7308x1349

Use of Legendre transforms in chemical
thermodynamics

(IUPAC Technical Report)

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 ther-
modynamic  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, respec-
tively. Further Legendre transforms can be used to introduce the chemical poten-
tial of any species, the gravitational potential, the electric potentials of phases, sur-
face  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  prob-
lems. Some of the transforms of the internal energy can also be regarded as trans-
forms 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

′

, trans-

formed 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

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.

R. A. ALBERTY

1350

CONTENTS

FUNDAMENTAL EQUATIONS OF THERMODYNAMICS FOR SYSTEMS WITHOUT
CHEMICAL REACTIONS
1.1

One-phase systems with N species

1.2

One-phase systems with one species

1.3

Other types of work

1.4

One-phase systems with N species and non-PV work

1.5

Phase equilibrium

FUNDAMENTAL EQUATIONS OF THERMODYNAMICS FOR SYSTEMS WITH
CHEMICAL REACTIONS
2.1

Components in chemical reaction systems

2.2

Gas reactions

2.3

Biochemical reactions

FUNDAMENTAL EQUATIONS OF THERMODYNAMICS FOR SYSTEMS WITH
GRAVITATIONAL WORK AND ELECTRICAL WORK
3.1

Systems with gravitational work

3.2

Systems with electrical work

FUNDAMENTAL EQUATIONS OF THERMODYNAMICS FOR SYSTEMS WITH OTHER
KINDS OF WORK
4.1

Systems with surface work

4.2

Systems with mechanical work

4.3

Systems with work of electric polarization

4.4

Systems with work of magnetic polarization

RECOMMENDATIONS

APPENDIX: FIELDS AND DENSITIES

NOMENCLATURE

REFERENCES

1. FUNDAMENTAL EQUATIONS OF THERMODYNAMICS FOR SYSTEMS WITHOUT
CHEMICAL REACTIONS

Thermodynamic properties such as the internal energy U, entropy S, temperature T, pressure P, and vol-
ume V behave like mathematical functions, and many relations between thermodynamic properties can
be obtained by simply using the operations of calculus. As various types of work in addition to PV work
are  included,  the  number  of  thermodynamic  properties  is  considerably  expanded.  Furthermore,
Legendre transforms can be used to define thermodynamic potentials in addition to U and S. This is
done to make it convenient to use certain intensive variables. Thermodynamic potentials are extensive
properties that, like the potential energy in mechanics, give information about the most stable state of
the system. When terms for non-PV work are introduced, the number of possible thermodynamic poten-
tials  increases  exponentially,  and  this  increases  difficulties  with  nomenclature  and  terminology.  This
Technical  Report  has  been  written  to  promote  discussion  of  these  problems  and  agreements  on  their
solution.

It is highly desirable that there be the widest possible general agreement about these basic mat-

ters because they affect nomenclature and terminology in various subfields of chemical thermodynam-
ics. If these subfields were sufficiently isolated, they could develop independent nomenclature and ter-
minology, but the subfields of thermodynamics are not isolated. Experiments often involve more than
one type of work other than PV work. Research on chemical equilibria may involve surfaces or phases
at different electric potentials. Electrochemistry may involve surfaces and chemical equilibria.

Use of Legendre transforms in chemical thermodynamics

1351

Biochemical applications may involve the coupling of reactions and mechanical work or the coupling
of reactions with the transport of ions between phases at different electric potentials.

From a mathematical point of view, there is a great deal in common between the thermodynamic

treatments of various types of work starting with the fundamental equations of thermodynamics.
Thermodynamics involves a network of inter-related equations, and so the nomenclature in any one
equation affects the nomenclature in many other equations. Different subfields of thermodynamics have
different needs and different traditions, but this report deals with the basic nomenclature and terminol-
ogy that underlies the treatment of all types of work in chemical thermodynamics.

Thermodynamics is such a large field that it is not possible to cover everything here. The inter-

esting phenomena of critical regions are not discussed. Many future developments in nomenclature are
to be expected. The emphasis of these recommendations is on the fundamental equations of thermody-
namics and the introduction of intensive variables with Legendre transforms.

1.1 One-phase systems with N species

The differential of the internal energy U of an open one-phase system involving only PV work and
changes in amounts of species is given by the fundamental equation [1].

(1.1-1)

where N is the number of species,

is the chemical potential of species i, and n

is the amount of

species i. This equation shows that the thermodynamic properties of the system behave like mathemat-
ical functions and can be differentiated and integrated. The variables on the right-hand side of a funda-
mental equation occur as conjugate pairs. The internal energy can be considered to be a function of S,
V, and {n

}, where {n

} is the set of amounts of species, and calculus yields the following relation

(1.1-2)

where j

≠

i. (Note that this convention is used throughout this report.) Comparison of eqs. 1.1-1 and

1.1-2 shows that

(1.1-3)

(1.1-4)

(1.1-5)

The variables in the differentials on the right-hand side of the fundamental equation have a special sig-
nificance and are referred to as natural variables [2–12]. For the system under consideration, the natu-
ral variables of U are S, V, and {n

}. The natural variables of a thermodynamic potential are important

because when a thermodynamic potential can be determined as a function of its natural variables, all of
the thermodynamic properties of the system can be calculated, as illustrated by eqs. 1.1-3 to
1.1-5. All of the thermodynamic properties can be calculated if U is determined as a function of S, V,
and {n

}, but not if U is determined as a function of T, P, and {n

}, or other set of N + 2 independent

variables. The natural variables are also important because they are used in the criterion for spontaneous
change and equilibrium based on a particular thermodynamic potential. The criterion for equilibrium

R. A. ALBERTY

1352

TdS

PdV

–

∑

V n

S n

S V n

= ∂

∂









+ ∂

∂









∂
∂







∑

,{ }

, ,

V n

= ∂

∂









{ }

S n

= − ∂

∂









{ }

S V n

= ∂

∂







, ,

for a one-phase system without chemical reactions is (dU)

S,V,{ni}

≤

0, which means that at constant S,

V, and {n

}, U can only decrease and is at a minimum at equilibrium. The natural variables of U are all

extensive. As terms for additional types of work are added to eq. 1.1-1, they should each involve the
differential of an extensive property (see Section 1.3).

Equation 1.1-1 can be integrated at constant values of the intensive properties to obtain

(1.1-6)

Alternatively, this equation can be viewed as a consequence of Euler’s theorem. A function f(x

, x

, …,x

)

is said to be homogeneous of degree n if

f(kx

, kx

, ...,kx

) = k

f(x

, x

, …,x

)

(1.1-7)

where k is a constant. For such a function, Euler’s theorem states that

(1.1-8)

The internal energy of an open system is homogeneous of degree one in terms of the extensive proper-
ties S, V, and {n

}, and so eq. 1.1-6 follows from Euler’s theorem. The integrated forms of fundamen-

tal equations are sometimes referred to as Euler equations, but it is better to call them integrated equa-
tions to avoid confusion with Euler’s theorem. It is important to remember that the intensive variables
for a system are not all independent, as discussed later in connection with the Gibbs–Duhem equation.

Equation 1.1-1 has the following mixed cross derivatives (Maxwell equations), which indicate

some of the relationships between the thermodynamic properties for this system:

(1.1-9)

(1.1-10)

(1.1-11)

(1.1-12)

Since S and V are often inconvenient natural variables from an experimental point of view,

Legendre transforms are used to define further thermodynamic potentials that have P as a natural vari-
able rather than V, T as a natural variable rather than S, and both T and P as natural variables. A
Legendre transform is a linear change in variables in which one or more products of conjugate variables
are subtracted from the internal energy to define a new thermodynamic potential [2–16]. The Legendre
transforms that introduce P, T, and T and P together as natural variables are

H = U + PV

(1.1-13)

A = U – TS

(1.1-14)

G = U + PV – TS

(1.1-15)

They define the enthalpy H, the Helmholtz energy A, and the Gibbs energy G. Legendre transforms are
also used in mechanics to obtain more convenient independent variables. The Lagrangian L is a func-

Use of Legendre transforms in chemical thermodynamics

1353

i i

–

∑

nf x x

…

(

)

= ∑

∂









∂









{ }

S n

V n

–

∂







= ∂

∂









{ }

S V n

V n

, ,

− ∂

∂







= ∂

∂









{ }

S V n

S n

, ,

∂







= ∂

∂







S V n

, ,

tion of coordinates and velocities, but it is often more convenient to define the Hamiltonian function H
with a Legendre transform because the Hamiltonian is a function of coordinates and momenta.
Quantum mechanics is based on the Hamiltonian rather than the Lagrangian function. The important
thing about Legendre transforms is that the new thermodynamic potentials defined in this way all con-
tain exactly the same information as U.

The fundamental equations for H, A, and G are obtained by taking the differentials of eqs. 1.1-13,

1.1-14, and 1.1-15, and substituting eq. 1.1-1. For example, the fundamental equation for G is

(1.1-16)

Each of the fundamental equations for the thermodynamic properties defined by Legendre transforms
provides more partial derivatives and more Maxwell equations. Some of the thermodynamic potentials
are also linked by Gibbs–Helmholtz equations:

(1.1-17)

(1.1-18)

The integrated forms of the fundamental equations for H, A, and G are

(1.1-19)

(1.1-20)

(1.1-21)

Thus, only the Gibbs energy of this system is made up of additive contributions from species. The nat-
ural variables of H, A, and G are indicated by the variables in parentheses. The corresponding criteria
of spontaneous change and equilibrium involve these natural variables; namely (dH)

S,P,{ni}

≤

(dA)

T,V,{ni}

≤

0, and (dG)

T,P,{ni}

≤

More Legendre transforms are possible for the system described by eq. 1.1-1 because chemical

potentials can be introduced as natural variables by use of Legendre transforms; that is described in
Sections 2.2 and 2.3.

Equations 1.1-13 to 1.1-15 define partial Legendre transforms. The complete transform is

obtained by subtracting all of the conjugate pairs for a system from U. For the system under discussion,
the complete Legendre transform [7,14] is

(1.1-22)

Taking the differential of the transformed internal energy U

′

and substituting eq. 1.1-1 yields

(1.1-23)

the differential of U

′

is zero because U

′

is equal to zero, as is evident from eqs. 1.1-6 and 1.1-22.

Equation 1.1-23 is referred to as the Gibbs–Duhem equation for the system. Since it gives a relation
between the intensive properties for the system, these properties are not independent for the system at
equilibrium. For the one-phase system, the number of independent intensive properties is N + 1. The

R. A. ALBERTY

1354

S T

V P

= −

+ ∑

G T

P n

= −

∂









{ }

(

/ )

A T

V n

= −

∂









{ }

( / )

H S P n

, ,

{ }

(

)

+ ∑

A T V n

, ,

{ }

(

)

= −

+ ∑

G T P n

, ,

{ }

(

)

= ∑

′ = +

−

− ∑

i i

= −

+ ∑

S T

V P

Gibbs–Duhem equation can be regarded as the source of the phase rule, according to which the num-
ber f of independent intensive variables is given by f = N – p + 2, where p is the number of phases, for
a system involving only PV work and chemical work, but no chemical reactions. If chemical reactions
are involved, the phase rule is f = N – R – p + 2 = C – p + 2, where R is the number of independent reac-
tions and C = N – R is the number of components. Components are discussed later in connection with
phase equilibria and chemical equilibria.

In view of the importance of natural variables, it is convenient to have a symbol for the number

of natural variables, just as it is convenient to have a symbol f for the number of independent intensive
variables (degrees of freedom), as given by the phase rule. It can be shown that the number D of natu-
ral variables (independent variables to describe the extensive state) for a system without chemical reac-
tions is given by

D = f + p = N + 2

(1.1-24)

For a system with chemical reactions,

D = f + p = N – R + 2 = C + 2

(1.1-25)

The number of thermodynamic potentials for a system is given by 2

, and the number of Legendre

transforms is 2

– 1. The number of thermodynamic potentials includes the potential that is equal to

zero and yields the Gibbs–Duhem equation. The number of Maxwell equations for each of the thermo-
dynamic potentials is D(D – 1)/2, and the number of Maxwell equations for all of the thermodynamic
potentials for a system is [D(D – 1)/2]2

[13]. When Legendre transforms are used to introduce two

new natural variables (T, P), then 2

= 2

= 4 thermodynamic potentials are related by Legendre trans-

forms, as we have seen with U, H, A, and G. There are four Maxwell equations.

This section has shown that intensive variables are introduced as natural variables only by use of

Legendre transforms. Since a Legendre transform defines a new thermodynamic potential, it is impor-
tant that the new thermodynamic property have its own symbol and name. The new thermodynamic
potentials contain all the information in U(S,V{n

}), and so the use of U, H, A, G, or other thermody-

namic potential in place of U is simply a matter of convenience.

1.2 One-phase systems with one species

In order to discuss the complete set of Legendre transforms for a system, we consider a one-phase sys-
tem with one species. The fundamental equation for U is

dU = TdS – PdV +

(1.2-1)

The integration of this fundamental equation at constant values of the intensive variables yields

U = TS – PV +

(1.2-2)

Since D = 3, there are 2

– 1 = 7 thermodynamic potentials defined by Legendre transforms and 24

Maxwell equations. There are not generally accepted symbols for all of these thermodynamic poten-
tials, and so a suggestion made by Callen [3] is utilized here. Callen pointed out that all conceivable
thermodynamic potentials can be represented by U followed by square brackets around a list of the
intensive variables introduced as natural variables by the Legendre transform defining the new thermo-
dynamic potential. For example, the thermodynamic potentials defined by Legendre transforms 1.1-13,
1.1-14, and 1.1-15 can be represented by U[P] = H, U[T] = A, and U[T,P] = G. The seven Legendre
transforms for a one-phase system with one species are

H = U + PV

(1.2-3)

A = U – TS

(1.2-4)

Use of Legendre transforms in chemical thermodynamics

1355

G = U + PV – TS

(1.2-5)

] = U –

(1.2-6)

U[P,

] = U + PV –

(1.2-7)

U[T,

] = U – TS –

(1.2-8)

U[T,P,

] = U + PV – TS –

n = 0

(1.2-9)

The first three Legendre transforms introduce P, T, and T and P together as natural variables. The last
four Legendre transforms introduce the chemical potential as a natural variable. Three of these ther-
modynamic potentials are frequently used in statistical mechanics, and there are generally accepted
symbols for the corresponding partition functions [12]: U[T] = A = – RTlnQ

where Q is the canonical

ensemble partition function; U[T,P] = G = – RTln

∆

, where

∆

is the isothermal-isobaric partition func-

tion; and U[T,

] = – RTln

Ξ,

where

is the grand canonical ensemble partition function. The last ther-

modynamic potential U[T,P,

] is equal to zero because it is the complete Legendre transform for the

system, and this Legendre transform leads to the Gibbs–Duhem equation.

Taking the differentials of the seven thermodynamic potentials defined in eqs. 1.2-3 to 1.2-9 and

substituting eq. 1.2-1 yields the fundamental equations for these seven thermodynamic potentials:

dH = TdS + VdP +

(1.2-10)

dA = – SdT – PdV +

(1.2-11)

dG = – SdT + VdP +

(1.2-12)

dU[

] = TdS – PdV – nd

(1.2-13)

dU[P,

] = TdS + VdP – nd

(1.2-14)

dU[T,

] = – SdT – PdV – nd

(1.2-15)

dU[T,P,

] = – SdT + VdP – nd

= 0

(1.2-16)

The last fundamental equation is the Gibbs–Duhem equation for the system, which shows that only two
of the three intensive properties are independent. Because of the Gibbs–Duhem equation, we can say that
the chemical potential of a pure substance is a function of temperature and pressure. The number f of
independent intensive variables is 2, and so D = f + p = 2 + 1 = 3. Each of these fundamental equations
yields D(D – 1)/2 = 3 Maxwell equations. The criteria of equilibrium provided by these thermodynamic
potentials are (dU)

S,V,n

≤

0, (dH)

S,P,n

≤

0, (dA)

T,V,n

≤

0, (dG)

T,P,n

≤

0, (dU[

])

S,V,

≤

0, (dU[P,

])

S,P,

≤

and (dU[T,

])

T,V,

≤

The integrated forms of the eight fundamental equations for this system are

U(S,V,n) = TS – PV +

(1.2-17)

H(S,P,n) = TS +

(1.2-18)

A(T,V,n) = – PV +

(1.2-19)

G(T,P,n) =

(1.2-20)

](S,V,

) = TS – PV

(1.2-21)

U[P,

](S,P,

) = TS

(1.2-22)

U[T,

](T,V,

) = – PV

(1.2-23)

U[T,P,

](T,P,

) = 0

(1.2-24)

The natural variables are shown in parentheses.

R. A. ALBERTY

1356

1.3 Other types of work

Table 1 shows a number of types of work terms that may be involved in a thermodynamic system [16].
The last column shows the form of work terms in the fundamental equation for the internal energy.
When there are no chemical reactions, the amounts of species are independent variables, but when there
are chemical reactions, the amounts of components are independent variables, as discussed in Section
2.1. In some cases, the extensive variables for other kinds of work are proportional to the amounts of
species so that these work terms are not independent of the chemical work terms. In these cases, the
other types of work do not introduce new natural variables for U.

Note that each type of work term in the fundamental equation for U is written in the form (inten-

sive variable)d(extensive variable). The intensive variables in the various work terms may be referred
to as generalized forces, and the extensive variables may be referred to as generalized displacements.
Several types of work terms may be involved in a single thermodynamic system. The first question that
must be considered before writing the fundamental equation for an actual system is the possible depend-
ence of the extensive variables m, Q

, L, A

, p, m on {n

}. It is not possible to give a general answer to

the question as to which extensive variables are independent because that depends on the system.

1.4 One-phase systems with N species and non-PV work

Here we consider an unspecified type of work XdY in which the extensive property Y is independent of
{n

}. Phase equilibrium and chemical equilibrium are discussed in Sections 1.5 and 2.1. The funda-

mental equation for U for the system considered is

dU = TdS – PdV +

+ XdY

(1.4-1)

Use of Legendre transforms in chemical thermodynamics

1357

Table 1 Conjugate pairs of variables in work terms for the fundamental equation for U

Type of work

Intensive variable

Extensive variable

Differential work in dU

Mechanical

Pressure-volume

–P

–PdV

Elastic

fdL

Gravitational

= gh

m =

∑

dm =

∑

ghM

Surface

Electromagnetic

Charge transfer

Electric polarization

E•dp

Magnetic polarization

B•dm

Chemical

Chemical: no reactions

(species)

Chemical: reactions

(components)

Here

= gh is the gravitational potential, g is the gravitational acceleration, h is height above the surface of the earth, m is mass,

is molar mass,

is the electric potential of the phase containing species i, Q

is the contribution of species i to the electric

charge of a phase, z

is the charge number, F is the Faraday constant, f is force of elongation, L is length in the direction of the

force,

is surface tension, A

is surface area, E is electric field strength, p is the electric dipole moment of the system, B is mag-

netic field strength (magnetic flux density), and m is the magnetic moment of the system. In some electrochemical systems,
Q

= Fz

so that dQ

= Fz

. The dots indicate scalar products of vectors. Some of the other work terms can be written in vec-

tor notation. Other types of work terms are possible, and some of the expressions for differential work are more complicated; for
example, the force on a solid may be represented by a tensor and

may be a centrifugal potential. The term

applies to flat

surfaces.

If we are only interested in specifying the chemical potential of one species (the s-th species), there are
D = 2

= 16 possible thermodynamic potentials. The following equations give the integrated equation

for U and the 15 Legendre transforms defining new thermodynamic potentials. In representing thermo-
dynamic potentials, Callen’s nomenclature [2,3] is extended by using H, A, and G with square brack-
ets, as well as U. The use of H[ ] indicates that P is introduced by the Legendre transform as well as the
indicated variables, the use of A[ ] indicates that T is introduced by the Legendre transform as well as
the indicated variables, and the use of G[ ] indicates that T and P are introduced as well as the indicat-
ed variables.

U = TS – PV +

+ XY = f(S,V,{n

},Y)

(1.4-2)

H = U + PV = TS +

+ XY = f(S,P,{n

},Y)

(1.4-3)

A = U – TS = – PV +

+ XY = f(T,V,{n

},Y)

(1.4-4)

G = U + PV – TS =

+ XY = f(T,P,{n

},Y)

(1.4-5)

U[X] = U – XY = TS – PV +

= f(S,V,{n

},X)

(1.4-6)

H[X] = H – XY = TS +

= f(S,P,{n

},X)

(1.4-7)

A[X] = A – XY = – PV +

= f(T,V,{n

},X)

(1.4-8)

G[X] = G – XY =

= f(T,P,{n

},X)

(1.4-9)

] = U – n

= TS – PV +

+ XY = f(S,V,{n

s,Y)

(1.4-10)

] = H – n

= TS +

+ XY = f(S,P,{n

,Y)

(1.4-11)

] = A – n

= – PV +

+ XY = f(T,V,{n

,Y)

(1.4-12)

] = G – n

+ XY = f(T,P,{n

,Y)

(1.4-13)

,X] = U – n

– XY = TS – PV +

= f(S,V,{n

,X)

(1.4-14)

,X] = H – n

– XY = TS +

= f(S,P,{n

,X)

(1.4-15)

,X] = A – n

– XY = – PV +

= f(T,V,{n

,X)

(1.4-16)

,X] = G – n

– XY =

= f(T,P,{n

,X)

(1.4-17)

where i

≠

s in eqs. 1.4-1 to 1.4-17. Note that this last thermodynamic potential would be a complete

Legendre transform if there was only one species present, and so it would lead to a Gibbs–Duhem equa-
tion in that case. When there are chemical reactions at equilibrium, we have to deal with components,
and this requires further interpretation of ni and

i in eqs. 1.4-10 to 1.4-18, which is discussed in

Sections 2.2 and 2.3.

1.5 Phase equilibrium

Fundamental equation eq. 1.1-16 for G for an open system can be used for a system with multiple phas-
es by counting each species in a different phase as a different species. This indicates that the system has
N + 2 natural variables, but in order to identify the natural variables for the system at equilibrium and
to write the criterion for equilibrium it is necessary to introduce the constraints, which are

for

each species in a two-phase system. The fundamental equation for a thermodynamic potential like G
can be written in terms of natural variables for the system at equilibrium by using the equilibrium con-
straints to reduce the number of terms. For example, consider a system with two species and two phas-
es. Substituting

1α

and

2α

2β

yields

R. A. ALBERTY

1358

dG = –SdT + VdP +

(1.5-1)

where n

= n

+ n

and n

= n

+ n

are the amounts of two components. Note that components

are conserved. This form of the fundamental equation indicates that there are four natural variables for
this system at equilibrium, which can be taken as T, P, n

, and n

. This is in agreement with D = f +

p = 2 + 2 = 4, where f = N – p + 2 = 2 – 2 + 2 = 2. However, there is a choice of natural variables, and
it may be more useful to write the fundamental equation for G as

dG = –SdT + VdP +

〈

〉

〈

〉

(1.5-2)

where n

= n

1α

+ n

2α

, n

= n

1β

+ n

2β

are the amounts of the two phases. The average chemical poten-

tials in the

and

phases are given by

(1.5-3)

(1.5-4)

Equation 1.5-1 indicates that the criterion for spontaneous change and equilibrium can be written

(dG)

T,P,nc1,nc2

≤

0, and eq. 1.5-2 indicates that the criterion can alternatively be written (dG)

T,P,n

≤

The fact that there are only two intensive properties for this system can also be understood by consid-
ering the two Gibbs–Duhem equations for the system. These equations for the separate phases both
involve dT, dP, d

, and d

. The quantity d

can be eliminated between these two equations, and the

resulting equation can be solved for d

as a function of T and P. Thus,

and

are both functions

of T and P.

The integrated forms of fundamental eqs. 1.5-1 and 1.5-2 are

(1.5-5)

(1.5-6)

In concluding Section 1, the following points are emphasized. Legendre transforms provide the

only means for introducing intensive variables into criteria for spontaneous change and equilibrium.
Intensive variables are introduced because they are often easier to control than extensive variables.
When various types of work terms are involved, the number of possible Legendre transforms is 2

– 1,

and the number of thermodynamic potentials that can be used in criteria of spontaneous change and
equilibrium is 2

, where D = f + p is the number of natural variables. Since each thermodynamic poten-

tial requires a symbol and name, there is a serious nomenclature problem. It is important to be aware of
all the Legendre transforms that can be applied to a given system. The problem of the dependence of
extensive variables in other types of work on {ni} has been pointed out.

2. FUNDAMENTAL EQUATIONS OF THERMODYNAMICS FOR SYSTEMS WITH
CHEMICAL REACTIONS

2.1 Components in chemical reaction systems

In a one-phase reaction system, the natural variables appear to be T, P, and {n

}, but because of the con-

straints of the chemical reactions, the {n

} are not independent variables for a closed reaction system.

The amounts of components are independent variables [13,17–19]. The conversion from amounts of
species to amounts of components in a chemical reaction system is illustrated by consideration of a
closed system in which the reaction

A + B = C

(2.1-1)

Use of Legendre transforms in chemical thermodynamics

1359

1 1

2 2

1 1

2 2

occurs. The fundamental equation for G is

dG = – SdT + VdP +

(2.1-2)

The equilibrium condition derived from this equation is

(2.1-3)

Using this constraint to eliminate

from the fundamental equation for G yields

(2.1-4)

where n

= n

+ n

and n

= n

+ n

are the amounts of components A and B. For this system, f =

N – R – p + 2 = 3 – 1 – 1 + 2 = 3, and D = f + p = 3 + 1 = 4. The natural variables can be chosen to be
T, P, n

, and n

, but this choice is not unique because

could have been eliminated using eq.

2.1-3. The amounts of components are constants for the system, and they need to be known in order to
calculate the equilibrium composition [20]. The criterion for spontaneous change and equilibrium can
be written (dG)

T,P,ncA,ncB

≤

0. When a system involves chemical reactions, the amounts of species at

any given time are given by

(2.1-5)

where n

is the initial amount of species i,

is the stoichiometric number of species i in reaction j,

is the extent of reaction j, and R is the number of independent reactions. Since there are N species and
R reactions, the number of components C is given by C = N – R.

In dealing with multireaction systems, it is useful to express eq. 2.1-5 in matrix form:

n = n

(2.1-6)

where n is the N

1 matrix of amounts of species, n

is the N

1 matrix of initial amounts of species,

is the N

R matrix of stoichiometric numbers, and

is the R

1 matrix of extents of reactions. The

fundamental equation for G for a multireaction system can be written

dG = – SdT + VdP +

dn = – SdT + VdP +

(2.1-7)

where

is the 1

N matrix of chemical potentials of species. This equation can be used to derive the

equilibrium condition [17]

= 0

(2.1-8)

where the zero matrix 0 is 1

R. Equation 2.1-7 is useful for thinking about the equilibrium conditions,

but to identify the natural variables and state the criterion of spontaneous change and equilibrium, it is
necessary to use components.

At equilibrium, there are R known equations of the type

(2.1-9)

Each equilibrium condition can be used to eliminate one chemical potential from eq. 2.1-1 so that

the fundamental equation for G stated in terms of natural variables for a chemical reaction system is

(2.1-10)

where

is the chemical potential of a species and n

is the amount of the component with that chem-

ical potential. The number of components in a system is unique, but different sets of components can

R. A. ALBERTY

1360

S T

V P

S T

V P

= −

(

)

(

)

+ ∑

ν ξ

ν µ

∑

S T

V P

= −

+ ∑

be chosen [19]. When a system is at chemical equilibrium, the chemical potential of a component is
given by

(2.1-11)

where j

≠

i. Beattie and Oppenheim [13] discuss their two theorems on chemical potentials: (1) “The

chemical potential of a component of a phase is independent of the choice of components”. (2) “The
chemical potential of a consitituent of a phase when considered to be a species is equal to its chemical
potential when considered to be a component”. Thus, for a chemical reaction system, the criterion for
equilibrium is (dG)

T,P,{ci}

≤

The number of components is equal to the rank of the conservation matrix A, which has a column

for each species and a row for each independent conservation equation [17]. The conservation matrix is
made up of the coefficients of the conservation equations for the system. In chemical reactions, atoms of
elements and electric charge are conserved, and sometimes groups of atoms are conserved as well, so that
conservation equations may not all be independent. If two elements always appear in the same ratio, they
can be considered to be a pseudoelement. There may be additional conservation equations that arise from
the mechanism of reaction. The independent conservation equations for a system are represented by

An = n

(2.1-12)

where A is the C

N conservation matrix, n is the N

1 matrix of amounts of species, and n

is the

1 matrix of amounts of components. The conservation matrix for a system is related to the stoi-

chiometric number matrix

= 0

(2.1-13)

where

is the N

R stoichiometric number matrix and the zero matrix is C

R. The stoichiometric num-

ber matrix

is in the null space of A and can be calculated by hand for small matrices or by computer for

large matrices. Alternatively, A

is the null space of

, where T indicates the transpose. Neither A nor

for a system are unique, but their row-reduced forms are unique for a given order of species.

The fundamental equation for G in terms of components (eq. 2.1-10) can be written in matrix

notation:

dG = – SdT + VdP +

d n

A d n

(2.1-14)

where

is the 1

C chemical potential matrix for the components and n

is the C

1 amount matrix

for components. The second form is obtained by use of eq. 2.1-12. Note that A is a transformation
matrix for the transformation from a set of N species to a set of C components. For a reaction system,
the number of degrees of freedom is given by f = N – R – p + 2 = C – p + 2, so that the intensive state
for a one-phase system is specified by f = C + 1 intensive variables. Since D = f + p = C + 2, the natu-
ral variables for a one-phase reaction system can be chosen to be T, P, and {n

For a reaction system, it may be useful to choose the chemical potential of a species to be an inde-

pendent variable. Some of the amounts of components can be replaced as natural variables by the chem-
ical potentials of the corresponding species by use of a Legendre transforms to define a transformed
Gibbs energy, as explained in Sections 2.2 and 2.3. It is evident from (dG)

T,P,{nci}

≤

0 that only C – 1

amounts of components can be replaced by specified chemical potentials because the description of the
state of the system must include at least one extensive variable.

2.2 Gas reactions

Under certain circumstances it is of interest to consider chemical equilibrium in a gaseous reaction sys-
tem when the chemical potential of a reacting species, represented here as B, is held constant. The cri-

Use of Legendre transforms in chemical thermodynamics

1361

T P n

= ∂

∂







, ,

terion for spontaneous change and equilibrium at specified T, P, and chemical potential of a species can
be derived by defining a new thermodynamic potential by using a Legendre transform in which a prod-
uct of conjugate variables is subtracted from the Gibbs energy. The conjugate variable to the chemical
potential of a species is the amount of the component corresponding with that species. The chemical
potential of species B can be introduced as a natural variable by making a Legendre transform of the
Gibbs energy of the form [21–23]

′

= G – n

(2.2-1)

where n

is the amount of component B in the system. The amount of component B in the system can

be expressed in terms of the amounts of B in the various species by

(2.2-2)

where N is the number of species in the system and N

is the number of component molecules of B in

a species molecule of i. Equation 2.2-2 gives the amount of the B component, and it is this amount that
has to be used in the Legendre transform because at chemical equilibrium, it is only the amounts of com-
ponents that are independent variables (see eq. 1.5-1). Substituting G =

and 2.2-2 in eq. 2.2-1

yields

(2.2-3)

where the transformed chemical potential

′

of species i is defined by

′

– N

(2.2-4)

When the chemical potential of B is specified, the contributions of other species to the transformed
Gibbs energy of the system are additive in their transformed chemical potentials

′

, as shown by eq.

2.2-3.

Eliminating

between eqs. 2.2-4 and 2.1-2 yields

(2.2-5)

Note that there is no longer a separate term for species B in the summation. Since the B component is
in a separate term, we can use the Legendre transform (2.2-1) to make

a natural variable. The dif-

ferential of G

′

in eq. 2.2-1 is taken, and eq. 2.2-5 is substituted to obtain

(2.2-6)

When the chemical potential of B is specified, species that differ only in the number of B mole-

cules that they contain become pseudoisomers, and they have the same transformed chemical potential
at equilibrium, just as isomers have the same chemical potential at equilibrium. The amounts of
pseudoisomers can be summed to yield n

′

, the amount of species in the pseudoisomer group; n

′

Thus, the specification of the chemical potential of a species leads to a reconceptualization of the equi-
librium calculation in terms of amounts n

′

of pseudoisomer groups, rather than amounts n

species. This

may lead to a considerable simplification of the fundamental equation because the number N

′

pseudoisomer groups may be considerably less than the number N of species. Thus, eq. 2.2-6 can be
written

(2.2-7)

R. A. ALBERTY

1362

N n

i i

c B

= ∑

′ =

−

(

)

∑

′

∑

i i

S T

V P

= −

′

∑

−

′ = −

−

′

∑

−

S T

V P

′ = −

−

′ ′

∑

′

S T

V P

For example, when the partial pressure of ethylene is specified, the successive isomer groups in

an entire homologous series become pseudoisomers and are represented by one term in the summation
in eq. 2.2-7.

Up to this point, the treatment has been completely general, but now we assume that the gases are

ideal so that

+ RT ln(P

)

(2.2-8)

where P

is the standard state pressure (1 bar = 0.1 MPa). In order to use P

as an independent variable

in the fundamental equation rather than

, d

in eq. 2.2-7 is replaced with the expression for the total

differential:

(2.2-9)

When the derivatives of

are taken and eq. 2.2-9 is substituted in eq. 2.2-7, we obtain

(2.2-10)

where N

′

is the number of pseudoisomer groups.

The transformed entropy of the system is given by

(2.2-11)

where S

–

is the molar entropy of B and n

is the total amount of B bound in the system. In eq. 2.2-10,

′

is the sum of the partial pressures of species other than B

(2.2-12)

Here N

–

(i) is the average number of B molecules in i and n

′

is the sum of the amounts of species

that differ only with respect to the number of B molecules that they contain. Thus, the natural variables
of G

′

before applying the equilibrium constraints are represented by G

′

(T, P

′

, P

, {n

′

}), where ni

′

the amount of pseudoisomer group i. After applying the equilibrium constraints, the criterion for equi-
librium is (dG

′

)

T,P

′

,PB,{nci}

≤ 0

. Note that eq. 2.2-10 shows that the transformed chemical potential of

species i is defined by

(2.2-13)

before chemical constraints are applied.

Equation 2.2-4 shows how to calculate the transformed chemical potential of a species.

Substituting eq. 2.2-8 in eq. 2.2-4 yields

(2.2-14)

where

′

– N

. For an actual calculation, eq. 2.2-14 for species i can be written in terms of

transformed Gibbs energies of formation.

∆

′

∆

′

+ RTln(P

)]

(2.2-15)

where

Use of Legendre transforms in chemical thermodynamics

1363

= ∂

∂









+ ∂

∂







′ =

′

′ −

(

)

′ ′

∑

′

S T

V P

n RT P

–

′ = −

n S

cB B

i n

′

∑

′

( )

′ = ∂ ′

∂ ′







′

T P P n

′ =

(

)

−

= ′ +

(

)

P P

∆

′

∆

– N

[

∆

+ RTln(P

)]

(2.2-16)

At chemical equilibrium, the various species binding B are pseudoisomers, and so the standard

transformed Gibbs energy of the isomer group can be calculated with the equation for isomer groups
[17].

∆

′

(pseudoisomer group) = – RTln{

exp[–

∆

′

/RT]}

(2.2-17)

where the summation includes all of the pseudoisomers in a group. It is important to note that the stan-
dard transformed Gibbs energy of the pseudoisomer group is not a weighted average of the standard
transformed Gibbs energies of formation of the pseudoisomers in the group because there is an entropy
of mixing term. It will always be more negative than any one of them; in other words, the pseudoiso-
mer group is more stable than any of the pseudoisomers. The mole fraction r

of the i-th pseudoisomers

in the pseudoisomer group is given by [24]

= exp{[

∆

′

(pseudoisomer group) –

∆

′

]/RT}

(2.2-18)

The standard transformed enthalpy of formation of the pseudoisomer group is a mole fraction weight-
ed average and is given by

∆

′

(pseudoisomer group) =

∆

′

(2.2-19)

This discussion has been based on making the chemical potential of one species a natural vari-

able, but it may be possible to make a Legendre transform involving more than one species. For exam-
ple, for a system of benzenoid polycyclic aromatic hydrocarbons, acetylene, and molecular hydrogen,
which has three components, it is possible to specify P(C

) and P(H

) [22]. If hydrogen atoms are

included in the system as a reactant (C = 4), it is possible to specify P(C

), P(H

), and P(H). In other

words, if species that are specified can be interconverted, it is not necessary that they be at equilibrium.

2.3 Biochemical reactions

This method of using a Legendre transformed Gibbs energy G

′

is especially useful in biochemistry

where it is convenient to study systems at a specified pH, and, in some cases, at a specified free con-
centration of Mg

or other cation that is bound by reactants. When the pH and pMg are specified, it is

as if the biochemical reaction was carried out in a reaction chamber connected to pH and pMg reser-
voirs through semipermeable membranes. In this case, the Gibbs energy G of the contents of the reac-
tion chamber is not minimized at equilibrium because the concentrations of certain species are held con-
stant. Hydrogen and magnesium are not conserved in the reaction chamber. This is the reason why bio-
chemical reactions are written in terms of sums of species, as in the hydrolysis of adenosine triphos-
phate (ATP) to adenosine diphosphate (ADP) and inorganic phosphate (P

), which is represented by the

biochemical equation

ATP + H

O = ADP + P

(2.3-1)

with apparent equilibrium constant K

′

defined by

(2.3-2)

where ATP, ADP, and P

represent sums of species and c

is the standard state concentration. This value

of K

′

depends on T, P, pH, pMg, and I (ionic strength). Note that K

′

and other thermodynamic proper-

ties like G, H, and

are taken to be functions of ionic strength so that concentrations can be used in

eq. 2.3-2. This means that pH = – lg[H

] and pMg = – lg[Mg

] because activity coefficients are incor-

porated in the thermodynamic properties.

R. A. ALBERTY

1364

′ =

[

]

[

][ ]

ADP P

ATP

The chemical potentials of hydrogen ion and the magnesium ion are introduced as natural vari-

ables by use of the Legendre transform [25–27].

′

= G – n

(H)

) – n

(Mg)

(Mg

)

(2.3-3)

The amount of the hydrogen component in the reaction chamber is given by

(2.3-4)

where N

(i) is the number of hydrogen atoms in species i and n

is the amount of i. The amount of the

magnesium component in the reaction chamber is given by

(2.3-5)

where N

(i) is the number of magnesium atoms in species i. The Gibbs energy of the material in the

reaction chamber is given by G =

. Substituting this and eqs. 2.3-4 and 2.3-5 in eq. 2.3-3 yields

′

, where the transformed chemical potential of species i is given by

′

– N

(H)

) – N

(Mg)

(Mg

)

(2.3-6)

Taking the differential of G

′

that is defined in eq. 2.3-3 and substituting eq. 1.1-16 leads to

(2.3-7)

where N

′

is the number of reactants (sums of species) in the system, and n

′

is the amount of reactant i

(sum of species). Some of the steps in the introduction of pH and pMg have been omitted, but they are
discussed elsewhere [27]. The transformed entropy S

′

of the system is defined by

(2.3-8)

This equation can be used to show that

(2.3-9)

where

(2.3-10)

Since H

′

= G

′

+ TS

′

= H – n

(H) H

–

) – n

(Mg) H

–

(Mg

)

(2.3-11)

This equation can be used to show that

′

–

′

(2.3-12)

where

(2.3-13)

The natural variables for G

′

are T, P, pH, pMg, and {n

′

}, and the criterion for equilibrium is

(dG

′

)

T,P,pH,pMg,{ni

′

}

≤

0. Equation 2.3-7 gives rise to many useful Maxwell equations.

Equation 2.3-7 can be used to derive eq. 2.3-2 for K

′

at specified T, P, pH, and pMg. The ionic

strength also has an effect on K

′

, but the ionic strength is not an independent variable in the same sense

as the other four. It is important to specify the ionic strength, just like it is important to specify the sol-
vent. The standard transformed Gibbs energy of reaction can be calculated using

Use of Legendre transforms in chemical thermodynamics

1365

i n

( ) = ∑

( )

i n

( )

= ∑

( )

dpH

dpMg

′ =

′

′ ′

∑

′

S T

V P

–

( )

ln(

)

(

)

ln(

)

′ = −

−

( ) (

)

(

) (

)

′ =

′ ′

∑

n S

i i

i S

′

= −

( )

−

(

)

( ) (

)

′ =

−

( )

−

(

)

i H

( )

∆

′

= – RTlnK

′

(2.3-14)

Other thermodynamic properties can be obtained by taking derivatives of this equation. Tables of

standard transformed Gibbs energies of formation at a specified pH and pMg and standard transformed
enthalpies of formation can be prepared and used like the usual tables of standard thermodynamic prop-
erties [28].

The Panel on Biochemical Thermodynamics of the IUBMB–IUPAC Joint Commission on

Biochemical Nomenclature has published Recommendations for Nomenclature and Tables in
Biochemical Thermodynamics [29]. Standard transformed formation properties of carbon dioxide in
aqueous solution at specified pH have been calculated [30,31]. Legendre transformed thermodynamic
potentials are also used in the study of binding and linkage by macromolecules [32,33].

3. FUNDAMENTAL EQUATIONS OF THERMODYNAMICS FOR SYSTEMS WITH
GRAVITATIONAL WORK AND ELECTRIC WORK

3.1 Systems with gravitational work

The effect of the gravitational potential

has been discussed earlier [34,35]. As shown in Table 1, grav-

itational work adds a term

dm =

ghM

to the fundamental equation for U and the fundamental

equation for G, where g is the gravitational acceleration and h is the height above the surface of the
earth. However, this term does not change the natural variables for G, which are T, P, and {n

Therefore, the fundamental equation for G (eq. 1.1-16) is unchanged. In order to bring in the gravita-
tional potential

, which is an external variable that is not affected by what happens in the system, the

chemical potential of a species in the system is taken as

+ RTlna

+ ghM

(3.1-1)

This is the definition of the activity a

of a species i in a system in which gravitational work is impor-

tant. In such a system, the definition of the standard chemical potential

has to include a statement

that

is the value at unit activity at the surface of the earth (h =0). For a given activity, the chemi-

cal potential

is a linear function of the height above the earth’s surface. The fundamental equation

for G is

dG = – SdT + VdP +

(

+ RTlna

+ ghM

)dn

(3.1-2)

Note that although the height above the surface of the earth is involved, the fundamental equation does
not give the derivative of G with respect to h. At equilibrium in an isothermal atmosphere, the chemi-
cal potential of each species is independent of height, and this leads to the barometric formula for an
ideal gas.

The height h can be introduced as a natural variable by use of the Legendre transform to define a

transformed Gibbs energy.

′

= G –

ghM

(3.1-3)

Taking the differential of G

′

and substituting eq. 3.1-2 yields

dG = – SdT + VdP +

(

+ RTlna

)dn

–

(3.1-4)

This indicates that it is reasonable to define a transformed chemical potential

′

(3.1-5)

where j

≠

i. Thus, eq. 3.1-1 can be written

R. A. ALBERTY

1366

′ = ∂ ′

∂







T P h n

, , ,

′

+ ghM

(3.1-6)

Since

is constant throughout an equilibrium system with changing gravitational potential,

′

varies

throughout the system. Other transformed thermodynamic properties of species i can be obtained by
taking derivatives of eq. 3.1-5. Since

= 0, where

is the stoichiometric number for i, for a chem-

ical reaction, there is no effect of a gravitational potential on the equilibrium constant for a chemical
reaction.

The potential

in a centrifugal field is given by [34]

(3.1-7)

where

is the angular velocity and r is the distance from the axis of rotation.

3.2 Systems with electric work

3.2a Fundamental equation for the Gibbs energy of a multiphase system with electric
work

In considering the thermodynamics of systems in which there are electric potential differences, the
activity a

of an ion is defined in terms of its chemical potential

i and the electric potential

of the

phase the ion is in [35–38]:

+ RTlna

+ z

(3.2a-1)

where

is the standard chemical potential of ion i in a phase with an electric potential of zero, F is

the Faraday constant, and z

is the charge number. The purpose of this definition is to introduce the

activity a

, which is more convenient than

in discussing experimental data. This shows that the chem-

ical potential of an ion is a function of

as well as a

. The activity has the same functional dependence

on intensive properties in the presence of electric potential differences as in their absence. The descrip-
tion of the state for the standard chemical potential

of species i has to include the statement that

is the same in the presence of an electric potential as in its absence.

The symbol

has been used in electrochemistry to describe ionic properties that depend on elec-

tric potential. It has the same physical meaning as

on the left-hand side of eq. 3.2a-1 and has been

referred to by electrochemists as the electrochemical potential. Thus,

+ RT lna

can be considered

to be the contribution to

independent of the electrical state of the phase in question. However,

in eq. 3.2a-1 is really the chemical potential in the sense of Gibbs in that it is a quantity that is

independent of phase at equilibrium.

The form of the fundamental equation for a system involving phases at different electric poten-

tials depends on the system. There is a fundamental equation for each phase, and the fundamental equa-
tion for the system is the sum of the fundamental equations for the various phases. This is illustrated
here by a system consisting of two aqueous phases separated by a semipermeable membrane. The two
phases contain ions A, B, and C, which are involved in the reaction A + B =C in each phase. The mem-
brane is permeable only by ion C. This system has been discussed by Alberty [39]. Since C can diffuse
through the membrane without counter ions, the membrane becomes polarized. When electric charge is
added to a conductor, as in this case when C diffuses through the membrane, the charge is concentrat-
ed on the surface of the conductor so that the bulk phases remain electrically neutral. If C is a cation
and some of it has diffused from the

side of the membrane to the

side of the membrane, the mem-

brane has a positively charged layer in the solution on the

side and a negatively charged layer of solu-

tion on the

side. These layers are formed in the charge relaxation time of about a nanosecond and

have a thickness of the Debye length [40], which is about 1 nm at an ionic strength I of 0.1 M. Many
biological membranes have capacitances of about 1

F cm

–2

, and in this case, the charge transfer per

square centimeter required to set up a potential difference of 0.1 V is 10

–12

mol of singly charged ions.

Use of Legendre transforms in chemical thermodynamics

1367

= −

2 2

As C diffuses through the membrane a difference in electric potential is set up that opposes the trans-
fer of more C ions, and so an equilibrium difference in electric potential across the membrane is
reached.

Since the bulk phases remain electrically neutral, even though they are at different electric poten-

tials, it is convenient to think of this system as having three phases,

, and a membrane phase con-

sisting of the membrane and thin layers of solution on either side with thickness of the order of 10 nm
(10 Debye lengths). When ion C diffuses through the membrane, these ions can be considered to come
from the thin layer on one side of the membrane and to go into the thin layer on the other side. This
transfer of the order of 10

–12

mol of C per cm

from one side to the other involves only a small frac-

tion of the C ions in the thin layers. Thus, the amounts of C in phases

and

are not altered by this

transfer, which leads to a large electric effect.

The fundamental equations for G for the three phases are

= –S

dT + V

dP +

(

3.2a-2

)

= –S

dT + V

dP +

(

3.2a-3

)

= (

−

)

(3.2a-4)

where G

is the Gibbs energy of the membrane including the thin layers of solution with diffuse ionic

gradients. The fundamental equation for the membrane is written in terms of the charge Q transferred
from the

side of the membrane to the

side; it could be written in terms of amounts of C in the thin

layers on either side of the membrane, but Q is used to emphasize that this quantity is independent of
n

and n

. In subsequent equations,

is taken as zero as a simplification. In writing eq. 3.2a-4, the

contribution of the entropy and volume of the membrane phase to the entropy and volume of the whole
system is neglected.

When ionic species are involved, there must be counter ions so that the bulk phases will be elec-

trically neutral. The inclusion of the counter ion in the fundamental equation for a phase increases the
number of species by one, but this brings in the electroneutrality condition so that the number of natu-
ral variables is not changed. In making equilibrium calculations, it is simpler to omit the counter ions
and the electroneutrality condition because the same equilibrium composition is obtained either way. It
has to be understood that when ions are involved, there are counter ions to make each of the bulk phas-
es electrically neutral. The membrane phase is an electrically neutral dipolar layer. Water is omitted in
writing the fundamental equations because its amounts in the three phases do not change.

The fundamental equation for G for the system is the sum of eqs. 3.2a-2 to 3.2a-4, which is

(3.2a-5)

where S = S

+ S

and V = V

+ V

. Because of the reaction in each phase, dn

= dn

= –dn

and

= dn

= –dn

. This leads to the equilibrium conditions

(3.2a-6)

(3.2a-7)

for the reactions in the two bulk phases:

+ B

= C

(3.2a-8)

+ B

= C

(3.2a-9)

The transfer of electric charge from the thin layer on the

side of the membrane to the thin layer on

the

side leads to the following equilibrium condition

R. A. ALBERTY

1368

S T

VdP

= −

(3.2a-10)

because the electric potential difference

reaches the value at equilibrium that is required to make

equal to

at the specified values of a

and a

. This condition corresponds with the reaction

= C

(3.2a-11)

In other words, the equilibration of C between the phases is accomplished by changing the electric
potential of the

phase, rather than changing the amount of C in the

phase.

The equilibrium conditions 3.2a-6, 3.2a-7, and 3.2a-10 can be used to derive the following three

equilibrium constant expressions by inserting equation 3.2a-1:

(3.2a-12)

(3.2a-13)

(3.2a-14)

Equation 3.2a-14 is the familiar relation for the membrane potential equation [3,10], except that

and a

are not independent variables. Note that the effect of the electric potential cancels in the

derivation of the expressions for K

and K

and that K

= K

. The equilibrium concentrations of C from

eqs. 3.2a-12 and 3.2a-13 can be substituted in eq. 3.2a-14 to obtain

(3.2a-15)

(3.2a-16)

This shows how a reaction between ions can produce a difference in electric potentials between phases
at equilibrium.

Substitution of the equilibrium conditions for the three reactions in the fundamental eq. 3.2a-5

yields

dG = –SdT + VdP +

(3.2a-17)

where n

= n

– n

, n

= n

– n

, n

= n

+ n

, n

= n

+ n

, and n

= n

. The use of n

to represent the amount of a component has been discussed in Section 1.5. Equation

3.2-17 indicates that there are six natural variables; D =6. The criterion for spontaneous change and
equilibrium for the system is

(dG)

T,P,ncA

,ncA

,ncC,Q

≤

(3.2a-18)

The chemical potential of A in the

phase is given by

(3.2a-19)

Use of Legendre transforms in chemical thermodynamics

1369

−

(

)

[

]

exp

−

(

)

[

]

exp

−







exp

−







exp

−

∂







T P n

, ,

3.2b Fundamental equation for the transformed Gibbs energy of a multiphase system
with electric work

The equilibrium relations of the preceding section were derived on the assumption that the charge trans-
ferred Q can be held constant, but that is not really practical from an experimental point of view. It is
better to consider the potential difference between the phases to be a natural variable. That is accom-
plished by use of the Legendre transform

′

= G –

(3.2b-1)

which defines the transformed Gibbs energy G

′

. Since

′

= dG –

dQ – Qd

(3.2b-2)

substituting eq. 3.2a-17 yields

′

= – SdT + VdP +

– Qd

(3.2b-3)

This indicates that there are six natural variables for the transformed Gibbs energy, the same as for the
Gibbs energy (eq. 3.2a-17). The criterion for spontaneous change and equilibrium is given by

(dG)

T,P,ncA

,ncA

,ncC,

≤

(3.2b-4)

This can be used to derive eqs. 3.2a-12 to 32.a-16. To learn more about the derivatives of the trans-
formed Gibbs energy, the chemical potentials of species are replaced by use of eq. 3.2a-1 to obtain

(3.2b-5)

Thus,

(3.2b-6)

This derivative is referred to as the transformed chemical potential of A in the

phase. Substituting this

relation in eq. 3.2a-1 yields

′

+ z

(3.2b-7)

which shows the relationship between the chemical potential and the transformed chemical potential.

Since this three-phase system has six natural variables at equilibrium (D = 6), the number of

intensive degrees of freedom f is given by f = D – p = 6 – 3 = 3, where p is the number of phases. This
is in accord with f = C – p + 3 = 3 – 3 + 3 = 3, where the electric potential is considered to be an inde-
pendent natural variable like T and P.

3.2c Thermodynamic properties of an ion in phases with different electric potentials

Equation 3.2a-5 can be written in a more general way as

dG = – SdT + VdP +

(3.2

c-1

)

Integration at constant values of the intensive variables yields

G =

∑

(3.2c-2)

The entropy of the system can be obtained by use of the following derivative:

R. A. ALBERTY

1370

′ = −

(

)

(

)

(

)

−

S T

V P

∂ ′

∂







′

T P n

, ,

(3.2c-3)

where {n

} represents the set of amounts of species in the

phase. Taking this derivative of G yields

(3.2

c-4

)

where S

–

is the partial molar entropy of i, since

is determined by Q, which is held constant.

Substituting eq. 3.2b-7 in eq. 3.2c-2 yields

(3.2

c-5

)

Taking the derivative in eq. 3.2c-3 yields

(

3.2c-6

)

where S

–

′

is the transformed molar entropy of i in the

phase. Comparing this equation with eq. 3.2c-4

shows that the molar entropy if a species is not affected by the electric potential of a phase: thus,
S

–

= S

–

′

and S = S

′

The corresponding molar enthalpy is obtained by use of the Gibbs–Helmholtz equation: H =

– T

[(

∂

(G/T)/

∂

. Applying this to eqs. 3.2c-1 and 3.2c-2 yields

H =

–

(3.2

c-7

)

where H

–

is the molar enthalpy of i, and

(

3.2c-8

)

where H

–

′

is the transformed molar enthalpy. Comparing eqs. 3.2c-7 and 3.2c-8 shows that

–

= H

–

′

+ Fz

(3.2c-9)

Thus, the molar enthalpy of an ion is affected by the electric potential of the phase in the same way as
the chemical potential (see eq. 3.2b-7).

3.2d Nomenclature of the electrochemical potential

A number of different treatments have been given of multiphase systems with electric potential differ-
ences between the phases, starting with Gibbs [1]. An early treatment was made by Guggenheim [41]
in which he used

and referred to it as the electrochemical potential. Later in his textbook,

Thermodynamics [35], he used the equivalent of eq. 3.2a-1 with

, and he referred to it as the electro-

chemical potential. In making recommendations about thermodynamic nomenclature for such systems,
IUPAC [42–44] has used

and referred to it as the electrochemical potential. The IUPAC

Recommendations for Quantities, Units, and Symbols in Physical Chemistry [45] has recommended

and this will continue in the next edition with the electrochemical potential defined as

+ RT ln a

+ z

(3.2d-1)

While the symbol

, has been used widely in electrochemistry, the symbol

(as defined in eq.

3.2a-1) has been used in the preceding three sections. The two symbols have the same meaning. The
important point is that this physical quantity is independent of phase at equilibrium. This aspect is par-
ticularly important when the effects of temperature and pressure are being discussed and when other
kinds of work, e.g., chemical and surface, are also involved. Since both conventions are currently used,
it is important to check which convention is being followed.

Use of Legendre transforms in chemical thermodynamics

1371

P n

= − ∂

∂









{ }

n S

+ ∑

∑

α α

β β

z n

i i

= ∑ ′

+ ∑ ′

∑

α α

β β

n S

= ∑

′ + ∑

′

α α

β β

n H

z n

i i

= ∑

′ + ∑

′ +

∑

4. FUNDAMENTAL EQUATIONS OF THERMODYNAMICS FOR SYSTEMS WITH OTHER
KINDS OF WORK

4.1 Systems with surface work

A number of treatments of the thermodynamics of systems with interfaces are available [34,35,47,48].
As an example of a system involving surface work, consider a binary liquid solution in contact with its
vapor or two immiscible binary solutions at equilibrium with variable surface area between the phases.
The fundamental equation for the Gibbs energy of the whole three-phase system is [49]

(4.1-1)

Here,

is the interfacial tension, and A

is the interfacial area. The superscripts

and

indicate the two

bulk liquid phases, and the

superscript indicates a property of the surface phase. If the three phases

are at equilibrium, the values of the chemical potentials are restricted by the following equilibrium con-
ditions:

, and

. In writing these conditions, the phase equilibria are

treated like chemical reactions. Since

and

, the superscripts on the

chemical potentials can be dropped when the system is at equilibrium. Thus, eq. 4.1-1 can be written
as

(4.1-2)

where the amounts of the two components are represented by n

and n

(4.1-3)

(4.1-4)

At equilibrium, the natural variables for G are indicated by G(T, P, n

, A

). The amounts of com-

ponents are independent variables because they are the amounts added to the system, but the amounts
of species in a phase are not independent variables because they are determined by the equilibrium.
Integration of eq. 4.1-2 at constant T, P, and composition yields

G =

(4.1-5)

There is a Gibbs–Duhem equation for each phase, including the interfacial phase, and the sum of

these three equations is the Gibbs–Duhem equation for the system, which can be obtained by making
the following Legendre transform to define a transformed Gibbs energy G

′

that has T, P,

, and

as its natural variables.

′

= G –

–

= 0

(4.1-6)

It is important that this equation contain n

and n

, rather than n

, n

, and n

because n

and n

are independent variables for the equilibrium system. Taking the differential of G

′

and substituting eq. 4.1-2 yields

0 = – SdT + VdP – n

– n

– A

(4.1-7)

This Gibbs–Duhem equation can be used to derive the Gibbs adsorption equation for a liquid–liq-

uid interface or a liquid–vapor interface. The equation for the system with a liquid–liquid interface is
quite complicated, but it reduces to the Gibbs adsorption derived by Bett, Rowlinson, and Saville [47]
for a liquid–vapor interface. This equation is

R. A. ALBERTY

1372

S T

V P

= −

S T

V P

S T

V P

= −

(

)

(

)

= −

(4.1-8)

where S

is the interfacial entropy, the adsorptions of the components are given by

and

, and x

and x

are the mole fractions of species 1 and 2 in the liquid phase. This indicates

that two derivatives can be determined experimentally.

(4.1-9)

and

(4.1-10)

Thus,

(4.1-11)

can be determined from measurements of the surface tension as a function of x

The IUPAC recommendations on the thermodynamic properties of surfaces [50] are based on

Legendre transforms, but they do not follow all of the conventions recommended here.

4.2 Systems with mechanical work

The thermodynamics of crystals is discussed very thoroughly by Wallace [51]. He shows that the fun-
damental equation for the Helmholtz energy of a crystal under stress is given by

(4.2-1)

where

is the tensor representing the applied stress and

is the Lagrangian strain parameter. The

applied stress is assumed to be uniform (i.e., constant on a given crystal surface), and the resulting strain
is homogeneous (i.e., uniform throughout the crystal).

Rather than going into the details of this subject here, we simply observe that the length L of a

solid subjected to a force f of extension is an extensive property, which is a natural variable of U, H, A,
and G. The fundamental equations for U and A can be written

dU = TdS + fdL

(4.2-2)

dA = – SdT + fdL

(4.2-3)

if PV work is negligible. This application of thermodynamics is of special interest to chemists in con-
nection with the properties of high polymers. For rubber, the tension is primarily an entropy effect. In
making stress-strain measurements, the change in the force f with temperature can be measured, but it
may be more convenient to hold the force constant and measure the length. In this case, it is convenient
to make the force a natural variable by making the Legendre transform

′

= A – fL = 0

(4.2-4)

This is a complete Legendre transform. Taking the differential of this equation and substituting
eq. 4.2-3 yields

0 = – SdT – Ldf

(4.2-5)

which is a Gibbs–Duhem equation. This yields the Maxwell equation

Use of Legendre transforms in chemical thermodynamics

1373

= −

(

)

−

[

]

−

(

)

∂
∂







∂
∂









(

)

−

[

]

P x

–

∂







−

(

)

∂
∂







T P

−

S T

= −

+ ∑ ττ η

(4.2-6)

Chemical work is coupled with mechanical work in muscle contraction.

4.3 Systems with work of electric polarization

The effect of work of electric transport on thermodynamics has been discussed in Section 3.2, but here
we are concerned with the work of producing electric polarization in a nonconductor by an electric
field. This topic has been discussed in a number of books [14,34]. In treating electric polarization, it is
of interest to consider electrically polarizable systems involving elongation work, but no PV work. The
fundamental equation for U for such a system is

dU = TdS + fdL + E

•

(4.3-1)

where E is the electric field strength and p is the dipole moment of the system. When it is not neces-
sary to consider pressure as a natural variable, we use the Helmholtz energy A, rather than the Gibbs
energy. The fundamental equation for A is

dA = – SdT + fdL + E

•

(4.3-2)

In discussions of thermoelectric, pyroelectric, and piezoelectric effects, it is advantageous to use the
transformed Helmholtz energy obtained with the Legendre transform

′

= A – fL – E

•

p = 0

(4.3-3)

This is a complete Legendre transform and yields a Gibbs–Duhem equation. Taking the differential of
A

′

and substituting 4.3-2 yields

0 = – SdT + VdP – Ldf – p

•

dE or 0 = – SdT + VdP – Ldf – pdE

(4.3-4)

where the last form applies to an isotropic system. This Gibbs–Duhem equation yields three Maxwell
equations [14]:

thermoelastic:

(4.3-5)

pyroelectric:

(4.3-6)

piezoelectric:

(4.3-7)

Note that Gibbs–Duhem equations are especially useful for obtaining Maxwell equations in which the
derivatives are with respect to intensive variables and the variables held constant are all intensive variables.

4.4 Systems with work of magnetic polarization

The effects of magnetic polarization on thermodynamics are discussed in several places in the literature
[14,34,52,53]. The fundamental equation for U for a system involving magnetic polarization is

R. A. ALBERTY

1374

∂







= ∂

∂









∂







= ∂

∂









E T

f E

∂









= ∂

∂









T f

f E

∂









= ∂

∂







T f

T E

dU = TdS + B

•

(4.4-1)

where B is the magnetic flux density and m is the magnetic moment of the system. It is assumed that
PV work is negligible. The corresponding fundamental equation for the Helmholtz energy is

dA = – SdT + B

•

dm or dA = – SdT + Bdm

(4.4-2)

In order to treat adiabatic demagnetization, it is advantageous to use the magnetic flux density as an
intensive variable. A transformed Helmholtz energy is defined by

′

= A – B

•

m = 0

(4.4-3)

This is the complete Legendre transform, and it yields the Gibbs–Duhem equation.

0 = – SdT – m

•

dB or 0 = – SdT – mdB

(4.4-4)

where the last form applies to an isotropic system. This yields the Maxwell equation

(4.4-5)

The effect of a magnetic field on a chemical reaction producing a paramagnetic species from a dia-
magnetic species is small and readily calculated.

5. RECOMMENDATIONS

These recommendations are based on the following definitions of the enthalpy H, Helmholtz
energy A, and Gibbs energy G: H = U + PV, A = U – TS, and G = U + PV – TS. We recommend
that these definitions not be altered. If the fundamental equation for U involves terms for work in
addition to PV work, they should be of the form (intensive property)d(extensive property). Thus,
the fundamental equations for U, H, A, and G involve the same non-PV work terms, which involve
the differentials of extensive properties.

Natural  variables  are  important  because  if  a  thermodynamic  potential  can  be  determined  as  a
function of its natural variables, all of the other thermodynamic properties of the system can be
calculated by taking partial derivatives. Natural variables are also important because they are held
constant  in  the  criterion  for  spontaneous  change  and  equilibrium.  It  is  important  to  distinguish
between  natural  variables  before  and  after  the  application  of  constraints  resulting  from  phase
equilibrium and chemical equilibrium. The criterion for equilibrium is stated in terms of the nat-
ural variables after all the constraints have been applied.

The chemical potential of species i is defined by

(5-1)

where j

≠

i and X

represent all of the independent extensive variables in non-PV and non-chem-

ical work involved. It is important to retain

for this purpose because

is the same throughout

a multiphase system at equilibrium, even if the phases are different states of matter and have dif-
ferent pressures or different electric potentials. When phase equilibrium and chemical equilibri-
um are involved, these derivatives can be written in terms of components rather than species.

In order to introduce the intensive variables of non-PV work as natural variables, it is necessary
to define thermodynamic potentials in addition to U, H, A, and G with Legendre transforms.
These Legendre transforms are of the form, U

′

= U –

(extensive property)(conjugate intensive

property), and so the transformed thermodynamic potential U

′

can always be represented by

Callen’s nomenclature as U[P

], where the P

are the intensive properties introduced as natural

Use of Legendre transforms in chemical thermodynamics

1375

∂







 =

∂









S V n

S P n

T V n

T P n

= ∂

∂







= ∂

∂







= ∂

∂







= ∂

∂







, ,

variables by the Legendre transform. This nomenclature can be extended by making Legendre
transforms of H, A, and G, and representing the transformed thermodynamic potentials by H[P

A[P

], and G[P

], where the P

are the intensive properties introduced as natural variables in addi-

tion to the intensive variables that have been introduced by the definitions of these thermody-
namic potentials. Examples are G[pH,pMg], G[

], A[f], G[r], and U[E]. Legendre transformed

thermodynamic potentials can be represented by U

′

, H

′

, A

′

, and G

′

, but it is necessary to specify

the intensive variables that have been introduced when this notation is used.

The transformed chemical potential of species i is defined by

(5-2)

where j

≠

i and the P

represents intensive variables that have been introduced by Legendre trans-

forms. The j is used to indicate that P

is not in the conjugate pair with extensive variables X

When phase equilibrium and chemical equilibrium are involved, these derivatives can be written
in terms of components rather than species.

The properties subscripted on partial derivatives are always natural variables. It is important to be
sure that natural variables are independent. The number D of natural variables is given by f + p,
where f is the number of independent intensive variables given by the phase rule and p is the num-
ber of phases.

6. APPENDIX: FIELDS AND DENSITIES

The variables in a fundamental equation are often classified as intensive variables and extensive vari-
ables, but there is a problem because a fundamental equation can be divided by volume, mass, or total
amount. When this is done, the fundamental equation is expressed entirely in terms of intensive vari-
ables. When a fundamental equation is written in this way, it is important to make a distinction between
two types of intensive variables because some of the intensive variables are uniform throughout a sys-
tem at equilibrium and others are not. Griffiths and Wheeler [54] recommended that a distinction be
made by referring to T, P,

, electric field strength, and magnetic field strength as “fields” and refer-

ring to extensive variables divided by volume, mass, or amount as “densities”. The important feature of
fields is that they have uniform values in a system at equilibrium. The pressure is an exception to this
statement when there are curved surfaces or when there are semipermeable membranes that lead to an
osmotic pressure at equilibrium. When fundamental equations are written for U, H, S, A, G, etc., some
of the variables are fields and others are extensive variables. When fundamental equations are divided
by volume, mass, or amount, some of the variables are fields and the others are densities.

7. NOMENCLATURE

Note: When primes are used on thermodynamic potentials, it is important to indicate in the context the
intensive variables that have been specified. This also applies when primes are used on equilibrium con-
stants, amounts, or numbers like the number of components, number of degrees of freedom, and stoi-
chiometric numbers. SI units are in parentheses.
a

activity of species i (dimensionless)

Helmholtz energy (J)

′

transformed Helmholtz energy (J)

R. A. ALBERTY

1376

′ = ∂ ′

∂







= ∂ ′

∂







= ∂ ′

∂







= ∂ ′

∂







S V n

X P

S P n

X P

T V n

X P

T P n

X P

, ,

surface area (m

)

conservation matrix (C

N) (dimensionless)

magnetic flux density (T)

magnitude of the magnetic flux density, B = |B| (T)

number of components (C = N – R) (dimensionless)

standard state concentration (mol L

–1

)

number of natural variables (dimensionless)

electric field strength (V m

–1

)

magnitude of the electric field strength, E = |E| (V m

–1

)

force (N)

number of independent intensive variables (degrees of freedom) (dimensionless)

Faraday (96 485 C mol

–1

)

acceleration of gravity (m s

–2

)

Gibbs energy (J)

′

transformed Gibbs energy (J)

∆

standard Gibbs energy of formation of species i (kJ mol

–1

)

∆

′

standard transformed Gibbs energy of formation of reactant i (kJ mol

–1

)

∆

standard Gibbs energy of reaction (J mol

–1

)

∆

′

standard transformed Gibbs energy of reaction at a specified pressure or concentration of a

species (J mol

–1

)

∆

Gibbs energy of reaction (J mol

–1

)

∆

′

transformed Gibbs energy of reaction at a specified pressure or concentration of a species

(J mol

–1

)

height above the surface of the earth (m)

enthalpy (J)

′

transformed enthalpy (J)

∆

reaction enthalpy (J mol

–1

)

–

partial molar enthalpy of i (J mol

–1

)

–

′

partial molar transformed enthalpy of i (J mol

–1

)

–

′

standard partial molar transformed enthalpy of i (J mol

–1

)

∆

′

standard transformed enthalpy of reaction at a specified concentration of a species (J mol

–1

)

∆

standard enthalpy of formation of species i (J mol

–1

)

∆

′

standard transformed enthalpy of formation of i at a specified concentration of a species

(J mol

–1

)

ionic strength (mol L

–1

)

equilibrium constant (dimensionless)

′

apparent equilibrium constant at specified concentration of a species (dimensionless)

elongation (m)

mass (kg)

magnetic dipole moment of the system (J T

–1

)

magnitude of the magnetic moment of the system (J T

–1

)

molar mass of species i (kg mol

–1

)

amount of species i (mol)

amount of component i (mol)

Use of Legendre transforms in chemical thermodynamics

1377

amount of species matrix (N

1) (mol)

amount of component matrix (C

1) (mol)

′

amount of reactant i (sum of species) (mol)

interfacial amount of species i (mol)

amount of B component (mol)

number of species when a single phase is involved and number of species in different

phases for a multiphase system (dimensionless)

′

number of reactants (pseudoisomer groups) (dimensionless)

(i)

number of hydrogen atoms in a molecule of i (dimensionless)

(i)

number of magnesium atoms in a molecule of i (dimensionless)

–

(i)

average number of B bound by a molecule of i (dimensionless)

pressure (bar)

partial pressure of i (bar)

standard state pressure (1 bar)

′

partial pressure of species other than the one with a specified pressure (bar)

intensive variable in Callen’s nomenclature (varies)

electric dipole moment of the system (C m)

magnitude of the dipole moment of the system (C m)

number of phases (dimensionless)

– lg([H

]/c

) (dimensionless)

pMg

– lg([Mg

]/c

) (dimensionless)

canonical ensemble partition function (dimensionless)

electric charge transferred (C)

equilibrium mole fraction of i within an isomer group or pseudoisomer group

(dimensionless)

gas constant (8.314 472 J K

–1

mol

–1

)

number of independent reactions (dimensionless)

entropy (J K

–1

)

∆

entropy of reaction (J K

–1

mol

–1

)

∆

′

transformed entropy of reaction (J K

–1

mol

–1

)

–

partial molar entropy of i (J K

–1

mol

–1

)

–

′

partial molar transformed entropy of i (J K

–1

mol

–1

)

–

′

standard partial molar transformed entropy of i (J K

–1

mol

–1

)

′

transformed entropy (J K

–1

)

temperature (K)

internal energy (J)

′

transformed internal energy (J)

U[P

]

Callen’s nomenclature for the transformed internal energy that has intensive variable P

a natural variable (J)

volume (m

)

extensive variable in Callen’s nomenclature (varies)

mole fraction of i (dimensionless)

number of protonic charges on ion i (dimensionless)

surface tension (N m

–1

)

R. A. ALBERTY

1378

adsorption of component i (n

) (mol m

–2

)

∆

isothermal-isobaric partition function (dimensionless)

Lagrangian strain parameter

chemical potential of species i (J mol

–1

)

standard chemical potential of species i (J mol

–1

)

′

transformed chemical potential of reactant i (J mol

–1

)

chemical potential matrix (1

N)(J mol

–1

)

component chemical potential matrix (1

C)(J mol

–1

)

electrochemical potential of i defined by eq. 3.2d-1 (J mol

–1

)

stoichiometric number of species i (dimensionless)

′

stoichiometric number of reactant (sum of species) i (dimensionless)

stoichiometric number matrix (N

R) (dimensionless)

applied stress tensor

extent of reaction (mol)

ground canonical partition functional (dimensionless)

electric potential of the phase containing species i (V, J C

–1

)

gravitational potential (J kg

–1

)

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