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# Internal energy and enthalpy

For a closed system (a system from which no matter can enter or exit), the first law of thermodynamics states that the change in internal energy ΔU of the system is equal to the amount of heat Q supplied to the system minus the amount of work W done by system on its surroundings. N.B. This, and subsequent equations, are not a mathematical expressions; the ' = ' sign is to be read as 'equivalent', not 'equals'. The expressions 'Q' and 'W' represent amounts of energy but energy in different forms, heat and work. It is fundamental to the science of heat that conversion between heat (Q) and work (W) is never 100% efficient, so the ' = ' sign cannot have its usual mathematical meaning. 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.g. shaft work performed by a compressor fan), which is called isochoric work: In this Section we will neglect the "other-work" contribution.

The internal energy, U, is a state function. In cyclical processes, such as the operation of a heat engine, state functions return to their initial values after completing one cycle. Then the differential, or infinitesimal increment, for the internal energy in an infinitesimal process is an exact differential dU. The symbol for exact differentials is the lowercase letter d.

In contrast, neither of the infintestimal increments δQ nor δW in an infinitesimal process represents the state of the system. Thus, infinitesimal increments of heat and work are inexact differentials. The lowercase Greek letter delta, δ, is the symbol for inexact differentials. 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.

The second law of thermodynamics observes that if heat is supplied to a system in which no irreversible processes take place and which has a well-defined temperature T, the increment of heat δQ and the temperature T form the exact differential and that S, the entropy of the working body, is a function of state. Likewise, with a well-defined pressure, P, behind the moving boundary, the work differential, δW, and the pressure, P, combine to form the exact differential with V the volume of the system, which is a state variable. 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 If V is constant and if P is constant with H the enthalpy defined by The enthalpy may be considered to be a function H (S,P) of its natural variables S and P. 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.

Chemical reactions

For a closed system in which a chemical reaction is of interest, the extent of reaction, denoted by ξ, states the degree of advancement of the reaction and is included as a further natural variable for internal energy and for enthalpy. This is written In practice, chemists often use tables of a special but unnamed thermodynamic potential that is not the enthalpy expressed in its natural variables; instead they use the enthalpy expressed as a function of temperature instead of entropy. This special potential is related to the natural form of the enthalpy H (S,P,ξ) by another partial Legendre transform, that makes its natural variables T, P, and ξ. The special unnamed potential is still usually called the enthalpy. It can be written This enthalpy is used to report the enthalpy change of reaction, also called the heat of reaction.

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