# Enthalpy

Enthalpy (symbolized H, also called heat content) is the sum of the internal energy of matter and the product of its volume multiplied by the pressure. Enthalpy is a quantifiable state function, and the total enthalpy of a system cannot be measured directly; the enthalpy change of a system is measured instead. Enthalpy is a thermodynamic potential, and is useful particularly for nearly-constant pressure process, where any energy input to the system must go into internal energy or the mechanical work of expanding the system. For a simple system, with a constant number of particles, the difference in enthalpy is the maximum amount of thermal energy derivable from a thermodynamic process in which the initial and final states are at the same pressure.

Enthalpy is defined by the following equation:

${\displaystyle H=U+pV\,}$

where (all units given in SI)

## Equations

From the first law of thermodynamics we have for a reversible process:

${\displaystyle dU=\delta Q-\delta W\,}$

where ${\displaystyle U}$ is the internal energy, ${\displaystyle \delta Q=TdS}$ is the energy added by heating and ${\displaystyle \delta W=PdV}$ is the work done by the system. Differentiating the expression for H  we have:

${\displaystyle dH=dU+(pdV+Vdp)\,}$
${\displaystyle =(TdS-pdV)+pdV+Vdp\,}$
${\displaystyle =TdS+Vdp\,}$

For a process which is not reversible, the entropy will be smaller than its equilibrium value so we may say that, in general,

${\displaystyle dH\leq TdS+Vdp\,}$

It is seen that if a thermodynamic process is isobaric (i.e. occurs at constant pressure), then dp = 0  and thus

${\displaystyle dH\leq \delta Q\,}$

The difference in enthalpy is the maximum thermal energy attainable from the system in an isobaric process. This explains why it is sometimes called the "heat content". In more mathematical terms, the integral of dH  over any isobar in state space is the maximum thermal energy attainable from the system. However, the enthalpy is a state function, which means that dH is an exact differential. It follows that the integral over any path will give the same value for the maximum work, a value which is only dependent upon the location of the beginning and end points in state space. It follows that, for a simple two dimensional system, the enthalpy is the maximum thermal energy attainable from any thermodynamic process in which the initial and final states are at the same pressure.

If, in addition the entropy is held constant as well, the above equation becomes:

${\displaystyle dH\leq 0\,}$

with the equality holding at equilibrium. It is seen that the enthalpy for a general system will continuously increase to its minimum value, which it maintains at equilbrium.

The total enthalpy of a system cannot be measured directly; the enthalpy change of a system is measured instead. Enthalpy change is defined by the following equation:

${\displaystyle \Delta H=H_{final}-H_{initial}\,}$

where

ΔH  is the enthalpy change
Hfinal  is the final enthalpy of the system, measured in joules. In a chemical reaction, Hfinal  is the enthalpy of the products.
Hinitial  is the initial enthalpy of the system, measured in joules. In a chemical reaction, Hinitial  is the enthalpy of the reactants.

For an exothermic reaction at constant pressure, the system's change in enthalpy is equal to the energy released in the reaction, including the energy retained in the system and lost through expansion against its surroundings. Similarly, for an endothermic reaction, the system's change in enthalpy is equal to the energy absorbed in the reaction, including the energy lost by the system and gained from compression from its surroundings.

In a more general form, the first law describes the internal energy with additional terms involving the chemical potential and the number of particles of various types. The differential statement for dH  is then:

${\displaystyle dH\leq TdS+Vdp+\sum _{i}\mu _{i}dN_{i}\,}$

where ${\displaystyle \mu _{i}}$ is the chemical potential for an i-type particle, and ${\displaystyle N_{i}}$ is the number of such particles. It is seen that not only must the Vdp  term be set to zero by requiring the pressures of the initial and final states to be the same, but the ${\displaystyle \mu _{i}dN_{i}}$ terms must be zero as well, by requiring that the particle numbers remain unchanged. Any further generalization will add even more terms whose extensive differential term must be set to zero in order for the interpretation of the enthalpy to hold.

## Standard enthalpy

Main article: Standard enthalpy.

The standard enthalpy change of reaction (denoted H° or Ho)is the enthalpy change that occurs in a system when 1 equivalent of matter is transformed by a chemical reaction under standard conditions.

A common standard enthalpy change is the standard enthalpy change of formation, which has been determined for a vast number of substances. The enthalpy change of any reaction under any conditions can be computed, given the standard enthalpy change of formation of all of the reactants and products. Other reactions with standard enthalpy change values include combustion (standard enthalpy change of combustion) and neutralisation (standard enthalpy change of neutralisation).