In thermodynamics, entropy is a numerical quantity that shows that many physical processes can go in only one direction in time. For example, cream and coffee can be mixed together, but cannot be "unmixed"; a piece of wood can be burned, but cannot be "unburned". The word 'entropy' has entered popular usage to refer a lack of order or predictability, or of a gradual decline into disorder. A more physical interpretation of thermodynamic entropy refers to spread of energy or matter, or to extent and diversity of microscopic motion.
If a movie that shows coffee being mixed or wood being burned is played in reverse, it would depict processes impossible in reality. Mixing coffee and burning wood are "irreversible". Irreversibility is described by a law of nature known as the second law of thermodynamics, which states that in an isolated system (a system not connected to any other system) which is undergoing change, entropy increases over time.
Entropy does not increase indefinitely. A body of matter and radiation eventually will reach an unchanging state, with no detectable flows, and is then said to be in a state of thermodynamic equilibrium. Thermodynamic entropy has a definite value for such a body and is at its maximum value. When bodies of matter or radiation, initially in their own states of internal thermodynamic equilibrium, are brought together so as to intimately interact and reach a new joint equilibrium, then their total entropy increases. For example, a glass of warm water with an ice cube in it will have a lower entropy than that same system some time later when the ice has melted leaving a glass of cool water. Such processes are irreversible: An ice cube in a glass of warm water will not spontaneously form from a glass of cool water. Some processes in nature are almost reversible. For example, the orbiting of the planets around the Sun may be thought of as practically reversible: A movie of the planets orbiting the Sun which is run in reverse would not appear to be impossible.
While the second law, and thermodynamics in general, is accurate in its predictions of intimate interactions of complex physical systems behave, scientists are not content with simply knowing how a system behaves, but want to know also why it behaves the way it does. The question of why entropy increases until equilibrium is reached was answered very successfully in 1877 by a famous scientist named Ludwig Boltzmann. The theory developed by Boltzmann and others, is known as statistical mechanics. Statistical mechanics is a physical theory which explains thermodynamics in terms of the statistical behavior of the atoms and molecules which make up the system. The theory not only explains thermodynamics, but also a host of other phenomena which are outside the scope of thermodynamics.
The concept of thermodynamic entropy arises from the second law of thermodynamics. This law of entropy increase quantifies the reduction in the capacity of an isolated compound thermodynamic system to do thermodynamic work on its surroundings, or indicates whether a thermodynamic process may occur. For example, whenever there is a suitable pathway, heat spontaneously flows from a hotter body to a colder one.
Thermodynamic entropy is only measured as a change in entropy () to a system containing a sub-system which undergoes heat transfer to its surroundings (inside the system of interest). It is based on the macroscopic relationship between heat flow into the sub-system and the temperature at which it occurs summed over the boundary of that sub-system.
Following the formalism of Clausius, the basic calculation can be mathematically stated as:
where is the increase or decrease in entropy, is the heat added to the system or subtracted from it, and is temperature. The 'equals' sign and the symbol imply that the heat transfer should be so small and slow that it scarcely changes the temperature .
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. In many cases, a 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. When applicable, entropy increase is the quantitative measure of that kind of a spontaneous process: how much energy has been effectively lost or become unavailable, by dispersing itself, or spreading itself out, as assessed at a specific temperature. For this assessment, when the temperature is higher, the amount of energy dispersed is assessed as 'costing' proportionately less. This is because a hotter body is generally more able to do thermodynamic work, other factors, such as internal energy, being equal. This is why a steam engine has a hot firebox.
The second law of thermodynamics deals only with changes of entropy (). The absolute entropy (S) of a system may be determined using the third law of thermodynamics, which specifies that the entropy of all perfectly crystalline substances is zero at the absolute zero of temperature. The entropy at another temperature is then equal to the increase in entropy on heating the system reversibly from absolute zero to the temperature of interest.
Thermodynamic entropy bears a close relationship to the concept of information entropy (H). Information entropy is a measure of the "spread" of a probability density or probability mass function. Thermodynamics makes no assumptions about the atomistic nature of matter, but when matter is viewed in this way, as a collection of particles constantly moving and exchanging energy with each other, and which may be described in a probabilistic manner, information theory may be successfully applied to explain the results of thermodynamics. The resulting theory is known as statistical mechanics.
An important concept in statistical mechanics is the idea of the microstate and the macrostate of a system. If we have a container of gas, for example, and we know the position and velocity of every molecule in that system, then we know the microstate of that system. If we only know the thermodynamic description of that system, the pressure, volume, temperature, and/or the entropy, then we know the macrostate of that system. What Boltzmann realized was that there are many different microstates that can yield the same macrostate, and, because the particles are colliding with each other and changing their velocities and positions, the microstate of the gas is always changing. But if the gas is in equilibrium, there seems to be no change in its macroscopic behavior: No changes in pressure, temperature, etc. Statistical mechanics relates the thermodynamic entropy of a macrostate to the number of microstates that could yield that macrostate. In statistical mechanics the entropy of the system is given by Ludwig Boltzmann's famous equation:
where S is the thermodynamic entropy, W is the number of microstates that may yield the macrostate, and is the Boltzmann constant. The natural logarithm of the number of microstates () is known as the information entropy of the system. This can be illustrated by a simple example:
If you flip two coins, you can have four different results. If H is heads and T is tails, we can have (H,H), (H,T), (T,H), and (T,T). We can call each of these a "microstate" for which we know exactly the results of the process. But what if we have less information? Suppose we only know the total number of heads?. This can be either 0, 1, or 2. We can call these "macrostates". Only microstate (T,T) will give macrostate zero, (H,T) and (T,H) will give macrostate 1, and only (H,H) will give macrostate 2. So we can say that the information entropy of macrostates 0 and 2 are ln(1) which is zero, but the information entropy of macrostate 1 is ln(2) which is about 0.69. Of all the microstates, macrostate 1 accounts for half of them.
It turns out that if you flip a large number of coins, the macrostates at or near half heads and half tails accounts for almost all of the microstates. In other words, for a million coins, you can be fairly sure that about half will be heads and half tails. The macrostates around a 50–50 ratio of heads to tails will be the "equilibrium" macrostate. A real physical system in equilibrium has a huge number of possible microstates and almost all of them are the equilibrium macrostate, and that is the macrostate you will almost certainly see if you wait long enough. In the coin example, if you start out with a very unlikely macrostate (like all heads, for example with zero entropy) and begin flipping one coin at a time, the entropy of the macrostate will start increasing, just as thermodynamic entropy does, and after a while, the coins will most likely be at or near that 50–50 macrostate, which has the greatest information entropy – the equilibrium entropy.
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 information entropy has been developed to describe any of several phenomena, depending on the field and the context in which it is being used. When it is applied to the problem of a large number of interacting particles, along with some other constraints, like the conservation of energy, and the assumption that all microstates are equally likely, the resultant theory of statistical mechanics is extremely successful in explaining the laws of thermodynamics.
Main article: Disgregation
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.
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. 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.
When the word 'entropy' was first defined and used in 1865, the very existence of atoms was still controversial, though it had long been speculated that temperature was due to the motion of microscopic constituents and that "heat" was the transferring of that motion 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:
Entropies can then be determined at other temperatures, by considering a series of reversible processes by which the temperature is raised from the absolute zero to the temperature in question.