For purposes of mathematical analysis of transfers, one thinks of fictive processes that are called reversible , with the temperature T of the system being hardly less than that of the surroundings, and the transfer taking place at an imperceptibly slow rate.
This equality is only valid for a fictive transfer in which there is no production of entropy, that is to say, in which there is no uncompensated entropy. The quantity T d S uncompensated was termed by Clausius the "uncompensated heat", though that does not accord with present-day terminology. Then one has. In non-equilibrium thermodynamics that approximates by assuming the hypothesis of local thermodynamic equilibrium, there is a special notation for this.
The transfer of energy as heat is assumed to take place across an infinitesimal temperature difference, so that the system element and its surroundings have near enough the same temperature T. Then one writes. The foregoing sign convention for work is used in the present article, but an alternate sign convention, followed by IUPAC, for work, is to consider the work performed on the system by its surroundings as positive.
This is the convention adopted by many modern textbooks of physical chemistry, such as those by Peter Atkins and Ira Levine, but many textbooks on physics define work as work done by the system. The work done by the system includes boundary work when the system increases its volume against an external force, such as that exerted by a piston and other work e.
The internal energy, U , is a state function. In cyclical processes, such as the operation of a heat engine, state functions of the working substance return to their initial values upon completion of a cycle. The differential, or infinitesimal increment, for the internal energy in an infinitesimal process is an exact differential d U.
The symbol for exact differentials is the lowercase letter d. Thus, infinitesimal increments of heat and work are inexact differentials.
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The integral of any inexact differential over the time it takes for a system to leave and return to the same thermodynamic state does not necessarily equal zero. In general, for homogeneous systems,. Associated with this differential equation is that the internal energy may be considered to be a function U S , V of its natural variables S and V. The internal energy representation of the fundamental thermodynamic relation is written.
The enthalpy representation of the fundamental thermodynamic relation is written. The internal energy representation and the enthalpy representation are partial Legendre transforms of one another. They contain the same physical information, written in different ways. Like the internal energy, the enthalpy stated as a function of its natural variables is a thermodynamic potential and contains all thermodynamic information about a body.
If a quantity Q of heat is added to a body while it does expansion work W on its surroundings, one has. In this scenario, the increase in enthalpy is equal to the quantity of heat added to the system.
Since many processes do take place at constant pressure, or approximately at atmospheric pressure, the enthalpy is therefore sometimes given the misleading name of 'heat content'. In terms of the natural variables S and P of the state function H , this process of change of state from state 1 to state 2 can be expressed as.
It is known that the temperature T S , P is identically stated by. Speculation on thermal energy or "heat" as a separate form of matter has a long history, see caloric theory , phlogiston and fire classical element. The modern understanding of thermal energy originates with Thompson 's mechanical theory of heat An Experimental Enquiry Concerning the Source of the Heat which is Excited by Friction , postulating a mechanical equivalent of heat.
The theory of classical thermodynamics matured in the s to s. John Tyndall 's Heat Considered as Mode of Motion was instrumental in popularising the idea of heat as motion to the English-speaking public. The theory was developed in academic publications in French, English and German. The process function Q was introduced by Rudolf Clausius in James Clerk Maxwell in his Theory of Heat outlines four stipulations for the definition of heat:. Use of "heat" as an abbreviated form of the specific concept of "quantity of energy transferred as heat" led to some terminological confusion by the early 20th century.
The generic meaning of "heat", even in classical thermodynamics, is just "thermal energy". Leonard Benedict Loeb in his Kinetic Theory of Gases makes a point of using "quanitity of heat" or "heat—quantity" when referring to Q : . The internal energy U X of a body in an arbitrary state X can be determined by amounts of work adiabatically performed by the body on its surroundings when it starts from a reference state O.
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Such work is assessed through quantities defined in the surroundings of the body. It is supposed that such work can be assessed accurately, without error due to friction in the surroundings; friction in the body is not excluded by this definition. The adiabatic performance of work is defined in terms of adiabatic walls, which allow transfer of energy as work, but no other transfer, of energy or matter. In particular they do not allow the passage of energy as heat.
According to this definition, work performed adiabatically is in general accompanied by friction within the thermodynamic system or body. For the definition of quantity of energy transferred as heat, it is customarily envisaged that an arbitrary state of interest Y is reached from state O by a process with two components, one adiabatic and the other not adiabatic. For convenience one may say that the adiabatic component was the sum of work done by the body through volume change through movement of the walls while the non-adiabatic wall was temporarily rendered adiabatic, and of isochoric adiabatic work.
Then the non-adiabatic component is a process of energy transfer through the wall that passes only heat, newly made accessible for the purpose of this transfer, from the surroundings to the body.
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The change in internal energy to reach the state Y from the state O is the difference of the two amounts of energy transferred. In this definition, for the sake of conceptual rigour, the quantity of energy transferred as heat is not specified directly in terms of the non-adiabatic process. It is defined through knowledge of precisely two variables, the change of internal energy and the amount of adiabatic work done, for the combined process of change from the reference state O to the arbitrary state Y.
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It is important that this does not explicitly involve the amount of energy transferred in the non-adiabatic component of the combined process. It is assumed here that the amount of energy required to pass from state O to state Y , the change of internal energy, is known, independently of the combined process, by a determination through a purely adiabatic process, like that for the determination of the internal energy of state X above. The rigour that is prized in this definition is that there is one and only one kind of energy transfer admitted as fundamental: energy transferred as work.
Energy transfer as heat is considered as a derived quantity.
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The uniqueness of work in this scheme is considered to guarantee rigor and purity of conception. The conceptual purity of this definition, based on the concept of energy transferred as work as an ideal notion, relies on the idea that some frictionless and otherwise non-dissipative processes of energy transfer can be realized in physical actuality.
The second law of thermodynamics, on the other hand, assures us that such processes are not found in nature. That heat is an appropriate and natural primitive for thermodynamics was already accepted by Carnot. Its continued validity as a primitive element of thermodynamical structure is due to the fact that it synthesizes an essential physical concept, as well as to its successful use in recent work to unify different constitutive theories. It is sometimes proposed that this traditional kind of presentation necessarily rests on "circular reasoning"; against this proposal, there stands the rigorously logical mathematical development of the theory presented by Truesdell and Bharatha This alternative approach admits calorimetry as a primary or direct way to measure quantity of energy transferred as heat.
It relies on temperature as one of its primitive concepts, and used in calorimetry.
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Such processes are not restricted to adiabatic transfers of energy as work. They include calorimetry, which is the commonest practical way of finding internal energy differences. It is calculated from the difference of the internal energies of the initial and final states of the system, and from the actual work done by the system during the process.
That internal energy difference is supposed to have been measured in advance through processes of purely adiabatic transfer of energy as work, processes that take the system between the initial and final states. In fact, the actual physical existence of such adiabatic processes is indeed mostly supposition, and those supposed processes have in most cases not been actually verified empirically to exist. Referring to conduction, Partington writes: "If a hot body is brought in conducting contact with a cold body, the temperature of the hot body falls and that of the cold body rises, and it is said that a quantity of heat has passed from the hot body to the cold body.
Referring to radiation, Maxwell writes: "In Radiation, the hotter body loses heat, and the colder body receives heat by means of a process occurring in some intervening medium which does not itself thereby become hot. Maxwell writes that convection as such "is not a purely thermal phenomenon". If, however, the convection is enclosed and circulatory, then it may be regarded as an intermediary that transfers energy as heat between source and destination bodies, because it transfers only energy and not matter from the source to the destination body.
In accordance with the first law for closed systems, energy transferred solely as heat leaves one body and enters another, changing the internal energies of each. Transfer, between bodies, of energy as work is a complementary way of changing internal energies. Though it is not logically rigorous from the viewpoint of strict physical concepts, a common form of words that expresses this is to say that heat and work are interconvertible. Cyclically operating engines, that use only heat and work transfers, have two thermal reservoirs, a hot and a cold one. They may be classified by the range of operating temperatures of the working body, relative to those reservoirs.