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Encyclopedia > Entropy
Ice melting - a classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice.[2]

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.[4] 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. Rudolf Clausius - physicist and mathematician Rudolf Julius Emanuel Clausius (January 2, 1822 â€“ August 24, 1888), was a German physicist and mathematician. ... Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ... In thermodynamics, a thermodynamic system is defined as that part of the universe that is under consideration. ... Chemical species is a common, general name for atoms, molecules, molecular fragments and ions as entities being subjected to a chemical process or to a measurement. ... This article needs to be cleaned up to conform to a higher standard of quality. ... The first law of thermodynamics, a generalized expression of the law of the conservation of energy, states: // Description Essentially, the First Law of Thermodynamics declares that energy is conserved for a closed system, with heat and work being the forms of energy transfer. ... James Prescott Joule (December 24, 1818&#8211;October 11, 1889) was an English physicist, born in Salford, near Manchester. ... Look up conservation of energy in Wiktionary, the free dictionary. ... For other uses, see Friction (disambiguation). ... A wave that loses amplitude is said to dissipate. ... Boltzmanns molecules (1896) shown at a rest position in a solid In thermodynamics, entropy is often associated with the amount of order, disorder, and or chaos in a thermodynamic system. ... The thermodynamic concept of entropy can be described qualitatively as a measure of energy dispersal at a specific temperature. ... In physics and chemistry, an extensive quantity (also referred to as an extensive variable) is a physical quantity whose value is proportional to the size of the system it describes. ... In thermodynamics, a state function, or state quantity, is a property of a system that depends only on the current state of the system, not on the way in which the system got to that state. ... Irreversibility is that property of an event which makes reverting back to the state before the occurrence of the event impossible. ... Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ...

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.[5] 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. For other uses, see Heat (disambiguation) In physics, heat, symbolized by Q, is energy transferred from one body or system to another due to a difference in temperature. ... In thermodynamics, a reversible process (or reversible cycle if the process is cyclic) is a process that can be reversed by means of infinitesimal changes in some property of the system. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Absolute zero is the lowest temperature that can be obtained in any macroscopic system. ... The thermodynamic free energy is a measure of the amount of mechanical (or other) work that can be extracted from a system, and is helpful in engineering applications. ...

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 thermodynamics, thermodynamic work is the quantity of energy transferred from one system to another. ... Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ... In thermodynamics, the Gibbs free energy is a thermodynamic potential which measures the useful work obtainable from a closed thermodynamic system at a constant temperature and pressure. ... In thermodynamics, the Helmholtz free energy is a thermodynamic potential which measures the â€œusefulâ€ work obtainable from a closed thermodynamic system at a constant temperature. ...

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.[6][7][8] Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In statistical mechanics, a microstate describes a specific detailed microscopic configuration of a system, that the system visits in the course of its thermal fluctuations. ... Not to be confused with information technology, information science, or informatics. ... Sigmund Freud - the central founder of psychodynamics Psychodynamics is the application of the principles of thermodynamics to psychology. ... In the natural sciences, thermoeconomics is the physics of economic value. ... This article is about evolution in biology. ...

Rudolf Clausius - originator of the concept of "entropy".
Main article: History of entropy

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. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Rudolf Clausius - physicist and mathematician Rudolf Julius Emanuel Clausius (January 2, 1822 â€“ August 24, 1888), was a German physicist and mathematician. ... The history of entropy, essentially, is the development of ideas set forth to theoretically understand why a certain amount of functionable energy released from combustion reactions is always lost to dissipation or friction, i. ... Lazare Carnot Comte Lazare Nicolas Marguerite Carnot (May 13, 1753â€”August 2, 1823) was a French politician, engineer, and mathematician. ... Sadi Carnot in the dress uniform of a student of the Ã‰cole polytechnique Nicolas LÃ©onard Sadi Carnot (June 1, 1796 - August 24, 1832) was a French physicist and military engineer who gave the first successful theoretical account of heat engines, now known as the Carnot cycle, thereby laying the... The caloric theory is an obsolete scientific theory that heat consists of a fluid called caloric that flows from hotter to colder bodies. ... For other uses, see Heat (disambiguation) In physics, heat, symbolized by Q, is energy transferred from one body or system to another due to a difference in temperature. ... In thermodynamics, motive power is an agency, as water or steam, used to impart motion. ... The second law of thermodynamics is an expression of the universal law of increasing entropy. ...

Carnot[attribution needed] 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.[9] Accordingly, Carnot[attribution needed] 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. Benjamin Thompson. ... The Carnot cycle is a particular thermodynamic cycle, modeled on the Carnot heat engine, studied by Nicolas LÃ©onard Sadi Carnot in the 1820s and expanded upon by Benoit Paul Ã‰mile Clapeyron in the 1830s and 40s. ...

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.[4] 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. Rudolf Clausius - physicist and mathematician Rudolf Julius Emanuel Clausius (January 2, 1822 â€“ August 24, 1888), was a German physicist and mathematician. ... Sir Isaac Newton FRS (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Ludwig Eduard Boltzmann (Vienna, Austrian Empire, February 20, 1844 â€“ Duino near Trieste, September 5, 1906) was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics. ... Josiah Willard Gibbs (February 11, 1839 â€“ April 28, 1903) was an American mathematical physicist who contributed much of the theoretical foundation that led to the development of chemical thermodynamics and was one of the founders of vector analysis. ... James Clerk Maxwell (13 June 1831 â€“ 5 November 1879) was a Scottish mathematician and theoretical physicist from Edinburgh, Scotland, UK. His most significant achievement was aggregating a set of equations in electricity, magnetism and inductance â€” eponymously named Maxwells equations â€” including an important modification (extension) of the AmpÃ¨res... Constantin CarathÃ©odory (Greek: ÎšÏ‰Î½ÏƒÏ„Î±Î½Ï„Î¯Î½Î¿Ï‚ ÎšÎ±ÏÎ±Î¸ÎµÎ¿Î´Ï‰ÏÎ®Ï‚) (September 13, 1873 â€“ February 2, 1950) was a Greek mathematician of the Modern Era. ...

## 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. ass hole ... Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Not to be confused with information technology, information science, or informatics. ...

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. The second law of thermodynamics is an expression of the universal law of increasing entropy. ...

### Macroscopic viewpoint (classical thermodynamics)

Conjugate variables
of thermodynamics
Pressure Volume
(Stress) (Strain)
Temperature Entropy
Chem. potential Particle no.

Thermodynamic System

Over time the temperature of the glass and its contents and the temperature of the room become equal. The entropy of the room has decreased as some of its energy has been dispersed to the ice and water. However, as calculated in the example, the entropy of the system of ice and water has increased more than the entropy of the surrounding room has decreased. In an isolated system such as the room and ice water taken together, the dispersal of energy from warmer to cooler always results in a net increase in entropy. Thus, when the 'universe' of the room and ice water system has reached a temperature equilibrium, the entropy change from the initial state is at a maximum. The entropy of the thermodynamic system is a measure of how far the equalization has progressed. Image File history File links Download high resolution version (931x818, 88 KB) Summary Creator: Libb Thims (user:wavesmikey) Date Created: 11/29/05 URL: http://www. ... Image File history File links Download high resolution version (931x818, 88 KB) Summary Creator: Libb Thims (user:wavesmikey) Date Created: 11/29/05 URL: http://www. ... In thermodynamics, an isolated system, as contrasted with a closed system, is a physical system that does not interact with its surroundings. ... Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ...

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 - the entropy increase will be entirely due to the mixing of the different substances.[10] In thermodynamics the Entropy of mixing is the entropy change when two different gases or two solvents mix or even when a solute is dissolved in a solvent. ...

From a macroscopic perspective, in classical thermodynamics the entropy is interpreted simply as a state function of a thermodynamic system: that is, a property depending only on the current state of the system, independent of how that state came to be achieved. The state function has the important property that, when multiplied by a reference temperature, it can be understood as a measure of the amount of energy in a physical system that cannot be used to do thermodynamic work; i.e., work mediated by thermal energy. More precisely, in any process where the system gives up energy ΔE, and its entropy falls by ΔS, a quantity at least TR ΔS of that energy must be given up to the system's surroundings as unusable heat (TR is the temperature of the system's external surroundings). Otherwise the process will not go forward. ass hole ... In thermodynamics, a state function, or state quantity, is a property of a system that depends only on the current state of the system, not on the way in which the system got to that state. ... Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ... In thermodynamics, work is the quantity of energy transferred from one system to another without an accompanying transfer of entropy. ... For other uses, see Heat (disambiguation) In physics, heat, symbolized by Q, is energy transferred from one body or system to another due to a difference in temperature. ...

In 1862, Clausius stated what he calls the “theorem respecting the equivalence-values of the transformations” or what is now known as the second law of thermodynamics, as such: The second law of thermodynamics is an expression of the universal law of increasing entropy. ...

The algebraic sum of all the transformations occurring in a cyclical process can only be positive, or, as an extreme case, equal to nothing.

Quantitatively, Clausius states the mathematical expression for this theorem is as follows. Let δQ be an element of the heat given up by the body to any reservoir of heat during its own changes, heat which it may absorb from a reservoir being here reckoned as negative, and T the absolute temperature of the body at the moment of giving up this heat, then the equation: Absolute zero is the lowest temperature that can be obtained in any macroscopic system. ...

$int frac{delta Q}{T} = 0$

must be true for every reversible cyclical process, and the relation:

$int frac{delta Q}{T} ge 0$

must hold good for every cyclical process which is in any way possible. This is the essential formulation of the second law and one of the original forms of the concept of entropy. It can be seen that the dimensions of entropy are energy divided by temperature, which is the same as the dimensions of Boltzmann's constant (kB) and heat capacity. The SI unit of entropy is "joule per kelvin" (J•K−1). In this manner, the quantity "ΔS" is utilized as a type of internal energy, which accounts for the effects of irreversibility, in the energy balance equation for any given system. In the Gibbs free energy equation, i.e. ΔG = ΔH - TΔS, for example, which is a formula commonly utilized to determine if chemical reactions will occur, the energy related to entropy changes TΔS is subtracted from the "total" system energy ΔH to give the "free" energy ΔG of the system, as during a chemical process or as when a system changes state. The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Look up si, Si, SI in Wiktionary, the free dictionary. ... The joule (IPA: or ) (symbol: J) is the SI unit of energy. ... For other uses, see Kelvin (disambiguation). ... Irreversibility is that property of an event which makes reverting back to the state before the occurrence of the event impossible. ... In thermodynamics, the Gibbs free energy is a thermodynamic potential which measures the useful work obtainable from a closed thermodynamic system at a constant temperature and pressure. ... For other uses, see Chemical reaction (disambiguation). ... In a scientific sense, a chemical process is a method or means of somehow changing one or more chemicals or chemical compounds. ...

### Microscopic definition of entropy (statistical mechanics)

In statistical thermodynamics the entropy is defined as the number of microscopic configurations that result in the observed macroscopic description of the thermodynamic system: In thermodynamics, statistical entropy is the modeling of the energetic function entropy using probability theory. ... Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In statistical mechanics, a microstate describes a specific detailed microscopic configuration of a system, that the system visits in the course of its thermal fluctuations. ... Macroscopic is commonly used to describe physical objects that are measurable and observable by the naked eye. ...

$S = k_B ln Omega !$

where

kB is Boltzmann's constant 1.38066×10−23 J K−1 and
$Omega !$ is the number of microstates corresponding to the observed thermodynamic macrostate.

This definition is considered to be the fundamental definition of entropy (as all other definitions can be mathematically derived from it, but not vice versa). The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... In statistical mechanics, a microstate describes a specific detailed microscopic configuration of a system, that the system visits in the course of its thermal fluctuations. ...

In Boltzmann's 1896 Lectures on Gas Theory, he showed that this expression gives a measure of entropy for systems of atoms and molecules in the gas phase, thus providing a measure for the entropy of classical thermodynamics.

In 1877, thermodynamicist Ludwig Boltzmann visualized a probabilistic way to measure the entropy of an ensemble of ideal gas particles, in which he defined entropy to be proportional to the logarithm of the number of microstates such a gas could 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. Ludwig Eduard Boltzmann (Vienna, Austrian Empire, February 20, 1844 â€“ Duino near Trieste, September 5, 1906) was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics. ... An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces. ... Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... SchrÃ¶dinger in 1933, when he was awarded the Nobel Prize in Physics Bust of SchrÃ¶dinger, in the courtyard arcade of the main building, University of Vienna, Austria. ...

Statistical mechanics explains entropy as the amount of uncertainty (or "mixedupness" in the phrase of Gibbs) which remains about a system, after its observable macroscopic properties have been taken into account. For a given set of macroscopic quantities, like temperature and volume, the entropy measures the degree to which the probability of the system is spread out over different possible quantum states. The more states available to the system with higher probability, and thus the greater the entropy. In essence, the most general interpretation of entropy is as a measure of our ignorance about a system. The equilibrium state of a system maximizes the entropy because we have lost all information about the initial conditions except for the conserved quantities; maximizing the entropy maximizes our ignorance about the details of the system.[11] Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Josiah Willard Gibbs (February 11, 1839 &#8211; April 28, 1903) was an American physical chemist. ... (LTE is an acronym for the progressive-instrumental rock band  Liquid Tension Experiment) In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann distribution. ...

On the molecular scale, the two definitions match up because adding heat to a system, which increases its classical thermodynamic entropy, also increases the system's thermal fluctuations, so giving an increased lack of information about the exact microscopic state of the system, i.e. an increased statistical mechanical entropy. For other uses, see Temperature (disambiguation). ...

### 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.[12] 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.[13][14] Entropy change also measures the mixing of substances as a summation of their relative quantities in the final mixture.[15] Willard Gibbs - founder of chemical thermodynamics In thermodynamics, chemical thermodynamics is the mathematical study of the interrelation of heat and work with chemical reactions or with a physical change of state within the confines of the laws of thermodynamics. ... Willard Gibbs - founder of chemical thermodynamics In thermodynamics, chemical thermodynamics is the mathematical study of the interrelation of heat and work with chemical reactions or with a physical change of state within the confines of the laws of thermodynamics. ... The second law of thermodynamics is an expression of the universal law of increasing entropy. ... In thermodynamics, an isolated system, as contrasted with a closed system, is a physical system that does not interact with its surroundings. ... For other uses, see Heat (disambiguation) In physics, heat, symbolized by Q, is energy transferred from one body or system to another due to a difference in temperature. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ...

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].[13] In thermodynamics, the Gibbs free energy is a thermodynamic potential which measures the useful work obtainable from a closed thermodynamic system at a constant temperature and pressure. ... In thermodynamics, the Gibbs free energy is a thermodynamic potential which measures the useful work obtainable from a closed thermodynamic system at a constant temperature and pressure. ... t In thermodynamics and molecular chemistry, the enthalpy or heat content (denoted as H or Î”H, or rarely as Ï‡) is a quotient or description of thermodynamic potential of a system, which can be used to calculate the useful work obtainable from a closed thermodynamic system under constant pressure. ...

### 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. The second law of thermodynamics is an expression of the universal law of increasing entropy. ... For a list of set rules, see Laws of science. ... The second law of thermodynamics is an expression of the universal law of increasing entropy. ... This article or section should include material from Parallel Path See also Perpetuum mobile as a musical term Perpetual motion machines (the Latin term perpetuum mobile is not uncommon) are a class of hypothetical machines which would produce useful energy in a way science cannot explain (yet). ... For other uses, see Chemical reaction (disambiguation). ...

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. Note: in the broadest sense, air conditioning can refer to any form of heating, ventilation, and air-conditioning. ...

### Entropy balance equation for open systems

In chemical engineering, the principles of thermodynamics are commonly applied to "open systems", i.e. those in which heat, work, and mass flow across the system boundary. In a system in which there are flows of both heat ($dot{Q}$) and work, i.e. $dot{W}_S$ (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 $dot{Q}/T$, where T is the absolute thermodynamic temperature of the system at the point of the heat flow. If, in addition, there are mass flows across the system boundaries, the total entropy of the system will also change due to this convected flow. Chemical engineering is the branch of engineering that deals with the application of physical science (e. ... An open system may refer to more than one thing: In the physical sciences, an open system (system theory) is a system that matter or energy can flow into and/or out of, in contrast to a closed system, which no energy or matter may enter or leave. ... For other uses, see Heat (disambiguation) In physics, heat, symbolized by Q, is energy transferred from one body or system to another due to a difference in temperature. ... In thermodynamics, work is the quantity of energy transferred from one system to another without an accompanying transfer of entropy. ... For other uses, see Mass (disambiguation). ... Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. ...

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.

To derive a generalized entropy balanced equation, we start with the general balance equation for the change in any extensive quantity Θ in a thermodynamic system, a quantity that may be either conserved, such as energy, or non-conserved, such as entropy. The basic generic balance expression states that dΘ/dt, i.e. the rate of change of Θ in the system, equals the rate at which Θ enters the system at the boundaries, minus the rate at which Θ leaves the system across the system boundaries, plus the rate at which Θ is generated within the system. Using this generic balance equation, with respect to the rate of change with time of the extensive quantity entropy S, the entropy balance equation for an open thermodynamic system is:[16] Image File history File links Download high-resolution version (946x652, 7 KB) Rasterized First_law_open_system. ... Image File history File links Download high-resolution version (946x652, 7 KB) Rasterized First_law_open_system. ... An unit operation is considered to be at steady-state with respect to an operation variable if that variable does not change with time. ... In physics and chemistry, an extensive quantity (also referred to as an extensive variable) is a physical quantity whose value is proportional to the size of the system it describes. ... Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ...

$frac{dS}{dt} = sum_{k=1}^K dot{M}_k hat{S}_k + frac{dot{Q}}{T} + dot{S}_{gen}$

where

$sum_{k=1}^K dot{M}_k hat{S}_k$ = the net rate of entropy flow due to the flows of mass into and out of the system (where $hat{S}$ = entropy per unit mass).
$frac{dot{Q}}{T}$ = the rate of entropy flow due to the flow of heat across the system boundary.
$dot{S}_{gen}$ = the rate of internal generation of entropy within the system.

Note, also, that if there are multiple heat flows, the term $dot{Q}/T$ is to be replaced by $sum dot{Q}_j/T_j$, where $dot{Q}_j$ is the heat flow and Tj is the temperature at the jth heat flow port into the system.

### Entropy in quantum mechanics (von Neumann entropy)

Main article: 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. Quantum statistical mechanics is the study of statistical ensembles of quantum mechanical systems. ... Quantum statistical mechanics is the study of statistical ensembles of quantum mechanical systems. ... For other persons named John Neumann, see John Neumann (disambiguation). ... A density matrix is a self-adjoint (or Hermitian) positive-semidefinite matrix, (possibly infinite dimensional), of trace one, that describes the statistical state of a quantum system. ...

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.[17]

### Standard textbook definitions

Note that textbook definitions are not always the most helpful definitions, but they are an important aspect of the culture surrounding the concept of entropy.

• Entropyenergy broken down in irretrievable heat.[18]
• 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.[19]
• the number of ways of arranging things in a system (times the Boltzmann's constant).[20]
• 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.[21]
• a direct measure of the randomness of a system.[22]
• a measure of energy dispersal at a specific temperature.[23]
• a measure of the partial loss of the ability of a system to perform work due to the effects of irreversibility.[24]
• an index of the tendency of a system towards spontaneous change.[25]
• 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.[26]
• a parameter representing the state of disorder of a system at the atomic, ionic, or molecular level.[27]
• a measure of disorder in the universe or of the availability of the energy in a system to do work.[28]

For other uses, see Heat (disambiguation) In physics, heat, symbolized by Q, is energy transferred from one body or system to another due to a difference in temperature. ... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... In thermodynamics, a microstate describes a specific detailed microscopic configuration of a system. ... A microstate is a sovereign state having a very small population or very little land area - usually both. ... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... In thermodynamics, a state function, or state quantity, is a property of a system that depends only on the current state of the system, not on the way in which the system got to that state. ... A microstate is a sovereign state having a very small population or very little land area - usually both. ... â€œRandomâ€ redirects here. ... The thermodynamic concept of entropy can be described qualitatively as a measure of energy dispersal at a specific temperature. ... Irreversibility is that property of an event which makes reverting back to the state before the occurrence of the event impossible. ...

## Approaches to understanding entropy

### Order and disorder

Entropy, historically, has often been associated with the amount of order, disorder, and/or chaos in a thermodynamic system. The traditional definition of entropy is that it refers to changes in the status quo of the system and is a measure of "molecular disorder" and the amount of wasted energy in a dynamical energy transformation from one state or form to another.[29] In this direction, a number of authors, in recent years, have derived exact entropy formulas to account for and measure disorder and order in atomic and molecular assemblies.[30][8][31][32] One of the simpler entropy order/disorder formulas is that derived in 1984 by thermodynamic physicist Peter Landsberg, which is based on a combination of thermodynamics and information theory arguments. Landsberg argues that when constraints operate on a system, such that it is prevented from entering one or more of its possible or permitted states, as contrasted with its forbidden states, the measure of the total amount of “disorder” in the system is given by the following expression:[31][32] Boltzmanns molecules (1896) shown at a rest position in a solid In thermodynamics, entropy, historically, has often been associated with the amount of order, disorder, and or chaos in a thermodynamic system. ... â€œRandomâ€ redirects here. ... For other uses, see Chaos (disambiguation). ... Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ... Thermodynamics (from the Greek Î¸ÎµÏÎ¼Î·, therme, meaning heat and Î´Ï…Î½Î±Î¼Î¹Ï‚, dynamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Not to be confused with information technology, information science, or informatics. ...

$Disorder=C_D/C_I,$

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

$Order=1-C_O/C_I,$

In which CD is the "disorder" capacity of the system, which is the entropy of the parts contained in the permitted ensemble, CI is the "information" capacity of the system, an expression similar to Shannon's channel capacity, and CO is the "order" capacity of the system.[8] This article does not cite any references or sources. ...

### Energy dispersal

The concept of entropy can be described qualitatively as a measure of energy dispersal at a specific temperature.[33] 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. The thermodynamic concept of entropy can be described qualitatively as a measure of energy dispersal at a specific temperature. ... ass hole ... Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ...

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.[34] 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.[35] 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.[36][12] The second law of thermodynamics is an expression of the universal law of increasing entropy. ... In thermodynamics, an isolated system, as contrasted with a closed system, is a physical system that does not interact with its surroundings. ... The first law of thermodynamics, a generalized expression of the law of the conservation of energy, states: // Description Essentially, the First Law of Thermodynamics declares that energy is conserved for a closed system, with heat and work being the forms of energy transfer. ... Peter William Atkins (b. ...

### Entropy and Information theory

In information theory, entropy is the measure of the amount of information that is missing before reception and is sometimes referred to as Shannon entropy.[37] Shannon entropy is a broad and general concept which finds applications in information theory as well as thermodynamics. It was originally devised by Claude Shannon in 1948 to 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 pi. 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 number of yes/no questions needed to determine the content of the message. Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ... This article or section is in need of attention from an expert on the subject. ... Not to be confused with information technology, information science, or informatics. ... Entropy of a Bernoulli trial as a function of success probability. ... Not to be confused with information technology, information science, or informatics. ... In physics the Maximum entropy school of thermodynamics (or more colloquially, the MaxEnt school of thermodynamics), initiated with two papers published in the Physical Review by Edwin T. Jaynes in 1957, views statistical mechanics as an inference process: a specific application of inference techniques rooted in information theory, which relate... Claude Shannon Claude Elwood Shannon (April 30, 1916 â€“ February 24, 2001), an American electrical engineer and mathematician, has been called the father of information theory,[1] and was the founder of practical digital circuit design theory. ...

The question of the link between information entropy and thermodynamic entropy is a hotly debated topic. Many authors argue that there is a link between the two,[38][39][40] while others will argue that they have absolutely nothing to do with each other.[41]

The expressions for the two entropies are very similar. The information entropy H for equal probabilities pi = p is:

$H=Kln(1/p),$

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

$S=kln(1/p),$

where p  is the probability of a system being in a particular microstate, given that it is in a particular macrostate, and k  is Boltzmann's constant. It can be seen that one may think of the thermodynamic entropy as Boltzmann's constant, divided by ln(2), times the number of yes/no questions that must be asked in order to determine the microstate of the system, given that we know the macrostate. The link between thermodynamic and information entropy was developed in a series of papers by Edwin Jaynes beginning in 1957.[42] Edwin Thompson Jaynes (July 5th, 1922 &#8211; April 30th, 1998) was Wayman Crow Distinguished Professor of Physics at Washington University in St. ...

The problem with linking thermodynamic entropy to information entropy is that in information entropy the entire body of thermodynamics which deals with the physical nature of entropy is missing. The second law of thermodynamics which governs the behavior of thermodynamic systems in equilibrium, and the first law which expresses heat energy as the product of temperature and entropy are physical concepts rather than informational concepts. If thermodynamic entropy is seen as including all of the physical dynamics of entropy as well as the equilibrium statistical aspects, then information entropy gives only part of the description of thermodynamic entropy. Some authors, like Tom Schneider, argue for dropping the word entropy for the H function of information theory and using Shannon's other term "uncertainty" instead.[43]

### Ice melting example

Main article: disgregation

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. In the history of thermodynamics, disgregation was defined in 1862 by Rudolf Clausius as the magnitude of the degree in which the molecules of a body are separated from each other. ... Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ... For other uses, see Heat (disambiguation) In physics, heat, symbolized by Q, is energy transferred from one body or system to another due to a difference in temperature. ... Standard enthalpy change of fusion of period three. ...

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. Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ...

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

Main article: Entropy and life

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."[45] Albert Lester Lehninger (February 17, 1917 - March 4, 1986) was an American biochemist, and is widely regarded as a pioneer in the field of bioenergetics. ... The thermodynamic free energy is a measure of the amount of mechanical (or other) work that can be extracted from a system, and is helpful in engineering applications. ... For other uses, see Heat (disambiguation) In physics, heat, symbolized by Q, is energy transferred from one body or system to another due to a difference in temperature. ...

Evolution related definitions:

• Negentropy - a shorthand colloquial phrase for negative entropy.[46]
• Ectropy - a measure of the tendency of a dynamical system to do useful work and grow more organized.[29]
• 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.

To meet Wikipedias quality standards, this article or section may require cleanup. ... Ectropy is the opposite of entropy. ... Syntropy is a term popularized by Buckminster Fuller but also developed by others to refer to an anti-entropy or negentropy. The following definition, referencing Fuller, can be found on a web site on Whole Systems: A tendency towards order and symmetrical combinations, designs of ever more advantageous and orderly... Extropianism, also reffered to as extropy, is a transhumanist philosophy characterized by a set of principles regarding extropy, defined by Dr. Max More in The Principles of Extropy. ... Ecological entropy is a measure of biodiversity in the study of biological ecology. ... Rainforests are among the most biodiverse ecosystems on earth Biodiversity is the variation of taxonomic life forms within a given ecosystem, biome or for the entire Earth. ... For the journal, see Ecology (journal). ...

### 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 is the only quantity in the physical sciences that picks a particular direction for time, sometimes called an arrow of time. ... This article or section does not cite its references or sources. ...

### 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. In physics, black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. ... The heat death is a possible final state of the universe, in which it has reached maximum entropy. ...

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. Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... Gravity is a force of attraction that acts between bodies that have mass. ... This article is about the astronomical body. ... Jacob David Bekenstein (born May 1, 1947) is a physicist who has contributed to the foundation of black hole thermodynamics and to other aspects of the connections between information and gravitation. ... Stephen William Hawking, CH, CBE, FRS, FRSA, (born 8 January 1942) is a British theoretical physicist. ...

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. For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ...

### 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.[47]
• 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.[48]
• Sackur-Tetrode entropy - the entropy of a monatomic classical ideal gas determined via quantum considerations.

Etymology: French calorie, from Latin calor (heat), from calere (to be warm). ... In thermodynamics, specifically in statistical mechanics, the Gibbs entropy is the usual statistical mechanical entropy of a thermodynamic system, where the summation is taken over the possible states of the system as a whole (typically a 6N-dimensional space, if the system contains N separate particles). ... In thermodynamics, specifically in statistical mechanics, the Boltzmann entropy is an approximation to the normal Gibbs entropy. ... The Tsallis entropy is a generalization of the standard Boltzmann-Gibbs entropy. ... In chemistry, the standard molar entropy is the entropy content of one mole of substance, under conditions of standard temperature and pressure. ... Black hole entropy is entropy carried by a black hole. ... For other uses, see Black hole (disambiguation). ... Residual entropy is physically significant entropy, which is present even after a substance is cooled arbitrarily close to absolute zero. ... Absolute zero is the lowest possible temperature where nothing could be colder, and no heat energy remains in a substance. ... In thermodynamics the Entropy of mixing is the entropy change when two different gases or two solvents mix or even when a solute is dissolved in a solvent. ... Water and steam are two different forms of the same chemical substance A chemical substance is a material with a definite chemical composition. ... 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. ... A polymer (from Greek: Ï€Î¿Î»Ï…, polu, many; and Î¼Î­ÏÎ¿Ï‚, meros, part) is a substance composed of molecules with large molecular mass composed of repeating structural units, or monomers, connected by covalent chemical bonds. ... 3-dimensional structure of hemoglobin, a globular protein. ... In physics, an entropic force acting in a system is a macroscopic force arising not as a result of an actual underlying microscopic force (such as electromagnetism), but as a statistical consequence of the whole systems tendency to increase its entropy. ... A thermodynamic free entropy is an entropic thermodynamic potential analogous to the free energy. ... Triacetone triperoxide (TATP) decomposes in an entropic explosion. ... The standard entropy change of vaporization is the increase in entropy when vaporizing a substance. ... (LTE is an acronym for the progressive-instrumental rock band  Liquid Tension Experiment) In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann distribution. ... For other uses, see Heat (disambiguation) In physics, heat, symbolized by Q, is energy transferred from one body or system to another due to a difference in temperature. ... Absolute zero is the lowest temperature that can be obtained in any macroscopic system. ... Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ... The Sackur-Tetrode equation is an expression for the entropy of a monatomic classical ideal gas which uses quantum considerations to arrive at an exact formula. ...

## 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.

In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of ergodic theory. ... The Lorenz attractor is an example of a non-linear dynamical system. ... In mathematics, in the area of ergodic theory, the topological entropy provides a way of defining the entropy in an iterated function map. ... In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ... In probability theory and information theory, the Kullback-Leibler divergence, or relative entropy, is a quantity which measures the difference between two probability distributions. ... In information theory, the RÃ©nyi entropy, a generalisation of Shannon entropy, is one of a family of functionals for quantifying the diversity, uncertainty or randomness of a system. ...

### 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: Sociology (from Latin: socius, companion; and the suffix -ology, the study of, from Greek Î»ÏŒÎ³Î¿Ï‚, lÃ³gos, knowledge [1]) is the systematic and scientific study of society, including patterns of social relationships, social action, and culture[2]. Areas studied in sociology can range from the analysis of brief contacts between anonymous... This article is about metaphor in literature and rhetoric. ...

• Entropology – the study or discussion of entropy or the name sometimes given to thermodynamics without differential equations.[5][49]
• Psychological entropy - the distribution of energy in the psyche, which tends to seek equilibrium or balance among all the structures of the psyche.[50]
• Economic entropy – a semi-quantitative measure of the irrevocable dissipation and degradation of natural materials and available energy with respect to economic activity.[51][52]
• 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.[53]
• Corporate entropy - energy waste as red tape and business team inefficiency, i.e. energy lost to waste.[54] (This definition is comparable to von Clausewitz's concept of friction in war.)

Thermodynamics (from the Greek Î¸ÎµÏÎ¼Î·, therme, meaning heat and Î´Ï…Î½Î±Î¼Î¹Ï‚, dynamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ... In psychology, psychodynamics is the study of the interrelationship of various parts of the mind, personality, or psyche as they relate to mental, emotional, or motivational forces especially at the subconscious level. ... In the natural sciences, thermoeconomics is the physics of economic value. ... Social entropy is a macrosociological systems theory. ... Red tape (or sometimes paperwork) is a derisive term for excessive regulation or rigid conformity to formal rules that is considered redundant or bureaucratic and hinders or prevents action or decision-making. ... Carl Philipp Gottfried von Clausewitz (June 1, 1780 â€“ November 16, 1831) was a Prussian soldier, military historian and influential military theorist. ... For other uses, see Friction (disambiguation). ...

## 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)

The second law of thermodynamics is an expression of the universal law of increasing entropy. ... Josiah Willard Gibbs (February 11, 1839 â€“ April 28, 1903) was an American mathematical physicist who contributed much of the theoretical foundation that led to the development of chemical thermodynamics and was one of the founders of vector analysis. ... For other persons named John Neumann, see John Neumann (disambiguation). ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Claude Shannon Claude Elwood Shannon (April 30, 1916 â€“ February 24, 2001), an American electrical engineer and mathematician, has been called the father of information theory,[1] and was the founder of practical digital circuit design theory. ... For other persons named John Neumann, see John Neumann (disambiguation). ...

 Geometrically frustrated magnet Maxwell's demon Multiplicity function Statistical mechanics Stirling's formula Thermodynamic potential Thermodynamic databases for pure substances

This article or section does not cite its references or sources. ... The Brownian ratchet is a thought experiment about an apparent perpetual motion machine postulated by Richard Feynman in a physics lecture at the California Institute of Technology on May 11, 1962 as an illustration of the laws of thermodynamics. ... For other uses, see Chaos Theory (disambiguation). ... In thermodynamics, a departure function is defined for any thermodynamic property as the difference between the property as computed for an ideal gas and the property of the species as it exists in the real world, for a specified temperature T and pressure P. Common departure functions include those for... t In thermodynamics and molecular chemistry, the enthalpy or heat content (denoted as H or Î”H, or rarely as Ï‡) is a quotient or description of thermodynamic potential of a system, which can be used to calculate the useful work obtainable from a closed thermodynamic system under constant pressure. ... Thermodynamic entropy provides a measure of certain aspects of energy in relation to absolute temperature. ... The entropy rate of a stochastic process is, informally, the time density of the average information in a stochastic process. ... // The word frustration was introduced to describe the situation where a system cannot simultaneously minimize the interaction energies between its components1. ... Maxwells demon is an 1867 thought experiment by the Scottish physicist James Clerk Maxwell, meant to raise questions about the possibility of violating the second law of thermodynamics. ... The multiplicity function Î©(n,N) is the same as the combinatoric function C(N,n). ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In mathematics, Stirlings approximation (or Stirlings formula) is an approximation for large factorials. ... In thermodynamics, thermodynamic potentials are parameters associated with a thermodynamic system and have the dimensions of energy. ... Thermodynamic databases contain information about thermodynamic properties for substances, the most important being enthalpy, entropy, and Gibbs free energy. ...

## References

1. ^ Note: In certain types of advanced system configurations, 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 a supercooled 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 adjoined systems, i.e. the adjacent hot and cold bodies, when thought of as one "universe", increases. Here are some further tutorials: Ice-meltingJCE example; Ice-melting and Entropy Change – example; Ice-melting and Entropy Change – discussions
2. ^ Clausius, Rudolf (1862). Communicated to the Naturforschende Gesellschaft of Zurich, Jan. 27th, 1862; published in the Vierteljahrschrift of 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. ^ Daintith, John (2005). Oxford Dictionary of Physics. Oxford University Press. ISBN 0-19-280628-9.
4. ^ a b 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.
5. ^ a b Perrot, Pierre (1998). A to Z of Thermodynamics. Oxford University Press. ISBN 0-19-856552-6.
6. ^ Avery, John (2003). Information Theory and Evolution. World Scientific. ISBN 981-238-399-9.
7. ^ Yockey, Hubert, P. (2005). Information Theory, Evolution, and the Origin of Life.. Cambridge University Press. ISBN 0-521-80293-8.
8. ^ a b c Brooks, Daniel, R.; Wiley, E.O. (1988). Entropy as Evolution – Towards a Unified Theory of Biology. University of Chicago Press. ISBN 0-226-07574-5.
9. ^ McCulloch, Richard, S. (1876). Treatise on the Mechanical Theory of Heat and its Applications to the Steam-Engine, etc.. D. Van Nostrand.
10. ^ See, e.g., Notes for a “Conversation About Entropy” for a brief discussion of both thermodynamic and "configurational" ("positional") entropy in chemistry.
11. ^ EntropyOrderParametersComplexity.pdf
12. ^ a b Atkins, Peter; Julio De Paula (2006). Physical Chemistry , 8th edition. Oxford University Press. ISBN 0-19-870072-5.
13. ^ a b Moore, J. W.; C. L. Stanistski, P. C. Jurs (2005). Chemistry, The Molecular Science ,. Brooks Cole. ISBN 0-534-42201-2.
14. ^ Jungermann, A.H. (2006). “Entropy and the Shelf Model: A Quantum Physical Approach to a Physical Property”. Journal of Chemical Education 83: 1686-1694
15. ^ Levine, I. N. (2002). Physical Chemistry, 5th edition. McGraw-Hill. ISBN 0-07-231808-2.
16. ^ Sandler, Stanley, I. (1989). Chemical and Engineering Thermodynamics. John Wiley & Sons. ISBN 0-471-83050-X.
17. ^ The information entropy of quantum mechanical states, Europhysics Letters 67, 700 (2004)
18. ^ 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.
19. ^ Baierlein, Ralph (2003). Thermal Physics. Cambridge University Press. ISBN 0-521-65838-1.
20. ^ Schroeder, Daniel, R. (2000). Thermal Physics. New York: Addison Wesley Longman. ISBN 0-201-38027-7.
21. ^ McGraw-Hill Concise Encyclopedia of Chemistry, 2004
22. ^ Chang, Raymond (1998). Chemistry, 6th Ed.. New York: McGraw Hill. ISBN 0-07-115221-0.
23. ^ Atkins, Peter; Julio De Paula (2006). Physical Chemistry , 8th edition. Oxford University Press. ISBN 0-19-870072-5.
24. ^ Cutnell, John, D.; Johnson, Kenneth, J. (1998). Physics, 4th edition. John Wiley and Sons, Inc.. ISBN 0-471-19113-2.
25. ^ Haynie, Donald, T. (2001). Biological Thermodynamics. Cambridge University Press. ISBN 0-521-79165-0.
26. ^ Oxford Dictionary of Science, 2005
27. ^ Barnes & Noble's Essential Dictionary of Science, 2004
28. ^ Gribbin's Encyclopedia of Particle Physics, 2000
29. ^ a b Haddad, Wassim M.; Chellaboina, VijaySekhar; Nersesov, Sergey G. (2005). Thermodynamics - A Dynamical Systems Approach. Princeton University Press. ISBN 0-691-12327-6.
30. ^ Callen, Herbert, B (2001). Thermodynamics and an Introduction to Thermostatistics, 2nd Ed.. John Wiley and Sons. ISBN 0-471-86256-8.
31. ^ a b Landsberg, P.T. (1984). “Is Equilibrium always an Entropy Maximum?” J. Stat. Physics 35: 159-69.
32. ^ a b Landsberg, P.T. (1984). “Can Entropy and “Order” Increase Together?” Physics Letters 102A:171-173
33. ^ Frank L. Lambert, A Student’s Approach to the Second Law and Entropy
34. ^ 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, University Chemistry Education - 2002 Papers, Royal Society of Chemistry.
35. ^ Frank L. Lambert, JCE 2002 (79) 187 [Feb Disorder--A Cracked Crutch for Supporting Entropy Discussions]
36. ^ Atkins, Peter (1984). The Second Law. Scientific American Library. ISBN 0-7167-5004-X.
37. ^ Balian, Roger (2003). Entropy – Protean Concept (PDF). Poincare Seminar 2: 119-45.
38. ^ Brillouin, Leon (1956). Science and Information Theory. name. ISBN 0-486-43918-6.
39. ^ Georgescu-Roegen, Nicholas (1971). The Entropy Law and the Economic Process. Harvard University Press. ISBN 0-674-25781-2.
40. ^ Chen, Jing (2005). The Physical Foundation of Economics - an Analytical Thermodynamic Theory. World Scientific. ISBN 981-256-323-7.
41. ^ Lin, Shu-Kun. (1999). “Diversity and Entropy.” Entropy (Journal), 1[1], 1-3.
42. ^ http://bayes.wustl.edu/etj/node1.html
43. ^ Schneider, Tom, Delila Syste (Deoxyrobonucleic 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.
44. ^ Entropy, Disorder and Life
45. ^ Lehninger, Albert (1993). Principles of Biochemistry, 2nd Ed.. Worth Publishers. ISBN 0-87901-711-2.
46. ^ Schrödinger, Erwin (1944). What is Life - the Physical Aspect of the Living Cell. Cambridge University Press. ISBN 0-521-42708-8.
47. ^ von Baeyer, Christian, H. (2003). Information - the New Language of Science. Harvard University Press. ISBN 0-674-01387-5.
48. ^ Serway, Raymond, A. (1992). Physics for Scientists and Engineers. Saunders Golden Subburst Series. ISBN 0-03-096026-6.
49. ^ 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 by Claude Levi-Strauss.)
50. ^ Hall, Calvin S.; Nordby, Vernon J. (1999). A Primer of Jungian Psychology. New York: Meridian. ISBN 0-452-01186-8.
51. ^ Georgescu-Roegen, Nicholas (1971). The Entropy Law and the Economic Process. Harvard University Press. ISBN 0-674-25781-2.
52. ^ Burley, Peter; Foster, John (1994). Economics and Thermodynamics – New Perspectives on Economic Analysis. Kluwer Academic Publishers. ISBN 0-7923-9446-1.
53. ^ Bailey, Kenneth, D. (1990). Social Entropy Theory. State University of New York Press. ISBN 0-7914-0056-5.
54. ^ DeMarco, Tom; Lister, Timothy (1999). Peopleware - Productive Projects and Teams, 2nd. Ed.. Dorset House Publishing Co.. ISBN 0-932633-43-9.
• P. Pluch Quantum Probability Theory, PhD Thesis, University of Klagenfurt (2006)

In physical chemistry, thermodynamics, chemistry and condensed matter physics, a critical point, also called a critical state, specifies the conditions (temperature, pressure) at which the liquid state of the matter ceases to exist. ... MIT redirects here. ... Claude L vi-Strauss (born November 28, 1908) is a French anthropologist who became one of the twentieth centurys greatest intellectuals by developing structuralism as a method of understanding human society and culture Biography Claude L vi-Strauss was born in Brussels and studied law and philosophy at the...

1. Ben-Naim, Arieh (2007). Entropy Demystified. World Scientific. ISBN 981-270-055-2.
2. Dugdale, J. S. (1996). Entropy and its Physical Meaning, 2nd Ed., Taylor and Francis (UK); CRC (US). ISBN 0748405690.
3. Fermi, Enrico (1937). Thermodynamics. Prentice Hall. ISBN 0-486-60361-X.
4. Kroemer, Herbert; Charles Kittel (1980). Thermal Physics, 2nd Ed., W. H. Freeman Company. ISBN 0-7167-1088-9.
5. Penrose, Roger (2005). The Road to Reality : A Complete Guide to the Laws of the Universe. ISBN 0-679-45443-8.
6. Reif, F. (1965). Fundamentals of statistical and thermal physics. McGraw-Hill. ISBN 0-07-051800-9.
7. Goldstein, Martin; Inge, F (1993). The Refrigerator and the Universe. Harvard University Press. ISBN 0-674-75325-9.

Enrico Fermi (September 29, 1901 â€“ November 28, 1954) was an Italian physicist most noted for his work on the development of the first nuclear reactor, and for his contributions to the development of quantum theory, particle physics and statistical mechanics. ... Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ...

Results from FactBites:

 A Students Approach to the Second Law and Entropy (2516 words) Entropy measures the spontaneous dispersal of energy: how much energy is spread out in a process, or how widely spread out it becomes — at a specific temperature. The equation for the entropy increase in the mixture uses the relative molar quantities of liquids that were mixed. That increased entropy of the solvent in a solution is the cause of the "colligative effects" that we study: (1) osmotic pressure, (2) boiling point elevation, and (3) freezing point depression.
 Entropy - Wikipedia, the free encyclopedia (3219 words) In thermodynamics, entropy, symbolized by S, is a state function of a thermodynamic system defined by the differential quantity dS = dQ / T, where dQ is the amount of heat absorbed in a reversible process in which the system goes from the one state to another, and T is the absolute temperature. Entropy is one of the factors that determines the free energy in the system and appears in the second law of thermodynamics. Entropy is said to be thermodynamically conjugate to temperature.
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