

= the rate of entropy flow due to the flow of heat across the system boundary.

= the rate of internal generation of entropy within the system.

Note, also, that if there are multiple heat flows, the term is to be replaced by where is the

heat flow and is the temperature at the jth heat flow port into the system.

Entropy in quantum mechanics (von Neumann entropy)

“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 thefirst place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, andmore 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 inphone-line signals [33]

In quantum statistical mechanics, the concept of entropy was developed by John von Neumann and is generallyreferred to as "von Neumann entropy", namely .where is the density matrix and Tr is the trace operator.This upholds the correspondence principle, because in the classical limit, i.e. whenever the classical notion ofprobability applies, this expression is equivalent to the familiar classical definition of entropy,

Von Neumann established a rigorous mathematical framework for quantum mechanics with his work MathematischeGrundlagen der Quantenmechanik. He provided in this work a theory of measurement, where the usual notion ofwave function collapse is described as an irreversible process (the so called von Neumann or projectivemeasurement). Using this concept, in conjunction with the density matrix he extended the classical concept ofentropy into the quantum domain.It is well known that a Shannon based definition of information entropy leads in the classical case to the Boltzmannentropy. It is tempting to regard the Von Neumann entropy as the corresponding quantum mechanical definition. Butthe latter is problematic from quantum information point of view. Consequently Stotland, Pomeransky, Bachmat andCohen have introduced a new definition of entropy that reflects the inherent uncertainty of quantum mechanicalstates. This definition allows to distinguish between the minimum uncertainty entropy of pure states, and the excessstatistical entropy of mixtures.[34]

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Approaches to understanding entropy

Order and disorderEntropy has often been loosely associated with the amount of order, disorder, and/or chaos in a thermodynamicsystem. The traditional qualitative description of entropy is that it refers to changes in the status quo of the systemand is a measure of "molecular disorder" and the amount of wasted energy in a dynamical energy transformationfrom one state or form to another.[35] In this direction, a number of authors, in recent years, have derived exactentropy formulas to account for and measure disorder and order in atomic and molecular assemblies.[36] [37] [38] [39]

One of the simpler entropy order/disorder formulas is that derived in 1984 by thermodynamic physicist PeterLandsberg, which is based on a combination of thermodynamics and information theory arguments. Landsbergargues that when constraints operate on a system, such that it is prevented from entering one or more of its possibleor permitted states, as contrasted with its forbidden states, the measure of the total amount of “disorder” in the systemis given by the following expression:[38] [39]

Similarly, the total amount of "order" in the system is given by:

In which CD is the "disorder" capacity of the system, which is the entropy of the parts contained in the permittedensemble, CI is the "information" capacity of the system, an expression similar to Shannon's channel capacity, andCO is the "order" capacity of the system.[37]

Energy dispersalThe concept of entropy can be described qualitatively as a measure of energy dispersal at a specific temperature.[40]

Similar terms have been in use from early in the history of classical thermodynamics, and with the development ofstatistical 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.[41] As the second lawof thermodynamics shows, in an isolated system internal portions at different temperatures will tend to adjust to asingle uniform temperature and thus produce equilibrium. A recently developed educational approach avoidsambiguous terms and describes such spreading out of energy as dispersal, which leads to loss of the differentialsrequired for work even though the total energy remains constant in accordance with the first law ofthermodynamics[42] (compare discussion in next section). Physical chemist Peter Atkins, for example, whopreviously wrote of dispersal leading to a disordered state, now writes that "spontaneous changes are alwaysaccompanied by a dispersal of energy".[26] [43]

Relating entropy to energy usefulness

Following on from the above, it is possible (in a thermal context) to regard entropy as an indicator or measure of theeffectiveness or usefulness of a particular quantity of energy.[44] This is because energy supplied at a hightemperature (i.e. with low entropy) tends to be more useful than the same amount of energy available at roomtemperature. Mixing a hot parcel of a fluid with a cold one produces a parcel of intermediate temperature, in whichthe overall increase in entropy represents a “loss” which can never be replaced.Thus, the fact that the entropy of the universe is steadily increasing, means that its total energy is becoming lessuseful: eventually, this will lead to the "heat death of the Universe".

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Ice melting exampleThe illustration for this article is a classic example in which entropy increases in a small "universe", athermodynamic system consisting of the "surroundings" (the warm room) and "system" (glass, ice, cold water). Inthis universe, some thermal energy δQ from the warmer room surroundings (at 298 K or 25 °C) will spread out to thecooler system of ice and water at its constant temperature T of 273 K (0 °C), the melting temperature of ice. Theentropy of the system will change by the amount dS = δQ/T, in this example δQ/273 K. (The thermal energy δQ forthis process is the energy required to change water from the solid state to the liquid state, and is called the enthalpyof fusion, i.e. the ΔH for ice fusion.) The entropy of the surroundings will change by an amount dS = −δQ/298 K. Soin 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 theentropy of the ice and water: the room temperature of 298 K is larger than 273 K and therefore the ratio, (entropychange), of δQ/298 K for the surroundings is smaller than the ratio (entropy change), of δQ/273 K for the ice+watersystem. To find the entropy change of our "universe", we add up the entropy changes for its constituents: thesurrounding room and the ice+water. The total entropy change is positive; this is always true in spontaneous eventsin a thermodynamic system and it shows the predictive importance of entropy: the final net entropy after such anevent 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 ofthe δQ/T over the continuous range, at many increments, in the initially cool to finally warm water can be found bycalculus. The entire miniature "universe", i.e. this thermodynamic system, has increased in entropy. Energy hasspontaneously become more dispersed and spread out in that "universe" than when the glass of ice water wasintroduced and became a "system" within it.Notice that the system will reach a point where the room, the glass and the contents of the glass will be at the sametemperature. In this situation, nothing else can happen: although thermal energy does exist in the room (in fact, theamount of thermal energy is the same as in the beginning, since it is a closed system), it is now unable to do usefulwork, as there is no longer a temperature gradient. Unless an external event intervenes (thus breaking the definitionof a closed system), the room is destined to remain in the same condition for all eternity. Therefore, following thesame reasoning but considering the whole universe as our "room", we reach a similar conclusion: that, at a certainpoint in the distant future, the whole universe will be a uniform, isothermic and inert body of matter, in which therewill be no available energy to do work. This condition is known as the "heat death of the Universe".

Entropy and adiabatic accessibilityA definition of entropy based entirely on the relation of adiabatic accessibility between equilibrium states was givenby E.H.Lieb and J. Yngvason in 1999.[45] This approach has several predecessors, including the pioneering work ofConstantin Carathéodory from 1909 [46] and the monograph by R. Giles from 1964.[47] In the setting of Lieb andYngvason one starts by picking, for a unit amount of the substance under consideration, two reference states and

such that the latter is adiabatically accessible from the former but not vice versa. Defining the entropies of thereference states to be 0 and 1 respectively the entropy of a state is defined as the largest number such that is adiabatically accessible from a composite state consisting of an amount in the state and a complementaryamount, , in the state . A simple but important result within this setting is that entropy is uniquelydetermined, apart from a choice of unit and an additive constant for each chemical element, by the followingproperties: It is monotonic with respect to the relation of adiabatic accessibility, additive on composite systems, andextensive under scaling.

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Standard textbook definitionsThe following is a list of additional definitions of entropy from a collection of textbooks.• a measure of energy dispersal at a specific temperature.[26]

• a measure of disorder in the universe or of the availability of the energy in a system to do work.[48]

Interdisciplinary applications of entropyAlthough the concept of entropy was originally a thermodynamic construct, it has been adapted in other fields ofstudy, including information theory, psychodynamics, thermoeconomics, and evolution.[37] [49] [50]

Thermodynamic and statistical mechanics concepts• Entropy unit - a non-S.I. unit of thermodynamic entropy, usually denoted "e.u." and equal to one calorie per

Kelvin per mole, or 4.184 Joules per Kelvin per mole.[51]

• 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.• 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.[52]

• Sackur-Tetrode entropy - the entropy of a monatomic classical ideal gas determined via quantumconsiderations.

Entropy and lifeFor nearly a century and a half, beginning with Clausius' 1863 memoir "On the Concentration of Rays of Heat andLight, and on the Limits of its Action", much writing and research has been devoted to the relationship betweenthermodynamic entropy and the evolution of life. The argument that life feeds on negative entropy or negentropy asasserted in the 1944 book What is Life? by physicist Erwin Schrödinger served as a further stimulus to this research.Recent writings have used the concept of Gibbs free energy to elaborate on this issue.[53]

In the 1982 textbook Principles of Biochemistry by American biochemist Albert Lehninger, for example, it is arguedthat the "order" produced within cells as they grow and divide is more than compensated for by the "disorder" theycreate in their surroundings in the course of growth and division. In short, according to Lehninger, "living organismspreserve their internal order by taking from their surroundings free energy, in the form of nutrients or sunlight, andreturning to their surroundings an equal amount of energy as heat and entropy."[54]

Evolution-related concepts:

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• Negentropy - a shorthand colloquial phrase for negative entropy.[55]

• Ectropy - a measure of the tendency of a dynamical system to do useful work and grow more organized.[35]

• Extropy – a metaphorical term defining the extent of a living or organizational system's intelligence, functionalorder, vitality, energy, life, experience, and capacity and drive for improvement and growth.

• Ecological entropy - a measure of biodiversity in the study of biological ecology.In a study titled “Natural selection for least action” published in the Proceedings of The Royal Society A., Ville Kailaand Arto Annila of the University of Helsinki describe how the second law of thermodynamics can be written as anequation of motion to describe evolution, showing how natural selection and the principle of least action can beconnected by expressing natural selection in terms of chemical thermodynamics. In this view, evolution explorespossible paths to level differences in energy densities and so increase entropy most rapidly. Thus, an organism servesas an energy transfer mechanism, and beneficial mutations allow successive organisms to transfer more energywithin their environment.[56]

CosmologySince a finite universe is an isolated system then, by the Second Law of Thermodynamics, its total entropy isconstantly increasing. It has been speculated, since the 19th century, that the universe is fated to a heat death inwhich all the energy ends up as a homogeneous distribution of thermal energy, so that no more work can beextracted from any source.If the universe can be considered to have generally increasing entropy, then—as Roger Penrose has pointedout—gravity plays an important role in the increase because gravity causes dispersed matter to accumulate into stars,which collapse eventually into black holes. The entropy of a black hole is proportional to the surface area of theblack hole's event horizon.[57] Jacob Bekenstein and Stephen Hawking have shown that black holes have themaximum possible entropy of any object of equal size. This makes them likely end points of all entropy-increasingprocesses, if they are totally effective matter and energy traps. Hawking has, however, recently changed his stanceon this aspect.The role of entropy in cosmology remains a controversial subject. Recent work has cast some doubt on the heatdeath hypothesis and the applicability of any simple thermodynamic model to the universe in general. Althoughentropy does increase in the model of an expanding universe, the maximum possible entropy rises much morerapidly, moving the universe further from the heat death with time, not closer. This results in an "entropy gap"pushing the system further away from the posited heat death equilibrium.[58] Other complicating factors, such as theenergy density of the vacuum and macroscopic quantum effects, are difficult to reconcile with thermodynamicalmodels, making any predictions of large-scale thermodynamics extremely difficult.The entropy gap is widely believed to have been originally opened up by the early rapid exponential expansion of theuniverse.

Information theoryIn information theory, entropy is the measure of the amount of information that is missing before reception and issometimes referred to as Shannon entropy.[59] Shannon entropy is a broad and general concept which findsapplications in information theory as well as thermodynamics. It was originally devised by Claude Shannon in 1948to study the amount of information in a transmitted message. The definition of the information entropy is, however,quite general, and is expressed in terms of a discrete set of probabilities :

In the case of transmitted messages, these probabilities were the probabilities that a particular message was actually transmitted, and the entropy of the message system was a measure of how much information was in the message. For the case of equal probabilities (i.e. each message is equally probable), the Shannon entropy (in bits) is just the

Entropy 14

number of yes/no questions needed to determine the content of the message.[19]

The question of the link between information entropy and thermodynamic entropy is a debated topic. While mostauthors argue that there is a link between the two,[60] [61] [62] a few argue that they have nothing to do with eachother.[63] [19]

The expressions for the two entropies are very similar. The information entropy H for equal probabilitiesis

where k is a constant which determines the units of entropy. For example, if the units are bits, then k = 1/ln(2). Thethermodynamic entropy S, from a statistical mechanical point of view, was first expressed by Boltzmann:

where p is the probability of a system being in a particular microstate, given that it is in a particular macrostate, andis Boltzmann's constant. It can be seen that one may think of the thermodynamic entropy as Boltzmann's

constant, divided by log(2), times the number of yes/no questions that must be asked in order to determine themicrostate of the system, given that we know the macrostate. The link between thermodynamic and informationentropy was developed in a series of papers by Edwin Jaynes beginning in 1957.[64]

There are many ways of demonstrating the equivalence of "information entropy" and "physics entropy", that is, theequivalence of "Shannon entropy" and "Boltzmann entropy". Nevertheless, some authors argue for dropping theword entropy for the H function of information theory and using Shannon's other term "uncertainty" instead.[65]

Mathematics• Kolmogorov-Sinai entropy - a mathematical type of entropy in dynamical systems related to measures of

partitions.[19]

• Relative entropy - is a natural distance measure from a "true" probability distribution P to an arbitraryprobability distribution Q.

• Rényi entropy - a generalized entropy measure for fractal systems.• Topological entropy - a way of defining entropy in an iterated function map in ergodic theory.• Volume entropy - a Riemannian invariant measuring the exponential rate of volume growth.

SociologyThe concept of entropy has also entered the domain of sociology, generally as a metaphor for chaos, disorder ordissipation of energy, rather than as a direct measure of thermodynamic or information entropy:• Corporate entropy - energy waste as red tape and business team inefficiency, i.e. energy lost to waste.[66] (This

definition is comparable to von Clausewitz's concept of friction in war.)• Economic entropy – a semi-quantitative measure of the irrevocable dissipation and degradation of natural

materials and available energy with respect to economic activity.[61] [67]

• Entropology – the study or discussion of entropy or the name sometimes given to thermodynamics withoutdifferential equations.[68] [69]

• Psychological entropy - the distribution of energy in the psyche, which tends to seek equilibrium or balanceamong all the structures of the psyche.[70]

• 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 (microsocietalvariables); also related to social equilibrium.[71]

Entropy 15

Notes[1] In complex systems of molecules, such as at the critical point of water or when salt is added to an ice-water mixture, entropy can either

increase or decrease depending on system parameters, such as temperature and pressure. For example, if the spontaneous crystallization of asupercooled liquid takes place under adiabatic conditions the entropy of the resulting crystal will be greater than that of the supercooled liquid(Denbigh, K. (1982). The Principles of Chemical Equilibrium, 4th Ed.). In general, however, when ice melts, the entropy of the two adjoinedsystems, the hot and cold bodies, increases. Here are some further tutorials: Ice-melting (http:/ / jchemed. chem. wisc. edu/ JCESoft/ CCA/CCA3/ MAIN/ ENTROPY/ PAGE1. HTM) – JCE example; Ice-melting and Entropy Change (http:/ / www. bartleby. com/ 64/ C004/ 024.html) – example; Ice-melting and Entropy Change (http:/ / www. ac. wwu. edu/ ~vawter/ PhysicsNet/ Topics/ ThermLaw2/ Entropy/InterptEntropy. html) – discussions

[2] Clausius, Rudolf (1862). Communicated to the Naturforschende Gesellschaft of Zurich, January 27, 1862; published in the Vierteljahrschriftof this Society, vol. vii. P. 48; in Poggendorff’s Annalen, May 1862, vol. cxvi. p. 73; in the Philosophical Magazine, S. 4. vol. xxiv. pp. 81,201; and in the Journal des Mathematiques of Paris, S. 2. vol. vii. P. 209.

[3] "Entropy" (http:/ / www. etymonline. com/ index. php?term=entropy). Online Etymology Dictionary. . Retrieved 2008-08-05.[4] A machine in this context includes engineered devices as well as biological organisms.[5] McGraw-Hill Concise Encyclopedia of Chemistry, 2004[6] Sandler S. I., Chemical and Engineering Thermodynamics, 3rd Ed. Wiley, New York, 1999 p91[7] McQuarrie D. A., Simon J. D., Physical Chemistry: A Molecular Approach, University Science Books, Sausalito 1997 pp 817.[8] Haynie, Donald, T. (2001). Biological Thermodynamics. Cambridge University Press. ISBN 0-521-79165-0.[9] Cutnell, John, D.; Johnson, Kenneth, J. (1998). Physics, 4th edition. John Wiley and Sons, Inc.. ISBN 0-471-19113-2.[10] Sethna, J. Statistical Mechanics Oxford University Press 2006 p78[11] Oxford Dictionary of Science, 2005[12] de Rosnay, Joel (1979). The Macroscope – a New World View (written by an M.I.T.-trained biochemist). Harper & Row, Publishers.

ISBN 0-06-011029-5.[13] Baierlein, Ralph (2003). Thermal Physics. Cambridge University Press. ISBN 0-521-65838-1.[14] Schroeder, Daniel, V. (2000). Introduction to Thermal Physics. New York: Addison Wesley Longman. ISBN 0-201-38027-7.[15] Chang, Raymond (1998). Chemistry, 6th Ed.. New York: McGraw Hill. ISBN 0-07-115221-0.[16] Barnes & Noble's Essential Dictionary of Science, 2004[17] Daintith, John (2005). Oxford Dictionary of Physics. Oxford University Press. ISBN 0-19-280628-9.[18] "Entropy production theorems and some consequences," Physical Review E; Saha, Arnab; Lahiri, Sourabh; Jayannavar, A. M; The

American Physical Society: 14 July 2009, p.1-10 (http:/ / link. aps. org/ doi/ 10. 1103/ PhysRevE. 80. 011117)[19] Frigg, R. and Werndl, C. "Entropy - A Guide for the Perplexed" (http:/ / charlottewerndl. net/ Entropy_Guide. pdf). In Probabilities in

Physics; Beisbart C. and Hartmann, S. Eds; Oxford University Press, Oxford, 2010[20] EntropyOrderParametersComplexity.pdf (http:/ / www. physics. cornell. edu/ sethna/ StatMech/ EntropyOrderParametersComplexity. pdf)[21] Jaynes, E. T., "The Gibbs Paradox," In Maximum Entropy and Bayesian Methods; Smith, C. R; Erickson, G. J; Neudorfer, P. O., Eds;

Kluwer Academic: Dordrecht, 1992, p.1-22 (http:/ / www. mdpi. org/ lin/ entropy/ cgibbs. pdf)[22] Ben-Naim, Arieh, On the So-Called Gibbs Paradox, and on the Real Paradox, Entropy, 9, 132-136, 2007 Link (http:/ / www. mdpi. org/

entropy/ papers/ e9030132. pdf)[23] "Carnot, Sadi (1796-1832)" (http:/ / scienceworld. wolfram. com/ biography/ CarnotSadi. html). Wolfram Research. 2007. . Retrieved

2010-02-24.[24] McCulloch, Richard, S. (1876). Treatise on the Mechanical Theory of Heat and its Applications to the Steam-Engine, etc.. D. Van Nostrand.[25] Clausius, Rudolf (1850). On the Motive Power of Heat, and on the Laws which can be deduced from it for the Theory of Heat. Poggendorff's

Annalen der Physick, LXXIX (Dover Reprint). ISBN 0-486-59065-8.[26] Atkins, Peter; Julio De Paula (2006). Physical Chemistry, 8th edition. Oxford University Press. ISBN 0-19-870072-5.[27] Moore, J. W.; C. L. Stanistski, P. C. Jurs (2005). Chemistry, The Molecular Science,. Brooks Cole. ISBN 0-534-42201-2.[28] Jungermann, A.H. (2006). “Entropy and the Shelf Model: A Quantum Physical Approach to a Physical Property”. Journal of Chemical

Education 83: 1686-1694[29] Levine, I. N. (2002). Physical Chemistry, 5th edition. McGraw-Hill. ISBN 0-07-231808-2.[30] (http:/ / www. grc. nasa. gov/ WWW/ k-12/ Numbers/ Math/ Mathematical_Thinking/ ideal_gases_under_constant. htm)[31] (http:/ / www. grc. nasa. gov/ WWW/ K-12/ airplane/ entropy. html)[32] Sandler, Stanley, I. (1989). Chemical and Engineering Thermodynamics. John Wiley & Sons. ISBN 0-471-83050-X.[33] M. Tribus, E.C. McIrvine, Energy and information (http:/ / math. library. wisc. edu/ reserves/ proxy/ Math801/ energy. pdf), Scientific

American, 224 (September 1971), 178–184.[34] The information entropy of quantum mechanical states (http:/ / arxiv. org/ abs/ quant-ph/ 0401021), Europhysics Letters 67, 700 (2004)[35] Haddad, Wassim M.; Chellaboina, VijaySekhar; Nersesov, Sergey G. (2005). Thermodynamics - A Dynamical Systems Approach. Princeton

University Press. ISBN 0-691-12327-6.[36] Callen, Herbert, B (2001). Thermodynamics and an Introduction to Thermostatistics, 2nd Ed.. John Wiley and Sons. ISBN 0-471-86256-8.[37] Brooks, Daniel, R.; Wiley, E.O. (1988). Evolution as Entropy– Towards a Unified Theory of Biology. University of Chicago Press.

ISBN 0-226-07574-5.

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[38] Landsberg, P.T. (1984). “Is Equilibrium always an Entropy Maximum?” J. Stat. Physics 35: 159-69.[39] Landsberg, P.T. (1984). “Can Entropy and “Order” Increase Together?” Physics Letters 102A:171-173[40] Frank L. Lambert, A Student’s Approach to the Second Law and Entropy (http:/ / www. entropysite. com/ students_approach. html)[41] Carson, E. M. and J. R. Watson (Department of Educational and Professional Studies, Kings College, London), Undergraduate students'

understandings of entropy and Gibbs Free energy (http:/ / www. rsc. org/ pdf/ uchemed/ papers/ 2002/ p2_carson. pdf), University ChemistryEducation - 2002 Papers, Royal Society of Chemistry.

[42] Frank L. Lambert, JCE 2002 (79) 187 [Feb] Disorder—A Cracked Crutch for Supporting Entropy Discussions (http:/ / jchemed. chem. wisc.edu/ HS/ Journal/ Issues/ 2002/ Feb/ abs187. html)

[43] Atkins, Peter (1984). The Second Law. Scientific American Library. ISBN 0-7167-5004-X.[44] Sandra Saary (Head of Science, Latifa Girls’ School, Dubai) (23 February 1993). "Book Review of “A Science Miscellany”" (http:/ /

DLMcN. com/ entropy2. html). Khaleej Times (Galadari Press, UAE): XI. .[45] Elliott H. Lieb, Jakob Yngvason: The Physics and Mathematics of the Second Law of Thermodynamics (http:/ / de. arxiv. org/ abs/ cond-mat/

9708200), Phys. Rep. 310, 1-96 (1999)[46] Constantin Carathéodory: Untersuchungen über die Grundlagen der Thermodynamik, Math. Ann., 67:355–386, 1909[47] Robin Giles: Mathematical Foundations of Thermodynamics", Pergamon, Oxford 1964[48] Gribbin's Encyclopedia of Particle Physics, 2000[49] Avery, John (2003). Information Theory and Evolution. World Scientific. ISBN 981-238-399-9.[50] Yockey, Hubert, P. (2005). Information Theory, Evolution, and the Origin of Life.. Cambridge University Press. ISBN 0-521-80293-8.[51] IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006-) " Entropy unit (http:/

/ goldbook. iupac. org/ E02151. html)".[52] Serway, Raymond, A. (1992). Physics for Scientists and Engineers. Saunders Golden Subburst Series. ISBN 0-03-096026-6.[53] Higgs, P. G., & Pudritz, R. E. (2009). “A thermodynamic basis for prebiotic amino acid synthesis and the nature of the first genetic code"

Accepted for publication in Astrobiology (http:/ / adsabs. harvard. edu/ cgi-bin/ bib_query?arXiv:0904. 0402)[54] Lehninger, Albert (1993). Principles of Biochemistry, 2nd Ed.. Worth Publishers. ISBN 0-87901-711-2.[55] Schrödinger, Erwin (1944). What is Life - the Physical Aspect of the Living Cell. Cambridge University Press. ISBN 0-521-42708-8.[56] Lisa Zyga (2008-08-11). "Evolution as Described by the Second Law of Thermodynamics" (http:/ / www. physorg. com/ news137679868.

html). Physorg.com. . Retrieved 2008-08-14.[57] von Baeyer, Christian, H. (2003). Information - the New Language of Science. Harvard University Press. ISBN 0-674-01387-5.Srednicki M

(August 1993). "Entropy and area". Phys. Rev. Lett. 71 (5): 666–669. doi:10.1103/PhysRevLett.71.666. PMID 10055336. Callaway DJE(April 1996). "Surface tension, hydrophobicity, and black holes: The entropic connection". Phys Rev E Stat Phys Plasmas Fluids RelatInterdiscip Topics 53 (4): 3738–3744. doi:10.1103/PhysRevE.53.3738. PMID 9964684.

[58] Stenger, Victor J. (2007). God: The Failed Hypothesis. Prometheus Books. ISBN 159-102-481-1.[59] Balian, Roger (2003). Entropy – Protean Concept (http:/ / www-spht. cea. fr/ articles_k2/ t03/ 193/ publi. pdf) (PDF). Poincaré Seminar 2:

119-45.[60] Brillouin, Leon (1956). Science and Information Theory. name. ISBN 0-486-43918-6.[61] Georgescu-Roegen, Nicholas (1971). The Entropy Law and the Economic Process. Harvard University Press. ISBN 0-674-25781-2.[62] Chen, Jing (2005). The Physical Foundation of Economics - an Analytical Thermodynamic Theory. World Scientific. ISBN 981-256-323-7.[63] Lin, Shu-Kun. (1999). “ Diversity and Entropy (http:/ / www. mdpi. com/ 1099-4300/ 1/ 1/ 1).” Entropy (Journal), 1[1], 1-3.[64] "Edwin T. Jaynes - Bibliography" (http:/ / bayes. wustl. edu/ etj/ node1. html). Bayes.wustl.edu. 1998-03-02. . Retrieved 2009-12-06.[65] Schneider, Tom, DELILA system (Deoxyribonucleic acid Library Language), (Information Theory Analysis of binding sites), Laboratory of

Mathematical Biology, National Cancer Institute, FCRDC Bldg. 469. Rm 144, P.O. Box. B Frederick, MD 21702-1201, USA.[66] DeMarco, Tom; Lister, Timothy (1999). Peopleware: Productive Projects and Teams, 2nd. Ed.. Dorset House Publishing Co..

ISBN 0-932633-43-9.[67] Burley, Peter; Foster, John (1994). Economics and Thermodynamics – New Perspectives on Economic Analysis. Kluwer Academic

Publishers. ISBN 0-7923-9446-1.[68] Perrot, Pierre (1998). A to Z of Thermodynamics. Oxford University Press. ISBN 0-19-856552-6.[69] Example: "Entropology, not anthropology, should be the word for the discipline that devotes itself to the study of the process of

disintegration in its most evolved forms." (In A World on Wane, London, 1961, pg. 397; translated by John Russell of Tristes Tropiques byClaude Lévi-Strauss.)

[70] Hall, Calvin S; Nordby, Vernon J. (1999). A Primer of Jungian Psychology. New York: Meridian. ISBN 0-452-01186-8.[71] Bailey, Kenneth, D. (1990). Social Entropy Theory. State University of New York Press. ISBN 0-7914....

Entropy 17

References70. Benjamin Gal-Or, “Cosmology, Physics and Philosophy”, Springer Verlag, 1981, 1983, 1987, ISBN0-387-90581-2, ISBN 0387965262.

Further reading• Ben-Naim, Arieh (2007). Entropy Demystified. World Scientific. ISBN 981-270-055-2.• Dugdale, J. S. (1996). Entropy and its Physical Meaning (2nd ed.). Taylor and Francis (UK); CRC (US).

ISBN 0748405690.• Fermi, Enrico (1937). Thermodynamics. Prentice Hall. ISBN 0-486-60361-X.• Gyftopoulos, E.P.; G.P. Beretta (1991, 2005, 2010). Thermodynamics. Foundations and Applications. Dover.

ISBN 0-486-43932-1.• Kroemer, Herbert; Charles Kittel (1980). Thermal Physics (2nd ed.). W. H. Freeman Company.

ISBN 0-7167-1088-9.• Penrose, Roger (2005). The Road to Reality: A Complete Guide to the Laws of the Universe. New York: A.A.

Knopf. ISBN 0-679-45443-8.• Reif, F. (1965). Fundamentals of statistical and thermal physics. McGraw-Hill. ISBN 0-07-051800-9.• Benjamin Gal-Or (1981, 1983, 1987). Cosmology, Physics and Philosophy. Springer Verlag.

ISBN 0-387-90581-2.• Goldstein, Martin; Inge, F (1993). The Refrigerator and the Universe. Harvard University Press.

ISBN 0-674-75325-9.• vonBaeyer; Hans Christian (1998). Maxwell's Demon: Why Warmth Disperses and Time Passes. Random House.

ISBN 0-679-43342-2.• Entropy for beginners

External links• Entropy - A Basic Understanding (http:/ / www. spiraxsarco. com/ resources/ steam-engineering-tutorials/

steam-engineering-principles-and-heat-transfer/ entropy-a-basic-understanding. asp) A primer for entropy from achemical perspective

• Interactive Shockwave Animation on Entropy (http:/ / www. 7stones. com/ Homepage/ Publisher/ entropy. html)• Max Jammer (1973). Dictionary of the History of Ideas: Entropy (http:/ / etext. lib. virginia. edu/ cgi-local/ DHI/

dhi. cgi?id=dv2-12)• Frank L. Lambert; entropysite.oxy.edu (http:/ / entropysite. oxy. edu/ ) – links to articles including simple

introductions to entropy for chemistry students (http:/ / www. entropysite. com/ students_approach. html) and forgeneral readers (http:/ / www. entropysimple. com/ ).

• Thermodynamics (http:/ / www. lightandmatter. com/ html_books/ 0sn/ ch05/ ch05. html) - a chapter from anonline textbook

• Entropy (http:/ / www. physnet. org/ modules/ pdf_modules/ m160. pdf) on Project PHYSNET (http:/ / www.physnet. org)

• Entropy (http:/ / www. mdpi. com/ journal/ entropy/ ) - an Open Access journal• An Intuitive Guide to the Concept of Entropy Arising in Various Sectors of Science (http:/ / en. wikibooks. org/

wiki/ An_Intuitive_Guide_to_the_Concept_of_Entropy_Arising_in_Various_Sectors_of_Science) - a wikibookon the interpretation of the concept of entropy.

Article Sources and Contributors 18

Article Sources and ContributorsEntropy  Source: http://en.wikipedia.org/w/index.php?oldid=414051489  Contributors: 129.132.139.xxx, 12seda78, 478jjjz, 99of9, A Macedonian, AThing, Aaronbrick, Abjad, Abtract,Addihockey10, Adhanali, Adodge, Aintsemic, Ajaxkroon, Alpt, Alrasheedan, AltheaJ, AngelOfSadness, Anonymous Dissident, Anville, Apostolos Margaritis, Arcfrk, Arcturus, Arjun r acharya,Arthena, Artur adib, Astavats, Astrobayes, Attilios, Atuin, AugPi, Aunt Entropy, Avjoska, Awaterl, Awickert, BJNartowt, BW95, Ballchef, Ballhausflip, Bargebum, BassBone, Bbanerje, Bduke,BenFrantzDale, Benjah-bmm27, Betacommand, Billgdiaz, BlckKnght, Bo Jacoby, Bobby1011, Bobo192, Bojack727, Boob12, Branxton, Brisvegas, Bryan Derksen, C1t1v151on, CYD, CalmerWaters, Camarks, Can't sleep, clown will eat me, Captain panda, Carmichael95, Causticorulos, Cbuckley, Chaiken, ChemGardener, Chenyu, Chester Markel, Chjoaygame, ChrisGriswold,Chrislewis.au, Christofurio, Chuayw2000, Ciacco, Cmbreuel, Complexica, Connelly, Constatin666999, Conversion script, Count Iblis, Craig Pemberton, Crasshopper, DARTH SIDIOUS 2,DGG, DJ Creature, DL5MDA, DLMcN, DVD R W, Dartbanks, Darth Panda, Dave souza, David Shear, DavidCary, DeadEyeArrow, Debzer, Dftb, Dhollm, Dick Chu, DinDraithou, Disdero,Djr32, Doetoe, Dolphin51, Dougbateman, Dougluce, Dr. Ebola, Dr.K., Drat, Dreg743, Drestros power, Drpriver, Dtguelph, Dysprosia, E David Moyer, EdJohnston, Edgar181, Edkarpov,Edsanville, Edzevallos, Egg, El C, Eleassar777, ElectricRay, Ellwyz, Emote, EnSamulili, Engwar, Enormousdude, Epbr123, Eric Hawthorne, Esowteric, Everyking, Evil saltine, Favonian,FilipeS, Fimbulfamb, Fox, FrankLambert, Freakofnurture, Fred t hamster, Fredrik, Frelke, Fresheneesz, G716, GT5162, Gail, Gaius Cornelius, Galor612, Galoubet, Garethb1961, Gary King,Gatewayofintrigue, Gene Nygaard, Geoff Plourde, Geoking66, Georgette2, Gerardw, Geschichte, Gianluigi, Giftlite, Giraffedata, Glen, Gmaster108, Gogo Dodo, Gonfer, Graeme Bartlett,Graham87, GregAsche, Gtxfrance, GuidoGer, Gurch, Gurudev23, H Padleckas, HEL, HGHSTROJAN, Hadal, Haeleth, Hagedis, Haham hanuka, Hai2410, Haipa Doragon, Hamiltondaniel,Hanspi, HappyCamper, HappyVR, Happysailor, Hdt83, Headbomb, Henry Flower, Heoigi, Herbee, Heron, Hkyriazi, Hmains, Homestarmy, Ht686rg90, Humanoid, ILikeMIDI, IRP, Ianml,Icairns, Imaginaryoctopus, Intgr, Inwind, Iris lorain, Itub, Ixtli, Izno, J'raxis, [email protected], J.delanoy, JHoffmueller, JabberWok, Jacobko, Jacobolus, Jani, Jdpipe, Jeffrey Mall, Jheald,Jiang, Jim62sch, Jitse Niesen, Jni, JohnBonham69, Jonathan48, Jorgenumata, Joriki, Josemald, Jschissel, Jsd, Juliancolton, Jwanders, K, Kafziel, Kahriman, Kaihsu, Karol Langner, Katzmik,Kbh3rd, Kbrose, Keenan Pepper, KillerChihuahua, Kissnmakeup, Kjoonlee, Kmarinas86, Knowledge Seeker, KnowledgeOfSelf, Kpedersen1, Kurykh, Kwiki, Lakinekaki, Lambiam,Larryisgood, Lateg, Laurascudder, LeBofSportif, Lea phys, Leafyplant, Lee J Haywood, Lerdsuwa, LidiaFourdraine, Light current, Ligulem, Linas, Linshukun, Locke9k, Lone Isle, Loom91,Looxix, Lordloihi, Lotje, LoveMonkey, Lseixas, Lumos3, MECU, MER-C, Macvienna, Mad540trix, Maghemite, MagnaMopus, Mani1, Marathoner, MarnetteD, Marskell, MartinSpacek,Massieu, Master Jay, Maurice Carbonaro, Mausy5043, Mbeychok, Mdd, Mean Free Path, Melaen, Memming, Mennato, Metamagician3000, Mgiganteus1, Michael C Price, Michael Hardy,Miguel de Servet, Mike Christie, MilesTerrex, Mjs, Mouse is back, Mouvement, Moveaway00, Ms2ger, Mschlindwein, Mwilso24, Mxn, Naddy, Nakon, NawlinWiki, Nbarth, Necron909,Neligterink, Netheril96, Nihiltres, Nikhil Sanjay Bapat, Nobleness of Mind, Nonsuch, NotableException, Numbo3, Nwbeeson, Obradovic Goran, Oceans and oceans, Oenus, Oleg Alexandrov,Olivier, Omnichic82, Omnipaedista, Omnist, Oneismany, Opabinia regalis, Otheus, PAR, Pachyphytum, Paganpan, Paisley, Pasquale.Carelli, Passwordwas1234, Paul August, Paul venter,Pbroks13, Pedrose, Pentasyllabic, Peterlin, Phil Boswell, Philip Trueman, 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