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Introduction 
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In thermodynamics, entropy (usual symbol S) is a measure of the number of specific ways in which a thermodynamic system may be arranged, often taken to be a measure of disorder, or a measure of progressing towards thermodynamic equilibrium. The entropy of an isolated system never decreases, because isolated systems spontaneously evolve towards thermodynamic equilibrium, the maximum entropy. Systems which are not isolated may decrease in entropy. Since entropy is a state function, the change in the entropy of a system is the same whether a process going from one defined state to another is reversible or irreversible, but irreversible processes increase the entropy of the environment.
The change in entropy (ΔS) was originally defined for a thermodynamically reversible process as
which is found from the uniform thermodynamic temperature (T) of a closed system dividing an incremental reversible transfer of heat into that system (dQ). The above definition is sometimes called the macroscopic definition of entropy because it can be used without regard to any microscopic picture of the contents of a system. In thermodynamics, entropy has been found to be more generally useful and it has several other formulations. Entropy was discovered when it was noticed to be a quantity that behaves as a function of state, as a consequence of the second law of thermodynamics. Entropy is an extensive property, but the entropy of a pure substance is usually given as an intensive property — either specific entropy (entropy per unit mass) or molar entropy (entropy per mole).
The absolute entropy (S rather than ΔS) was defined later, using either statistical mechanics or the third law of thermodynamics.
In the modern microscopic interpretation of entropy in statistical mechanics, entropy is the amount of additional information needed to specify the exact physical state of a system, given its thermodynamic specification. Understanding the role of thermodynamic entropy in various processes requires understanding how and why that information changes as the system evolves from its initial condition. It is often said that entropy is an expression of the disorder, or randomness of a system, or of our lack of information about it. The second law is now often seen as an expression of the fundamental postulate of statistical mechanics via the modern definition of entropy. Entropy has the dimension of energy divided by temperature, which has a unit of joules per kelvin (J/K) in the International System of Units.
The analysis which led to the concept of entropy began with the work of French mathematician Lazare Carnot who in his 1803 paper Fundamental Principles of Equilibrium and Movement proposed that in any machine the accelerations and shocks of the moving parts 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 which posited that in all heatengines whenever "caloric", or what is now known as heat, falls through a temperature difference, work or motive power can be produced from the actions of the "fall of caloric" between a hot and cold body. He made the analogy with that of how water falls in a water wheel. This was an early insight into the second law of thermodynamics.^{[1]} Carnot based his views of heat partially on the early 18th century "Newtonian hypothesis" that both heat and light were types of indestructible forms of matter, which are attracted and repelled by other matter, and partially on the contemporary views of Count Rumford who showed (1789) that heat could be created by friction as when cannon bores are machined.^{[2]} Carnot reasoned that if the body of the working substance, such as a body of steam, is returned to its original state at the end of a complete engine cycle, that "no change occurs in the condition of the working body".
The first law of thermodynamics, formalized based on the heatfriction experiments of James Joule in 1843, deals with the concept of energy, which is conserved in all processes; the first law, however, is unable to quantify the effects of friction and dissipation.
In the 1850s and 1860s, German physicist Rudolf Clausius objected to the supposition 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.^{[3]} Clausius described entropy as the transformationcontent, i.e. dissipative energy use, of a thermodynamic system or working body of chemical species during a change of state.^{[3]} 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, Josiah Willard Gibbs, and James Clerk Maxwell gave entropy a statistical basis. In 1877 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. Carathéodory linked entropy with a mathematical definition of irreversibility, in terms of trajectories and integrability.
Any method involving the notion of entropy, the very existence of which depends on the second law of thermodynamics, will doubtless seem to many farfetched, and may repel beginners as obscure and difficult of comprehension.
There are two related definitions of entropy: the thermodynamic definition and the statistical mechanics definition. Historically, the classical thermodynamics definition developed first, and it has more recently been extended in the area of nonequilibrium thermodynamics. Entropy was defined from a classical thermodynamics viewpoint, in which the details of the system's constituents are not directly considered, with their behavior only showing up in macroscopically averaged properties, e.g. heat capacity. Later, thermodynamic entropy was more generally defined from a statistical thermodynamics viewpoint, in which the detailed constituents — modeled at first classically, e.g. Newtonian particles constituting a gas, and later quantummechanically (photons, phonons, spins, etc.) — were explicitly considered.
There are many thermodynamic properties that are functions of state. This means that at a particular thermodynamic state (which should not be confused with the microscopic state of a system), these properties have a certain value. Often, if two properties have a particular value, then the state is determined and the other properties values are set. For instance, an ideal gas, at a particular temperature and pressure, has a particular volume according to the ideal gas equation. As another instance, a pure substance of single phase at a particular uniform temperature and pressure (and thus a particular state) is at not only a particular volume but also at a particular entropy.^{[5]} That entropy is a function of state is one reason it is useful. In the Carnot cycle, the working fluid returns to the same state at a particular stage of the cycle, hence the line integral of any state function, such as entropy, over the cycle is zero.
Entropy is defined for a reversible process and for a system that, at all times, can be treated as being at a uniform state and thus at a uniform temperature. Reversibility is an ideal that some real processes approximate and that is often presented in study exercises. For a reversible process, entropy behaves as a conserved quantity and no change occurs in total entropy. More specifically, total entropy is conserved in a reversible process and not conserved in an irreversible process.^{[6]} One has to be careful about system boundaries. For example, in the Carnot cycle, while the heat flow from the hot reservoir to the cold reservoir represents an increase in entropy, the work output, if reversibly and perfectly stored in some energy storage mechanism, represents a decrease in entropy that could be used to operate the heat engine in reverse and return to the previous state, thus the total entropy change is still zero at all times if the entire process is reversible. Any process that does not meet the requirements of a reversible process must be treated as an irreversible process, which is usually a complex task. An irreversible process increases entropy.^{[7]}
Heat transfer situations require two or more nonisolated systems in thermal contact. In irreversible heat transfer, heat energy is irreversibly transferred from the higher temperature system to the lower temperature system, and the combined entropy of the systems increases. Each system, by definition, must have its own absolute temperature applicable within all areas in each respective system in order to calculate the entropy transfer. Thus, when a system at higher temperature T_{H} transfers heat dQ to a system of lower temperature T_{C}, the former loses entropy dQ/T_{H} and the latter gains entropy dQ/T_{C}. The combined entropy change is dQ/T_{C}dQ/T_{H} which is positive, reflecting an increase in the combined entropy. When calculating entropy, the same requirement of having an absolute temperature for each system in thermal contact exchanging heat also applies to the entropy change of an isolated system having no thermal contact.
The concept of entropy arose from Rudolf Clausius's study of the Carnot cycle.^{[8]} In a Carnot cycle, heat () is absorbed from a 'hot' reservoir, isothermally at the higher temperature , and given up isothermally to a 'cold' reservoir, , at a lower temperature, . According to Carnot's principle, work can only be done when there is a temperature difference, and the work should be some function of the difference in temperature and the heat absorbed. Carnot did not distinguish between and , since he was working under the incorrect hypothesis that caloric theory was valid, and hence heat was conserved (the incorrect assumption that and were equal) when, in fact, .^{[9]} Through the efforts of Clausius and Kelvin, it is now known that the maximum work that can be done is the product of the Carnot efficiency and the heat absorbed at the hot reservoir: In order to derive the Carnot efficiency, , Kelvin had to evaluate the ratio of the work done to the heat absorbed in the isothermal expansion with the help of the CarnotClapeyron equation which contained an unknown function, known as the Carnot function. The fact that the Carnot function could be the temperature, measured from zero, was suggested by Joule in a letter to Kelvin, and this allowed Kelvin to establish his absolute temperature scale.^{[10]} It is also known that the work is the difference in the heat absorbed at the hot reservoir and rejected at the cold one: Since the latter is valid over the entire cycle, this gave Clausius the hint that at each stage of the cycle, work and heat would not be equal, but rather their difference would be a state function that would vanish upon completion of the cycle. The state function was called the internal energy and it became the first law of thermodynamics.^{[11]}
Now equating the two expressions gives
If we allow to incorporate the algebraic sign, this becomes a sum and implies that there is a function of state which is conserved over a complete cycle. Clausius called this state function entropy. One can see that entropy was discovered through mathematics rather than through laboratory results. It is a mathematical construct and has no easy physical analogy. This makes the concept somewhat obscure or abstract, akin to how the concept of energy arose.
Then Clausius asked what would happen if there would be less work done than that predicted by Carnot's principle. The righthand side of the first equation would be the upper bound of the work, which would now be converted into an inequality When the second equation is used to express the work as a difference in heats, we get or So more heat is given off to the cold reservoir than in the Carnot cycle. If we denote the entropies by for the two states, then the above inequality can be written as a decrease in the entropy . The wasted heat implies that irreversible processes must have prevented the cycle from carrying out maximum work.
Conjugate variables of thermodynamics  

Pressure  Volume 
(Stress)  (Strain) 
Temperature  Entropy 
Chemical potential  Particle number 
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. Clausius created the term entropy in 1865 as an extensive thermodynamic variable was shown to be useful in characterizing the Carnot cycle. Heat transfer along the isotherm steps of the Carnot cycle was found to be proportional to the temperature of a system (known as its absolute temperature). This relationship was expressed in increments of entropy equal to the ratio of incremental heat transfer divided by temperature, which was found to vary in the thermodynamic cycle but eventually return to the same value at the end of every cycle. Thus it was found to be a function of state, specifically a thermodynamic state of the system. Clausius wrote that he "intentionally formed the word Entropy as similar as possible to the word Energy", basing the term on the Greek ἡ τροπή [tropé], transformation.^{[12]}^{[note 1]}
While Clausius based his definition on a reversible process, there are also irreversible processes that change entropy. Following the second law of thermodynamics, entropy of an isolated system always increases. The difference between an isolated system and closed system is that heat may not flow to and from an isolated system, but heat flow to and from a closed system is possible. Nevertheless, for both closed and isolated systems, and indeed, also in open systems, irreversible thermodynamics processes may occur.
According to the Clausius equality, for a reversible cyclic process: . This means the line integral is path independent.
So we can define a state function S called entropy, which satisfies:
With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the 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 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. In any process where the system gives up energy ΔE, and its entropy falls by ΔS, a quantity at least T_{R} ΔS of that energy must be given up to the system's surroundings as unusable heat (T_{R} is the temperature of the system's external surroundings). Otherwise the process will not go forward. In classical thermodynamics, the entropy of a system is defined only if it is in thermodynamic equilibrium.
The statistical definition was developed by Ludwig Boltzmann in the 1870s by analyzing the statistical behavior of the microscopic components of the system. Boltzmann showed 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 deeper understanding of its nature.
The interpretation of entropy in statistical mechanics is the measure of uncertainty, or mixedupness in the phrase of Gibbs, which remains about a system after its observable macroscopic properties, such as temperature, pressure and volume, have been taken into account. For a given set of macroscopic variables, the entropy measures the degree to which 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 the system including the position and velocity of every molecule. The more such states available to the system with appreciable probability, the greater the entropy. In statistical mechanics, entropy is 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).^{[13]}^{[14]}^{[15]} This definition describes the entropy as being proportional to the natural logarithm of the number of possible microscopic configurations of the individual atoms and molecules of the system (microstates) which could give rise to the observed macroscopic state (macrostate) of the system. The constant of proportionality is the Boltzmann constant.
Specifically, entropy is a logarithmic measure of the number of states with significant probability of being occupied:
where k_{B} is the Boltzmann constant, equal to 1.38065×10^{−23} J K^{−1}. The summation is over all the possible microstates of the system, and p_{i} is the probability that the system is in the ith microstate.^{[16]} This definition assumes that the basis set of states has been picked so that there is no information on their relative phases. In a different basis set, the more general expression is
where is the density matrix and is the matrix logarithm. This density matrix formulation is not needed in cases of thermal equilibrium so long as the basis states are chosen to be energy eigenstates. For most practical purposes, this can be taken as the fundamental definition of entropy since all other formulas for S can be mathematically derived from it, but not vice versa.
In what has been called the fundamental assumption of statistical thermodynamics or the fundamental postulate in statistical mechanics, the occupation of any microstate is assumed to be equally probable (i.e. P_{i}=1/Ω where Ω is the number of microstates); this assumption is usually justified for an isolated system in equilibrium.^{[17]} Then the previous equation reduces to:
In thermodynamics, such a system is one in which the volume, number of molecules, and internal energy are fixed (the microcanonical ensemble).
The most general interpretation of entropy is as a measure of our uncertainty 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 variables; maximizing the entropy maximizes our ignorance about the details of the system.^{[18]} This uncertainty is not of the everyday subjective kind, but rather the uncertainty inherent to the experimental method and interpretative model.
The interpretative model has a central role in determining entropy. The qualifier "for a given set of macroscopic variables" above has deep implications: if two observers use different sets of macroscopic variables, they will 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 of thermodynamics to observer A. In other words: the set of macroscopic variables one chooses must include everything that may change in the experiment, otherwise one might see decreasing entropy!^{[19]}
Entropy can be defined for any Markov processes with reversible dynamics and the detailed balance property.
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.
Entropy is the abovementioned unexpected and, to some, obscure integral that arises directly from the Carnot cycle. It is reversible heat divided by temperature. It is, remarkably, a function of state and it is fundamental and very useful.
In a thermodynamic system, pressure, density, and temperature tend to become uniform over time because this equilibrium state has higher probability (more possible combinations of microstates) than any other; see statistical mechanics. As an example, for a glass of ice water in air at room temperature, the difference in temperature between a warm room (the surroundings) and cold glass of ice and water (the system and not part of the room), begins to be equalized as portions of the thermal energy from the warm surroundings spread to the cooler system of ice and water. 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.
Thermodynamic entropy is a nonconserved state function that is of great importance in the sciences of physics and chemistry.^{[13]}^{[20]} Historically, the concept of entropy evolved in order to explain why some processes (permitted by conservation laws) occur spontaneously while their time reversals (also permitted by conservation laws) do not; systems tend to progress in the direction of increasing entropy.^{[21]}^{[22]} For isolated systems, entropy never decreases.^{[20]} This fact has several important consequences in science: first, it prohibits "perpetual motion" machines; and second, it implies the arrow of entropy has the same direction as the arrow of time. Increases in entropy correspond to irreversible changes in a system, because some energy is expended as waste heat, limiting the amount of work a system can do.^{[13]}^{[14]}^{[23]}^{[24]}
Unlike many other functions of state, entropy cannot be directly observed but must be calculated. Entropy can be calculated for a substance as the standard molar entropy from absolute zero (also known as absolute entropy) or as a difference in entropy from some other reference state which is defined as zero entropy. Entropy has the dimension of energy divided by temperature, which has a unit of joules per kelvin (J/K) in the International System of Units. While these are the same units as heat capacity, the two concepts are distinct.^{[25]} Entropy is not a conserved quantity: for example, in an isolated system with nonuniform temperature, heat might irreversibly flow and the temperature become more uniform such that entropy increases. The second law of thermodynamics, states that a closed system has entropy which may increase or otherwise remain constant. Chemical reactions cause changes in entropy and entropy plays an important role in determining in which direction a chemical reaction spontaneously proceeds.
One dictionary definition of entropy is that it is "a measure of thermal energy per unit temperature that is not available for useful work". For instance, a substance at uniform temperature is at maximum entropy and cannot drive a heat engine. A substance at nonuniform temperature is at a lower entropy (than if the heat distribution is allowed to even out) and some of the thermal energy can drive a heat engine.
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 change will be entirely due to the mixing of the different substances. At a statistical mechanical level, this results due to the change in available volume per particle with mixing.^{[26]}
The second law of thermodynamics states that in general the total entropy of any system will not decrease other than by increasing the entropy of some other system. Hence, in a system isolated from its environment, the entropy of that system will tend not to decrease. It follows that heat will not flow from a colder body to a hotter body without the application of work (the imposition of order) to the colder body. Secondly, it is impossible for any device operating on a cycle to produce net work from a single temperature reservoir; the production of net work requires flow of heat from a hotter reservoir to a colder reservoir, or a single expanding reservoir undergoing adiabatic cooling, which performs adiabatic work. As a result, there is no possibility of a perpetual motion system. 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.
It follows from the second law of thermodynamics that the entropy of a system that is not isolated may decrease. An air conditioner, for example, may cool the air in a room, thus reducing the entropy of the air of that system. The heat expelled from the room (the system), which the air conditioner transports and discharges to the outside air, will always make a bigger contribution 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 of thermodynamics.
In mechanics, the second law in conjunction with the fundamental thermodynamic relation places limits on a system's ability to do useful work.^{[27]} The entropy change of a system at temperature T absorbing an infinitesimal amount of heat δq in a reversible way, is given by δq/T. More explicitly, an energy T_{R} S is not available to do useful work, where T_{R} is the temperature of the coldest accessible reservoir or heat sink external to the system. For further discussion, see Exergy.
Statistical mechanics demonstrates that entropy is governed by probability, thus allowing for a decrease in disorder even in an isolated system. Although this is possible, such an event has a small probability of occurring, making it unlikely.^{[28]}
The entropy of a system depends on its internal energy and the external parameters, such as the volume. In the thermodynamic limit this fact leads to an equation relating the change in the internal energy to changes in the entropy and the external parameters. This relation is known as the fundamental thermodynamic relation. If the volume 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 the change from one state of thermal equilibrium to another with infinitesimally larger entropy and volume happens in a nonquasistatic way (so during this change the system may be very far out of thermal equilibrium and then the entropy, 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 relations between heat capacities.
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 an isolated system – the combination of a subsystem under study and its surroundings – increases during all spontaneous chemical and physical processes. The Clausius equation of δq_{rev}/T = ΔS introduces the measurement of entropy change, ΔS. Entropy change describes the direction and quantifies the magnitude of simple changes such as heat transfer between systems – always from hotter to cooler spontaneously.
The thermodynamic entropy therefore has the dimension of energy divided by temperature, and the unit joule per kelvin (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 many processes it is useful to specify the entropy as an intensive property independent of the size, as a specific entropy characteristic of the type of system studied. Specific entropy may be expressed relative to a unit of mass, typically the kilogram (unit: Jkg^{−1}K^{−1}). Alternatively, in chemistry, it is also referred to one mole of substance, in which case it is called the molar entropy with a unit of Jmol^{−1}K^{−1}.
Thus, when one mole of substance at about 0K is warmed by its surroundings to 298K, the sum of the incremental values of q_{rev}/T constitute each element's or compound's standard molar entropy, an indicator of the amount of energy stored by a substance at 298K.^{[29]}^{[30]} Entropy change also measures the mixing of substances as a summation of their relative quantities in the final mixture.^{[31]}
Entropy is equally essential in predicting the extent and direction of complex chemical reactions. For such applications, ΔS must be incorporated in an expression that includes both the system and its surroundings, ΔS_{universe} = ΔS_{surroundings} + ΔS _{system}. This expression becomes, via some steps, the Gibbs free energy equation for reactants and products in the system: ΔG [the Gibbs free energy change of the system] = ΔH [the enthalpy change] −T ΔS [the entropy change].^{[29]}
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 () and work, i.e. (shaft work) and P(dV/dt) (pressurevolume 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 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.
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 nonconserved, 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:^{[32]}
where
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.
The fundamental equation of thermodynamics for a system containing n constituent species, with the ith species having N_{i} particles is, with additional terms:
U is internal energy, T is temperature, P is pressure, V is volume, and N_{i} are the chemical potential and number of molecules of the chemical, and Q are electric potential and charge, v and p are velocity and momentum.
Solving for the change in entropy we get:
There is a minuscule change in internal energy for any change in entropy (ds will change by 1/T*dU). But in theory, the entropy of a system can be changed without changing its energy. That is done by keeping all variables constant, including temperature (isothermally) and entropy (adiabatically). That is easy to see, but typically the energy of the system will change. e.g. You can attempt to keep volume constant but you will always do work on the system, and work changes the energy.^{[33]} Other potentials such as the gravitational potential can also be taken into account.
For certain simple transformations in systems of constant composition, the entropy changes are given by simple formulas.^{[34]}
For the expansion (or compression) of an ideal gas from an initial volume and pressure to a final volume and pressure at any constant temperature, the change in entropy is given by:
Here is the number of moles of gas and is the ideal gas constant. These equations also apply for expansion into a finite vacuum or a throttling process, where the temperature, internal energy and enthalpy for an ideal gas remain constant.
For heating (or cooling) of any system (gas, liquid or solid) at constant pressure from an initial temperature to a final temperature , the entropy change is
provided that the constantpressure molar heat capacity (or specific heat) C_{P} is constant and that no phase transition occurs in this temperature interval.
Similarly at constant volume, the entropy change is
where the constantvolume heat capacity C_{v} is constant and there is no phase change.
At low temperatures near absolute zero, heat capacities of solids quickly drop off to near zero, so the assumption of constant heat capacity does not apply.^{[35]}
Since entropy is a state function, the entropy change of any process in which temperature and volume both vary is the same as for a path divided into two steps  heating at constant volume and expansion at constant temperature. For an ideal gas, the total entropy change is^{[36]}
Similarly if the temperature and pressure of an ideal gas both vary,
Reversible phase transitions occur at constant temperature and pressure. The reversible heat is the enthalpy change for the transition, and the entropy change is the enthalpy change divided by the thermodynamic temperature. For fusion (melting) of a solid to a liquid at the melting point T_{m}, the entropy of fusion is
Similarly for vaporization of a liquid to a gas at the boiling point T_{b}, the entropy of vaporization is
As a fundamental aspect of thermodynamics and physics, several different approaches to entropy beyond that of Clausius and Boltzmann are valid.
The following is a list of additional definitions of entropy from a collection of textbooks:
In Boltzmann's definition, entropy is a measure of the number of possible microscopic states (or microstates) of a system in thermodynamic equilibrium. Consistent with the Boltzmann definition, the second law of thermodynamics needs to be reworded as such that entropy increases over time, though the underlying principle remains the same.
Entropy has often been loosely associated with the amount of order, disorder, and/or chaos in a thermodynamic system. The traditional qualitative description 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. In this direction, several recent authors have derived exact entropy formulas to account for and measure disorder and order in atomic and molecular assemblies.^{[39]}^{[40]}^{[41]} One of the simpler entropy order/disorder formulas is that derived in 1984 by thermodynamic physicist Peter Landsberg, based on a combination of thermodynamics and information theory arguments. He 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:^{[40]}^{[41]}
Similarly, the total amount of "order" in the system is given by:
In which C_{D} is the "disorder" capacity of the system, which is the entropy of the parts contained in the permitted ensemble, C_{I} is the "information" capacity of the system, an expression similar to Shannon's channel capacity, and C_{O} is the "order" capacity of the system.^{[39]}
The concept of entropy can be described qualitatively as a measure of energy dispersal at a specific temperature.^{[42]} Similar terms have been in use from early in the history of classical thermodynamics, and with the development of statistical thermodynamics and quantum theory, entropy changes have been described in terms of the mixing or "spreading" of the total energy of each constituent of a system over its particular quantized energy levels.
Ambiguities in the terms disorder and chaos, which usually have meanings directly opposed to equilibrium, contribute to widespread confusion and hamper comprehension of entropy for most students.^{[43]} 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^{[44]} (compare discussion in next section). 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".^{[45]}
Following on from the above, it is possible (in a thermal context) to regard entropy as an indicator or measure of the effectiveness or usefulness of a particular quantity of energy.^{[46]} This is because energy supplied at a high temperature (i.e. with low entropy) tends to be more useful than the same amount of energy available at room temperature. Mixing a hot parcel of a fluid with a cold one produces a parcel of intermediate temperature, in which the 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 less useful: eventually, this will lead to the "heat death of the Universe".
A definition of entropy based entirely on the relation of adiabatic accessibility between equilibrium states was given by E.H.Lieb and J. Yngvason in 1999.^{[47]} This approach has several predecessors, including the pioneering work of Constantin Carathéodory from 1909 ^{[48]} and the monograph by R. Giles from 1964.^{[49]} In the setting of Lieb and Yngvason 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 the reference 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 complementary amount, , in the state . A simple but important result within this setting is that entropy is uniquely determined, apart from a choice of unit and an additive constant for each chemical element, by the following properties: It is monotonic with respect to the relation of adiabatic accessibility, additive on composite systems, and extensive under scaling.
In quantum statistical mechanics, the concept of entropy was developed by John von Neumann and is generally referred to as "von Neumann entropy",
where ρ is the density matrix and Tr is the trace operator.
This upholds the correspondence principle, because in the classical limit, when the phases between the basis states used for the classical probabilities are purely random, this expression is equivalent to the familiar classical definition of entropy,
i.e. in such a basis the density matrix is diagonal.
Von Neumann established a rigorous 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 function collapse is described as an irreversible process (the socalled 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.
I thought of calling it 'information', but the word was overly used, so I decided to call it 'uncertainty'. [...] 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.'
When viewed in terms of information theory, the entropy state function is simply the amount of information (in the Shannon sense) that would be needed to specify the full microstate of the system. This is left unspecified by the macroscopic description.
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.^{[51]} 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 p_{i} so that
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 the average amount of information in a 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.^{[16]}
The question of the link between information entropy and thermodynamic entropy is a debated topic. While most authors argue that there is a link between the two,^{[52]}^{[53]}^{[54]}^{[55]}^{[56]} a few argue that they have nothing to do with each other.^{[16]}
The expressions for the two entropies are similar. The information entropy H for equal probabilities p_{i} = p = 1/n is
where k is a constant which determines the units of entropy.^{[57]} There are many ways of demonstrating the equivalence of "information entropy" and "physics entropy", that is, the equivalence of "Shannon entropy" and "Boltzmann entropy". Nevertheless, some authors argue for dropping the word entropy for the H function of information theory and using Shannon's other term "uncertainty" instead.^{[58]}
Although the concept of entropy was originally a thermodynamic construct, it has been adapted in other fields of study, including information theory, psychodynamics, thermoeconomics/ecological economics, and evolution.^{[39]}^{[59]}^{[60]}
Entropy is the only quantity in the physical sciences that seems to imply a particular direction of progress, sometimes called an arrow of time. As time progresses, the second law of thermodynamics states that the entropy of an isolated system never decreases. Hence, from this perspective, entropy measurement is thought of as a kind of clock.
Since a finite universe is an isolated system, the Second Law of Thermodynamics states that its total entropy is constantly increasing. It has been speculated, since the 19th century, that the universe is fated to a heat death in which all the energy ends up as a homogeneous distribution of thermal energy, so that no more work can be extracted from any source.
If the universe can be considered to have generally increasing entropy, then – as Sir 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. The entropy of a black hole is proportional to the surface area of the black hole's event horizon.^{[62]} 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 entropyincreasing processes, if they are totally effective matter and energy traps. Hawking has, however, recently changed his stance on this aspect, in a paper which largely redefined the event horizons of black holes and suggested that the escape of energy from black holes might be possible due to quantum activity.^{[63]}
The role of entropy in cosmology remains a controversial subject since the time of Ludwig Bolzmann. Recent work has cast some 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, 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.^{[64]} 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 largescale thermodynamics extremely difficult.^{[65]}
The entropy gap is widely believed to have been originally opened up by the early rapid exponential expansion of the universe.
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