structural:summarization, linguistic:paraphrasing, stylistic:tone
Heat and enthalpy

For a closed system (a system from which no matter can enter or exit), one version of the first law of thermodynamics states that the change in internal energy Δ U of the system is equal to the amount of heat Q supplied to the system minus the amount of thermodynamic work W done by system on its surroundings. 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.

This formula can be re-written so as to express a definition of quantity of energy transferred as heat, based purely on the concept of adiabatic work, if it is supposed that Δ U is defined and measured solely by processes of adiabatic work:

The thermodynamic work done by the system is through mechanisms defined by its thermodynamic state variables, for example, its volume V, not through variables that necessarily involve mechanisms in the surroundings. The latter are such as shaft work, and include isochoric work.

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.

In contrast, neither of the infinitesimal increments δ Q nor δ W in an infinitesimal process represents the change in a state function 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 in a process where the system leaves and then returns to the same thermodynamic state does not necessarily equal zero.

As recounted above, in the section headed heat and entropy, the second law of thermodynamics observes that if heat is supplied to a system in a reversible process, the increment of heat δ Q and the temperature T form the exact differential

and that S, the entropy of the working body, is a state function. Likewise, with a well-defined pressure, P, behind a slowly moving (quasistatic) boundary, the work differential, δ W, 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 systems of uniform pressure and temperature without composition change,

Associated with this differential equation is the concept 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 as

If V is constant

and if P is constant

with the enthalpy H 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. 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 only expansion work W on its surroundings, one has

If this is constrained to happen at constant pressure, i.e. with Δ P = 0, the expansion work W done by the body is given by W = P Δ V; recalling the first law of thermodynamics, one has

Consequently, by substitution one has

In this scenario, the increase in enthalpy is equal to the quantity of heat added to the system. This is the basis of the determination of enthalpy changes in chemical reactions by calorimetry. Since many processes do take place at constant atmospheric pressure, the enthalpy is sometimes given the misleading name of 'heat content' or heat function, while it actually depends strongly on the energies of covalent bonds and intermolecular forces.

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

Consequently,

In this case, the integral specifies a quantity of heat transferred at constant pressure.