Entropy is a Science concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the microscopic description of nature in statistical physics, and to the principles of information theory. It has found far-ranging applications in chemistry and physics, in biological systems and their relation to life, in cosmology, economics, sociology, weather science, climate change and information systems including the transmission of information in telecommunication.
Entropy is central to the second law of thermodynamics, which states that the entropy of an isolated system left to spontaneous evolution cannot decrease with time. As a result, isolated systems evolve toward thermodynamic equilibrium, where the entropy is highest. A consequence of the second law of thermodynamics is that certain processes are irreversible.
The thermodynamic concept was referred to by Scottish scientist and engineer William Rankine in 1850 with the names thermodynamic function and heat-potential. In 1865, German physicist Rudolf Clausius, one of the leading founders of the field of thermodynamics, defined it as the quotient of an infinitesimal amount of heat to the instantaneous temperature. He initially described it as transformation-content, in German Verwandlungsinhalt, and later coined the term entropy from a Greek word for transformation.Brush, S.G. (1976). The Kind of Motion We Call Heat: a History of the Kinetic Theory of Gases in the 19th Century, Book 2, Statistical Physics and Irreversible Processes, Elsevier, Amsterdam, , pp. 576–577.
Austrian physicist Ludwig Boltzmann explained entropy as the measure of the number of possible microscopic arrangements or states of individual atoms and molecules of a system that comply with the macroscopic condition of the system. He thereby introduced the concept of statistical disorder and probability distributions into a new field of thermodynamics, called statistical mechanics, and found the link between the microscopic interactions, which fluctuate about an average configuration, to the macroscopically observable behaviour, in form of a simple law, with a proportionality constant, the Boltzmann constant, which has become one of the defining universal constants for the modern International System of Units.
The first law of thermodynamics, deduced from the heat-friction experiments of James Joule in 1843, expresses the concept of energy and its conservation in all processes; the first law, however, is unsuitable to separately 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 that 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. On : Poggendorff's Annalen der Physik und Chemie. He described his observations as a dissipative use of energy, resulting in a transformation-content (Verwandlungsinhalt in German), of a thermodynamic system or working body of chemical species during a change of state. That was in contrast to earlier views, based on the theories of Isaac Newton, that heat was an indestructible particle that had mass. Clausius discovered that the non-usable energy increases as steam proceeds from inlet to exhaust in a steam engine. From the prefix en-, as in 'energy', and from the Greek word τροπή tropē, which is translated in an established lexicon as turning or changeLiddell, H. G., Scott, R. (1843/1978). A Greek–English Lexicon, revised and augmented edition, Oxford University Press, Oxford UK, , pp. 1826–1827. and that he rendered in German as Verwandlung, a word often translated into English as transformation, in 1865 Clausius coined the name of that property as entropy. The word was adopted into the English language in 1868.
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 as proportional to the natural logarithm of the number of microstates such a gas could occupy. The proportionality constant in this definition, called the Boltzmann constant, has become one of the defining universal constants for the modern International System of Units (SI). Henceforth, the essential problem in statistical thermodynamics has been to determine the distribution of a given amount of energy E over N identical systems. Constantin Carathéodory, a Greek mathematician, linked entropy with a mathematical definition of irreversibility, in terms of trajectories and integrability.
In more detail, Clausius explained his choice of "entropy" as a name as follows:
I prefer going to the ancient languages for the names of important scientific quantities, so that they may mean the same thing in all living tongues. I propose, therefore, to call S the entropy of a body, after the Greek word "transformation". I have designedly coined the word entropy to be similar to energy, for these two quantities are so analogous in their physical significance, that an analogy of denominations seems to me helpful.Leon Cooper added that in this way "he succeeded in coining a word that meant the same thing to everybody: nothing".
In contrast, an irreversible process increases the total entropy of the system and surroundings. Any process that happens quickly enough to deviate from the thermal equilibrium cannot be reversible; the total entropy increases, and the potential for maximum work to be done during the process is lost.
It is known that a work produced by an engine over a cycle equals to a net heat absorbed over a cycle.. Thus, with the sign convention for a heat transferred in a thermodynamic process ( for an absorption and for a dissipation) we get:Since this equality holds over an entire Carnot cycle, it gave Clausius the hint that at each stage of the cycle the difference between a work and a net heat would be conserved, rather than a net heat itself. Which means there exists a state function with a change of . It is called an internal energy and forms a central concept for the first law of thermodynamics.
Finally, comparison for both the representations of a work output in a Carnot cycle gives us:.Similarly to the derivation of internal energy, this equality implies existence of a state function with a change of and which is conserved over an entire cycle. Clausius called this state function entropy.
In addition, the total change of entropy in both thermal reservoirs over Carnot cycle is zero too, since the inversion of a heat transfer direction means a sign inversion for the heat transferred during isothermal stages:Here we denote the entropy change for a thermal reservoir by , where is either for a hot reservoir or for a cold one.
If we consider a heat engine which is less effective than Carnot cycle (i.e., the work produced by this engine is less than the maximum predicted by Carnot's theorem), its work output is capped by Carnot efficiency as:Substitution of the work as the net heat into the inequality above gives us:or in terms of the entropy change :A Carnot cycle and an entropy as shown above prove to be useful in the study of any classical thermodynamic heat engine: other cycles, such as an Otto cycle, Diesel cycle or Brayton cycle, could be analysed from the same standpoint. Notably, any machine or cyclic process converting heat into work (i.e., heat engine) that is claimed to produce an efficiency greater than the one of Carnot is not viable — due to violation of the second law of thermodynamics.
For further analysis of sufficiently discrete systems, such as an assembly of particles, statistical thermodynamics must be used. Additionally, descriptions of devices operating near the limit of Matter wave, e.g. Solar cell, have to be consistent with quantum statistics.
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 for irreversible processes. The difference between an isolated system and closed system is that energy may not flow to and from an isolated system, but energy 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 theorem, for a reversible cyclic thermodynamic process: which means the line integral is State function. Thus we can define a state function , called entropy:Therefore, thermodynamic entropy has the dimension of energy divided by temperature, and the unit joule per kelvin (J/K) in the International System of Units (SI).
To find the entropy difference between any two states of the system, the integral must be evaluated for some reversible path between the initial and final states. Since an entropy is a state function, the entropy change of the system for an irreversible path is the same as for a reversible path between the same two states. However, the heat transferred to or from the surroundings is different as well as its entropy change.
We can calculate the change of entropy only by integrating the above formula. To obtain the absolute value of the entropy, we consider the third law of thermodynamics: perfect crystals at the absolute zero have an entropy .
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 of energy to the surrounding at the temperature , its entropy falls by and at least of that energy must be given up to the system's surroundings as a heat. Otherwise, this process cannot go forward. In classical thermodynamics, the entropy of a system is defined if and only if it is in a thermodynamic equilibrium (though a chemical equilibrium is not required: for example, the entropy of a mixture of two moles of hydrogen and one mole of oxygen in standard conditions is well-defined).
The interpretation of entropy in statistical mechanics is the measure of uncertainty, disorder, 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 momentum of every molecule. The more such states are available to the system with appreciable probability, the greater the entropy. In statistical mechanics, entropy is a measure of the number of ways a system can be arranged, often taken to be a measure of "disorder" (the higher the entropy, the higher the disorder).
The Boltzmann constant, and therefore entropy, have dimensions of energy divided by temperature, which has a unit of per kelvin (J⋅K−1) in the International System of Units (or kg⋅m2⋅s−2⋅K−1 in terms of base units). The entropy of a substance is usually given as an intensive property — either entropy per unit mass (SI unit: J⋅K−1⋅kg−1) or entropy per unit amount of substance (SI unit: J⋅K−1⋅mol−1).
Specifically, entropy is a logarithmic measure for the system with a number of states, each with a probability of being occupied (usually given by the Boltzmann distribution):where is the Boltzmann constant and the summation is performed over all possible microstates of the system. Frigg, R. and Werndl, C. "Entropy – A Guide for the Perplexed" . In Probabilities in Physics; Beisbart C. and Hartmann, S. (eds.); Oxford University Press, Oxford, 2010.
In case states are defined in a continuous manner, the summation is replaced by an integral over all possible states, or equivalently we can consider the expected value of the logarithm of the probability that a microstate is occupied:This definition assumes the basis states to be picked in a way that there is no information on their relative phases. In a general case the expression is:where is a density matrix, is a Trace class and is a matrix logarithm. The density matrix formalism is not required if the system is in thermal equilibrium so long as the basis states are chosen to be Quantum state of the Hamiltonian. For most practical purposes it can be taken as the fundamental definition of entropy since all other formulae for can be derived from it, but not vice versa.
In what has been called the fundamental postulate in statistical mechanics, among system microstates of the same energy (i.e., degenerate microstates) each microstate is assumed to be populated with equal probability , where is the number of microstates whose energy equals that of the system. Usually, this assumption is justified for an isolated system in a thermodynamic equilibrium. Then in case of an isolated system the previous formula reduces to:In thermodynamics, such a system is one with a fixed volume, number of molecules, and internal energy, called a microcanonical ensemble.
The most general interpretation of entropy is as a measure of the extent of uncertainty about a system. The equilibrium state of a system maximizes the entropy because it does not reflect all information about the initial conditions, except for the conserved variables. 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 when two observers use different sets of macroscopic variables. For example, consider observer A using variables , , and observer B using variables , , , . If observer B changes variable , then observer A will see a violation of the second law of thermodynamics, since he does not possess information about variable and its influence on the system. In other words, one must choose a complete set of macroscopic variables to describe the system, i.e. every independent parameter that may change during experiment.
Entropy can also be defined for any 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.
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 equalisation has progressed.
Thermodynamic entropy is a non-conserved state function that is of great importance in the sciences of physics and chemistry. Historically, the concept of entropy evolved to explain why some processes (permitted by conservation laws) occur spontaneously while their T-symmetry (also permitted by conservation laws) do not; systems tend to progress in the direction of increasing entropy.
Unlike many other functions of state, entropy cannot be directly observed but must be calculated. Absolute standard molar entropy of a substance can be calculated from the measured temperature dependence of its heat capacity. The molar entropy of ions is obtained as a difference in entropy from a reference state defined as zero entropy. The second law of thermodynamics states that the entropy of an isolated system must increase or remain constant. Therefore, entropy is not a conserved quantity: for example, in an isolated system with non-uniform temperature, heat might irreversibly flow and the temperature become more uniform such that entropy increases. Chemical reactions cause changes in entropy and system entropy, in conjunction with enthalpy, 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" in a cyclic process. For instance, a substance at uniform temperature is at maximum entropy and cannot drive a heat engine. A substance at non-uniform 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 is no net exchange of heat or work – the entropy change is 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.
Furthermore, it has been shown that the definitions of entropy in statistical mechanics is the only entropy that is equivalent to the classical thermodynamics entropy under the following postulates:
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, always makes a bigger contribution to the entropy of the environment than 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. The entropy change of a system at temperature absorbing an infinitesimal amount of heat in a reversible way, is given by . More explicitly, an energy is not available to do useful work, where 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.
The applicability of a second law of thermodynamics is limited to systems in or sufficiently near equilibrium state, so that they have defined entropy. Some inhomogeneous systems out of thermodynamic equilibrium still satisfy the hypothesis of local thermodynamic equilibrium, so that entropy density is locally defined as an intensive quantity. For such systems, there may apply a principle of maximum time rate of entropy production. It states that such a system may evolve to a steady state that maximises its time rate of entropy production. This does not mean that such a system is necessarily always in a condition of maximum time rate of entropy production; it means that it may evolve to such a steady state.
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 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: J⋅kg−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 J⋅mol−1⋅K−1.
Thus, when one mole of substance at about is warmed by its surroundings to , the sum of the incremental values of constitute each element's or compound's standard molar entropy, an indicator of the amount of energy stored by a substance at . Entropy change also measures the mixing of substances as a summation of their relative quantities in the final mixture.
Entropy is equally essential in predicting the extent and direction of complex chemical reactions. For such applications, must be incorporated in an expression that includes both the system and its surroundings: Via additional steps this expression becomes the equation of Gibbs free energy change for reactants and products in the system at the constant pressure and temperature :where is the enthalpy change and is the entropy change.
+ | + | Spontaneous at high T | Ice melting |
– | – | Spontaneous at low T | Water freezing |
– | + | Spontaneous at all T | Propane combustion |
+ | – | Non-spontaneous at all T | Ozone formation |
To derive a generalised entropy balanced equation, we start with the general balance equation for the change in any extensive quantity 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 , i.e. the rate of change of in the system, equals the rate at which enters the system at the boundaries, minus the rate at which leaves the system across the system boundaries, plus the rate at which is generated within the system. For an open thermodynamic system in which heat and work are transferred by paths separate from the paths for transfer of matter, using this generic balance equation, with respect to the rate of change with time of the extensive quantity entropy , the entropy balance equation is:The overdots represent derivatives of the quantities with respect to time.where is the net rate of entropy flow due to the flows of mass into and out of the system with entropy per unit mass , is the rate of entropy flow due to the flow of heat across the system boundary and is the rate of entropy generation within the system, e.g. by chemical reactions, , internal heat transfer or Friction such as viscosity.
In case of multiple heat flows the term is replaced by , where is the heat flow through -th port into the system and is the temperature at the -th port.
The nomenclature "entropy balance" is misleading and often deemed inappropriate because entropy is not a conserved quantity. In other words, the term is never a known quantity but always a derived one based on the expression above. Therefore, the open system version of the second law is more appropriately described as the "entropy generation equation" since it specifies that:with zero for reversible process and positive values for irreversible one.
Similarly at constant volume, the entropy change is:where the constant-volume molar heat capacity 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.
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:Similarly if the temperature and pressure of an ideal gas both vary:
In Boltzmann's analysis in terms of constituent particles, entropy is a measure of the number of possible microscopic states (or microstates) of a system in thermodynamic equilibrium.
Here, is the "disorder" capacity of the system, which is the entropy of the parts contained in the permitted ensemble, is the "information" capacity of the system, an expression similar to Shannon's channel capacity, and is the "order" capacity of the system.
Ambiguities in the terms disorder and chaos, which usually have meanings directly opposed to equilibrium, contribute to widespread confusion and hamper comprehension of entropy for most students. As the second law of thermodynamics shows, in an isolated system internal portions at different temperatures 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 (compare discussion in next section). Physical chemist Peter Atkins, in his textbook Physical Chemistry, introduces entropy with the statement that "spontaneous changes are always accompanied by a dispersal of energy or matter and often both".
As the entropy of the universe is steadily increasing, its total energy is becoming less useful. Eventually, this is theorised to lead to the heat death of the universe.
This upholds the correspondence principle, because in the classical limit, when the phases between the basis states are purely random, this expression is equivalent to the familiar classical definition of entropy for states with classical probabilities :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 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.
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 size of information of 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 binary questions needed to determine the content of the message.
Most researchers consider information entropy and thermodynamic entropy directly linked to the same concept, while others argue that they are distinct.
The process of measurement goes as follows. First, a sample of the substance is cooled as close to absolute zero as possible. At such temperatures, the entropy approaches zerodue to the definition of temperature. Then, small amounts of heat are introduced into the sample and the change in temperature is recorded, until the temperature reaches a desired value (usually 25 °C). The obtained data allows the user to integrate the equation above, yielding the absolute value of entropy of the substance at the final temperature. This value of entropy is called calorimetric entropy.
Entropy has been proven useful in the analysis of base pair sequences in DNA. Many entropy-based measures have been shown to distinguish between different structural regions of the genome, differentiate between coding and non-coding regions of DNA, and can also be applied for the recreation of evolutionary trees by determining the evolutionary distance between different species.
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 . The entropy of a black hole is proportional to the surface area of the black hole's event horizon. 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.
The role of entropy in cosmology remains a controversial subject since the time of Ludwig Boltzmann. 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.
Current theories suggest the entropy gap to have been originally opened up by the early rapid exponential expansion of the universe.. In honor of John Wheeler's 90th birthday.
In economics, Georgescu-Roegen's work has generated the term 'entropy pessimism'. Since the 1990s, leading ecological economist and steady-state theorist Herman Daly – a student of Georgescu-Roegen – has been the economics profession's most influential proponent of the entropy pessimism position.
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