Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state 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 farranging 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.
The thermodynamic concept was referred to by Scottish scientist and engineer William Rankine in 1850 with the names thermodynamic function and heatpotential.
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 transformationcontent, in German Verwandlungsinhalt, and later coined the term entropy from a Greek word for transformation. Referring to microscopic constitution and structure, in 1862, Clausius interpreted the concept as meaning disgregation.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.A consequence of entropy is that certain processes are irreversible or impossible, aside from the requirement of not violating the conservation of energy, the latter being expressed in the first law of thermodynamics. Entropy is central to the second law of thermodynamics, which states that the entropy of isolated systems left to spontaneous evolution cannot decrease with time, as they always arrive at a state of thermodynamic equilibrium, where the entropy is highest.
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 behavior, in form of a simple law, with a proportionality constant, the Boltzmann constant, that has become one of the defining universal constants for the modern International System of Units (SI).
In 1948, Bell Labs scientist Claude Shannon developed similar statistical concepts of measuring microscopic uncertainty and multiplicity to the problem of random losses of information in telecommunication signals. Upon John von Neumann's suggestion, Shannon named this entity of missing information in analogous manner to its use in statistical mechanics as entropy, and gave birth to the field of information theory. This description has been identified as a universal definition of the concept of entropy.
The first law of thermodynamics, deduced from the heatfriction 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 transformationcontent ( 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 nonusable 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. "Sucht man für S einen bezeichnenden Namen, so könnte man, ähnlich wie von der Gröſse U gesagt ist, sie sey der Wärme und Werkinhalt des Körpers, von der Gröſse S sagen, sie sey der Verwandlungsinhalt des Körpers. Da ich es aber für besser halte, die Namen derartiger für die Wissenschaft wichtiger Gröſsen aus den alten Sprachen zu entnehmen, damit sie unverändert in allen neuen Sprachen angewandt werden können, so schlage ich vor, die Gröſse S nach dem griechischen Worte ἡ τροπή, die Verwandlung, die Entropie des Körpers zu nennen. Das Wort Entropie habei ich absichtlich dem Worte Energie möglichst ähnlich gebildet, denn die beiden Gröſsen, welche durch diese Worte benannt werden sollen, sind ihren physikalischen Bedeutungen nach einander so nahe verwandt, daſs eine gewisse Gleichartigkeit in der Benennung mir zweckmäſsig zu seyn scheint." (p. 390). 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."
Here $W$ is work done by the Carnot heat engine, $Q\_\backslash text\{H\}$ is heat to the engine from the hot reservoir, and $\backslash frac\{T\_\backslash text\{C\}\}\{T\_\backslash text\{H\}\}Q\_\backslash text\{H\}$ is heat to the cold reservoir from the engine. To derive the Carnot efficiency, which is (a number less than one), Kelvin had to evaluate the ratio of the work output to the heat absorbed during the isothermal expansion with the help of the Carnot–Clapeyron equation, which contained an unknown function called the Carnot function. The possibility that the Carnot function could be the temperature as measured from a zero point of temperature was suggested by Joule in a letter to Kelvin. This allowed Kelvin to establish his absolute temperature scale.
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 the change of a state function that would vanish upon completion of the cycle. The state function was called the internal energy, that is central to the first law of thermodynamics.
Now equating () and () gives, for the engine per Carnot cycle,.
This implies that there is a function of state whose change is and this state function is conserved over a complete Carnot cycle, like other state function such as the internal energy. Clausius called this state function entropy. One can see that entropy was discovered through mathematics rather than through laboratory experimental 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. This equation shows an entropy change per Carnot cycle is zero. In fact, an entropy change in the both thermal reservoirs per Carnot cycle is also zero since that change is simply expressed by reverting the sign of each term in the equation () according to the fact that, for example, for heat transfer from the hot reservoir to the engine, the engine receives the heat while the hot reservoir loses the same amount of the heat;
where we denote an entropy change for a thermal reservoir by , for i as either H (Hot reservoir) or C (Cold reservoir), by considering the abovementioned signal convention of heat for the engine.
Clausius then asked what would happen if less work is produced by the system than that predicted by Carnot's principle for the same thermal reservoir pair and the same heat transfer from the hot reservoir to the engine . In this case, the righthand side of the equation () would be the upper bound of the work output by the system, and the equation would now be converted into an inequality$$W\; <\; \backslash left(1\; \; \backslash frac\{T\_\backslash text\{C\}\}\{T\_\backslash text\{H\}\}\backslash right)\; Q\_\backslash text\{H\}$$When the equation () is used to express the work as a net or total heat exchanged in a cycle, we get$$Q\_\backslash text\{H\}+Q\_\backslash text\{C\}<\backslash left(1\backslash frac\{T\_\backslash text\{C\}\}\{T\_\backslash text\{H\}\}\backslash right)Q\_\backslash text\{H\}$$or$$Q\_\backslash text\{C\}>\backslash frac\{T\_\backslash text\{C\}\}\{T\_\backslash text\{H\}\}Q\_\backslash text\{H\}$$by considering the sign convention of heat where > 0 is heat that is from the hot reservoir and is absorbed by the engine and < 0 is the waste heat given off to the cold reservoir from the engine. So, more heat is given up to the cold reservoir than in the Carnot cycle. The above inequality $Q\_\backslash text\{H\}+Q\_\backslash text\{C\}<\backslash left(1\backslash frac\{T\_\backslash text\{C\}\}\{T\_\backslash text\{H\}\}\backslash right)Q\_\backslash text\{H\}$ can be written as$$\backslash frac\{Q\_\backslash text\{H\}\}\{T\_\backslash text\{H\}\}\; +\; \backslash frac\{Q\_\backslash text\{C\}\}\{T\_\backslash text\{C\}\}\; <\; 0.$$If we, again, denote an entropy change for a thermal reservoir by , for i as either H (Hot reservoir) or C (Cold reservoir), by considering the abovementioned signal convention of heat for the engine, then$$\backslash Delta\; S\_\backslash text\{r,H\}+\; \backslash Delta\; S\_\backslash text\{r,C\}\; >\; 0$$or telling that the magnitude of the entropy earned by the cold reservoir is greater than the entropy lost by the hot reservoir. The net entropy change in the engine per its thermodynamic cycle is zero, so the net entropy change in the engine and both the thermal reservoirs per cycle increases if work produced by the engine is less than the work achieved by a Carnot engine in the equation ().
The Carnot cycle and Carnot efficiency as shown in the equation () are useful because they define the upper bound of the possible work output and the efficiency of any classical thermodynamic heat engine. Other cycles, such as the Otto cycle, Diesel cycle and Brayton cycle, can be analyzed from the standpoint of the Carnot cycle. Any machine or cyclic process that converts heat to work and is claimed to produce an efficiency greater than the Carnot efficiency is not viable because it violates the second law of thermodynamics.
For very small numbers of particles in the system, statistical thermodynamics must be used. The efficiency of devices such as photovoltaic cells requires an analysis from the standpoint of quantum mechanics.
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 process: $\backslash oint\; \backslash frac\{\backslash delta\; Q\_\backslash text\{rev\}\}\{T\}\; =\; 0$. This means the line integral $\backslash int\_L\; \backslash frac\{\backslash delta\; Q\_\backslash text\{rev\}\}\{T\}$ is State function.
So we can define a state function called entropy, which satisfies $d\; S\; =\; \backslash frac\{\backslash delta\; Q\_\backslash text\{rev\}\}\{T\}$.
To find the entropy difference between any two states of a system, the integral must be evaluated for some reversible path between the initial and final states.
Since 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, and the entropy change of, the surroundings is different.We can only obtain the change of entropy by integrating the above formula. To obtain the absolute value of the entropy, 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 Δ E, and its entropy falls by Δ S, a quantity at least T_{R} Δ S of that energy must be given up to the system's surroundings as heat ( T_{R} is the temperature of the system's external surroundings). Otherwise the process cannot go forward. In classical thermodynamics, the entropy of a system is defined only if it is in physical thermodynamic equilibrium. (But chemical equilibrium is not required: the entropy of a mixture of two moles of hydrogen and one mole of oxygen at 1 bar pressure and 298 K is welldefined.)
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 velocity 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⋅m^{2}⋅s^{−2}⋅K^{−1} in terms of base units). The entropy of a substance is usually given as an intensive propertyeither 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 of the number of system states with significant probability of being occupied:
($p\_i$ is the probability that the system is in $i$th state, usually given by the Boltzmann distribution; if states are defined in a continuous manner, the summation is replaced by an integral over all possible states) or, equivalently, the expected value of the logarithm of the probability that a microstate is occupied
where k_{B} is the Boltzmann constant, equal to . 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. 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 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 $\backslash widehat\{\backslash rho\}$ is the density matrix, $\backslash operatorname\{Tr\}$ is trace and $\backslash log$ 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, among system microstates of the same energy (degenerate microstates) each microstate is assumed to be populated with equal probability; this assumption is usually justified for an isolated system in equilibrium.
Then for an isolated system p_{ i} = 1/Ω, where Ω is the number of microstates whose energy equals the system's energy, and the previous equation reduces toIn thermodynamics, such a system is one in which the volume, number of molecules, and internal energy are fixed (the microcanonical ensemble).
For a given thermodynamic system, the excess entropy is defined as the entropy minus that of an ideal gas at the same density and temperature, a quantity that is always negative because an ideal gas is maximally disordered.
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: if two observers use different sets of macroscopic variables, they see 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.
Entropy can 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.
In a thermodynamic system, pressure and temperature tend to become uniform over time because the equilibrium state has higher probability (more possible 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 the warm room (the surroundings) and the cold glass of ice and water (the system and not part of the room) decreases 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, of which the entropy has increased.
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.
Historically, the concept of entropy evolved to explain why some processes (permitted by conservation laws) occur spontaneously while their Tsymmetry (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 nonuniform 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 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 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 $T$ absorbing an infinitesimal amount of heat $\backslash delta\; q$ in a reversible way, is given by $\backslash delta\; q\; /\; T$. 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.
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 maximizes 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.
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 nonquasistatic way (so during this change the system may be very far out of thermal equilibrium and then the wholesystem 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.
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: 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 $q\_\{\backslash text\{rev\}\}\; /\; T$ 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, $\backslash Delta\; S$ must be incorporated in an expression that includes both the system and its surroundings, $\backslash Delta\; S\_\{\backslash text\{universe\}\}\; =\; \backslash Delta\; S\_\{\backslash text\{surroundings\}\}\; +\; \backslash Delta\; S\_\{\backslash text\{system\}\}$. This expression becomes, via some steps, the Gibbs free energy equation for reactants and products in the system: $\backslash Delta\; G$ the $=\; \backslash Delta\; H$ the $\; T\backslash ,\backslash Delta\; S$ the.
To derive a generalized entropy balanced equation, we start with the general balance equation for the change in any extensive quantity $\backslash theta$ 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 $d\backslash theta/dt$, i.e. the rate of change of $\backslash theta$ in the system, equals the rate at which $\backslash theta$ enters the system at the boundaries, minus the rate at which $\backslash theta$ leaves the system across the system boundaries, plus the rate at which $\backslash theta$ 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 $t$ of the extensive quantity entropy $S$, the entropy balance equation is:
The overdots represent derivatives of the quantities with respect to time.
where
If there are multiple heat flows, the term $\backslash dot\{Q\}/T$ is replaced by $\backslash sum\; \backslash dot\{Q\}\_j/T\_j,$ where $\backslash dot\{Q\}\_j$ is the heat flow and $T\_j$ is the temperature at the $j$th heat flow port into the system.
Note that the nomenclature "entropy balance" is misleading and often deemed inappropriate because entropy is not a conserved quantity. In other words, the term $\backslash dot\{S\}\_\backslash text\{gen\}$ 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 $\backslash dot\{S\}\_\backslash text\{gen\}\; \backslash ge\; 0$, with zero for reversible processes or greater than zero for irreversible ones.
Here $n$ is the amount of gas (in moles) and $R$ 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.
Similarly at constant volume, the entropy change is
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.
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.
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 leads to the heat death of the universe.
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,
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.
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.
which is the Boltzmann entropy formula, where $k$ is the Boltzmann constant, which may be interpreted as the thermodynamic entropy per nat. Some authors argue for dropping the word entropy for the $H$ function of information theory and using Shannon's other term, "uncertainty", instead.Schneider, Tom, DELILA system (Deoxyribonucleic acid Library Language), (Information Theory Analysis of binding sites), Laboratory of Mathematical Biology, National Cancer Institute, Frederick, MD
The resulting relation describes how entropy changes $dS$ when a small amount of energy $dQ$ is introduced into the system at a certain temperature $T$.
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 entropybased measures have been shown to distinguish between different structural regions of the genome, differentiate between coding and noncoding 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 entropyincreasing 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, GeorgescuRoegen's work has generated the term 'entropy pessimism'. Since the 1990s, leading ecological economist and steadystate theorist Herman Daly – a student of GeorgescuRoegen – has been the economics profession's most influential proponent of the entropy pessimism position.

