First law of thermodynamics
The first law of thermodynamics is a version of the law of conservation of energy, adapted for thermodynamic systems. The internal energy of an isolated system is constant and energy can be transformed from one form to another, but cannot be created or destroyed. The first law is often formulated by stating that the change in the internal energy of a closed system is equal to the amount of heat supplied to the system, minus the amount of work done by the system on its surroundings.
The process of development of the first law of thermodynamics was by way of many tries and mistakes of investigation, over a period of about half a century. The first full statements of the law came in 1850 from Rudolf Clausius and by William Rankine; Rankine's statement was perhaps not quite as clear and distinct as was Clausius'. A main aspect of the struggle was to deal with the previously proposed caloric theory of heat.
Germain Hess in 1840 stated a conservation law for the so-called 'heat of reaction' for chemical reactions. His law was later recognized as a consequence of the first law of thermodynamics, but Hess's statement was not explicitly concerned with the relation between energy exchanges by heat and work.
According to Truesdell (1980), Julius Robert von Mayer in 1841 made a statement that meant that "in a process at constant pressure, the heat used to produce expansion is universally interconvertible with work", but this is not a general statement of the first law.
Original statements 
The first explicit statement of the first law of thermodynamics, by Rudolf Clausius in 1850, referred to cyclic thermodynamic processes.
- In all cases in which work is produced by the agency of heat, a quantity of heat is consumed which is proportional to the work done; and conversely, by the expenditure of an equal quantity of work an equal quantity of heat is produced.
Clausius also stated the law in another form, referring to the existence of a function of state of the system, the internal energy, and expressed it in terms of a differential equation for the increments of a thermodynamic process. This equation may described as follows:
- In a thermodynamic process involving a closed system, the increment in the internal energy is equal to the difference between the heat accumulated by the system and the work done by it.
Conceptually revised statement 
The original nineteenth century statements of the first law of thermodynamics appeared in a conceptual framework in which transfer of energy as heat was taken as a primitive notion, not defined or constructed by the theoretical development of the framework, but rather presupposed as prior to it and already accepted. The primitive notion of heat was taken as empirically established, especially through calorimetry regarded as a subject in its own right, prior to thermodynamics. Jointly primitive with this notion of heat were the notions of empirical temperature and thermal equilibrium. This framework did not presume a concept of energy in general, but regarded it as derived or synthesized from the prior notions of heat and work. This framework might be called the "thermodynamic" approach.
In the early twentieth century, an alternative approach became preferred by many writers, an approach that might be called the "mechanical approach". This approach regards energy, defined by mechanics, as a primitive notion. It derives the notions of transfer of energy as heat, and of temperature, in its theoretical development. It regards calorimetry as a derived theory. It has an early origin in the nineteenth century, for example in the work of Helmholtz, but also in the work of many others.
The revised statement of the law takes the notions of adiabatic mechanical work, and of non-adiabatic transfer of energy, as empirically or theoretically established primitive notions. It rests on the primitive notion of walls, especially adiabatic walls, presupposed as physically established. Two systems separated by an adiabatic wall can come to their respective internal mechanical and material thermodynamic equilibrium states completely independently of one another. Energy can pass such walls as only as adiabatic work, reversibly or irreversibly.
The revised statement of the law postulates that a change in the internal energy of a system due to an arbitrary process of interest, that takes the system from its specified initial to its specified final state of internal thermodynamic equilibrium, can be determined through the physical existence of a reference process, for those specified states, that occurs purely through stages of adiabatic work.
The revised statement is then
- For a closed system, in any arbitrary process of interest that takes it from an initial to a final state of internal thermodynamic equilibrium, the change of internal energy is the same as that for a reference adiabatic work process that links those two states. This is so regardless of the path of the process of interest, and regardless of whether it is an adiabatic or a non-adiabatic process.
This statement is much less less close to the empirical basis than are the original statements, but is often regarded as conceptually parsimonious in that it rests only on the concepts of adiabatic work and of non-adiabatic processes, not on the concepts of transfer of energy as heat and of empirical temperature that are presupposed by the original statements. It is often regarded as theoretically preferable because of this conceptual parsimony.
The first law of thermodynamics was expressed in two ways by Clausius. One way referred to cyclic processes and the inputs and outputs of the system, but did not refer to increments in the internal state of the system. The other way referred to any incremental change in the internal state of the system, and did not expect the process to be cyclic. A cyclic process is one which can be repeated indefinitely often and still eventually leave the system in its original state.
In each repetition of a cyclic process, the work done by the system is proportional to the heat consumed by the system. In a cyclic process in which the system does work on its surroundings, it is necessary that some heat be taken in by the system and some be put out, and the difference is the heat consumed by the system in the process. The constant of proportionality is universal and independent of the system and was measured by James Joule in 1845 and 1847, who described it as the mechanical equivalent of heat.
In any incremental process, the change in the internal energy is considered due to a combination of heat added to the system and work done by the system. Taking as an infinitesimal (differential) change in internal energy, one writes
where and are infinitesimal amounts of heat supplied to the system by its surroundings and work done by the system on its surroundings, respectively. This sign convention is implicit in Clausius' statement of the law given above, and is consistent with the use of thermodynamics to study heat engines which provide useful work, which is regarded as positive.
In modern style of teaching science, however, it is conventional to use the IUPAC convention by which the first law is formulated in terms of the work done on the system. With this alternate sign convention for work, the first law for a closed system may be written:
This convention follows physicists such as Max Planck, and considers all net energy transfers to the system as positive and all net energy transfers from the system as negative, irrespective of any use for the system as an engine or other device.
When a system expands in a quasistatic process, the work done by the system on the environment is the product, P dV, of pressure, P, and volume change, dV, whereas the work done on the system is -P dV. Using either sign convention for work, the change in internal energy of the system is:
Work and heat are expressions of actual physical processes which supply or remove energy, while is a mathematical abstraction that keeps account of the exchanges of energy that befall the system. Thus the term heat for means that amount of energy added or removed by conduction of heat or by thermal radiation, rather than referring to a form of energy within the system. Likewise, work energy for means "that amount of energy gained or lost as the result of work". Internal energy is a property of the system whereas work done and heat supplied are not. A significant result of this distinction is that a given internal energy change can be achieved by, in principle, many combinations of heat and work.
The internal energy of a system is not uniquely defined. It is defined only up to an arbitrary additive constant of integration, which can be adjusted to give arbitrary reference zero levels. This non-uniqueness is in keeping with the abstract mathematical nature of the internal energy. The internal energy is stated relative to a conventionally chosen standard reference state of the system.
Various statements of the law for closed systems 
The law is of very great importance and generality and is consequently thought of from several points of view. Most careful textbook statements of the law express it for closed systems. It is stated in several ways, sometimes even by the same author.
For the thermodynamics of closed systems, the distinction between transfers of energy as work and as heat is central and is within the scope of the present article. For the thermodynamics of open systems, such a distinction is beyond the scope of the present article, but some limited comments are made on it in the section below headed 'First law of thermodynamics for open systems'.
There are two main ways of stating a law of thermodynamics, physically or mathematically. They should be logically coherent and consistent with one another.
An example of a physical statement is that of Planck (1897/1903):
- It is in no way possible, either by mechanical, thermal, chemical, or other devices, to obtain perpetual motion, i.e. it is impossible to construct an engine which will work in a cycle and produce continuous work, or kinetic energy, from nothing.
This physical statement is restricted neither to closed systems nor to systems with states that are strictly defined only for thermodynamic equilibrium; it has meaning also for open systems and for systems with states that are not in thermodynamic equilibrium.
An example of a mathematical statement is that of Crawford (1963):
- For a given system we let ΔE kin = large-scale mechanical energy, ΔE pot = large-scale potential energy, and ΔE tot = total energy. The first two quantities are specifiable in terms of appropriate mechanical variables, and by definition
- For any finite process, whether reversible or irreversible,
- The first law in a form that involves the principle of conservation of energy more generally is
- Here Q and W are heat and work added, with no restrictions as to whether the process is reversible, quasistatic, or irreversible.[Warner, Am. J. Phys., 29, 124 (1961)]
This statement by Crawford, for W, uses the sign convention of IUPAC, not that of Clausius. Though it does not explicitly say so, this statement refers to closed systems, and to internal energy U defined for bodies in states of thermodynamic equilibrium, which possess well-defined temperatures.
The history of statements of the law for closed systems has two main periods, before and after the work of Bryan (1907), of Carathéodory (1909), and the approval of Carathéodory's work given by Born (1921). The earlier traditional versions of the law for closed systems are nowadays often considered to be out of date.
Carathéodory's celebrated presentation of equilibrium thermodynamics refers to closed systems, which are allowed to contain several phases connected by internal walls of various kinds of impermeability and permeability, explicitly including walls that are permeable only to heat. Carathéodory's version of the first law of thermodynamics was stated in an axiom which refrained from defining or mentioning temperature or quantity of heat transferred. That axiom stated that the internal energy of a phase in equilibrium is a function of state, that the sum of the internal energies of the phases is the total internal energy of the system, and that the value of the total internal energy of the system is changed by the amount of work done adiabatically on it, considering work as a form of energy. That article considered this statement to be an expression of the law of conservation of energy for such systems. This version is nowadays widely accepted as authoritative, but is stated in slightly varied ways by different authors. The Carathéodory statement of the law in axiomatic form does not mention heat or temperature, but the equilibrium states to which it refers are explicitly defined by variable sets that necessarily include "non-deformation variables", such as pressures, which, within reasonable restrictions, can be rightly interpreted as empirical temperatures, and the walls connecting the phases of the system are explicitly defined as possibly impermeable to heat or permeable only to heat. According to Münster (1970), "A somewhat unsatisfactory aspect of Carathéodory's theory is that a consequence of the Second Law must be considered at this point [in the statement of the first law], i.e. that it is not always possible to reach any state 2 from any other state 1 by means of an adiabatic process." Münster instances that no adiabatic process can reduce the internal energy of a system at constant volume. Carathéodory's paper asserts that its statement of the first law corresponds exactly to Joule's experimental arrangement, regarded as an instance of adiabatic work. It does not point out that Joule's experimental arrangement performed essentially irreversible work, through friction of paddles in a liquid, or passage of electric current through a resistance inside the system, driven by motion of a coil and inductive heating, or by an external current source, which can access the system only by the passage of electrons, and so is not strictly adiabatic, because electrons are a form of matter, which cannot penetrate adiabatic walls. The paper goes on to base its main argument on the possibility of quasi-static adiabatic work, which is essentially reversible. The paper asserts that it will avoid reference to Carnot cycles, and then proceeds to base its argument on cycles of forward and backward quasi-static adiabatic stages, with isothermal stages of zero magnitude.
Some respected modern statements of the first law for closed systems assert the existence of internal energy as a function of state defined in terms of adiabatic work and accept the Carathéodory idea that heat is not defined in its own right, that is to say calorimetrically or as due to temperature difference; they define heat as a residual difference between change of internal energy and work done on the system, when that work does not account for the whole of the change of internal energy and the system is not adiabatically isolated.
Sometimes the existence of the internal energy is made explicit but work is not explicitly mentioned in the statement of the first postulate of thermodynamics. Heat supplied is then defined as the residual change in internal energy after work has been taken into account, in a non-adiabatic process.
A respected modern author states the first law of thermodynamics as "Heat is a form of energy", which explicitly mentions neither internal energy nor adiabatic work. Heat is defined as energy transferred by thermal contact with a reservoir, which has a temperature, and is generally so large that addition and removal of heat do not alter its temperature. A current student text on chemistry defines heat thus: "heat is the exchange of thermal energy between a system and its surroundings caused by a temperature difference." The author then explains how heat is defined or measured by calorimetry, in terms of heat capacity, specific heat capacity, molar heat capacity, and temperature.
A respected text disregards the Carathéodory's exclusion of mention of heat from the statement of the first law for closed systems, and admits heat calorimetrically defined along with work and internal energy. Another respected text defines heat exchange as determined by temperature difference, but also mentions that the Born (1921) version is "completely rigorous". These versions follow the traditional approach that is now considered out of date, exemplified by that of Planck (1897/1903).
Evidence for the first law of thermodynamics for closed systems 
The first law of thermodynamics for closed systems was originally induced from empirically observed evidence, however, it is now taken to be the definition of heat via the law of conservation of energy and the definition of work in terms of changes in the external parameters of a system. The original discovery of the law was gradual over a period of perhaps half a century or more, and some early studies were in terms of cyclic processes. The following is an account in terms of changes of state through compound processes that are not necessarily cyclic. This account first considers processes for which the first law is easily verified because of their simplicity, namely adiabatic processes (in which no heat is transferred) and adynamic processes (in which no work is transferred).
Adiabatic processes 
Given a closed system in an initial state, if work is done on the system in an adiabatic (i.e. no heat transfer) way, and given the final state after a process, the amount of work required to be transferred to the system is the same, irrespective of how this work is performed. The work done on the system is defined and measured by changes in mechanical or quasi-mechanical variables external to the system. Physically, adiabatic transfer of energy as work requires the existence of adiabatic enclosures.
For instance, in Joule's experiment, the initial system is a tank of water with a paddle wheel inside. If we isolate thermally the tank and move the paddle wheel with a pulley and a weight we can relate the increase in temperature with the height descended by the mass. Now the system is returned to its initial state, isolated again, and the same amount of work is done on the tank using different devices (an electric motor, a chemical battery, a spring,...). In every case, the amount of work can be measured independently. The return to the initial state is not conducted by doing adiabatic work on the system. The evidence shows that the final state of the water (in particular, its temperature and volume) is the same in every case. It is irrelevant if the work is electrical, mechanical, chemical,... or if done suddenly or slowly, as long as it is performed in an adiabatic way, that is to say, without heat transfer into or out of the system.
Evidence of this kind shows that to increase the temperature of the water in the tank, the qualitative kind of adiabatically performed work does not matter. No qualitative kind of adiabatic work has ever been observed to decrease the temperature of the water in the tank.
A change from one state to another, for example an increase of both temperature and volume, may be conducted in several stages, for example by externally supplied electrical work on a resistor in the body, and adiabatic expansion allowing the body to do work on the surroundings. It needs to be shown that the time order of the stages, and their relative magnitudes, does not affect the amount of adiabatic work that needs to be done for the change of state. According to one respected scholar: "Unfortunately, it does not seem that experiments of this kind have ever been carried out carefully. ... We must therefore admit that the statement which we have enunciated here, and which is equivalent to the first law of thermodynamics, is not well founded on direct experimental evidence."
This kind of evidence, of independence of sequence of stages, combined with the above-mentioned evidence, of independence of qualitative kind of work, would show the existence of a very important state variable that corresponds with adiabatic work, but not that such a state variable represented a conserved quantity. For the latter, another step of evidence is needed, which may be related to the concept of reversibility, as mentioned below.
That very important state variable was first recognized and denoted by Clausius in 1850, but he did not then name it, and he defined it in terms not only of work but also of heat transfer in the same process. It was also independently recognized in 1850 by Rankine, who also denoted it ; and in 1851 by Kelvin who then called it "mechanical energy", and later "intrinsic energy". In 1865, after some hestitation, Clausius began calling his state function "energy". In 1882 it was named as the internal energy by Helmholtz. If only adiabatic processes were of interest, and heat could be ignored, the concept of internal energy would hardly arise or be needed. The relevant physics would be largely covered by the concept of potential energy, as was intended in the 1847 paper of Helmholtz on the principle of conservation of energy, though that did not deal with forces that cannot be described by a potential, and thus did not fully justify the principle; moreover that paper was very critical of the early work of Joule which had by then been performed. A great merit of the internal energy concept is that it frees thermodynamics from a restriction to cyclic processes, and allows a treatment in terms of thermodynamic states.
In an adiabatic process, adiabatic work takes the system either from a reference state with internal energy to an arbitrary one with internal energy , or from the state to the state :
Except under the special, and strictly speaking, fictional, condition of reversibility, only one of the processes or is empirically feasible by a simple application of externally supplied work. The reason for this is given as the second law of thermodynamics and is not considered in the present article.
The fact of such irreversibility may be dealt with in two main ways, according to different points of view. since the work of Bryan (1907),
- To deal with it nowadays, the most accepted way, followed by Carathéodory, is to rely on the previously established concept of quasi-static processes, as follows. Actual physical processes of transfer of energy as work are always at least to some degree irreversible. The irreversibility is often due to mechanisms known as dissipative, that transform bulk kinetic energy into internal energy. Examples are friction and viscosity. If the process is performed more slowly, the frictional or viscous dissipation is less. In the limit of infinitely slow performance, the dissipation tends to zero and then the limiting process, though fictional rather than actual, is notionally reversible, and is called quasi-static. Throughout the course of the fictional limiting quasi-static process, the internal intensive variables of the system are equal to the external intensive variables, those which describe the reactive forces exerted by the surroundings. This can be taken to justify the formula
- Another way to deal with it is to allow that experiments with processes of heat transfer to or from the system may be used to justify the formula just above. Moreover, it deals to some extent with the problem of lack of direct experimental evidence that the time order of stages of a process does not matter in the determination of internal energy. This way does not provide theoretical purity in terms of adiabatic work processes, but is empirically feasible, and is in accord with experiments actually done, such as the Joule experiments mentioned just above, and with older traditions.
The formula above allows that to go by processes of quasi-static adiabatic work from the state to the state we can take a path that goes through the reference state , since the quasi-static adiabatic work is independent of the path
This kind of empirical evidence, coupled with theory of this kind, largely justifies the following statement:
- For all adiabatic processes between two specified states of a closed system of any nature, the net work done is the same regardless the details of the process, and determines a state function called internal energy, ."
Adynamic processes 
A complementary observable aspect of the first law is about heat transfer. Adynamic transfer of energy as heat can be measured empirically by changes in the surroundings of the system of interest by calorimetry. This again requires the existence of adiabatic enclosure of the entire process, system and surroundings, though the separating wall between the surroundings and the system is thermally conductive or radiatively permeable, not adiabatic. A calorimeter can rely on measurement of sensible heat, which requires the existence of thermometers and measurement of temperature change in bodies of known sensible heat capacity under specified conditions; or it can rely on the measurement of latent heat, through measurement of masses of material which change phase, at temperatures fixed by the occurrence of phase changes under specified conditions in bodies of known latent heat of phase change. The calorimeter can be calibrated by adiabatically doing externally determined work on it. The most accurate method is by passing an electric current from outside through a resistance inside the calorimeter. The calibration allows comparison of calorimetric measurement of quantity of heat transferred with quantity of energy transferred as work. According to one textbook, "The most common device for measuring is an adiabatic bomb calorimeter." According to another textbook, "Calorimetry is widely used in present day laboratories." According to one opinion, "Most thermodynamic data come from calorimetry..." According to another opinion, "The most common method of measuring “heat” is with a calorimeter."
When the system evolves with transfer of energy as heat, without energy being transferred as work, in an adynamic process, the heat transferred to the system is equal to the increase in its internal energy:
General case for reversible processes 
Heat transfer is practically reversible when it is driven by practically negligibly small temperature gradients. Work transfer is practically reversible when it occurs so slowly that there are no frictional effects within the system; frictional effects outside the system should also be zero if the process is to be globally reversible. For a particular reversible process in general, the work done reversibly on the system, , and the heat transferred reversibly to the system, are not required to occur respectively adiabatically or adynamically, but they must belong to the same particular process defined by its particular reversible path, , through the space of thermodynamic states. Then the work and heat transfers can occur and be calculated simultaneously.
Putting the two complementary aspects together, the first law for a particular reversible process can be written
This combined statement is the expression the first law of thermodynamics for reversible processes for closed systems.
In particular, if no work is done on a thermally isolated closed system we have
This is one aspect of the law of conservation of energy and can be stated:
- The internal energy of an isolated system remains constant.
General case for irreversible processes 
If, in a process of change of state of a closed system, the energy transfer is not under a practically zero temperature gradient and practically frictionless, then the process is irreversible. Then the heat and work transfers may be difficult to calculate, and irreversible thermodynamics is called for. Nevertheless, the first law still holds and provides a check on the measurements and calculations of the work done irreversibly on the system, , and the heat transferred irreversibly to the system, , which belong to the same particular process defined by its particular irreversible path, , through the space of thermodynamic states.
This means that the internal energy is a function of state and that the internal energy change between two states is a function only of the two states.
Overview of the weight of evidence for the law 
The first law of thermdynamics is very general and makes so many predictions that they can hardly all be directly tested by experiment. Nevertheless, very very many of its predictions have been found empirically accurate. And very importantly, no accurately and properly conducted experiment has ever detected a violation of the law. Consequently, within its scope of applicability, the law is so reliably established, that, nowadays, rather than experiment being considered as testing the accuracy of the law, it is far more practical and realistic to think of the law as testing the accuracy of experiment. An experimental result that seems to violate the law may be assumed to be inaccurate or wrongly conceived, for example due to failure to consider an important physical factor.
State functional formulation for infinitesimal processes 
When the heat and work transfers in the equations above are infinitesimal in magnitude, they are often denoted by δ, rather than exact differentials denoted by "d", as a reminder that heat and work do not describe the state of any system. The integral of an inexact differential depends upon the particular path taken through the space of thermodynamic parameters while the integral of an exact differential depends only upon the initial and final states. If the initial and final states are the same, then the integral of an inexact differential may or may not be zero, but the integral of an exact differential will always be zero. The path taken by a thermodynamic system through a chemical or physical change is known as a thermodynamic process.
For a homogeneous system, with a well-defined temperature and pressure, the expression for dU can be written in terms of exact differentials, if the work that the system does is equal to its pressure times the infinitesimal increase in its volume. Here one assumes that the changes are quasistatic, so slow that there is at each instant negligible departure from thermodynamic equilibrium within the system. In other words, δW = -PdV where P is pressure and V is volume. As such a quasistatic process in a homogeneous system is reversible, the total amount of heat added to a closed system can be expressed as δQ =TdS where T is the temperature and S the entropy of the system. Therefore, for closed, homogeneous systems:
The above equation is known as the fundamental thermodynamic relation, for which the independent variables are taken as S and V, with respect to which T and P are partial derivatives of U. While this has been derived for quasistatic changes, it is valid in general, as U can be considered as a thermodynamic state function of the independent variables S and V.
E.g., suppose that the system is initially in a state of thermal equilibrium defined by S and V, and then the system is suddenly perturbed so that thermal equilibrium breaks down and no temperature and pressure can be defined. Then the system settles down again to a state of thermal equilibrium, defined by an entropy and a volume which differ infinitesimally from the initial values. The infinitesimal difference in internal energy between the initial and final state will then satisfy the above equation. The work done and heat added to the system will then not satisfy the above expressions, they will instead satisfy the inequalities: δQ < TdS' and δW < PdV'.
In the case of a closed system in which the particles of the system are of different types and, because chemical reactions may occur, their respective numbers are not necessarily constant, the expression for dU becomes:
where dNi is the (small) increase in amount of type-i particles in the reaction, and μi is known as the chemical potential of the type-i particles in the system. If dNi is expressed in kg then μi is expressed in J/kg. The statement of the first law, using exact differentials is now:
If the system has more external mechanical variables than just the volume that can change, the fundamental thermodynamic relation generalizes to:
Here the Xi are the generalized forces corresponding to the external variables xi. The parameters Xi are independent of the size of the system and are called intensive parameters and the xi are proportional to the size and called extensive parameters.
For an open system, there can be transfers of particles as well as energy into or out of the system during a process. For this case, the first law of thermodynamics still holds, in the form that the internal energy is a function of state and the change of internal energy in a process is a function only of its initial and final states, as noted in the section below headed First law of thermodynamics for open systems.
A useful idea from mechanics is that the energy gained by a particle is equal to the force applied to the particle multiplied by the displacement of the particle while that force is applied. Now consider the first law without the heating term: dU = -PdV. The pressure P can be viewed as a force (and in fact has units of force per unit area) while dVis the displacement (with units of distance times area). We may say, with respect to this work term, that a pressure difference forces a transfer of volume, and that the product of the two (work) is the amount of energy transferred out of the system as a result of the process. If one were to make this term negative then this would be the work done on the system.
It is useful to view the TdS term in the same light: here the temperature is known as a "generalized" force (rather than an actual mechanical force) and the entropy is a generalized displacement.
Similarly, a difference in chemical potential between groups of particles in the system drives a chemical reaction that changes the numbers of particles, and the corresponding product is the amount of chemical potential energy transformed in process. For example, consider a system consisting of two phases: liquid water and water vapor. There is a generalized "force" of evaporation which drives water molecules out of the liquid. There is a generalized "force" of condensation which drives vapor molecules out of the vapor. Only when these two "forces" (or chemical potentials) are equal will there be equilibrium, and the net rate of transfer will be zero.
The two thermodynamic parameters which form a generalized force-displacement pair are termed "conjugate variables". The two most familiar pairs are, of course, pressure-volume, and temperature-entropy.
Spatially inhomogeneous systems 
Classical thermodynamics is initially focused on closed homogeneous systems (e.g. Planck 1897/1903), which might be regarded as 'zero-dimensional' in the sense that they have no spatial variation. But it is desired to study also systems with distinct internal motion and spatial inhomogeneity. For such systems, the principle of conservation of energy is expressed in terms not only of internal energy as defined for homogeneous systems, but also in terms of kinetic energy and potential energies of parts of the inhomogeneous system with respect to each other and with respect to long-range external forces. How the total energy of a system is allocated between these three more specific kinds of energy varies according to the purposes of different writers; this is because these components of energy are to some extent mathematical artefacts rather than actually measured physical quantities. For any closed homogeneous component of an inhomogeneous closed system, if denotes the total energy of that component system, one may write
Potential energy can be exchanged with the surroundings of the system when the surroundings impose a force field, such as gravitational or electromagnetic, on the system.
A compound system consisting of two interacting closed homogeneous component subsystems has a potential energy of interaction between the subsystems. Thus, in an obvious notation, one may write
The quantity in general lacks an assignment to either subsystem in a way that is not arbitrary, and this stands in the way of a general non-arbitrary definition of transfer of energy as work. On occasions, authors make their various respective arbitrary assignments.
The distinction between internal and kinetic energy is hard to make in the presence of turbulent motion within the system, as friction gradually dissipates macroscopic kinetic energy of localised bulk flow into molecular random motion of molecules that is classified as internal energy. The rate of dissipation by friction of kinetic energy of localised bulk flow into internal energy, whether in turbulent or in streamlined flow, is an important quantity in non-equilibrium thermodynamics. This is a serious difficulty for attempts to define entropy for time-varying spatially inhomogeneous systems.
First law of thermodynamics for open systems 
For the first law of thermodynamics, there is no trivial passage of physical conception from the closed system view to an open system view. For closed systems, the concepts of an adiabatic enclosure and of an adiabatic wall are fundamental. Matter and internal energy cannot permeate or penetrate such a wall. For an open system, there is a wall that allows penetration by matter. In general, matter in motion will carry with it some internal energy, and some potential energy changes will accompany the motion. An open system is not adiabatically enclosed. By definition therefore, adiabatic work cannot be done on an open system. Since the revised and more rigorous definition of the internal energy of a closed system rests upon the possibility of processes by which adiabatic work takes the system from one state to another, this leaves a problem for the definition of internal energy for an open system, for which adiabatic work is not possible. This problem is usually solved by recourse to the principle of conservation of energy. This principle allows a composite isolated system to be derived from two other component non-interacting isolated systems, in such a way that the total energy of the composite isolated system is equal to the sum of the total energies of the two component isolated systems. If then the component isolated systems are subject to a thermodynamic operation that creates a pathway for matter to diffuse and energy to be transferred between them, the result can be viewed as an open system and its surroundings. The internal energies of the open system and of its surroundings can be measured on the two component isolated systems, considered as closed systems as above, before the operation of linking them through a permeable pathway. In this case, the law of conservation of energy requires that, for a process in which internal energy is transferred between the system and its surrroundings,
where ΔUs and ΔUo denote the changes in internal energy of the system and of its surroundings respectively. This is a statement of the first law of thermodynamics for an open system, that fits well with the conceptually revised and rigorous statement of the law stated above. For the thermodynamic operation of adding two systems with internal energies U1 and U2 , to produce a new system with internal energy U, one may write U = U1 + U2; if the reference states for U1 and U2 are fixed, then the reference state for U must be specified accordingly.
Also of course
where ΔNs and ΔNo denote the changes in mole number of a component substance of the system and of its surroundings respectively. This is a statement of the law of conservation of mass.
In contrast to the case of closed systems, for open systems, in the presence of diffusion, there is no unconstrained and unconditional physical distinction between convective transfer of internal energy by bulk flow of matter, the transfer of internal energy without transfer of matter (usually called heat conduction), and change of various potential energies. The older traditional way and the conceptually revised (Carathéodory) way agree that there is no physically unique definition of heat and work transfer processes between open systems. Following the work of Josiah Willard Gibbs, the ideas of heat and work transfer for closed systems are superseded for open systems by the ideas of transfer of kinetic energy of bulk flow, of bulk potential and of internal energies, and of entropy.
An example is evaporation. One may consider an open system consisting of a collection of vapour in a controlled volume, enclosed except where it is allowed to receive more vapour from or to condense into its parent liquid, which may be considered as another open system in open contact with the vapour system. The process might be a mechanical increase in the controlled volume of the vapour. Some work will be done by the vapour in the mechanical part of the process, but also some of the parent liquid will evaporate and enter the vapour collection which is the system. Some internal energy will accompany the vapour that enters the system, but it will not make sense to try to uniquely identify part of that internal energy as heat and part of it as work. Consequently, the energy transfer of the process as a whole, though having a component of mechanical work, cannot be uniquely split into heat and work transfers to or from the open system. The component of total energy transfer that accompanies the transfer of vapour into the system is customarily called 'latent heat of evaporation', but this is a quirk of historical language usage, not in strict compliance with the thermodynamic definition of transfer of energy as heat. In this example, kinetic energy of bulk flow and potential energy with respect to long-range external forces such as gravity are both considered to be zero. The first law of thermodynamics refers to the change of internal energy of the open system.
There are some cases in which a process for an open system can, for particular purposes, be considered as if it were for a closed system. In an open system, by definition hypothetically or potentially, matter can pass between the system and its surroundings. But when, in a particular case, the process of interest involves only hypothetical or potential but no actual passage of matter, the process can be considered as if it were for a closed system.
See also 
- Truesdell, C.A. (1980).
- Hess, H. (1840). Thermochemische Untersuchungen, Annalen der Physik und Chemie (Poggendorff, Leipzig) 126(6): 385-404 .
- Truesdell, C.A. (1980), pp. 157-158.
- Mayer, Robert (1841). Paper: 'Remarks on the Forces of Nature"; as quoted in: Lehninger, A. (1971). Bioenergetics - the Molecular Basis of Biological Energy Transformations, 2nd. Ed. London: The Benjamin/Cummings Publishing Company.
- Clausius, R. (1850). Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen, Annalen der Physik und Chemie (Poggendorff, Leipzig), 155 (3): 368-394, particularly on page 373 , translation here taken from Truesdell, C.A. (1980), pp. 188-189.
- Clausius, R. (1850). Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen, Annalen der Physik und Chemie (Poggendorff, Leipzig), 155 (3): 368-394, page 384 .
- Bailyn, M. (1994), p. 79.
- Helmholtz, H. (1847). Ueber die Erhaltung der Kraft. Eine physikalische Abhandlung, G. Reimer (publisher), Berlin, read on 23 July in a session of the Physikalischen Gesellschaft zu Berlin. Reprinted in Helmholtz, H. von (1882), Wissenschaftliche Abhandlungen, Band 1, J.A. Barth, Leipzig. Translated and edited by J. Tyndall, in Scientific Memoirs, Selected from the Transactions of Foreign Academies of Science and from Foreign Journals. Natural Philosophy (1853), volume 7, edited by J. Tyndall, W. Francis, published by Taylor and Francis, London, pp. 114–162, reprinted as volume 7 of Series 7, The Sources of Science, edited by H. Woolf, (1966), Johnson Reprint Corporation, New York, and again in Brush, S.G., The Kinetic Theory of Gases. An Anthology of Classic Papers with Historical Commentary, volume 1 of History of Modern Physical Sciences, edited by N.S. Hall, Imperial College Press, London, ISBN 1-86094-347-0, pp. 89–110.
- Bailyn, (1994), p. 82.
- Pippard, A.B. (1957/1966), p. 15. According to Herbert Callen, in his most widely cited text, Pippard's text gives a "scholarly and rigorous treatment"; see Callen, H.B. (1960/1985), p. 485. It is also recommended by Münster, A. (1970), p. 376.
- Quantities, Units and Symbols in Physical Chemistry (IUPAC Green Book) See Sec. 2.11 Chemical Thermodynamics
- Planck, M.(1897/1903). Treatise on Thermodynamics, translated by A. Ogg, Longmans, Green & Co., London., p. 43
- Münster, A. (1970).
- Kirkwood, J.G., Oppenheim, I. (1961), pp. 31–33.
- Planck, M.(1897/1903), p. 40.
- Crawford, F.H. (1963), pp. 106–107.
- Bryan, G.H. (1907), p. 47.
- C. Carathéodory (1909). "Untersuchungen über die Grundlagen der Thermodynamik". Mathematische Annalen 67: 355–386. A partly reliable translation is to be found at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA. doi:10.1007/BF01450409.
- Born, M. (1921). Kritische Betrachtungen zur traditionellen Darstellung der Thermodynamik, Physik. Zeitschr. 22: 218–224.
- Buchdahl, H.A. (1966), p. 34.
- Münster, A. (1970), pp. 23–24.
- Reif, F. (1965), p. 122.
- Haase, R. (1971), pp. 24–25.
- Pippard, A.B. (1957/1966), p. 14.
- Reif, F. (1965), p. 82.
- Adkins, C.J. (1968/1983), p. 31.
- Callen, H.B. (1960/1985), pp. 13, 17.
- Kittel, C. Kroemer, H. (1980). Thermal Physics, (first edition by Kittel alone 1969), second edition, W.H. Freeman, San Francisco, ISBN 0-7167-1088-9, pp. 49, 227.
- Tro, N.J. (2008). Chemistry. A Molecular Approach, Pearson/Prentice Hall, Upper Saddle River NJ, ISBN 0-13-100065-9, p. 246.
- Kirkwood, J.G., Oppenheim, I. (1961), pp. 17–18. Kirkwood & Oppenheim 1961 is recommended by Münster, A. (1970), p. 376. It is also cited by Eu, B.C. (2002), Generalized Thermodynamics, the Thermodynamics of Irreversible Processes and Generalized Hydrodynamics, Kluwer Academic Publishers, Dordrecht, ISBN 1-4020-0788-4, pp. 18, 29, 66.
- Guggenheim, E.A. (1949/1967). Thermodynamics. An Advanced Treatment for Chemists and Physicists, (first edition 1949), fifth edition 1967, North-Holland, Amsterdam, pp. 9–10. Guggenheim 1949/1965 is recommended by Buchdahl, H.A. (1966), p. 218. It is also recommended by Münster, A. (1970), p. 376.
- Planck, M.(1897/1903).
- Cropper, W.H. (1986). Rudolf Clausius and the road to entropy, Am. J. Phys., 54: 1068–1074.
- Truesdell, C.A. (1980), pp. 161–162.
- Buchdahl, H.A. (1966), p. 43.
- Maxwell, J. C. (1871). Theory of Heat, Longmans, Green, and Co., London, p. 150.
- Planck, M. (1897/1903), Section 71, p. 52.
- Bailyn, M. (1994), p. 95.
- Adkins, C.J. (1968/1983), p. 35.
- Atkins, P., de Paula, J. (1978/2010). Physical Chemistry, (first edition 1978), ninth edition 2010, Oxford University Press, Oxford UK, ISBN 978-0-19-954337-3, p. 54.
- Kondepudi, D. (2008). Introduction to Modern Thermodynamics, Wiley, Chichester, ISBN 978-0-470-01598-8, p. 63.
- Gislason, E.A., Craig, N.C. (2005). Cementing the foundations of thermodynamics:comparison of system-based and surroundings-based definitions of work and heat, J. Chem. Thermodynamics 37: 954–966.
- Rosenberg, R.M. (2010). From Joule to Caratheodory and Born: A conceptual evolution of the first law of thermodynamics, J. Chem. Edu., 87: 691–693.
- Bailyn, M. (1994), 254-256.
- Glansdorff, P., Prigogine, I. (1971), page 8.
- Tisza, L. (1966), p. 91.
- Denbigh, K.G. (1951), p. 50.
- Thomson, William (1852 a). "On a Universal Tendency in Nature to the Dissipation of Mechanical Energy" Proceedings of the Royal Society of Edinburgh for April 19, 1852 [This version from Mathematical and Physical Papers, vol. i, art. 59, pp. 511.]
- Thomson, W. (1852 b). On a universal tendency in nature to the dissipation of mechanical energy, Philosophical Magazine 4: 304-306.
- Helmholtz, H. (1869/1871). Zur Theorie der stationären Ströme in reibenden Flüssigkeiten, Verhandlungen des naturhistorisch-medizinischen Vereins zu Heidelberg, Band V: 1-7. Reprinted in Helmholtz, H. (1882), Wissenschaftliche Abhandlungen, volume 1, Johann Ambrosius Barth, Leipzig, pages 223-230 
- Münster A. (1970), Sections 14, 15, pp. 45–51.
- Landsberg, P.T. (1978), p. 78.
- Münster, A. (1970), p. 46.
- Tisza, L. (1966), p. 41.
- Beattie, J.A., Oppenheim, I. (1979), p. 296.
- Callen H.B. (1960/1985), p. 54.
- Tisza, L. (1966), p. 110.
- Tisza, L. (1966), p. 111.
- Denbigh, K.G. (1951), p. 56. Denbigh states in a footnote that he is indebted to correspondence with Professor E.A. Guggenheim and with Professor N.K. Adam. From this, Denbigh concludes "It seems, however, that when a system is able to exchange both heat and matter with its environment, it is impossible to make an unambiguous distinction between energy transported as heat and by the migration of matter, without already assuming the existence of the 'heat of transport'."
- Fitts, D.D. (1962). Nonequilibrium Thermodynamics. Phenomenological Theory of Irreversible Processes in Fluid Systems, McGraw-Hill, New York, p. 28.
- Denbigh, K. (1954/1971), pp. 81–82.
- Münster, A. (1970), p. 50.
- Haase, R. (1963/1969), p. 15.
- Haase, R. (1971), p. 20.
- Smith, D.A. (1980). Definition of heat in open systems, Aust. J. Phys., 33: 95–105.
- Bailyn, M. (1994), p. 308.
Cited sources 
- Adkins, C.J. (1968/1983). Equilibrium Thermodynamics, (first edition 1968), third edition 1983, Cambridge University Press, ISBN 0-521-25445-0.
- Bailyn, M. (1994). A Survey of Thermodynamics, American Institute of Physics Press, New York, ISBN 0-88318-797-3.
- Bryan, G.H. (1907). Thermodynamics. An Introductory Treatise dealing mainly with First Principles and their Direct Applications, B.G. Teubner, Leipzig.
- Buchdahl, H.A. (1966), The Concepts of Classical Thermodynamics, Cambridge University Press, London.
- Callen, H.B. (1960/1985), Thermodynamics and an Introduction to Thermostatistics, (first edition 1960), second edition 1985, John Wiley & Sons, New York, ISBN 0–471–86256–8.
- Crawford, F.H. (1963). Heat, Thermodynamics, and Statistical Physics, Rupert Hart-Davis, London, Harcourt, Brace & World, Inc..
- Denbigh, K.G. (1951). The Thermodynamics of the Steady State, Methuen, London, Wiley, New York.
- Denbigh, K. (1954/1971). The Principles of Chemical Equilibrium. With Applications in Chemistry and Chemical Engineering, third edition, Cambridge University Press, Cambridge UK.
- Glansdorff, P., Prigogine, I., (1971). Thermodynamic Theory of Structure, Stability and Fluctuations, Wiley, London, ISBN 0-471-30280-5.
- Haase, R. (1963/1969). Thermodynamics of Irreversible Processes, English translation, Addison-Wesley Publishing, Reading MA.
- Haase, R. (1971). Survey of Fundamental Laws, chapter 1 of Thermodynamics, pages 1–97 of volume 1, ed. W. Jost, of Physical Chemistry. An Advanced Treatise, ed. H. Eyring, D. Henderson, W. Jost, Academic Press, New York, lcn 73–117081.
- Kirkwood, J.G., Oppenheim, I. (1961). Chemical Thermodynamics, McGraw-Hill Book Company, New York.
- Landsberg, P.T. (1978). Thermodynamics and Statistical Mechanics, Oxford University Press, Oxford UK, ISBN 0-19-851142-6.
- Münster, A. (1970), Classical Thermodynamics, translated by E.S. Halberstadt, Wiley–Interscience, London, ISBN 0-471-62430-6.
- Pippard, A.B. (1957/1966). Elements of Classical Thermodynamics for Advanced Students of Physics, original publication 1957, reprint 1966, Cambridge University Press, Cambridge UK.
- Planck, M.(1897/1903). Treatise on Thermodynamics, translated by A. Ogg, Longmans, Green & Co., London.
- Reif, F. (1965). Fundamentals of Statistical and Thermal Physics, McGraw-Hill Book Company, New York.
- Tisza, L. (1966). Generalized Thermodynamics, M.I.T. Press, Cambridge MA.
- Truesdell, C.A. (1980). The Tragicomical History of Thermodynamics, 1822-1854, Springer, New York, ISBN 0-387-90403-4.
Further reading 
- Goldstein, Martin, and Inge F. (1993). The Refrigerator and the Universe. Harvard University Press. ISBN 0-674-75325-9. OCLC 32826343. Chpts. 2 and 3 contain a nontechnical treatment of the first law.
- Çengel Y.A. and Boles M. (2007). Thermodynamics: an engineering approach. McGraw-Hill Higher Education. ISBN 0-07-125771-3. Chapter 2.