In thermodynamics, a numerical measure of disorder; symbol S, units J/K (joules per kelvin). As a system becomes increasingly disordered, its entropy increases. For example, the entropy of a system comprising a drop of ink and a tank of water increases when the drop of ink is added to the water and disperses through it, since dispersed ink is highly disordered. Entropy can never decrease, which in the ink-in-water example amounts to the observation that the particles of ink never spontaneously gather themselves back into a single drop.
Formally, an increase in entropy is equal to the quantity of heat added to a system divided by its temperature (in kelvin) at a constant temperature. For instance the increase in entropy of 1 kg of ice melting to water at 273 K (0ºC) is 1223 J/K. Freezing water to ice decreases the entropy of the water, but at the expense of increasing the entropy of the whole system, such as the refrigerator and room containing it. Entropy expresses a direction in time for processes which might otherwise appear to be reversible on grounds of energy conservation alone. Only for reversible processes in which no heat is added or removed is entropy constant. The second law of thermodynamics states that for all processes entropy is either constant or increases.
In thermodynamics, entropy is an extensive state function that accounts for the effects of irreversibility in thermodynamic systems, particularly in heat engines during an engine cycle. Spontaneous changes occur with an increase in entropy. In simple terms, entropy change is related to either a change to a more ordered or disordered state at a microscopic level, which is an early visualisation of the motional energy of molecules, and to the idea dissipation of energy via intermolecular molecular frictions and collisions. In recent years, entropy, from a non-mathematical perspective, has been interpreted in terms of the "dispersal" of energy.
Quantitatively, entropy, symbolized by S, is defined by the differential quantity dS = δQ / T, where δQ is the amount of heat absorbed in a reversible process in which the system goes from one state to another, and T is the absolute temperature. Entropy is one of the factors that determines the free energy of the system.
When a system's energy is defined as the sum of its "useful" energy, (e.g. that energy which cannot be used for external work, then entropy may be (most concretely) visualized as the "scrap" or "useless" energy whose energetic prevalance over the total energy of a system is directly proportional to the absolute temperature of the considered system, as is the case with the Gibbs free energy or Helmholtz free energy relations.
In terms of statistical mechanics, the entropy describes the number of the possible microscopic configurations of the system. The statistical definition of entropy is generally thought to be the more fundamental definition, from which all other important properties of entropy follow.
History
The short history of entropy begins with the work of mathematician Lazare Carnot who in his 1803 work Fundamental Principles of Equilibrium and Movement postulated that in any machine the accelerations and shocks of the moving parts all represent losses of moment of activity.
Definition and description of entropy
The entropy of a thermodynamic system can be interpreted in two distinct, but compatible, ways, i.e.
Macroscopic viewpoint (thermodynamics)
In a thermodynamic system, a "universe" consisting of "surroundings" and "systems" and made up of quantities of matter, its pressure differences, density differences, and temperature differences all tend to equalize over time. In the ice melting example, 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 heat energy from the warm surroundings become spread out to the cooler system of ice and water. The entropy of the room has decreased and 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.
A special case of entropy increase, the entropy of mixing, occurs when two or more different substances are mixed. If the substances are at the same temperature and pressure, there will be no net exchange of heat or work - the entropy increase will be entirely due to the mixing of the different substances.
From a macroscopic perspective, in classical thermodynamics the entropy is interpreted simply as a state function of a thermodynamic system: that is, a property depending only on the current state of the system, independent of how that state came to be achieved. The state function has the important property that, when multiplied by a reference temperature, it can be understood as a measure of the amount of energy in a physical system that cannot be used to do thermodynamic work; More precisely, in any process where the system gives up energy ΔE, and its entropy falls by ΔS, a quantity at least TR ΔS of that energy must be given up to the system's surroundings as unusable heat (TR is the temperature of the system's external surroundings). Let δQ be an element of the heat given up by the body to any reservoir of heat during its own changes, heat which it may absorb from a reservoir being here reckoned as negative, and T the absolute temperature of the body at the moment of giving up this heat, then the equation:
must be true for every reversible cyclical process, and the relation:
must hold good for every cyclical process which is in any way possible. It can be seen that the dimensions of entropy are energy divided by temperature, which is the same as the dimensions of Boltzmann's constant (k) and heat capacity. In this manner, the quantity "ΔS" is utilized as a type of internal ordering energy, which accounts for the effects of irreversibility, in the energy balance equation for any given system. ΔG = ΔH - TΔS, for example, which is a formula commonly utilized to determine if chemical reactions will occur, the energy related to entropy changes TΔS is subtracted from the "total" system energy ΔH to give the "free" energy ΔG of the system, as during a chemical process or as when a system changes state.
Microscopic viewpoint (statistical mechanics)
From a microscopic perspective, in statistical thermodynamics the entropy is a measure of the number of microscopic configurations that are capable of yielding the observed macroscopic description of the thermodynamic system:
where Ω is the number of microscopic configurations, and kB is Boltzmann's constant. In Boltzmann's 1896 Lectures on Gas Theory, he showed that this expression gives a measure of entropy for systems of atoms and molecules in the gas phase, thus providing a measure for the entropy of classical thermodynamics.
In 1877, thermodynamicist Ludwig Boltzmann visualized a probabilistic way to measure the entropy of an ensemble of ideal gas particles, in which he defined entropy to be proportional to the logarithm of the number of microstates such a gas could occupy.
Statistical mechanics explains entropy as the amount of uncertainty (or "mixedupness" in the phrase of Gibbs) which remains about a system, after its observable macroscopic properties have been taken into account. For a given set of macroscopic quantities, like temperature and volume, the entropy measures the degree to which the probability of the system is spread out over different possible quantum states. The more states available to the system with higher probability, and thus the greater the entropy. In essence, the most general interpretation of entropy is as a measure of our ignorance about a system. The equilibrium state of a system maximizes the entropy because we have lost all information about the initial conditions except for the conserved quantities; maximizing the entropy maximizes our ignorance about the details of the system.
On the molecular scale, the two definitions match up because adding heat to a system, which increases its classical thermodynamic entropy, also increases the system's thermal fluctuations, so giving an increased lack of information about the exact microscopic state of the system, i.e.
The second law
An important law of physics, the second law of thermodynamics, states that the total entropy of any isolated thermodynamic system tends to increase over time, approaching a maximum value;
In general, according to the second law, the entropy of a system that is not isolated may decrease. The heat, however, involved in operating the air conditioner always makes a bigger contribution to the entropy of the environment than the decrease of the entropy of the air. Thus the total entropy of the room and the environment increases, in agreement with the second law.
Entropy balance equation for open systems
In chemical engineering, the principles of thermodynamics are commonly applied to "open systems", i.e. (shaft work) and P(dV/dt) (pressure-volume work), across the system boundaries, the heat flow, but not the work flow, causes a change in the entropy of the system. This rate of entropy change is , where 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 non-conserved, such as entropy. 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:
where
= the net rate of entropy flow due to the flows of mass into and out of the system (where = entropy per unit mass). = the rate of entropy flow due to the flow of heat across the system boundary. = the rate of internal generation of entropy within the system.Note, also, that if there are multiple heat flows, the term is to be replaced by , where is the heat flow and Tj is the temperature at the jth heat flow port into the system.
Approaches to understanding entropy
Order and disorder
Entropy, historically, has often been associated with the amount of order, disorder, and or chaos in a thermodynamic system. The traditional definition of entropy is that it refers to changes in the status quo of the system and is a measure of "molecular disorder" and the amount of wasted energy in a dynamical energy transformation from one state or form to another. In this direction, a number of authors, in recent years, have derived exact entropy formulas to account for and measure disorder and order in atomic and molecular assemblies. One of the simpler entropy order/disorder formulas is that derived in 1984 by thermodynamic physicist Peter Landsberg, which is based on a combination of thermodynamics and information theory arguments. Landsberg argues that when constraints operate on a system, such that it is prevented from entering one or more of its possible or permitted states, as contrasted with its forbidden states, the measure of the total amount of “disorder” in the system is given by the following expression:
Similarly, the total amount of "order" in the system is given by:
In which CD is the "disorder" capacity of the system, which is the entropy of the parts contained in the permitted ensemble, CI is the "information" capacity of the system, an expression similar to Shannon's channel capacity, and CO is the "order" capacity of the system.
Energy dispersal
The concept of entropy can be described qualitatively as a measure of energy dispersal at a specific temperature. 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.
Information theory
The concept of entropy in information theory describes how much randomness (or, alternatively, "uncertainty") there is in a signal or random event. The entropy in statistical mechanics can be considered to be a specific application of Shannon entropy, according to a viewpoint known as Maximum entropy thermodynamics. The statistical mechanical entropy is then proportional to the minimum number of yes/no questions you have to ask in order to determine the microstate, given that you know the macrostate.
Information theory Definitions:
Information entropy - an information theory type of entropy related to noise in phone line signals or the amount of uncertainty about an event associated with a given probability distribution. Rényi entropy - a generalisation of information entropy; Binary entropy function - the entropy of a Bernoulli trial with probability of success p; Joint entropy - is the measure how much entropy is contained in a joint system of two random variables;Ice melting example
The illustration for this article is a classic example in which entropy increases in a small 'universe', a thermodynamic system consisting of the 'surroundings' (the warm room) and 'system' (glass, ice, cold water). In this universe, some heat energy δQ from the warmer room surroundings (at 298 K or 25 C) will spread out to the cooler system of ice and water at its constant temperature T of 273 K (0 C), the melting temperature of ice. The entropy of the system will change by the amount dS = δQ/T, in this example δQ/273 K. the ΔH for ice fusion.) The entropy of the surroundings will change by an amount dS = -δQ/298 K. So in this example, the entropy of the system increases, whereas the entropy of the surroundings decreases.
It is important to realize that the decrease in the entropy of the surrounding room is less than the increase in the entropy of the ice and water: the room temperature of 298 K is larger than 273 K and therefore the ratio, (entropy change), of δQ/298 K for the surroundings is smaller than the ratio (entropy change), of δQ/273 K for the ice+water system. To find the entropy change of our 'universe', we add up the entropy changes for its constituents: the surrounding room, and the ice+water. this is always true in spontaneous events in a thermodynamic system and it shows the predictive importance of entropy: the final net entropy after such an event is always greater than was the initial entropy. this thermodynamic system, has increased in entropy.
Topics in Entropy
Entropy and life
For over a century and a half, beginning with Clausius' 1863 memoir "On the Concentration of Rays of Heat and Light, and on the Limits of its Action", much writing and research has been devoted to the relationship between thermodynamic entropy and the evolution of life. In other cases, some creationists who have shown a thorough misunderstanding of entropy, have argued that entropy rules out evolution. In short, according to Lehninger, "living organisms preserve their internal order by taking from their surroundings free energy, in the form of nutrients or sunlight, and returning to their surroundings an equal amount of energy as heat and entropy."
Evolution related definitions:
Negentropy - a shorthand colloquial phrase for negative entropy.The arrow of time
Entropy is the only quantity in the physical sciences that "picks" a particular direction for time, sometimes called an arrow of time. As we go "forward" in time, the Second Law of Thermodynamics tells us that the entropy of an isolated system can only increase or remain the same;
Entropy and cosmology
We have previously mentioned that a finite universe may be considered an isolated system. As such, it may be subject to the Second Law of Thermodynamics, so that its total entropy is constantly increasing.
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.
The role of entropy in cosmology remains a controversial subject. Although entropy does increase in the model of an expanding universe, the maximum possible entropy rises much more rapidly and leads to an "entropy gap", thus pushing the system further away from equilibrium with each time increment. Gibbs entropy - the usual statistical mechanical entropy of a thermodynamic system. Boltzmann entropy - a type of Gibbs entropy, which neglects internal statistical correlations in the overall particle distribution. Tsallis entropy - a generalization of the standard Boltzmann-Gibbs entropy. Standard molar entropy - is the entropy content of one mole of substance, under conditions of standard temperature and pressure. Black hole entropy - is the entropy carried by a black hole, which is proportional to the surface area of the black hole. Residual entropy - the entropy present after a substance is cooled arbitrarily close to absolute zero. Entropy of mixing - the change in the entropy when two different chemical substances or components are mixed. Loop entropy - is the entropy lost upon bringing together two residues of a polymer within a prescribed distance. Conformational entropy - is the entropy associated with the physical arrangement of a polymer chain that assumes a compact or globular state in solution. Free entropy - an entropic thermodynamic potential analogous to the free energy. Entropy change – a change in entropy dS between two equilibrium states is given by the heat transferred dQrev divided by the absolute temperature T of the system in this interval. Sackur-Tetrode entropy - the entropy of a monatomic classical ideal gas determined via quantum considerations.
Other relations
Generalized entropy
Many generalizations of entropy have been studied, two of which, Tsallis and Rényi entropies, are widely used and the focus of active research.
The Rényi entropy is an information measure for fractal systems.
The Tsallis entropy is employed in Tsallis statistics to study nonextensive thermodynamics.
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