PHYSICAL
AND BIOPHYSICAL CHEMISTRY DIVISION

COMMISSION ON THERMODYNAMICS

**Use of Legendre transforms in chemical thermodynamics (IUPAC Technical
Report) **

Robert A. Alberty

Department of Chemistry, Massachusetts Institute of
Technology, Cambridge, MA, USA

**Abstract:** The fundamental equation of thermodynamics for the
internal energy *U* may include terms for various types of work
and involves only differentials of extensive variables. The fundamental
equation for *U* 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 *U* 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 biochemistry
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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