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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.
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:.
The summation is over all the possible microstates of the system, and p i is the probability that the system is in the i -th microstate. In a different basis set, the more general expression is.
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.
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.
The interpretative model has a central role in determining entropy. The qualifier "for a given set of macroscopic variables" above has deep implications: 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.
Entropy can be defined for any Markov processes with reversible dynamics and the detailed balance property.
Entropy arises directly from the Carnot cycle. It can also be described as the reversible heat divided by temperature.
Entropy is a fundamental function of state. In a thermodynamic system , pressure, density, and temperature tend to become uniform over time because the equilibrium state has higher probability more possible combinations of microstates than any other state.
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 equalize 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. In other words, 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 non-conserved state function that is of great importance in the sciences of physics and chemistry.
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.
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.
While these are the same units as heat capacity , the two concepts are distinct. 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 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. Hence, in a system isolated from its environment, the entropy of that system tends not to decrease.
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, 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.
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. The applicability of a second law of thermodynamics is limited to systems which are near or in equilibrium state.
One of the guiding principles for such systems is the maximum entropy production principle. The entropy of a system depends on its internal energy and its external parameters, such as its volume.
In the thermodynamic limit, this fact leads to an equation relating the change in the internal energy U to changes in the entropy and the external parameters.
This relation is known as the fundamental thermodynamic relation. If external pressure P bears on the volume V as the only external parameter, this relation is:.
Since both internal energy and entropy are monotonic functions of temperature T , implying that 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 non-quasistatic 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.
Entropy change describes the direction and quantifies the magnitude of simple changes such as heat transfer between systems — always from hotter to cooler spontaneously.
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: Entropy is equally essential in predicting the extent and direction of complex chemical reactions.
This expression becomes, via some steps, the Gibbs free energy equation for reactants and products in the system: In chemical engineering , the principles of thermodynamics are commonly applied to " open systems ", i.
If there are mass flows across the system boundaries, they also influence the total entropy of the system. This account, in terms of heat and work, is valid only for cases in which the work and heat transfers are by paths physically distinct from the paths of entry and exit of matter from 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 t of the extensive quantity entropy S , the entropy balance equation is: For certain simple transformations in systems of constant composition, the entropy changes are given by simple formulas.
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.
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 . 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. Consistent with the Boltzmann definition, the second law of thermodynamics needs to be re-worded 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 or disorder , or of 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.
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: The concept of entropy can be described qualitatively as a measure of energy dispersal at a specific temperature.
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.
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 , 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".
Following on from the above, it is possible in a thermal context to regard lower entropy as an indicator or measure of the effectiveness or usefulness of a particular quantity of energy.
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: A definition of entropy based entirely on the relation of adiabatic accessibility between equilibrium states was given by E.
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 ",.
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,.
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. Conversation between Claude Shannon and John von Neumann regarding what name to give to the attenuation in phone-line signals .
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.
It was originally devised by Claude Shannon in 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. 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,      a few argue that they have nothing to do with each other.
The Shannon entropy in nats is:. There are many ways of demonstrating the equivalence of "information entropy" and "physics entropy", that is, the equivalence of "Shannon entropy" and "Boltzmann entropy".
Entropy of a substance can be measured, although in an indirect way. 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 zero—due to the definition of temperature. 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 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 in large systems over significant periods of time.
Hence, from this perspective, entropy measurement is thought of as a clock in these conditions. Since a finite universe is an isolated system, the second law of thermodynamics states that its total entropy is continually 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 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.
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. Romanian American economist Nicholas Georgescu-Roegen , a progenitor in economics and a paradigm founder of ecological economics , made extensive use of the entropy concept in his magnum opus on The Entropy Law and the Economic Process.
From Wikipedia, the free encyclopedia. This article is about entropy in thermodynamics. For other uses, see Entropy disambiguation.
Not to be confused with Enthalpy. For a more accessible and less technical introduction to this topic, see Introduction to entropy. The classical Carnot heat engine.
Classical Statistical Chemical Quantum thermodynamics. Zeroth First Second Third. Conjugate variables in italics. Free energy Free entropy.
History General Heat Entropy Gas laws. Caloric theory Theory of heat. 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.
Second law of thermodynamics. Thermodynamic Equilibrium and Non-equilibrium thermodynamics. Entropy order and disorder.
I thought of calling it "information", but the word was overly used, so I decided to call it "uncertainty".
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.
Entropy information theory , Entropy in thermodynamics and information theory , and Entropic uncertainty. Poker sites are careful to ensure that player information is always secured at all stages, from the moment you make a transaction to when your account details are stored on their servers.
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