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In 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 natural logarithm of the number of microstates such a gas could occupy.

Henceforth, the essential problem in statistical thermodynamics , i. There are two related definitions of entropy: Historically, the classical thermodynamics definition developed first.

In the classical thermodynamics viewpoint, the system is composed of very large numbers of constituents atoms, molecules and the state of the system is described by the average thermodynamic properties of those constituents; the details of the system's constituents are not directly considered, but their behavior is described by macroscopically averaged properties, e.

The early classical definition of the properties of the system assumed equilibrium. The classical thermodynamic definition of entropy has more recently been extended into the area of non-equilibrium thermodynamics.

Later, the thermodynamic properties, including entropy, were given an alternative definition in terms of the statistics of the motions of the microscopic constituents of a system — modeled at first classically, e.

Newtonian particles constituting a gas, and later quantum-mechanically photons, phonons , spins, etc. The statistical mechanics description of the behavior of a system is necessary as the definition of the properties of a system using classical thermodynamics become an increasingly unreliable method of predicting the final state of a system that is subject to some process.

There are many thermodynamic properties that are functions of state. This means that at a particular thermodynamic state which should not be confused with the microscopic state of a system , these properties have a certain value.

Often, if two properties of the system are determined, then the state is determined and the other properties' values can also be determined.

For instance, a quantity of gas at a particular temperature and pressure has its state fixed by those values and thus has a specific volume that is determined by those values.

As another instance, a system composed of a pure substance of a single phase at a particular uniform temperature and pressure is determined and is thus a particular state and is at not only a particular volume but also at a particular entropy.

In the Carnot cycle, the working fluid returns to the same state it had at the start of the cycle, hence the line integral of any state function, such as entropy, over this reversible cycle is zero.

Entropy is conserved for a reversible process. A reversible process is one that does not deviate from thermodynamic equilibrium, while producing the maximum work.

Any process which happens quickly enough to deviate from thermal equilibrium cannot be reversible. In these cases energy is lost to heat, total entropy increases, and the potential for maximum work to be done in the transition is also lost.

More specifically, total entropy is conserved in a reversible process and not conserved in an irreversible process. An irreversible process increases entropy.

The concept of entropy arose from Rudolf Clausius 's study of the Carnot cycle. According to Carnot's principle, work can only be produced by the system when there is a temperature difference, and the work should be some function of the difference in temperature and the heat absorbed Q H.

Carnot did not distinguish between Q H and Q C , since he was using the incorrect hypothesis that caloric theory was valid, and hence heat was conserved the incorrect assumption that Q H and Q C were equal when, in fact, Q H is greater than Q C.

The possibility that the Carnot function could be the temperature as measured from a zero 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 a state function that would vanish upon completion of the cycle.

The state function was called the internal energy and it became the first law of thermodynamics. This implies that there is a function of state which is conserved over a complete cycle of the Carnot cycle.

Clausius called this state function entropy. One can see that entropy was discovered through mathematics rather than through laboratory results.

It is a mathematical construct and has no easy physical analogy. This makes the concept somewhat obscure or abstract, akin to how the concept of energy arose.

Clausius then asked what would happen if there should be less work produced by the system than that predicted by Carnot's principle. The right-hand side of the first equation would be the upper bound of the work output by the system, which would now be converted into an inequality.

So more heat is given up to the cold reservoir than in the Carnot cycle. The entropy that leaves the system is greater than the entropy that enters the system, implying that some irreversible process prevents the cycle from producing the maximum amount of work predicted by the Carnot equation.

The Carnot cycle and efficiency are useful because they define the upper bound of the possible work output and the efficiency of any classical thermodynamic system.

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 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 require an analysis from the standpoint of quantum mechanics.

The thermodynamic definition of entropy was developed in the early s by Rudolf Clausius and essentially describes how to measure the entropy of an isolated system in thermodynamic equilibrium with its parts.

Clausius created the term entropy as an extensive thermodynamic variable that was shown to be useful in characterizing the Carnot cycle.

Heat transfer along the isotherm steps of the Carnot cycle was found to be proportional to the temperature of a system known as its absolute temperature.

This relationship was expressed in increments of entropy equal to the ratio of incremental heat transfer divided by temperature, which was found to vary in the thermodynamic cycle but eventually return to the same value at the end of every cycle.

Thus it was found to be a function of state , specifically a thermodynamic state of the system. 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 heat may not flow to and from an isolated system, but heat flow to and from a closed system is possible.

Nevertheless, for both closed and isolated systems, and indeed, also in open systems, irreversible thermodynamics processes may occur. According to the Clausius equality , for a reversible cyclic process: Clausius coined the name entropy Entropie for S in 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.

We can only obtain the change of entropy by integrating the above formula. From a macroscopic perspective, in classical thermodynamics the entropy is interpreted as a state function of a thermodynamic system: Otherwise the process cannot go forward.

In classical thermodynamics, the entropy of a system is defined only if it is in thermodynamic equilibrium.

The statistical definition was developed by Ludwig Boltzmann in the s by analyzing the statistical behavior of the microscopic components of the system.

Boltzmann showed that this definition of entropy was equivalent to the thermodynamic entropy to within a constant number which has since been known as Boltzmann's constant.

In summary, the thermodynamic definition of entropy provides the experimental definition of entropy, while the statistical definition of entropy extends the concept, providing an explanation and a deeper understanding of its nature.

The interpretation of entropy in statistical mechanics is the measure of uncertainty, or mixedupness in the phrase of Gibbs , which remains about a system after its observable macroscopic properties, such as temperature, pressure and volume, have been taken into account.

For a given set of macroscopic variables, the entropy measures the degree to which the probability of the system is spread out over different possible microstates.

In contrast to the macrostate, which characterizes plainly observable average quantities, a microstate specifies all molecular details about the system including the position and velocity of every molecule.

The more such states available to the system with appreciable probability, the greater the entropy. In statistical mechanics, entropy is a measure of the number of ways in which a system may be arranged, often taken to be a measure of "disorder" the higher the entropy, the higher the disorder.

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. In Boltzmann's Lectures on Gas Theory , he showed that this expression gives a measure of entropy for systems of atoms and molecules in the gas phase, thus providing a measure for the entropy of classical thermodynamics.

Entropy 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.

The second law of thermodynamics requires that, in general, the total entropy of any system can't decrease other than by increasing the entropy of some other system.

Hence, in a system isolated from its environment, the entropy of that system tends not to decrease. It follows that heat can't flow from a colder body to a hotter body without the application of work the imposition of order to the colder body.

Secondly, it is impossible for any device operating on a cycle to produce net work from a single temperature reservoir; the production of net work requires flow of heat from a hotter reservoir to a colder reservoir, or a single expanding reservoir undergoing adiabatic cooling , which performs adiabatic work.

As a result, there is no possibility of a perpetual motion system. It follows that a reduction in the increase of entropy in a specified process, such as a chemical reaction , means that it is energetically more efficient.

It follows from the second law of thermodynamics that the entropy of a system that is not isolated may decrease. An air conditioner , for example, may cool the air in a room, thus reducing the entropy of the air of that system.

The heat expelled from the room the system , which the air conditioner transports and discharges to the outside air, 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.

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 [45]. 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.

In Boltzmann's definition, entropy is a measure of the number of possible microscopic states or microstates of a system in thermodynamic equilibrium.

Consistent with the Boltzmann definition, the second law of thermodynamics needs to be 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: 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.

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 [53] 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 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.

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