First law of thermodynamics

The original 19th-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 also took as primitive the notion of transfer of energy as work. 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. By one author, this framework has been called the "thermodynamic" approach.[8]

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.[9]

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.[10] This equation may be 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.

Because of its definition in terms of increments, the value of 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 customarily stated relative to a conventionally chosen standard reference state of the system.

The concept of internal energy is considered by Bailyn to be of "enormous interest". Its quantity cannot be immediately measured, but can only be inferred, by differencing actual immediate measurements. Bailyn likens it to the energy states of an atom, that were revealed by Bohr's energy relation hν = E_n'' − E_n'. In each case, an unmeasurable quantity (the internal energy, the atomic energy level) is revealed by considering the difference of measured quantities (increments of internal energy, quantities of emitted or absorbed radiative energy).[11]

In 1907, George H. Bryan wrote about systems between which there is no transfer of matter (closed systems): "Definition. When energy flows from one system or part of a system to another otherwise than by the performance of mechanical work, the energy so transferred is called heat."[12] This definition may be regarded as expressing a conceptual revision, as follows. This was systematically expounded in 1909 by Constantin Carathéodory, whose attention had been drawn to it by Max Born. Largely through Born's[13] influence, this revised conceptual approach to the definition of heat came to be preferred by many twentieth-century writers. It might be called the "mechanical approach".[14]

Energy can also be transferred from one thermodynamic system to another in association with transfer of matter. Born points out that in general such energy transfer is not resolvable uniquely into work and heat moieties. In general, when there is transfer of energy associated with matter transfer, work and heat transfers can be distinguished only when they pass through walls physically separate from those for matter transfer.

The "mechanical" approach postulates the law of conservation of energy. It also postulates that energy can be transferred from one thermodynamic system to another adiabatically as work, and that energy can be held as the internal energy of a thermodynamic system. It also postulates that energy can be transferred from one thermodynamic system to another by a path that is non-adiabatic, and is unaccompanied by matter transfer. Initially, it "cleverly" (according to Bailyn) refrains from labelling as 'heat' such non-adiabatic, unaccompanied transfer of energy. It rests on the primitive notion of walls, especially adiabatic walls and non-adiabatic walls, defined as follows. Temporarily, only for purpose of this definition, one can prohibit transfer of energy as work across a wall of interest. Then walls of interest fall into two classes, (a) those such that arbitrary systems separated by them remain independently in their own previously established respective states of internal thermodynamic equilibrium; they are defined as adiabatic; and (b) those without such independence; they are defined as non-adiabatic.[15]

This approach derives the notions of transfer of energy as heat, and of temperature, as theoretical developments, not taking them as primitives. It regards calorimetry as a derived theory. It has an early origin in the nineteenth century, for example in the work of Helmholtz,[16] but also in the work of many others.[8]

The first law of thermodynamics for a closed system 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 an incremental change in the internal state of the system, and did not expect the process to be cyclic.

A cyclic process is one that can be repeated indefinitely often, returning the system to its initial state. Of particular interest for single cycle of a cyclic process are the net work done, and the net heat taken in (or 'consumed', in Clausius' statement), by the system.

In a cyclic process in which the system does net work on its surroundings, it is observed to be physically necessary not only that heat be taken into the system, but also, importantly, that some heat leave the system. The difference is the heat converted by the cycle into work. In each repetition of a cyclic process, the net work done by the system, measured in mechanical units, is proportional to the heat consumed, measured in calorimetric units.

The constant of proportionality is universal and independent of the system and in 1845 and 1847 was measured by James Joule, who described it as the mechanical equivalent of heat.

In a general process, the change in the internal energy of a closed system is equal to net energy added as heat to the system minus the thermodynamic work done by the system, both being measured in mechanical units. Taking ${\displaystyle \Delta U}$ as a change in internal energy, one writes

${\displaystyle \Delta U=Q~-~W~~~~{\text{(sign convention of Clausius and generally in this article)}},}$

where ${\displaystyle Q}$ denotes the net quantity of heat supplied to the system by its surroundings and ${\displaystyle W}$ denotes the net work done by the system. This sign convention is implicit in Clausius' statement of the law given above. It originated with the study of heat engines that produce useful work by consumption of heat; the key performance indicator of any heat engine is its thermal efficiency, which is the quotient of the net work done and the heat supplied to the system (disregarding waste heat given off). Thermal efficiency must be positive, which is the case if net work done and heat supplied are both of the same sign; by convention both are given the positive sign.

Nowadays, however, writers often use the IUPAC convention by which the first law is formulated with thermodynamic work done on the system by its surroundings having a positive sign. With this now often used sign convention for work, the first law for a closed system may be written:[23]

${\displaystyle \Delta U=Q+W~~~~{\text{(sign convention of IUPAC)}}.}$

(This convention follows physicists such as Max Planck,[24] 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.)

Continuing in the Clausius sign convention for work, when a system expands in a quasistatic process, the thermodynamic work done by the system on the surroundings is the product, ${\displaystyle P~\mathrm {d} V}$, of pressure, ${\displaystyle P}$, and volume change, ${\displaystyle \mathrm {d} V}$, whereas the thermodynamic work done on the system by the surroundings is ${\displaystyle -P\,\mathrm {d} V}$. Using either sign convention for work, the change in internal energy of the system is:

${\displaystyle \mathrm {d} U=\delta Q-P\,\mathrm {d} V~~~~{\text{(quasi-static process)}},}$

where ${\displaystyle \delta Q}$ denotes the infinitesimal amount of heat supplied to the system from its surroundings and ${\displaystyle \delta }$ denotes an inexact differential.

Work and heat are expressions of actual physical processes of supply or removal of energy, while the internal energy ${\displaystyle U}$ is a mathematical abstraction that keeps account of the exchanges of energy that befall the system. Thus the term 'heat' for ${\displaystyle Q}$ means "that amount of energy added or removed as heat in the thermodynamic sense", rather than referring to a form of energy within the system. Likewise, the term 'work energy' for ${\displaystyle W}$ means "that amount of energy gained or lost through thermodynamic 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 ${\displaystyle \Delta U}$ can be achieved by different combinations of heat and work. (This may be signaled by saying that heat and work are path dependent, while change in internal energy depends only on the initial and final states of the process. It is necessary to bear in mind that thermodynamic work is measured by change in the system, not necessarily the same as work measured by forces and distances in the surroundings;[25] this distinction is noted in the term 'isochoric work' (at constant volume).)

In an adiabatic process, there is transfer of energy as work but not as heat. For all adiabatic process that takes a system from a given initial state to a given final state, irrespective of how the work is done, the respective eventual total quantities of energy transferred as work are one and the same, determined just by the given initial and final states. 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 the tank thermally, and move the paddle wheel with a pulley and a weight, we can relate the increase in temperature with the distance descended by the mass. Next, 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."[17] Another expression of this view is "... no systematic precise experiments to verify this generalization directly have ever been attempted."[41]

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 an 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 important state variable was first recognized and denoted ${\displaystyle U}$ 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 ${\displaystyle U}$ ; 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 ${\displaystyle U}$ "energy". In 1882 it was named as the internal energy by Helmholtz.[42] 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 critical of the early work of Joule that had by then been performed.[43] 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 ${\displaystyle O}$ with internal energy ${\displaystyle U(O)}$ to an arbitrary one ${\displaystyle A}$ with internal energy ${\displaystyle U(A)}$, or from the state ${\displaystyle A}$ to the state ${\displaystyle O}$:

${\displaystyle U(A)=U(O)-W_{O\to A}^{\mathrm {adiabatic} }\,\,\mathrm {or} \,\,U(O)=U(A)-W_{A\to O}^{\mathrm {adiabatic} }\,.}$

Except under the special, and strictly speaking, fictional, condition of reversibility, only one of the processes  ${\displaystyle \mathrm {adiabatic} ,\,O\to A}$   or  ${\displaystyle \mathrm {adiabatic} ,\,{A\to O}\,}$  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), the most accepted way to deal with it nowadays, followed by Carathéodory,[19][22][44] is to rely on the previously established concept of quasi-static processes,[45][46][47] 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 that describe the reactive forces exerted by the surroundings.[48] This can be taken to justify the formula

  ${\displaystyle W_{A\to O}^{\text{adiabatic, quasi-static}}=-W_{O\to A}^{\text{adiabatic, quasi-static}}\,.}$

  (1)

• 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 (1) 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 (1) above allows that to go by processes of quasi-static adiabatic work from the state ${\displaystyle A}$ to the state ${\displaystyle B}$ we can take a path that goes through the reference state ${\displaystyle O}$, since the quasi-static adiabatic work is independent of the path

${\displaystyle -W_{A\to B}^{\mathrm {adiabatic,\,quasi-static} }=-W_{A\to O}^{\mathrm {adiabatic,\,quasi-static} }-W_{O\to B}^{\mathrm {adiabatic,\,quasi-static} }=W_{O\to A}^{\mathrm {adiabatic,\,quasi-static} }-W_{O\to B}^{\mathrm {adiabatic,\,quasi-static} }=-U(A)+U(B)=\Delta U}$

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, ${\displaystyle U}$.

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 that 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 transferring an externally determined amount of heat into it, for instance from a resistive electrical heater inside the calorimeter through which a precisely known electric current is passed at a precisely known voltage for a precisely measured period of time. The calibration allows comparison of calorimetric measurement of quantity of heat transferred with quantity of energy transferred as (surroundings-based)[25] work. According to one textbook, "The most common device for measuring ${\displaystyle \Delta U}$ is an adiabatic bomb calorimeter."[49] According to another textbook, "Calorimetry is widely used in present day laboratories."[50] According to one opinion, "Most thermodynamic data come from calorimetry...".[25]

When the system evolves with transfer of energy as heat, without energy being transferred as work, in an adynamic process,[51] the heat transferred to the system is equal to the increase in its internal energy:

${\displaystyle Q_{A\to B}^{\mathrm {adynamic} }=\Delta U\,.}$

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 reversible in the strict thermodynamic sense. For a particular reversible process in general, the work done reversibly on the system, ${\displaystyle W_{A\to B}^{\mathrm {path} \,P_{0},\,\mathrm {reversible} }}$, and the heat transferred reversibly to the system, ${\displaystyle Q_{A\to B}^{\mathrm {path} \,P_{0},\,\mathrm {reversible} }}$ are not required to occur respectively adiabatically or adynamically, but they must belong to the same particular process defined by its particular reversible path, ${\displaystyle P_{0}}$, 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

${\displaystyle -W_{A\to B}^{\mathrm {path} \,P_{0},\,\mathrm {reversible} }+Q_{A\to B}^{\mathrm {path} \,P_{0},\,\mathrm {reversible} }=\Delta U\,.}$

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

${\displaystyle \Delta U=0\,}$.

This is one aspect of the law of conservation of energy and can be stated:

The internal energy of an isolated system remains constant.

If, in a process of change of state of a closed system, the energy transfer is not under a practically zero temperature gradient, practically frictionless, and with nearly balanced forces, then the process is irreversible. Then the heat and work transfers may be difficult to calculate with high accuracy, although the simple equations for reversible processes still hold to a good approximation in the absence of composition changes. Importantly, the first law still holds and provides a check on the measurements and calculations of the work done irreversibly on the system, ${\displaystyle W_{A\to B}^{\mathrm {path} \,P_{1},\,\mathrm {irreversible} }}$, and the heat transferred irreversibly to the system, ${\displaystyle Q_{A\to B}^{\mathrm {path} \,P_{1},\,\mathrm {irreversible} }}$, which belong to the same particular process defined by its particular irreversible path, ${\displaystyle P_{1}}$, through the space of thermodynamic states.

${\displaystyle -W_{A\to B}^{\mathrm {path} \,P_{1},\,\mathrm {irreversible} }+Q_{A\to B}^{\mathrm {path} \,P_{1},\,\mathrm {irreversible} }=\Delta U\,.}$

This means that the internal energy ${\displaystyle U}$ is a function of state and that the internal energy change ${\displaystyle \Delta U}$ between two states is a function only of the two states.

The first law of thermodynamics is so general that its predictions cannot all be directly tested. In many properly conducted experiments it has been precisely supported, and never violated. Indeed, 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 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 account for an important physical factor. Thus, some may regard it as a principle more abstract than a law.

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 in general possible. According to Max Born, the transfer of matter and energy across an open connection "cannot be reduced to mechanics".[65] 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 work transfer), and change of various potential energies.[66][67][68] 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.[69][70][71][72][73][74]

In particular, between two otherwise isolated open systems an adiabatic wall is by definition impossible.[75] This problem is 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. Two previously isolated systems can be subjected to the thermodynamic operation of placement between them of a wall permeable to matter and energy, followed by a time for establishment of a new thermodynamic state of internal equilibrium in the new single unpartitioned system.[76] The internal energies of the initial two systems and of the final new system, considered respectively as closed systems as above, can be measured.[63] Then the law of conservation of energy requires that[77][78]

${\displaystyle \Delta U_{s}+\Delta U_{o}=0\,,}$

where ΔU_s and ΔU_o 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 a transfer between two otherwise isolated open systems,[79] 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; the reference states for U, U₁ and U₂ should be specified accordingly, maintaining also that the internal energy of a system be proportional to its mass, so that the internal energies are extensive variables.[63][80]

There is a sense in which this kind of additivity expresses a fundamental postulate that goes beyond the simplest ideas of classical closed system thermodynamics; the extensivity of some variables is not obvious, and needs explicit expression; indeed one author goes so far as to say that it could be recognized as a fourth law of thermodynamics, though this is not repeated by other authors.[81][82]

Also of course[77][78]

${\displaystyle \Delta N_{s}+\Delta N_{o}=0\,,}$

where ΔN_s and ΔN_o 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.

A system connected to its surroundings only through contact by a single permeable wall, but otherwise isolated, is an open system. If it is initially in a state of contact equilibrium with a surrounding subsystem, a thermodynamic process of transfer of matter can be made to occur between them if the surrounding subsystem is subjected to some thermodynamic operation, for example, removal of a partition between it and some further surrounding subsystem. The removal of the partition in the surroundings initiates a process of exchange between the system and its contiguous surrounding subsystem.

An example is evaporation. One may consider an open system consisting of a collection of liquid, enclosed except where it is allowed to evaporate into or to receive condensate from its vapor above it, which may be considered as its contiguous surrounding subsystem, and subject to control of its volume and temperature.

A thermodynamic process might be initiated by a thermodynamic operation in the surroundings, that mechanically increases in the controlled volume of the vapor. Some mechanical work will be done within the surroundings by the vapor, but also some of the parent liquid will evaporate and enter the vapor collection which is the contiguous surrounding subsystem. Some internal energy will accompany the vapor that leaves 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 that accompanies the transfer of matter between the system and its surrounding subsystem 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 vapor into the surrounding subsystem is customarily called 'latent heat of evaporation', but this use of the word heat is a quirk of customary historical language, 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, between its initial and final states of internal equilibrium.

An open system can be in contact equilibrium with several other systems at once.[19][83][84][85][86][87][88][89]

This includes cases in which there is contact equilibrium between the system, and several subsystems in its surroundings, including separate connections with subsystems through walls that are permeable to the transfer of matter and internal energy as heat and allowing friction of passage of the transferred matter, but immovable, and separate connections through adiabatic walls with others, and separate connections through diathermic walls impermeable to matter with yet others. Because there are physically separate connections that are permeable to energy but impermeable to matter, between the system and its surroundings, energy transfers between them can occur with definite heat and work characters. Conceptually essential here is that the internal energy transferred with the transfer of matter is measured by a variable that is mathematically independent of the variables that measure heat and work.[90]

With such independence of variables, the total increase of internal energy in the process is then determined as the sum of the internal energy transferred from the surroundings with the transfer of matter through the walls that are permeable to it, and of the internal energy transferred to the system as heat through the diathermic walls, and of the energy transferred to the system as work through the adiabatic walls, including the energy transferred to the system by long-range forces. These simultaneously transferred quantities of energy are defined by events in the surroundings of the system. Because the internal energy transferred with matter is not in general uniquely resolvable into heat and work components, the total energy transfer cannot in general be uniquely resolved into heat and work components.[91] Under these conditions, the following formula can describe the process in terms of externally defined thermodynamic variables, as a statement of the first law of thermodynamics:

${\displaystyle \Delta U_{0}\,=\,Q\,-\,W\,-\,\sum _{i=1}^{m}\Delta U_{i}\,\,\,\,\,{\text{(suitably defined surrounding subsystems, general process, quasi-static or irreversible),}}}$

(3)

where ΔU₀ denotes the change of internal energy of the system, and ΔU_i denotes the change of internal energy of the ith of the m surrounding subsystems that are in open contact with the system, due to transfer between the system and that ith surrounding subsystem, and Q denotes the internal energy transferred as heat from the heat reservoir of the surroundings to the system, and W denotes the energy transferred from the system to the surrounding subsystems that are in adiabatic connection with it. The case of a wall that is permeable to matter and can move so as to allow transfer of energy as work is not considered here.

If the system is described by the energetic fundamental equation, U₀ = U₀(S, V, N_j), and if the process can be described in the quasi-static formalism, in terms of the internal state variables of the system, then the process can also be described by a combination of the first and second laws of thermodynamics, by the formula

${\displaystyle \mathrm {d} U_{0}\,=\,T\,\mathrm {d} S\,-\,P\,\mathrm {d} V\,+\,\sum _{j=1}^{n}\mu _{j}\,\mathrm {d} N_{j}}$

(4)

where there are n chemical constituents of the system and permeably connected surrounding subsystems, and where T, S, P, V, N_j, and μ_j, are defined as above.[92]

For a general natural process, there is no immediate term-wise correspondence between equations (3) and (4), because they describe the process in different conceptual frames.

Nevertheless, a conditional correspondence exists. There are three relevant kinds of wall here: purely diathermal, adiabatic, and permeable to matter. If two of those kinds of wall are sealed off, leaving only one that permits transfers of energy, as work, as heat, or with matter, then the remaining permitted terms correspond precisely. If two of the kinds of wall are left unsealed, then energy transfer can be shared between them, so that the two remaining permitted terms do not correspond precisely.

For the special fictive case of quasi-static transfers, there is a simple correspondence.[93] For this, it is supposed that the system has multiple areas of contact with its surroundings. There are pistons that allow adiabatic work, purely diathermal walls, and open connections with surrounding subsystems of completely controllable chemical potential (or equivalent controls for charged species). Then, for a suitable fictive quasi-static transfer, one can write

${\displaystyle \delta Q\,=\,T\,\mathrm {d} S-T\textstyle {\sum _{i}}s_{i}\,dN_{i}\,{\text{ and }}\delta W\,=\,P\,\mathrm {d} V\,\,\,\,\,\,{\text{(suitably defined surrounding subsystems, quasi-static transfers of energy)}},}$

where ${\displaystyle dN_{i}}$ is the added amount of species ${\displaystyle i}$ and ${\displaystyle s_{i}}$ is the corresponding molar entropy.[94]

For fictive quasi-static transfers for which the chemical potentials in the connected surrounding subsystems are suitably controlled, these can be put into equation (4) to yield

${\displaystyle \mathrm {d} U_{0}\,=\,\delta Q\,-\,\delta W\,+\,\sum _{j=1}^{n}h_{j}\,\mathrm {d} N_{j}\,\,\,\,\,\,\,{\text{(suitably defined surrounding subsystems, quasi-static transfers)}}.}$

(5)

where ${\displaystyle h_{i}}$ is the molar enthalpy of species ${\displaystyle i}$.[72][95][96]

The transfer of energy between an open system and a single contiguous subsystem of its surroundings is considered also in non-equilibrium thermodynamics. The problem of definition arises also in this case. It may be allowed that the wall between the system and the subsystem is not only permeable to matter and to internal energy, but also may be movable so as to allow work to be done when the two systems have different pressures. In this case, the transfer of energy as heat is not defined.

The first law of thermodynamics for any process on the specification of equation (3) can be defined as

${\displaystyle \mathrm {\Delta } U\,=\,\Delta Q\,-\,p\Delta V\,+\,\sum _{j=1}^{n}h_{j}\,\mathrm {\Delta } N_{j}\,\,\,\,\,\,\,.}$

(6)

where ΔU denotes the change of internal energy of the system, Δ Q denotes the internal energy transferred as heat from the heat reservoir of the surroundings to the system, p Δ V denotes the work of the system and ${\displaystyle h_{i}}$ is the molar enthalpy of species ${\displaystyle i}$, coming into the system from the surrounding that is in contact with the system.

Formula (6) is valid in general case, both for quasi-static and for irreversible processes. The situation of the quasi-static process is considered in the previous Section, which in our terms defines

${\displaystyle \mathrm {d} U=T\,dS-\,p\Delta V\,+\sum _{\alpha }\,\mu _{\alpha }\,\Delta N_{\alpha },}$

(7)

${\displaystyle \mathrm {d} S={\frac {\Delta E}{T}},\quad \Delta E=\Delta Q+\sum _{\alpha }\,\eta _{\alpha }\,\Delta N_{\alpha }.}$

(8)

To describe deviation of the thermodynamic system from equilibrium, in addition to fundamental variables that are used to fix the equilibrium state, as was described above, a set of variables ${\displaystyle \xi _{1},\xi _{2},\ldots }$ that are called internal variables have been introduced, which allows[97][98][99] to formulate for the general case

${\displaystyle \mathrm {d} U=T\,dS-\,p\Delta V\,+\sum _{j}\,\Xi _{j}\,\Delta \xi _{j}+\sum _{\alpha }\,\mu _{\alpha }\,\Delta n_{\alpha },}$

(9)

${\displaystyle \mathrm {d} S={\frac {\Delta E}{T}},\quad \Delta E=\Delta Q-\sum _{j}\,\Xi _{j}\,\Delta \xi _{j}+\sum _{\alpha }\,\eta _{\alpha }\,\Delta n_{\alpha }.}$

(10)

Methods for study of non-equilibrium processes mostly deal with spatially continuous flow systems. In this case, the open connection between system and surroundings is usually taken to fully surround the system, so that there are no separate connections impermeable to matter but permeable to heat. Except for the special case mentioned above when there is no actual transfer of matter, which can be treated as if for a closed system, in strictly defined thermodynamic terms, it follows that transfer of energy as heat is not defined. In this sense, there is no such thing as 'heat flow' for a continuous-flow open system. Properly, for closed systems, one speaks of transfer of internal energy as heat, but in general, for open systems, one can speak safely only of transfer of internal energy. A factor here is that there are often cross-effects between distinct transfers, for example that transfer of one substance may cause transfer of another even when the latter has zero chemical potential gradient.

Usually transfer between a system and its surroundings applies to transfer of a state variable, and obeys a balance law, that the amount lost by the donor system is equal to the amount gained by the receptor system. Heat is not a state variable. For his 1947 definition of "heat transfer" for discrete open systems, the author Prigogine carefully explains at some length that his definition of it does not obey a balance law. He describes this as paradoxical.[100]

The situation is clarified by Gyarmati, who shows that his definition of "heat transfer", for continuous-flow systems, really refers not specifically to heat, but rather to transfer of internal energy, as follows. He considers a conceptual small cell in a situation of continuous-flow as a system defined in the so-called Lagrangian way, moving with the local center of mass. The flow of matter across the boundary is zero when considered as a flow of total mass. Nevertheless, if the material constitution is of several chemically distinct components that can diffuse with respect to one another, the system is considered to be open, the diffusive flows of the components being defined with respect to the center of mass of the system, and balancing one another as to mass transfer. Still there can be a distinction between bulk flow of internal energy and diffusive flow of internal energy in this case, because the internal energy density does not have to be constant per unit mass of material, and allowing for non-conservation of internal energy because of local conversion of kinetic energy of bulk flow to internal energy by viscosity.

Gyarmati shows that his definition of "the heat flow vector" is strictly speaking a definition of flow of internal energy, not specifically of heat, and so it turns out that his use here of the word heat is contrary to the strict thermodynamic definition of heat, though it is more or less compatible with historical custom, that often enough did not clearly distinguish between heat and internal energy; he writes "that this relation must be considered to be the exact definition of the concept of heat flow, fairly loosely used in experimental physics and heat technics."[101] Apparently in a different frame of thinking from that of the above-mentioned paradoxical usage in the earlier sections of the historic 1947 work by Prigogine, about discrete systems, this usage of Gyarmati is consistent with the later sections of the same 1947 work by Prigogine, about continuous-flow systems, which use the term "heat flux" in just this way. This usage is also followed by Glansdorff and Prigogine in their 1971 text about continuous-flow systems. They write: "Again the flow of internal energy may be split into a convection flow ρuv and a conduction flow. This conduction flow is by definition the heat flow W. Therefore: j[U] = ρuv + W where u denotes the [internal] energy per unit mass. [These authors actually use the symbols E and e to denote internal energy but their notation has been changed here to accord with the notation of the present article. These authors actually use the symbol U to refer to total energy, including kinetic energy of bulk flow.]"[102] This usage is followed also by other writers on non-equilibrium thermodynamics such as Lebon, Jou, and Casas-Vásquez,[103] and de Groot and Mazur.[104] This usage is described by Bailyn as stating the non-convective flow of internal energy, and is listed as his definition number 1, according to the first law of thermodynamics.[73] This usage is also followed by workers in the kinetic theory of gases.[105][106][107] This is not the ad hoc definition of "reduced heat flux" of Haase.[108]

In the case of a flowing system of only one chemical constituent, in the Lagrangian representation, there is no distinction between bulk flow and diffusion of matter. Moreover, the flow of matter is zero into or out of the cell that moves with the local center of mass. In effect, in this description, one is dealing with a system effectively closed to the transfer of matter. But still one can validly talk of a distinction between bulk flow and diffusive flow of internal energy, the latter driven by a temperature gradient within the flowing material, and being defined with respect to the local center of mass of the bulk flow. In this case of a virtually closed system, because of the zero matter transfer, as noted above, one can safely distinguish between transfer of energy as work, and transfer of internal energy as heat.[109]

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