Thermodynamic entropy

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 (${\displaystyle \Delta S}$) 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:^[3]

${\displaystyle {\rm {\delta }}S={\frac {{\rm {\delta }}q}{T}}.}$

where ${\displaystyle \delta S}$ is the increase or decrease in entropy, ${\displaystyle \delta q}$ is the heat added to the system or subtracted from it, and ${\displaystyle T}$ is temperature. The 'equals' sign and the symbol ${\displaystyle \delta }$ imply that the heat transfer should be so small and slow that it scarcely changes the temperature ${\displaystyle T}$.

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,

${\displaystyle {{\rm {\delta }}S}\geq {\frac {{\rm {\delta }}q}{T}}.}$

According to the first law of thermodynamics, which deals with the conservation of energy, the loss ${\displaystyle \delta q}$ 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 (${\displaystyle \Delta S}$). 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.^[4] 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.^[5]

Statistical mechanics and information entropy

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:

${\displaystyle S=k_{\text{B}}\,\ln W}$

where S is the thermodynamic entropy, W is the number of microstates that may yield the macrostate, and ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant. The natural logarithm of the number of microstates (${\displaystyle \ln W}$) 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.

Ice melting provides an example of entropy increasing.

Classical calculation of 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, ${\displaystyle \Delta S}$, was described in macroscopic terms that could be directly measured, such as volume, temperature, or pressure. However, today the classical equation of entropy, ${\displaystyle \Delta S={\frac {q_{\mathrm {rev} }}{T}}}$ can be explained, part by part, in modern terms describing how molecules are responsible for what is happening:

• ${\displaystyle \Delta S}$ is the change in entropy of a system (some physical substance of interest) after some motional energy ("heat") has been transferred to it by fast-moving molecules. So, ${\displaystyle \Delta S=S_{\mathrm {final} }-S_{\mathrm {initial} }}$.
• Then, ${\displaystyle \Delta S=S_{\mathrm {final} }-S_{\mathrm {initial} }={\frac {q_{\mathrm {rev} }}{T}}}$, 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 is 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, less-exactly 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 ${\displaystyle {\frac {q_{\mathrm {rev} }}{T}}={\frac {6008\,\mathrm {J} }{273\,\mathrm {K} }}}$, or 22 J/K.
  • When the temperature is not at the melting or boiling point of a substance no intermolecular bond-breaking 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 ${\displaystyle {\frac {q_{\mathrm {rev} }}{T}}}$ 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, ${\displaystyle C_{p}}$), multiplied by the integral of ${\displaystyle {\frac {dT}{T}}}$ from ${\displaystyle T_{\mathrm {initial} }}$ to ${\displaystyle T_{\mathrm {final} }}$, is directly given by ${\displaystyle \Delta S=C_{p}\ln {\frac {T_{\mathrm {final} }}{T_{\mathrm {initial} }}}}$.

Thermodynamic entropy

• A measure of energy unavailable for work: This is an often-repeated phrase which, although it is true, requires considerable clarification to be understood. It is only true for cyclic reversible processes, and is in this sense misleading. By "work" is meant moving an object, for example, lifting a weight, or bringing a flywheel up to speed, or carrying a load up a hill. To convert heat into work, using a coal-burning steam engine, for example, one must have two systems at different temperatures, and the amount of work you can extract depends on how large the temperature difference is, and how large the systems are. If one of the systems is at room temperature, and the other system is much larger, and near absolute zero temperature, then almost ALL of the energy of the room temperature system can be converted to work. If they are both at the same room temperature, then NONE of the energy of the room temperature system can be converted to work. Entropy is then a measure of how much energy cannot be converted to work, given these conditions. 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 temperature of the hot system, is the amount of energy that cannot converted to work.

• An indicator of irreversibility: fitting closely with the 'unavailability of energy' interpretation is the 'irreversibility' interpretation. Spontaneous thermodynamic processes are irreversible, in the sense that they do not spontaneously undo themselves. Thermodynamic processes artificially imposed by agents in the surroundings of a body also have irreversible effects on the body. For example, when James Prescott Joule used a device that delivered a measured amount of mechanical work from the surroundings through a paddle that stirred a body of water, the energy transferred was received by the water as heat. There was scarce expansion of the water doing thermodynamic work back on the surroundings. The body of water showed no sign of returning the energy by stirring the paddle in reverse. The work transfer appeared as heat, and was not recoverable without a suitably cold reservoir in the surroundings. Entropy gives a precise account of such irreversibility.

• Dispersal: Edward A. Guggenheim proposed an ordinary language interpretation of entropy that may be rendered as "dispersal of modes of microscopic motion throughout their accessible range".^[7][8] Later, along with a criticism of the idea of entropy as 'disorder', the dispersal interpretation was advocated by Frank L. Lambert,^[6][9] and is used in some student textbooks.^[10]

The interpretation properly refers to dispersal in abstract microstate spaces, but it may be loosely visualised in some simple examples of spatial spread of matter or energy. If a partition is removed from between two different gases, the molecules of each gas spontaneously disperse as widely as possible into their respectively newly accessible volumes; this may be thought of as mixing. If a partition, that blocks heat transfer between two bodies of different temperatures, is removed so that heat can pass between the bodies, then energy spontaneously disperses or spreads as heat from the hotter to the colder.

Beyond such loose visualizations, in a general thermodynamic process, considered microscopically, spontaneous dispersal occurs in abstract microscopic phase space. According to Newton's and other laws of motion, phase space provides a systematic scheme for the description of the diversity of microscopic motion that occurs in bodies of matter and radiation. The second law of thermodynamics may be regarded as quantitatively accounting for the intimate interactions, dispersal, or mingling of such microscopic motions. In other words, entropy may be regarded as measuring the extent of diversity of motions of microscopic constituents of bodies of matter and radiation in their own states of internal thermodynamic equilibrium.

Information entropy and statistical mechanics

• 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, and arbitrary interpretations of, 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, in a convenient though arbitrary interpretation, "disorder" may be sharply defined as the Shannon entropy of the probability distribution of microstates given a particular macrostate,^[11]: 379 in which case the connection of "disorder" to thermodynamic entropy is straightforward, but arbitrary and not immediately obvious to anyone unfamiliar with information theory.

• Missing information: The idea that information entropy is a measure of how much one does not know about a system is quite useful.

If, instead of using the natural logarithm to define information entropy, we instead use the base 2 logarithm, then the information entropy is roughly equal to the average number of (carefully chosen ^[12]) yes/no questions that would have to be asked to get complete information about the system under study. In the introductory example of two flipped coins, the information entropy for the macrostate which contains one head and one tail, one would only need one question to determine its exact state, (e.g. is the first one heads?") and instead of expressing the entropy as ln(2) one could say, equivalently, that it is Log₂(2) which equals the number of questions we would need to ask: One. When measuring entropy using the natural logarithm (ln), the unit of information entropy is called a "nat", but when it is measured using the base-2 logarithm, the unit of information entropy is called a "bit". This is just a difference in units, much like the difference between inches and centimeters. (1 nat = e bits). Thermodynamic entropy is equal to Boltzmann's constant times the information entropy expressed in nats. The information entropy expressed in bits is equal to the number of yes–no questions that need to be answered in order to determine the microstate from the macrostate.

The concepts of "disorder" and "spreading" can be analyzed with this information entropy concept in mind. For example, if we take a new deck of cards out of the box, it is arranged in "perfect order" (spades, hearts, diamonds, clubs, each suit beginning with the ace and ending with the king), we may say that we then have an "ordered" deck with an information entropy of zero. If we thoroughly shuffle the deck, the information entropy will be about 225.6 bits: We will need to ask about 225.6 questions, on average, to determine the exact order of the shuffled deck. We can also say that the shuffled deck has become completely "disordered" or that the ordered cards have been "spread" throughout the deck. But information entropy does not say that the deck needs to be ordered in any particular way. If we take our shuffled deck and write down the names of the cards, in order, then the information entropy becomes zero. If we again shuffle the deck, the information entropy would again be about 225.6 bits, even if by some miracle it reshuffled to the same order as when it came out of the box, because even if it did, we would not know that. So the concept of "disorder" is useful if, by order, we mean maximal knowledge and by disorder we mean maximal lack of knowledge. The "spreading" concept is useful because it gives a feeling to what happens to the cards when they are shuffled. The probability of a card being in a particular place in an ordered deck is either 0 or 1, in a shuffled deck it is 1/52. The probability has "spread out" over the entire deck. Analogously, in a physical system, entropy is generally associated with a "spreading out" of mass or energy.

The connection between thermodynamic entropy and information entropy is given by Boltzmann's equation, which says that S = k_B ln W. If we take the base-2 logarithm of W, it will yield the average number of questions we must ask about the microstate of the physical system in order to determine its macrostate.^[13]