Thermodynamics is a branch of natural science concerned with heat and its relation to energy and work. It defines macroscopic variables (such as temperature, internal energy, entropy, and pressure) that characterize materials and radiation, and explains how they are related and by what laws they change with time. Thermodynamics describes the average behavior of very large numbers of microscopic constituents, and its laws can be derived from statistical mechanics.
Thermodynamics applies to a wide variety of topics in science and engineering—such as engines, phase transitions, chemical reactions, transport phenomena, and even black holes. Results of thermodynamic calculations are essential for other fields of physics and for chemistry, chemical engineering, aerospace engineering, mechanical engineering, cell biology, biomedical engineering, and materials science—and useful in other fields such as economics. ξ1 ξ2
Much of the empirical content of thermodynamics is contained in the four laws. The first law asserts the existence of a quantity called the internal energy of a system, which is distinguishable from the kinetic energy of bulk movement of the system and from its potential energy with respect to its surroundings. The first law distinguishes transfers of energy between closed systems as heat and as work.Crawford, F.H. (1963). Heat, Thermodynamics, and Statistical Physics, Rupert HartDavis, London, Harcourt, Brace & World, Inc., pp. 106–107.Haase, R. (1963/1969). Thermodynamics of Irreversible Processes, translated in English, AddisonWesley, Reading MA, pp. 10–11.Münster, A. (1970). The second law concerns two quantities called temperature and entropy. Entropy expresses the limitations, arising from what is known as irreversibility, on the amount of thermodynamic work that can be delivered to an external system by a thermodynamic process. ξ3 Temperature, whose properties are also partially described by the zeroth law of thermodynamics, quantifies the direction of energy flow as heat between two systems in thermal contact and quantifies the commonsense notions of "hot" and "cold".
Historically, thermodynamics developed out of a desire to increase the efficiency of early , particularly through the work of French physicist Nicolas Léonard Sadi Carnot (1824) who believed that the efficiency of heat engines was the key that could help France win the Napoleonic Wars. ξ4 Irishborn British physicist Lord Kelvin was the first to formulate a concise definition of thermodynamics in 1854:
Initially, the thermodynamics of heat engines concerned mainly the thermal properties of their 'working materials', such as steam. This concern was then linked to the study of energy transfers in chemical processes, for example to the investigation, published in 1840, of the heats of chemical reactionsHess, H. (1840). der Physik (Leipzig) 125.langEN Thermochemische Untersuchungen, Annalen der Physik und Chemie (Poggendorff, Leipzig) 126(6): 385–404. by Germain Hess, which was not originally explicitly concerned with the relation between energy exchanges by heat and work. Chemical thermodynamics studies the role of entropy in .Gibbs, Willard, J. (1876). Transactions of the Connecticut Academy, III, pp. 108–248, Oct. 1875 – May 1876, and pp. 343–524, May 1877 – July 1878.Duhem, P.M.M. (1886). Le Potential Thermodynamique et ses Applications, Hermann, Paris.Guggenheim, E.A. (1933). Modern Thermodynamics by the Methods of J.W. Gibbs, Methuen, London.Guggenheim, E.A. (1949/1967) ξ5 ξ6 ξ7 Also, statistical thermodynamics, or statistical mechanics, gave explanations of macroscopic thermodynamics by statistical predictions of the collective motion of particles based on the mechanics of their microscopic behavior.
Thermodynamics arose from the study of energy transfers that can be strictly resolved into two distinct components, heat and work, specified by macroscopic variables.Bridgman, P.W. (1943). The Nature of Thermodynamics, Harvard University Press, Cambridge MA, p. 48.Partington, J.R. (1949), page 118.
Thermodynamic equilibrium is one of the most important concepts for thermodynamics.Tisza, L. (1966), p. 18. The temperature of a system in thermodynamic equilibrium is well defined, and is perhaps the most characteristic quantity of thermodynamics. As the systems and processes of interest are taken further from thermodynamic equilibrium, their exact thermodynamical study becomes more difficult. Relatively simple approximate calculations, however, using the variables of equilibrium thermodynamics, are of much practical value in engineering. In many important practical cases, such as heat engines or refrigerators, the systems consist of many subsystems at different temperatures and pressures. In practice, thermodynamic calculations deal effectively with these complicated dynamic systems provided the equilibrium thermodynamic variables are nearly enough welldefined.
Basic for thermodynamics are the concepts of system and surroundings.Kondepudi, D. (2008). Introduction to Modern Thermodynamics, Wiley, Chichester, ISBN 9780470015988. Includes local equilibrium thermodynamics. The surroundings of a thermodynamic system consist of physical devices and of other thermodynamic systems that can interact with it. An example of a thermodynamic surrounding is a heat bath, which is considered to be held at a prescribed temperature, regardless of the interactions it might have with the system.
There are two fundamental kinds of physical entity in thermodynamics, states of a system, and of a systems. This allows two fundamental approaches to thermodynamic reasoning, that in terms of states of a system, and that in terms of cyclic processes of a system. Also necessary for thermodynamic reasoning are .
A thermodynamic system can be defined in terms of its states. In this way, a thermodynamic system is a macroscopic physical object, explicitly specified in terms of macroscopic physical and chemical variables that describe its macroscopic properties. The macroscopic state variables of thermodynamics have been recognized in the course of empirical work in physics and chemistry.
A thermodynamic system can also be defined in terms of the processes it can undergo. Of particular interest are cyclic processes. This was the way of the founders of thermodynamics in the first three quarters of the nineteenth century.
A thermodynamic operation is a conceptual step that changes the definition of a system or its surroundings. For example, the partition between two thermodynamic systems can be removed so as to produce a single system. There is a sense in which Maxwell's demon if he existed would be able to violate the laws or of thermodynamics because he is permitted to perform thermodynamic operations, which are permitted to be unnatural.
For thermodynamics and statistical thermodynamics to apply to a process in a body, it is necessary that the atomic mechanisms of the process fall into just two classes:
The rapid atomic mechanisms mediate the macroscopic changes that are of interest for thermodynamics and statistical thermodynamics, because they quickly bring the system near enough to thermodynamic equilibrium. "When intermediate rates are present, thermodynamics and statistical mechanics cannot be applied." Such intermediate rate atomic processes do not bring the system near enough to thermodynamic equilibrium in the time frame of the macroscopic process of interest. This separation of time scales of atomic processes is a theme that recurs throughout the subject.
For example, classical thermodynamics is characterized by its study of materials that have equations of state or characteristic equations. They express relations between macroscopic mechanical variables and temperature that are reached much more rapidly than the progress of any imposed changes in the surroundings, and are in effect variables of state for thermodynamic equilibrium. They express the constitutive peculiarities of the material of the system. A classical material can usually be described by a function that makes pressure dependent on volume and temperature, the resulting pressure being established much more rapidly than any imposed change of volume or temperature.Planck, M. 1923/1926, page 5.Partington, p. 121.Adkins, pp. 19–20.Haase, R. (1971), pages 11–16.
The present article takes a gradual approach to the subject, starting with a focus on cyclic processes and thermodynamic equilibrium, and then gradually beginning to further consider nonequilibrium systems.
Thermodynamic facts can often be explained by viewing macroscopic objects as assemblies of very many microscopic or objects that obey Hamiltonian dynamics.Balescu, R. (1975). Equilibrium and Nonequilibrium Statistical Mechanics, WileyInterscience, New York, ISBN 0471046000.Schrödinger, E. (1946/1967). Statistical Thermodynamics. A Course of Seminar Lectures, Cambridge University Press, Cambridge UK. The microscopic or atomic objects exist in species, the objects of each species being all alike. Because of this likeness, statistical methods can be used to account for the macroscopic properties of the thermodynamic system in terms of the properties of the microscopic species. Such explanation is called statistical thermodynamics; also often it is also referred to by the term 'statistical mechanics', though this term can have a wider meaning, referring to 'microscopic objects', such as economic quantities, that do not obey Hamiltonian dynamics.
Later designs implemented a steam release valve that kept the machine from exploding. By watching the valve rhythmically move up and down, Papin conceived of the idea of a piston and a cylinder engine. He did not, however, follow through with his design. Nevertheless, in 1697, based on Papin's designs, engineer Thomas Savery built the first engine, followed by Thomas Newcomen in 1712. Although these early engines were crude and inefficient, they attracted the attention of the leading scientists of the time.
The concepts of heat capacity and latent heat, which were necessary for development of thermodynamics, were developed by professor Joseph Black at the University of Glasgow, where James Watt worked as an instrument maker. Watt consulted with Black on tests of his steam engine, but it was Watt who conceived the idea of the external condenser, greatly raising the steam engine's efficiency.The Newcomen engine was improved from 1711 until Watt's work, making the efficiency comparison subject to qualification, but the increase from the Newcomen 1765 version was on the order of 100%. Drawing on all the previous work led Sadi Carnot, the "father of thermodynamics", to publish Reflections on the Motive Power of Fire (1824), a discourse on heat, power, energy and engine efficiency. The paper outlined the basic energetic relations between the Carnot engine, the Carnot cycle, and motive power. It marked the start of thermodynamics as a modern science.
The first thermodynamic textbook was written in 1859 by William Rankine, originally trained as a physicist and a civil and mechanical engineering professor at the University of Glasgow. ξ8 The first and second laws of thermodynamics emerged simultaneously in the 1850s, primarily out of the works of William Rankine, Rudolf Clausius, and William Thomson (Lord Kelvin).
The foundations of statistical thermodynamics were set out by physicists such as James Clerk Maxwell, Ludwig Boltzmann, Max Planck, Rudolf Clausius and J. Willard Gibbs.
From 1873 to '76, the American mathematical physicist Josiah Willard Gibbs published a series of three papers, the most famous being "On the equilibrium of heterogeneous substances". Gibbs showed how thermodynamic processes, including , could be graphically analyzed. By studying the energy, entropy, volume, chemical potential, temperature and pressure of the thermodynamic system, one can determine if a process would occur spontaneously. ξ9 Chemical thermodynamics was further developed by Pierre Duhem, Gilbert N. Lewis, Merle Randall, and E. A. Guggenheim, who applied the mathematical methods of Gibbs.
The components of the word thermodynamic are derived from the Greek words therme, meaning "heat," and dynamis, meaning "power" (Haynie claims that the word was coined around 1840). Oxford English Dictionary, Oxford University Press, Oxford UK.
Pierre Perrot claims that the term thermodynamics was coined by James Joule in 1858 to designate the science of relations between heat and power. Joule, however, never used that term, but used instead the term perfect thermodynamic engine in reference to Thomson’s 1849 phraseology.
By 1858, thermodynamics, as a functional term, was used in William Thomson's paper An Account of Carnot's Theory of the Motive Power of Heat.Kelvin, William T. (1849) "An Account of Carnot's Theory of the Motive Power of Heat – with Numerical Results Deduced from Regnault's Experiments on Steam." Transactions of the Edinburg Royal Society, XVI. January 2. Scanned Copy
In the account in terms of equilibrium states of a system, a state of thermodynamic equilibrium in a simple system (as defined below in this article), with no externally imposed force field, is spatially homogeneous.
In the classical account strictly and purely in terms of cyclic processes, the spatial interior of the 'working body' of a cyclic process is not considered; the 'working body' thus does not have a defined internal thermodynamic state of its own because no assumption is made that it should be in thermodynamic equilibrium; only its inputs and outputs of energy as heat and work are considered.Serrin, J. (1986). An outline of thermodynamical structure, Chapter 1, pp. 3–32 in Serrin, J., editor, New Perspectives in Thermodynamics, Springer–Verlag, Berlin, ISBN 3540159312. It is of course possible, and indeed common, for the account in terms of equilibrium states of a system to describe cycles composed of indefinitely many equilibrium states.
Classical thermodynamics was originally concerned with the transformation of energy in cyclic processes, and the exchange of energy between closed systems defined only by their equilibrium states. For these, the distinction between transfers of energy as heat and as work was central.
As classical thermodynamics developed, the distinction between heat and work became less central. This was because there was more interest in open systems, for which the distinction between heat and work is not simple, and is beyond the scope of the present article. Alongside amount of heat transferred as a fundamental quantity, entropy, considered below, was gradually found to be a more generally applicable concept, especially when chemical reactions are of interest. Massieu in 1869 considered entropy as the basic dependent thermodynamic variable, with energy potentials and the reciprocal of thermodynamic temperature as fundamental independent variables. can be useful in presentday nonequilibrium thermodynamics. In 1875, in the work of Josiah Willard Gibbs, the basic thermodynamic quantities were energy potentials, such as internal energy, as dependent variables, and entropy, considered as a fundamental independent variable.Callen, H.B. (1960/1985), Chapter 6, pages 131–152.
All actual physical processes are to some degree irreversible. Classical thermodynamics can consider irreversible processes, but its account in exact terms is restricted to variables that refer only to initial and final states of thermodynamic equilibrium, or to rates of input and output that do not change with time. For example, classical thermodynamics can consider longtimeaverage rates of flows generated by continually repeated irreversible cyclic processes. Also it can consider irreversible changes between equilibrium states of systems consisting of several phases (as defined below in this article), or with removable or replaceable partitions. But for systems that are described in terms of equilibrium states, it considers neither flows, nor spatial inhomogeneities in simple systems with no externally imposed force fields such as gravity. In the account in terms of equilibrium states of a system, descriptions of irreversible processes refer only to initial and final static equilibrium states; rates of progress are not considered.Callen, H.B. (1960/1985), p. 13.Landsberg, P.T. (1978). Thermodynamics and Statistical Mechanics, Oxford University Press, Oxford UK, ISBN 0198511426, p. 1.
For processes that involve only suitably small and smooth spatial inhomogeneities and suitably small changes with time, a good approximation can be found through the assumption of local thermodynamic equilibrium. Within the large or global region of a process, for a suitably small local region, this approximation assumes that a quantity known as the entropy of the small local region can be defined in a particular way. That particular way of definition of entropy is largely beyond the scope of the present article, but here it may be said that it is entirely derived from the concepts of classical thermodynamics; in particular, neither flow rates nor changes over time are admitted into the definition of the entropy of the small local region. It is assumed without proof that the instantaneous global entropy of a nonequilibrium system can be found by adding up the simultaneous instantaneous entropies of its constituent small local regions. Local equilibrium thermodynamics considers processes that involve the timedependent production of entropy by dissipative processes, in which kinetic energy of bulk flow and chemical potential energy are converted into internal energy at timerates that are explicitly accounted for. Timevarying bulk flows and specific diffusional flows are considered, but they are required to be dependent variables, derived only from material properties described only by static macroscopic equilibrium states of small local regions. The independent state variables of a small local region are only those of classical thermodynamics.
For generalized or extended thermodynamics, the definition of the quantity known as the entropy of a small local region is in terms beyond those of classical thermodynamics; in particular, flow rates are admitted into the definition of the entropy of a small local region. The independent state variables of a small local region include flow rates, which are not admitted as independent variables for the small local regions of local equilibrium thermodynamics.
Outside the range of classical thermodynamics, the definition of the entropy of a small local region is no simple matter. For a thermodynamic account of a process in terms of the entropies of small local regions, the definition of entropy should be such as to ensure that the second law of thermodynamics applies in each small local region. It is often assumed without proof that the instantaneous global entropy of a nonequilibrium system can be found by adding up the simultaneous instantaneous entropies of its constituent small local regions. For a given physical process, the selection of suitable independent local nonequilibrium macroscopic state variables for the construction of a thermodynamic description calls for qualitative physical understanding, rather than being a simply mathematical problem concerned with a uniquely determined thermodynamic description. A suitable definition of the entropy of a small local region depends on the physically insightful and judicious selection of the independent local nonequilibrium macroscopic state variables, and different selections provide different generalized or extended thermodynamical accounts of one and the same given physical process. This is one of the several good reasons for considering entropy as an epistemic physical variable, rather than as a simply material quantity. According to a respected author: "There is no compelling reason to believe that the classical thermodynamic entropy is a measurable property of nonequilibrium phenomena, ..."Grandy, W.T., Jr (2008), passim and p. 123.
In theoretical studies, it is often convenient to consider the simplest kind of thermodynamic system. This is defined variously by different authors.Gibbs J.W. (1875), pp. 115–116.Bryan, G.H. (1907), p. 5.Haase, R. (1971), p. 13.Bailyn, M. (1994), p. 145. For the present article, the following definition is convenient, as abstracted from the definitions of various authors. A region of material with all intensive properties continuous in space and time is called a phase. A simple system is for the present article defined as one that consists of a single phase of a pure chemical substance, with no interior partitions.
Within a simple isolated thermodynamic system in thermodynamic equilibrium, in the absence of externally imposed force fields, all properties of the material of the system are spatially homogeneous.Bailyn, M. (1994), Section 6.11. Much of the basic theory of thermodynamics is concerned with homogeneous systems in thermodynamic equilibrium.Planck, M. (1897/1903), passim.
Most systems found in nature or considered in engineering are not in thermodynamic equilibrium, exactly considered. They are changing or can be triggered to change over time, and are continuously and discontinuously subject to flux of matter and energy to and from other systems.. For example, according to Callen, "in absolute thermodynamic equilibrium all radioactive materials would have decayed completely and nuclear reactions would have transmuted all nuclei to the most stable isotopes. Such processes, which would take cosmic times to complete, generally can be ignored.". Such processes being ignored, many systems in nature are close enough to thermodynamic equilibrium that for many purposes their behaviour can be well approximated by equilibrium calculations.
Many thermodynamic processes can be modeled by compound or composite systems, consisting of several or many contiguous component simple systems, initially not in thermodynamic equilibrium, but allowed to transfer mass and energy between them. Natural thermodynamic processes are described in terms of a tendency towards thermodynamic equilibrium within simple systems and in transfers between contiguous simple systems. Such natural processes are irreversible.Guggenheim, E.A. (1949/1967), §1.12.
This statement implies that thermal equilibrium is an equivalence relation on the set of under consideration. Systems are said to be in thermal equilibrium with each other if spontaneous molecular thermal energy exchanges between them do not lead to a net exchange of energy. This law is tacitly assumed in every measurement of temperature. For two bodies known to be at the same temperature, deciding if they are in thermal equilibrium when put into thermal contact does not require actually bringing them into contact and measuring any changes of their observable properties in time.Moran, Michael J. and Howard N. Shapiro, 2008. Fundamentals of Engineering Thermodynamics. 6th ed. Wiley and Sons: 16. In traditional statements, the law provides an empirical definition of temperature and justification for the construction of practical thermometers. In contrast to absolute thermodynamic temperatures, empirical temperatures are measured just by the mechanical properties of bodies, such as their volumes, without reliance on the concepts of energy, entropy or the first, second, or third laws of thermodynamics.Planck, M. (1897/1903), p. 1. Empirical temperatures lead to calorimetry for heat transfer in terms of the mechanical properties of bodies, without reliance on mechanical concepts of energy.
The physical content of the zeroth law has long been recognized. For example, Rankine in 1853 defined temperature as follows: "Two portions of matter are said to have equal temperatures when neither tends to communicate heat to the other."Rankine, W.J.M. (1953). Proc. Roy. Soc. (Edin.), 20(4). Maxwell in 1872 stated a "Law of Equal Temperatures".Maxwell, J.C. (1872), page 32. He also stated: "All Heat is of the same kind."Maxwell, J.C. (1872), page 57. Planck explicitly assumed and stated it in its customary presentday wording in his formulation of the first two laws.Planck, M. (1897/1903), pp. 1–2. By the time the desire arose to number it as a law, the other three had already been assigned numbers, and so it was designated the zeroth law.
The first law of thermodynamics asserts the existence of a state variable for a system, the internal energy, and tells how it changes in thermodynamic processes. The law allows a given internal energy of a system to be reached by any combination of heat and work. It is important that internal energy is a variable of state of the system (see Thermodynamic state) whereas heat and work are variables that describe processes or changes of the state of systems.
The first law observes that the internal energy of an isolated system obeys the principle of conservation of energy, which states that energy can be transformed (changed from one form to another), but cannot be created or destroyed.Helmholtz, H. von, (1847). Ueber die Erhaltung der Kraft, G. Reimer, Berlin.Joule, J.P. (1847). On matter, living force, and heat, Manchester Courier, May 5 and May 12, 1847.Partington, J.R. (1949), page 150.Kondepudi & Prigogine (1998), pages 3132.
The second law of thermodynamics is an expression of the universal principle of dissipation of kinetic and potential energy observable in nature. The second law is an observation of the fact that over time, differences in temperature, pressure, and chemical potential tend to even out in a physical system that is isolated from the outside world. Entropy is a measure of how much this process has progressed. The entropy of an isolated system that is not in equilibrium tends to increase over time, approaching a maximum value at equilibrium.
In classical thermodynamics, the second law is a basic postulate applicable to any system involving heat energy transfer; in statistical thermodynamics, the second law is a consequence of the assumed randomness of molecular chaos. There are many versions of the second law, but they all have the same effect, which is to explain the phenomenon of irreversibility in nature.
The third law of thermodynamics is a statistical law of nature regarding entropy and the impossibility of reaching absolute zero of temperature. This law provides an absolute reference point for the determination of entropy. The entropy determined relative to this point is the absolute entropy. Alternate definitions are, "the entropy of all systems and of all states of a system is smallest at absolute zero," or equivalently "it is impossible to reach the absolute zero of temperature by any finite number of processes".
Absolute zero is −273.15 °C (degrees Celsius), or −459.67 °F (degrees Fahrenheit) or 0 K (kelvin).
Types of transfers permitted  in a thermodynamic process  for a type of partition !type of partition !colspan="3"  type of transfer 
!Mass and energy !Work !Heat  
The volume can be the region surrounding a single atom resonating energy, as Max Planck defined in 1900; it can be a body of steam or air in a steam engine, such as Sadi Carnot defined in 1824; it can be the body of a tropical cyclone, such as Kerry Emanuel theorized in 1986 in the field of atmospheric thermodynamics; it could also be just one nuclide (i.e. a system of ) as hypothesized in quantum thermodynamics.
Anything that passes across the boundary needs to be accounted for in a proper transfer balance equation. Thermodynamics is largely about such transfers.
Boundary sectors are of various characters: rigid, flexible, fixed, moveable, actually restrictive, and fictive or not actually restrictive. For example, in an engine, a fixed boundary sector means the piston is locked at its position; then no pressurevolume work is done across it. In that same engine, a moveable boundary allows the piston to move in and out, permitting pressurevolume work. There is no restrictive boundary sector for the whole earth including its atmosphere, and so roughly speaking, no pressurevolume work is done on or by the whole earth system. Such a system is sometimes said to be diabatically heated or cooled by radiation.Goody, R.M., Yung, Y.L. (1989). Atmospheric Radiation. Theoretical Basis, second edition, Oxford University Press, Oxford UK, ISBN 0195051343, p. 5Wallace, J.M., Hobbs, P.V. (2006). Atmospheric Science. An Introductory Survey, second edition, Elsevier, Amsterdam, ISBN 9780127329512, p. 292.
Thermodynamics distinguishes classes of systems by their boundary sectors.
Engineering and natural processes are often described as composites of many different component simple systems, sometimes with unchanging or changing partitions between them. A change of partition is an example of a thermodynamic operation.
The approach through states of thermodynamic equilibrium of a system requires a full account of the state of the system as well as a notion of process from one state to another of a system, but may require only an idealized or partial account of the state of the surroundings of the system or of other systems.
The method of description in terms of states of thermodynamic equilibrium has limitations. For example, processes in a region of turbulent flow, or in a burning gas mixture, or in a Knudsen gas may be beyond "the province of thermodynamics".Glansdorff, P., Prigogine, I., (1971). Thermodynamic Theory of Structure, Stability and Fluctuations, WileyInterscience, London, ISBN 0471302805, page 15.Haase, R., (1971), page 16.Eu, B.C. (2002), p. 13. This problem can sometimes be circumvented through the method of description in terms of cyclic or of timeinvariant flow processes. This is part of the reason why the founders of thermodynamics often preferred the cyclic process description.
Approaches through processes of timeinvariant flow of a system are used for some studies. Some processes, for example JouleThomson expansion, are studied through steadyflow experiments, but can be accounted for by distinguishing the steady bulk flow kinetic energy from the internal energy, and thus can be regarded as within the scope of classical thermodynamics defined in terms of equilibrium states or of cyclic processes.Adkins, C.J. (1968/1975), pp. 46–49. Other flow processes, for example , are essentially defined by the presence of differential flows or diffusion so that they cannot be adequately accounted for in terms of equilibrium states or classical cyclic processes.Adkins, C.J. (1968/1975), p. 172.Lebon, G., Jou, D., CasasVázquez, J. (2008), pp. 37–38.
The notion of a cyclic process does not require a full account of the state of the system, but does require a full account of how the process occasions transfers of matter and energy between the principal system (which is often called the working body) and its surroundings, which must include at least two heat reservoirs at different known and fixed temperatures, one hotter than the principal system and the other colder than it, as well as a reservoir that can receive energy from the system as work and can do work on the system. The reservoirs can alternatively be regarded as auxiliary idealized component systems, alongside the principal system. Thus an account in terms of cyclic processes requires at least four contributory component systems. The independent variables of this account are the amounts of energy that enter and leave the idealized auxiliary systems. In this kind of account, the working body is often regarded as a "black box",Buchdahl, H.A. (1966). The Concepts of Classical Thermodynamics, Cambridge University Press, London, pp. 117–118. and its own state is not specified. In this approach, the notion of a properly numerical scale of empirical temperature is a presupposition of thermodynamics, not a notion constructed by or derived from it.
If a system is simple as defined above, and is in thermodynamic equilibrium, and is not subject to an externally imposed force field, such as gravity, electricity, or magnetism, then it is homogeneous, that is say, spatially uniform in all respects.Guggenheim, E.A. (1949/1967), p. 6.
In a sense, a homogeneous system can be regarded as spatially zerodimensional, because it has no spatial variation.
If a system in thermodynamic equilibrium is homogeneous, then its state can be described by a few physical variables, which are mostly classifiable as and .Lavenda, B.H. (1978). Thermodynamics of Irreversible Processes, Macmillan, London, ISBN 0333216164, p. 12.
An intensive variable is one that is unchanged with the thermodynamic operation of scaling of a system.
An extensive variable is one that simply scales with the scaling of a system, without the further requirement used just below here, of additivity even when there is inhomogeneity of the added systems.
Examples of extensive thermodynamic variables are total mass and total volume. Under the above definition, entropy is also regarded as an extensive variable. Examples of intensive thermodynamic variables are temperature, pressure, and chemical concentration; intensive thermodynamic variables are defined at each spatial point and each instant of time in a system. Physical macroscopic variables can be mechanical, material, or thermal. Temperature is a thermal variable; according to Guggenheim, "the most important conception in thermodynamics is temperature."
Intensive variables have the property that if any number of systems, each in its own separate homogeneous thermodynamic equilibrium state, all with the same respective values of all of their intensive variables, regardless of the values of their extensive variables, are laid contiguously with no partition between them, so as to form a new system, then the values of the intensive variables of the new system are the same as those of the separate constituent systems. Such a composite system is in a homogeneous thermodynamic equilibrium. Examples of intensive variables are temperature, chemical concentration, pressure, density of mass, density of internal energy, and, when it can be properly defined, density of entropy.Guggenheim, E.A. (1949/1967), p. 19. In other words, intensive variables are not altered by the thermodynamic operation of scaling.
For the immediately present account just below, an alternative definition of extensive variables is considered, that requires that if any number of systems, regardless of their possible separate thermodynamic equilibrium or nonequilibrium states or intensive variables, are laid side by side with no partition between them so as to form a new system, then the values of the extensive variables of the new system are the sums of the values of the respective extensive variables of the individual separate constituent systems. Obviously, there is no reason to expect such a composite system to be in a homogeneous thermodynamic equilibrium. Examples of extensive variables in this alternative definition are mass, volume, and internal energy. They depend on the total quantity of mass in the system.Guggenheim, E.A. (1949/1967), pp. 18–19. In other words, although extensive variables scale with the system under the thermodynamic operation of scaling, nevertheless the present alternative definition of an extensive variable requires more than this: it requires also its additivity regardless of the inhomogeneity (or equality or inequality of the values of the intensive variables) of the component systems.
Though, when it can be properly defined, density of entropy is an intensive variable, for inhomogeneous systems, entropy itself does not fit into this alternative classification of state variables.Grandy, W.T., Jr (2008), Chapter 5, pp. 59–68.Kondepudi & Prigogine (1998), pp. 116–118. The reason is that entropy is a property of a system as a whole, and not necessarily related simply to its constituents separately. It is true that for any number of systems each in its own separate homogeneous thermodynamic equilibrium, all with the same values of intensive variables, removal of the partitions between the separate systems results in a composite homogeneous system in thermodynamic equilibrium, with all the values of its intensive variables the same as those of the constituent systems, and it is reservedly or conditionally true that the entropy of such a restrictively defined composite system is the sum of the entropies of the constituent systems. But if the constituent systems do not satisfy these restrictive conditions, the entropy of a composite system cannot be expected to be the sum of the entropies of the constituent systems, because the entropy is a property of the composite system as a whole. Therefore, though under these restrictive reservations, entropy satisfies some requirements for extensivity defined just above, entropy in general does not fit the immediately present definition of an extensive variable.
Being neither an intensive variable nor an extensive variable according to the immediately present definition, entropy is thus a standout variable, because it is a state variable of a system as a whole. A nonequilibrium system can have a very inhomogeneous dynamical structure. This is one reason for distinguishing the study of equilibrium thermodynamics from the study of nonequilibrium thermodynamics.
The physical reason for the existence of extensive variables is the timeinvariance of volume in a given inertial reference frame, and the strictly local conservation of mass, momentum, angular momentum, and energy. As noted by Gibbs, entropy is unlike energy and mass, because it is not locally conserved. The standout quantity entropy is never conserved in real physical processes; all real physical processes are irreversible.Guggenheim, E.A. (1949/1967), Section 1.12, pp. 12–13. The motion of planets seems reversible on a short time scale (millions of years), but their motion, according to Newton's laws, is mathematically an example of deterministic chaos. Eventually a planet suffers an unpredictable collision with an object from its surroundings, outer space in this case, and consequently its future course is radically unpredictable. Theoretically this can be expressed by saying that every natural process dissipates some information from the predictable part of its activity into the unpredictable part. The predictable part is expressed in the generalized mechanical variables, and the unpredictable part in heat.
Other state variables can be regarded as conditionally 'extensive' subject to reservation as above, but not extensive as defined above. Examples are the Gibbs free energy, the Helmholtz free energy, and the enthalpy. Consequently, just because for some systems under particular conditions of their surroundings such state variables are conditionally conjugate to intensive variables, such conjugacy does not make such state variables extensive as defined above. This is another reason for distinguishing the study of equilibrium thermodynamics from the study of nonequilibrium thermodynamics. In another way of thinking, this explains why heat is to be regarded as a quantity that refers to a process and not to a state of a system.
A system with no internal partitions, and in thermodynamic equilibrium, can be inhomogeneous in the following respect: it can consist of several socalled 'phases', each homogeneous in itself, in immediate contiguity with other phases of the system, but distinguishable by their having various respectively different physical characters, with discontinuity of intensive variables at the boundaries between the phases; a mixture of different chemical species is considered homogeneous for this purpose if it is physically homogeneous.Planck, M. (1897/1903), p. 65. For example, a vessel can contain a system consisting of water vapour overlying liquid water; then there is a vapour phase and a liquid phase, each homogeneous in itself, but still in thermodynamic equilibrium with the other phase. For the immediately present account, systems with multiple phases are not considered, though for many thermodynamic questions, multiphase systems are important.
In one way, the system is considered to be connected to the surroundings by some kind of more or less separating partition, and allowed to reach equilibrium with the surroundings with that partition in place. Then, while the separative character of the partition is kept unchanged, the conditions of the surroundings are changed, and exert their influence on the system again through the separating partition, or the partition is moved so as to change the volume of the system; and a new equilibrium is reached. For example, a system is allowed to reach equilibrium with a heat bath at one temperature; then the temperature of the heat bath is changed and the system is allowed to reach a new equilibrium; if the partition allows conduction of heat, the new equilibrium is different from the old equilibrium.
In the other way, several systems are connected to one another by various kinds of more or less separating partitions, and to reach equilibrium with each other, with those partitions in place. In this way, one may speak of a 'compound system'. Then one or more partitions is removed or changed in its separative properties or moved, and a new equilibrium is reached. The JouleThomson experiment is an example of this; a tube of gas is separated from another tube by a porous partition; the volume available in each of the tubes is determined by respective pistons; equilibrium is established with an initial set of volumes; the volumes are changed and a new equilibrium is established.Planck, M. (1897/1903), Section 70, pp. 48–50.Guggenheim, E.A. (1949/1967), Section 3.11, pp. 92–92.Sommerfeld, A. (1952/1956), Section 1.5 C, pp. 23–25.Callen, H.B. (1960/1985), Section 6.3.. Another example is in separation and mixing of gases, with use of chemically semipermeable membranes.Planck, M. (1897/1903), Section 236, pp. 211–212.
Several commonly studied thermodynamic processes are:
It is sometimes of interest to study a process in which several variables are controlled, subject to some specified constraint. In a system in which a chemical reaction can occur, for example, in which the pressure and temperature can affect the equilibrium composition, a process might occur in which temperature is held constant but pressure is slowly altered, just so that chemical equilibrium is maintained all the way. There is a corresponding process at constant temperature in which the final pressure is the same but is reached by a rapid jump. Then it can be shown that the volume change resulting from the rapid jump process is smaller than that from the slow equilibrium process. The work transferred differs between the two processes.
A cyclic process of a system requires in its surroundings at least two heat reservoirs at different temperatures, one at a higher temperature that supplies heat to the system, the other at a lower temperature that accepts heat from the system. The early work on thermodynamics tended to use the cyclic process approach, because it was interested in machines that converted some of the heat from the surroundings into mechanical power delivered to the surroundings, without too much concern about the internal workings of the machine. Such a machine, while receiving an amount of heat from a higher temperature reservoir, always needs a lower temperature reservoir that accepts some lesser amount of heat, the difference in amounts of heat being converted to work.Truesdell, C.A. (1980).Kondepudi, D. (2008). Introduction to Modern Thermodynamics, Wiley, Chichester, ISBN 9780470015988, Sections 3.1,3.2, pp. 97–108. Later, the internal workings of a system became of interest, and they are described by the states of the system. Nowadays, instead of arguing in terms of cyclic processes, some writers are inclined to derive the concept of absolute temperature from the concept of entropy, a variable of state.
A thermodynamic reservoir is a system so large that it does not appreciably alter its state parameters when brought into contact with the test system. It is used to impose a particular value of a state parameter upon the system. For example, a pressure reservoir is a system at a particular pressure, which imposes that pressure upon any test system that it is mechanically connected to. The Earth's atmosphere is often used as a pressure reservoir.
Conjugate variables are pairs of thermodynamic concepts, with the first being akin to a "force" applied to some thermodynamic system, the second being akin to the resulting "displacement," and the product of the two equalling the amount of energy transferred. The common conjugate variables are:
The five most well known potentials are:
where $T$ is the temperature, $S$ the entropy, $p$ the pressure, $V$ the volume, $\backslash mu$ the chemical potential, $N$ the number of particles in the system, and $i$ is the count of particles types in the system.
Thermodynamic potentials can be derived from the energy balance equation applied to a thermodynamic system. Other thermodynamic potentials can also be obtained through Legendre transformation.
In 1909, Constantin Carathéodory presented a purely mathematical axiomatic formulation, a description often referred to as geometrical thermodynamics, and sometimes said to take the "mechanical approach" to thermodynamics. The Carathéodory formulation is restricted to equilibrium thermodynamics and does not attempt to deal with nonequilibrium thermodynamics, forces that act at a distance on the system, or surface tension effects.Turner, L.A. (1962). Simplification of Carathéodory's treatment of thermodynamics, Am. J. Phys. 30: 781–786. Moreover, Carathéodory's formulation does not deal with materials like water near 4 °C, which have a density extremum as a function of temperature at constant pressure.Turner, L.A. (1962). Further remarks on the zeroth law, Am. J. Phys. 30: 804–806.Thomsen, J.S., Hartka, T.J., (1962). Strange Carnot cycles; thermodynamics of a system with a density maximum, Am. J. Phys. 30: 26–33, 30: 388–389. Carathéodory used the law of conservation of energy as an axiom from which, along with the contents of the zeroth law, and some other assumptions including his own version of the second law, he derived the first law of thermodynamics. Consequently one might also describe Carathėodory's work as lying in the field of energetics,Duhem, P. (1911). Traité d'Energetique'', GautierVillars, Paris. which is broader than thermodynamics. Carathéodory presupposed the law of conservation of mass without explicit mention of it.
Since the time of Carathėodory, other influential axiomatic formulations of thermodynamics have appeared, which like Carathéodory's, use their own respective axioms, different from the usual statements of the four laws, to derive the four usually stated laws.Callen, H.B. (1960/1985).Truesdell, C., Bharatha, S. (1977). The Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines, Rigorously Constructed upon the Foundation Laid by S. Carnot and F. Reech, Springer, New York, ISBN 0387079718.Wright, P.G. (1980). Conceptually distinct types of thermodynamics, Eur. J. Phys. 1: 81–84.
Many axiomatic developments assume the existence of states of thermodynamic equilibrium and of states of thermal equilibrium. States of thermodynamic equilibrium of compound systems allow their component simple systems to exchange heat and matter and to do work on each other on their way to overall joint equilibrium. Thermal equilibrium allows them only to exchange heat. The physical properties of glass depend on its history of being heated and cooled and, strictly speaking, glass is not in thermodynamic equilibrium.
According to Herbert Callen's widely cited 1985 text on thermodynamics: "An essential prerequisite for the measurability of energy is the existence of walls that do not permit transfer of energy in the form of heat.".Callen, H.B. (1960/1985), p. 16. According to Werner Heisenberg's mature and careful examination of the basic concepts of physics, the theory of heat has a selfstanding place.Heisenberg, W. (1958). Physics and Philosophy, Harper & Row, New York, pp. 98–99.
From the viewpoint of the axiomatist, there are several different ways of thinking about heat, temperature, and the second law of thermodynamics. The Clausius way rests on the empirical fact that heat is conducted always down, never up, a temperature gradient. The Kelvin way is to assert the empirical fact that conversion of heat into work by cyclic processes is never perfectly efficient. A more mathematical way is to assert the existence of a function of state called the entropy that tells whether a hypothesized process occurs spontaneously in nature. A more abstract way is that of Carathéodory that in effect asserts the irreversibility of some adiabatic processes. For these different ways, there are respective corresponding different ways of viewing heat and temperature.
The Clausius–Kelvin–Planck way This way prefers ideas close to the empirical origins of thermodynamics. It presupposes transfer of energy as heat, and empirical temperature as a scalar function of state. According to Gislason and Craig (2005): "Most thermodynamic data come from calorimetry..."Gislason, E.A., Craig, N.C. (2005). Cementing the foundations of thermodynamics:comparison of systembased and surroundingsbased definitions of work and heat, J. Chem. Thermodynamics 37: 954–966. According to Kondepudi (2008): "Calorimetry is widely used in present day laboratories."Kondepudi, D. (2008). Introduction to Modern Thermodynamics, Wiley, Chichester, ISBN 9780470015988, p. 63. In this approach, what is often currently called the zeroth law of thermodynamics is deduced as a simple consequence of the presupposition of the nature of heat and empirical temperature, but it is not named as a numbered law of thermodynamics. Planck attributed this point of view to Clausius, Kelvin, and Maxwell. Planck wrote (on page 90 of the seventh edition, dated 1922, of his treatise) that he thought that no proof of the second law of thermodynamics could ever work that was not based on the impossibility of a perpetual motion machine of the second kind. In that treatise, Planck makes no mention of the 1909 Carathéodory way, which was well known by 1922. Planck for himself chose a version of what is just above called the Kelvin way.Planck, M. (1922/1927). The development by Truesdell and Bharatha (1977) is so constructed that it can deal naturally with cases like that of water near 4 °C.
The way that assumes the existence of entropy as a function of state This way also presupposes transfer of energy as heat, and it presupposes the usually stated form of the zeroth law of thermodynamics, and from these two it deduces the existence of empirical temperature. Then from the existence of entropy it deduces the existence of absolute thermodynamic temperature.
The Carathéodory way This way presupposes that the state of a simple onephase system is fully specifiable by just one more state variable than the known exhaustive list of mechanical variables of state. It does not explicitly name empirical temperature, but speaks of the onedimensional "nondeformation coordinate". This satisfies the definition of an empirical temperature, that lies on a onedimensional manifold. The Carathéodory way needs to assume moreover that the onedimensional manifold has a definite sense, which determines the direction of irreversible adiabatic process, which is effectively assuming that heat is conducted from hot to cold. This way presupposes the often currently stated version of the zeroth law, but does not actually name it as one of its axioms. According to one author, Carathéodory's principle, which is his version of the second law of thermodynamics, does not imply the increase of entropy when work is done under adiabatic conditions (as was noted by PlanckPlanck, M. (1926). Über die Begründung des zweiten Hauptsatzes der Thermodynamik, Sitzungsberichte der Preußischen Akademie der Wissenschaften, physikalischmathematischen Klasse, pp. 453–463.). Thus Carathéodory's way leaves unstated a further empirical fact that is needed for a full expression of the second law of thermodynamics.Münster, A. (1970). Classical Thermodynamics, translated by E.S. Halberstadt, Wiley–Interscience, London, ISBN 0471624306, p 41.
Gradually, the laws of thermodynamics came to be used to explain phenomena that occur outside the experimental laboratory. For example, phenomena on the scale of the earth's atmosphere cannot be reproduced in a laboratory experiment. But processes in the atmosphere can be modeled by use of thermodynamic ideas, extended well beyond the scope of laboratory equilibrium thermodynamics.Iribarne, J.V., Godson, W.L. (1973/1989). Atmospheric thermodynamics, second edition, reprinted 1989, Kluwer Academic Publishers, Dordrecht, ISBN 9027712964.Peixoto, J.P., Oort, A.H. (1992). Physics of climate, American Institute of Physics, New York, ISBN 0883187124North, G.R., Erukhimova, T.L. (2009). Atmospheric Thermodynamics. Elementary Physics and Chemistry, Cambridge University Press, Cambridge UK, ISBN 9780521899635. A parcel of air can, near enough for many studies, be considered as a closed thermodynamic system, one that is allowed to move over significant distances. The pressure exerted by the surrounding air on the lower face of a parcel of air may differ from that on its upper face. If this results in rising of the parcel of air, it can be considered to have gained potential energy as a result of work being done on it by the combined surrounding air below and above it. As it rises, such a parcel usually expands because the pressure is lower at the higher altitudes that it reaches. In that way, the rising parcel also does work on the surrounding atmosphere. For many studies, such a parcel can be considered nearly to neither gain nor lose energy by heat conduction to its surrounding atmosphere, and its rise is rapid enough to leave negligible time for it to gain or lose heat by radiation; consequently the rising of the parcel is near enough adiabatic. Thus the adiabatic gas law accounts for its internal state variables, provided that there is no precipitation into water droplets, no evaporation of water droplets, and no sublimation in the process. More precisely, the rising of the parcel is likely to occasion friction and turbulence, so that some potential and some kinetic energy of bulk converts into internal energy of air considered as effectively stationary. Friction and turbulence thus oppose the rising of the parcel.Holton, J.R. (2004). An Introduction of Dynamic Meteorology, fourth edition, Elsevier, Amsterdam, ISBN 9780123540157.Mak, M. (2011). Atmospheric Dynamics, Cambridge University Press, Cambridge UK, ISBN 9780521195737.
The following titles are more technical:

