Introduction to entropy
Entropy is an important concept in the branch of physics known as thermodynamics. The idea of "irreversibility" is central to the understanding of entropy. Everyone has an intuitive understanding of irreversibility. If one watches a movie of everyday life running forward and in reverse, it is easy to distinguish between the two. The movie running in reverse shows impossible things happening – water jumping out of a glass into a pitcher above it, smoke going down a chimney, water in a glass freezing to form ice cubes, crashed cars reassembling themselves, and so on. The intuitive meaning of expressions such as "you can't unscramble an egg", or "you can't take the cream out of the coffee" is that these are irreversible processes. No matter how long you wait, the cream won't jump out of the coffee into the creamer.
Thermodynamics  

The classical Carnot heat engine  


 
In thermodynamics, one says that the "forward" processes – pouring water from a pitcher, smoke going up a chimney, etc. – are "irreversible": they cannot happen in reverse. All real physical processes involving systems in everyday life, with many atoms or molecules, are irreversible. For an irreversible process in an isolated system (a system not subject to outside influence), the thermodynamic state variable known as entropy is never decreasing. In everyday life, there may be processes in which the increase of entropy is practically unobservable, almost zero. In these cases, a movie of the process run in reverse will not seem unlikely. For example, in a 1second video of the collision of two billiard balls, it will be hard to distinguish the forward and the backward case, because the increase of entropy during that time is relatively small. In thermodynamics, one says that this process is practically "reversible", with an entropy increase that is practically zero. The statement of the fact that the entropy of an isolated system never decreases is known as the second law of thermodynamics.
Classical thermodynamics is a physical theory which describes a "system" in terms of the thermodynamic variables of the system or its parts. Some thermodynamic variables are familiar: temperature, pressure, volume. Entropy is a thermodynamic variable which is less familiar and not as easily understood. A "system" is any region of space containing matter and energy: A cup of coffee, a glass of ice water, an automobile, an egg. Thermodynamic variables do not give a "complete" picture of the system. Thermodynamics makes no assumptions about the microscopic nature of a system and does not describe nor does it take into account the positions and velocities of the individual atoms and molecules which make up the system. Thermodynamics deals with matter in a macroscopic sense; it would be valid even if the atomic theory of matter were wrong. This is an important quality, because it means that reasoning based on thermodynamics is unlikely to require alteration as new facts about atomic structure and atomic interactions are found. The essence of thermodynamics is embodied in the four laws of thermodynamics.
Unfortunately, thermodynamics provides little insight into what is happening at a microscopic level. Statistical mechanics is a physical theory which explains thermodynamics in microscopic terms. It explains thermodynamics in terms of the possible detailed microscopic situations the system may be in when the thermodynamic variables of the system are known. These are known as "microstates" whereas the description of the system in thermodynamic terms specifies the "macrostate" of the system. Many different microstates can yield the same macrostate. It is important to understand that statistical mechanics does not define temperature, pressure, entropy, etc. They are already defined by thermodynamics. Statistical mechanics serves to explain thermodynamics in terms of microscopic behavior of the atoms and molecules in the system.[1]^{:329, 333}
In statistical mechanics, the entropy of a system is described as a measure of how many different microstates there are that could give rise to the macrostate that the system is in. The entropy of the system is given by Ludwig Boltzmann's famous equation:
where is the entropy of the macrostate, is Boltzmann's constant, and is the total number of possible microstates that might yield the macrostate. The concept of irreversibility stems from the idea that if you have a system in an "unlikely" macrostate ( is relatively small) it will soon move to the "most likely" macrostate (with larger ) and the entropy S will increase. A glass of warm water with an ice cube in it is unlikely to just happen, it must have been recently created, and the system will move to a more likely macrostate in which the ice cube is partially or entirely melted and the water is cooled. Statistical mechanics shows that the number of microstates which give ice and warm water is much smaller than the number of microstates that give the reduced ice mass and cooler water.
Explanation
The concept of thermodynamic entropy arises from the second law of thermodynamics. This law of entropy increase quantifies the reduction in the capacity of a system for change or determines whether a thermodynamic process may occur. For example, heat always flows from a region of higher temperature to one with lower temperature until temperature becomes uniform.
Entropy is calculated in two ways, the first is the entropy change () to a system containing a subsystem which undergoes heat transfer to its surroundings (inside the system of interest). It is based on the macroscopic relationship between heat flow into the subsystem and the temperature at which it occurs summed over the boundary of that subsystem. The second calculates the absolute entropy (S) of a system based on the microscopic behaviour of its individual particles. This is based on the natural logarithm of the number of microstates possible in a particular macrostate () called the thermodynamic probability. Roughly, it gives the probability of the system's being in that state. In this sense it effectively defines entropy independently from its effects due to changes which may involve heat, mechanical, electrical, chemical energies etc. but also includes logical states such as information.
Following the formalism of Clausius, the first calculation can be mathematically stated as:[2]
Where is the increase or decrease in entropy, is the heat added to the system or subtracted from it, and is temperature. The equal sign indicates that the change is reversible, because Clausius shows a proportional relationship between entropy and the energy flow, in a system, the heat energy can be transformed into work, and work can be transformed into heat through a cyclical process.[3] If the temperature is allowed to vary, the equation must be integrated over the temperature path. This calculation of entropy change does not allow the determination of absolute value, only differences. In this context, the Second Law of Thermodynamics may be stated that for heat transferred over any valid process for any system, whether isolated or not,
According to the first law of thermodynamics, which deals with the conservation of energy, the loss of heat will result in a decrease in the internal energy of the thermodynamic system. Thermodynamic entropy provides a comparative measure of the amount of decrease in internal energy and the corresponding increase in internal energy of the surroundings at a given temperature. A simple and more concrete visualization of the second law is that energy of all types changes from being localized to becoming dispersed or spread out, if it is not hindered from doing so. Entropy change is the quantitative measure of that kind of a spontaneous process: how much energy has flowed or how widely it has become spread out at a specific temperature.
The second calculation defines entropy in absolute terms and comes from statistical mechanics. The entropy of a particular macrostate is defined to be Boltzmann's constant times the natural logarithm of the number of microstates corresponding to that macrostate, or mathematically
where the variables are defined as before.
The macrostate of a system is what we know about the system, for example the temperature, pressure, and volume of a gas in a box. For each set of values of temperature, pressure, and volume there are many arrangements of molecules which result in those values. The number of arrangements of molecules which could result in the same values for temperature, pressure and volume is the number of microstates.
The concept of entropy has been developed to describe any of several phenomena, depending on the field and the context in which it is being used. Information entropy takes the mathematical concepts of statistical thermodynamics into areas of probability theory unconnected with heat and energy.
Example of increasing entropy
Ice melting provides an example in which entropy increases in a small system, a thermodynamic system consisting of the surroundings (the warm room) and the entity of glass container, ice and water which has been allowed to reach thermodynamic equilibrium at the melting temperature of ice. In this system, some heat (δQ) from the warmer surroundings at 298 K (25 °C; 77 °F) transfers to the cooler system of ice and water at its constant temperature (T) of 273 K (0 °C; 32 °F), the melting temperature of ice. The entropy of the system, which is δQ/T, increases by δQ/273 K. The heat δQ for this process is the energy required to change water from the solid state to the liquid state, and is called the enthalpy of fusion, i.e. ΔH for ice fusion.
It is important to realize that the entropy of the surrounding room decreases less than the entropy of the ice and water increases: 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 and water system. 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.
As the temperature of the cool water rises to that of the room and the room further cools imperceptibly, the sum of the δQ/T over the continuous range, “at many increments”, in the initially cool to finally warm water can be found by calculus. The entire miniature ‘universe’, i.e. this thermodynamic system, has increased in entropy. Energy has spontaneously become more dispersed and spread out in that ‘universe’ than when the glass of ice and water was introduced and became a 'system' within it.
Origins and uses
Originally, entropy was named to describe the "waste heat," or more accurately, energy loss, from heat engines and other mechanical devices which could never run with 100% efficiency in converting energy into work. Later, the term came to acquire several additional descriptions, as more was understood about the behavior of molecules on the microscopic level. In the late 19th century, the word "disorder" was used by Ludwig Boltzmann in developing statistical views of entropy using probability theory to describe the increased molecular movement on the microscopic level. That was before quantum behavior came to be better understood by Werner Heisenberg and those who followed. Descriptions of thermodynamic (heat) entropy on the microscopic level are found in statistical thermodynamics and statistical mechanics.
For most of the 20th century, textbooks tended to describe entropy as "disorder", following Boltzmann's early conceptualisation of the "motional" (i.e. kinetic) energy of molecules. More recently, there has been a trend in chemistry and physics textbooks to describe entropy as energy dispersal.[4] Entropy can also involve the dispersal of particles, which are themselves energetic. Thus there are instances where both particles and energy disperse at different rates when substances are mixed together.
The mathematics developed in statistical thermodynamics were found to be applicable in other disciplines. In particular, information sciences developed the concept of information entropy, which lacks the Boltzmann constant inherent in thermodynamic entropy.
Heat and entropy
At a microscopic level, kinetic energy of molecules is responsible for the temperature of a substance or a system. “Heat” is the kinetic energy of molecules being transferred: when motional energy is transferred from hotter surroundings to a cooler system, fastermoving molecules in the surroundings collide with the walls of the system which transfers some of their energy to the molecules of the system and makes them move faster.
 Molecules in a gas like nitrogen at room temperature at any instant are moving at an average speed of nearly 500 miles per hour (210 m/s), repeatedly colliding and therefore exchanging energy so that their individual speeds are always changing. Assuming an idealgas model, average kinetic energy increases linearly with absolute temperature, so the average speed increases as the square root of temperature.
 Thus motional molecular energy (‘heat energy’) from hotter surroundings, like fastermoving molecules in a flame or violently vibrating iron atoms in a hot plate, will melt or boil a substance (the system) at the temperature of its melting or boiling point. That amount of motional energy from the surroundings that is required for melting or boiling is called the phasechange energy, specifically the enthalpy of fusion or of vaporization, respectively. This phasechange energy breaks bonds between the molecules in the system (not chemical bonds inside the molecules that hold the atoms together) rather than contributing to the motional energy and making the molecules move any faster – so it does not raise the temperature, but instead enables the molecules to break free to move as a liquid or as a vapor.
 In terms of energy, when a solid becomes a liquid or a vapor, motional energy coming from the surroundings is changed to ‘potential energy‘ in the substance (phase change energy, which is released back to the surroundings when the surroundings become cooler than the substance's boiling or melting temperature, respectively). Phasechange energy increases the entropy of a substance or system because it is energy that must be spread out in the system from the surroundings so that the substance can exist as a liquid or vapor at a temperature above its melting or boiling point. When this process occurs in a 'universe' that consists of the surroundings plus the system, the total energy of the 'universe' becomes more dispersed or spread out as part of the greater energy that was only in the hotter surroundings transfers so that some is in the cooler system. This energy dispersal increases the entropy of the 'universe'.
The important overall principle is that ”Energy of all types changes from being localized to becoming dispersed or spread out, if not hindered from doing so. Entropy (or better, entropy change) is the quantitative measure of that kind of a spontaneous process: how much energy has been transferred/T or how widely it has become spread out at a specific temperature."
Classical calculation of entropy
When entropy was first defined and used in 1865 the very existence of atoms was still controversial and there was no concept that temperature was due to the motional energy of molecules or that “heat” was actually the transferring of that motional molecular energy from one place to another. Entropy change, , was described in macroscopic terms that could be directly measured, such as volume, temperature, or pressure. However, today the classical equation of entropy, can be explained, part by part, in modern terms describing how molecules are responsible for what is happening:
 is the change in entropy of a system (some physical substance of interest) after some motional energy (“heat”) has been transferred to it by fastmoving molecules. So, .
 Then, , the quotient of the motional energy (“heat”) q that is transferred "reversibly" (rev) to the system from the surroundings (or from another system in contact with the first system) divided by T, the absolute temperature at which the transfer occurs.
 “Reversible” or “reversibly” (rev) simply means that T, the temperature of the system, has to stay (almost) exactly the same while any energy is being transferred to or from it. That’s easy in the case of phase changes, where the system absolutely must stay in the solid or liquid form until enough energy is given to it to break bonds between the molecules before it can change to a liquid or a gas. For example, in the melting of ice at 273.15 K, no matter what temperature the surroundings are – from 273.20 K to 500 K or even higher, the temperature of the ice will stay at 273.15 K until the last molecules in the ice are changed to liquid water, i.e., until all the hydrogen bonds between the water molecules in ice are broken and new, lessexactly fixed hydrogen bonds between liquid water molecules are formed. This amount of energy necessary for ice melting per mole has been found to be 6008 joules at 273 K. Therefore, the entropy change per mole is , or 22 J/K.
 When the temperature isn't at the melting or boiling point of a substance no intermolecular bondbreaking is possible, and so any motional molecular energy (“heat”) from the surroundings transferred to a system raises its temperature, making its molecules move faster and faster. As the temperature is constantly rising, there is no longer a particular value of “T” at which energy is transferred. However, a "reversible" energy transfer can be measured at a very small temperature increase, and a cumulative total can be found by adding each of many small temperature intervals or increments. For example, to find the entropy change from 300 K to 310 K, measure the amount of energy transferred at dozens or hundreds of temperature increments, say from 300.00 K to 300.01 K and then 300.01 to 300.02 and so on, dividing the q by each T, and finally adding them all.
 Calculus can be used to make this calculation easier if the effect of energy input to the system is linearly dependent on the temperature change, as in simple heating of a system at moderate to relatively high temperatures. Thus, the energy being transferred “per incremental change in temperature” (the heat capacity, ), multiplied by the integral of from to , is directly given by .
Introductory descriptions of entropy
 As a measure of disorder: Traditionally, 20th century textbooks have introduced entropy as order and disorder so that it provides "a measurement of the disorder or randomness of a system". It has been argued that ambiguities in the terms used (such as "disorder" and "chaos") contribute to widespread confusion and can hinder comprehension of entropy for most students. On the other hand, "disorder" may be very clearly defined as the Shannon entropy of the probability distribution of microstates given a particular macrostate,[1]^{:379} in which case the connection of "disorder" to thermodynamic entropy is straightforward, but not immediately obvious to anyone unfamiliar with information theory.
 Energy dispersal: A more recent formulation associated with Frank L. Lambert describes entropy as energy dispersal.[4] As with "disorder", the meaning of the term "dispersal" must be taken in a very specific way, which is quite different than the lay meaning of "dispersal". While an increase in entropy is often associated with a spatial reduction in the concentration of the energy density, and never with an increase, counterexamples exist which illustrate that the concept of "dispersal" is not immediately obvious. Most counterexamples may be included in the concept of "dispersal" when the "space" in which the dispersal occurs includes the space of quantum energy levels versus population numbers, but this reduces the effectiveness of the spreading concept as an introduction to the concept of entropy.
 As a measure of energy unavailable for work: This is an oftenrepeated phrase which requires considerable clarification in order to be understood. It is not true except for cyclic reversible processes and is in this sense misleading. Given a container of gas, ALL of its internal energy may be converted to work. (More accurately, the amount of work that may be converted can be made arbitrarily close to the total internal energy.) More precisely, for an isolated system comprising two closed systems at different temperatures, in the process of reaching equilibrium the amount of entropy lost by the hot system, multiplied by the thermodynamic temperature of the hot system, is a measure of the amount of energy that is unavailable for work. As a description of the fundamental nature of entropy, it can be misleading in this sense.
See also
References
 Callen, Herbert B. (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed.). New York: John Wiley & Sons. ISBN 0471862568.
 I. Klotz, R. Rosenberg, Chemical Thermodynamics – Basic Concepts and Methods, 7th ed., Wiley (2008), p. 125
 Entropy and second law of thermodynamics (Chapter 4)
 Entropy Sites — A Guide Content selected by Frank L. Lambert
Further reading
 Goldstein, Martin and Inge F. (1993). The Refrigerator and the Universe: Understanding the Laws of Energy. Harvard Univ. Press. ISBN 9780674753259. Chapters 4–12 touch on entropy.