**Chemistry International**

Vol. 24, No. 1

**January 2002**

**Highlights
from ***Pure and Applied Chemistry*

*Use of Legendre
Transforms in Chemical Thermodynamics (IUPAC Technical Report) *

by Robert
A. Alberty

*Pure and Applied Chemistry*, Vol. 73, No. 8, pp. 1349-1380
(2001)

The fundamental
equation of thermodynamics for the internal energy m
may include terms for various types of work and involves only differentials
of extensive variables. The
fundamental equation for m yields intensive
variables as partial derivatives of the internal energy with respect
to other extensive properties. In addition to the terms from the combined
first and second laws for a system involving *PV* work, the fundamental
equation for the internal energy may involve terms for chemical work,
gravitational work, work of electric transport, elongation work, surface
work, work of electric and magnetic polarization, and other kinds of
work.

Fundamental
equations for other thermodynamic potentials can be obtained by use
of Legendre transforms that define these other thermodynamic potentials
in terms of m minus conjugate pairs of intensive
and extensive variables involved in one or more work terms. The independent
variables represented by differentials in a fundamental equation are
referred to as natural variables. The natural variables of a thermodynamic
potential are important because if a thermodynamic potential can be
determined as a function of its natural variables, all of the thermodynamic
properties of the system can be obtained by taking partial derivatives
of the thermodynamic potential with respect to the natural variables.
The natural variables are also important because they are held constant
in the criterion for spontaneous change and equilibrium based on that
thermodynamic potential. By use of Legendre transforms any desired set
of natural variables can be obtained. The enthalpy *H*, Helmholtz
energy *A*, and Gibbs energy *G* are defined by Legendre transforms
that introduce *P*, *T*, and *P* and *T* together
as natural variables, respectively.

Further
Legendre transforms can be used to introduce the chemical potential
of any species, the gravitational potential, the electric potentials
of phases, surface tension, force of elongation, electric field strength,
magnetic field strength, and other intensive variables as natural variables.
The large number of transformed thermodynamic potentials that can be
defined raises serious nomenclature problems. Some of the transforms
of the internal energy can also be regarded as transforms of *H*,
*A*, or *G*. Since transforms of *U*, *H*, *A*,
and *G* are useful, they can be referred to as the transformed
internal energy *U´*, transformed enthalpy *H´*,
transformed Helmholtz energy *A´*, and transformed Gibbs energy
*G´* in a context where it is clear what additional intensive
natural variables have been introduced. The chemical potential m_{i}
of a species is an especially important intensive property because its
value is uniform throughout a multiphase system at equilibrium even
though the phases may be different states of matter or be at different
pressures, gravitational potentials, or electric potentials. When the
chemical potential of a species is held constant, a Legendre transform
can be used to define a transformed Gibbs energy, which is minimized
at equilibrium at a specified chemical potential of that species. For
example, transformed chemical potentials are useful in bio-chemistry
because it is convenient to use pH as an independent variable. Recommendations
are made to clarify the use of transformed thermodynamic potentials
of systems and transformed chemical potentials of species.

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