# Auxiliary Functions

```Auxiliary Functions
Introduction
• The study of thermodynamics provides the criteria for equilibrium in
systems and the determination of the effect, on an equilibrium state, of
changes in the external influences acting on the system.
• This provision is determined by the practicality of the equations of state
for the system.
• To achieve this, it is desirable to develop simple expressions which contain
convenient choice of independent variables which can be easily controlled
and measured.
• It’s is often found that some required thermodynamic expressions which
are not easily amenable to experimental measurement are related in a
simpler manner to some measurable quantities.
• New thermodynamic functions are introduced and their properties and
interrelationships are examined:
Helmholtz and Gibbs Free Energies
• The Helmholtz free energy, A, is defined as
π΄ = π + ππ
(5.1)
• The Gibbs free energy, G, is defined as
πΊ = π + ππ − ππ
= π» − ππ
(5.2)
• For a system undergoing a change of state from 1 to 2,
πΊ2 − πΊ1 = π»2 − π»1 − π2π2 − π1π1
= π2 − π1 + π2π2 − π1π1 − (π2π2 − π1π1)
• For a closed system, the first law gives,
π2 − π1 = π − π€
and thus
πΊ2 − πΊ1 = π − π€ + π2π2 − π1π1 − (π2π2 − π1π1)
(5.6)
• If the processes is carried out such that π1 = π2 = π, the constant
temperature of the heat reservoir which supplies or withdraws heat from
the system, and also if π1 = π2 = π, the constant pressure at which the
system undergoes a change in volume , then
(πΊ2 − πΊ1) = π − π€ + π π2 − π1 − π(π2 − π1)
• If the system preforms chemical or electrical work in addition to π − π
work, the work terms are included in π€, i.e.
π€ = π€′ + π π2 − π1
in which π€′ is the sum of all non π − π forms of work done. Substituting into
the above eqn. gives,
πΊ2 − πΊ1 = π − π€′ − π π2 − π1
and from the second law as,
π ≤ π(π2 − π1)
then,
π€′ ≤ −(πΊ2 − πΊ1)
(5.7)
The equality can be written as,
−π€ ′ = πΊ2 − πΊ1 + πβππππ
• Thus, for an isothermal, isobaric process, during which no form of work
other than P-V work is performed i.e. π€′ = 0,
πΊ2 − πΊ1 + πβππππ = 0
(5.8)
Such a process can only occur spontaneously (with an increase in entropy) if
the Gibbs free energy decreases. As πππππ = 0 is a criterion for a reversible
process, thermodynamic equilibrium requires that,
ππΊ = 0
(5.9)
• Thus for a system undergoing a process at constant π and constant π, the
Gibbs free energy can only decrease or remain constant, and the
attainment of equilibrium in the system coincides with the system having
the minimum value of πΊ consistent with the values of π and π.
• Since
ππ = πππ − πππ
and
π» = π + ππ
and thus
ππ» = πππ + πππ
(5.10)
π΄ = π − ππ
and thus
ππ΄ = −πππ − πππ
(5.11)
πΊ = π» − ππ
and thus
ππΊ = −πππ + πππ
(5.12)
Variation of Composition and Size of System
• If the size and composition of a system can vary during a process, the
specification of only two variables will be no longer sufficient to fix the
state of the system.
• It is therefore necessary to specify the number of moles within the
system.
• Gibbs free energy is thus a function of π, π and the numbers of moles of
all the species present in the system:
πΊ = πΊ(π, π, ππ , ππ, ππ, … )
(5.13)
Where ππ, ππ, ππ, … . are the number of moles of the species π, π, π, … present
in the system and the state of the system is only fixed when all the
independent variables are fixed. Differentiation of (5.13) gives:
ππΊ
ππΊ
ππΊ
ππΊ =
ππ +
ππ +
πππ +
ππ π, ππ, ππ …
ππ π, ππ, ππ, …
πππ π, π, ππ, ππ, …
ππΊ
πππ
πππ π, π , ππ, ππ, …
+β―
(5.14)
• If the numbers of moles of all of the individual species remain constant
during the process the above eqn. simplifies to eqn. (5.12) i.e.
ππΊ = −πππ + πππ
from which it is seen that
ππΊ
ππ
π , ππ, ππ, …
= −π
and
ππΊ
ππ
Substituting into eqn. (5.14) gives
π, ππ, ππ, …
π=π
ππΊ = −πππ + πππ +
π=1
=π
ππΊ
πππ
π, π, ππ, …
πππ
(5.15)
where
π=π
π=1
ππΊ
πππ
π, π, ππ, …
πππ
is the sum of π terms, one for each species of the π species, each of which is
determined by partial differentiation of πΊ w.r.t the number of moles of the πth
species at constant π, π, and ππ, where ππ represents the number of moles of
every species other than the πth species.
Chemical Potential
• The term ππΊ πππ π, π, ππ… is called the chemical potential of the species π,
designated ο­π , i.e.
ππΊ
πππ π, π, ππ, …
= ο­π
(5.16)
• The chemical potential of π in a homogeneous phase is the rate of increase
of πΊ with ππ when the species π is added to the system at constant
π, π and number of moles of all the other species.
• If the system is large enough that the addition of 1 mole of π, at constant π
and π, does not measurably change the composition of the system, then
ο­π is the increase in πΊ of the system caused by the addition.
• Eqn (5.15) can thus be written as
ππΊ = −πππ + πππ + π1 πππππ
(5.17)
•
•
Eqn (5.17) can be applied to open systems which exchange matter and heat
with their surroundings, and to closed systems which undergo changes in
composition caused by chemical reactions.
Equations (3.12), ππ = πππ − πππ, (5.10), ππ» = πππ + πππ, and (5.11),
ππ΄ = −πππ − πππ, can be made applicable to open systems by including the
terms describing the dependences on composition of, respectively, π, π» and
π΄:
π
ππ = πππ − πππ +
1
ππ
πππ
π , π , ππ, …
πππ
(5.18)
π
ππ» = πππ − πππ +
1
ππ»
πππ
π , π , ππ, …
πππ
(5.19)
π
ππ΄ = −πππ + πππ +
1
ππ΄
πππ
π, π , ππ, …
πππ
(5.20)
• Inspection of Eqns. (5.16), (5.18), (5.19) and (5.20) shows that:
ππΊ
ππ
ππ»
ππ΄
= ππ =
=
=
πππ π, π, ππ
πππ π, π , ππ
πππ π, π, ππ
πππ π, π , ππ
(5.21)
and hence the complete set of equations is:
ππ = πππ − πππ +
πππππ
(5.22)
ππ» = πππ + πππ +
πππππ
(5.23)
ππ΄ = −πππ − πππ +
πππππ
(5.24)
ππΊ = −πππ + πππ +
πππππ
(5.25)
• The first law gives
ππ = πΏπ − πΏπ€
which on comparison with equation (5.22), ππ = πππ − πππ + πππππ ,
indicates that, for a closed system undergoing a process involving a reversible
change in composition (e.g. a reversible chemical reaction),
πΏπ = πππ
and
πΏπ€ = πππ +
π’ππππ
The term π’ππππ is thus the chemical work done by the system which was
denoted as π€′ in Eq. (5.7), π€′ ≤ −(πΊ2 − πΊ1) .
Thermodynamic Relations
The following relations are obtained from Eqns. (5.22)-(5.25):
π=
ππ
ππ
π , ππππ
=
ππ»
ππ
π , ππππ
(5.26)
ππ
π=−
ππ
ππ΄
=−
ππ
π, ππππ
π, ππππ
(5.27)
ππ»
π=
ππ
ππΊ
=
ππ
π, ππππ
π, ππππ
(5.28)
π=−
ππ΄
ππ
π , ππππ
=−
ππΊ
ππ
π , ππππ
(5.29)
Maxwell’s Equations
• If π is a state function and π₯ and π¦ are chosen as the independent
variables in a closed system of fixed composition then,
π = π(π₯, π¦)
and by differentiation,
ππ
ππ
ππ =
ππ₯ +
ππ¦
ππ₯ π¦
ππ¦ π₯
• If the partial derivative ππ ππ₯ π¦ is itself a function of π₯ and π¦, being given
by ππ ππ₯ π¦ = πΏ(π₯, π¦), and similarly ππ ππ¦ π₯ = π π₯, π¦ then
ππ = πΏππ₯ + πππ¦
Thus
π ππ
ππ¦ ππ₯
and
π¦ π₯
ππΏ
=
ππ¦
π₯
π ππ
ππ
=
ππ₯ ππ¦ π₯ π¦
ππ₯ π¦
But, as π is a state function, the change in π is independent of the order of
differentiation, and thus,
π ππ
π ππ
π2π
=
=
ππ¦ ππ₯ π¦ π₯
ππ₯ ππ¦ π₯ π¦ ππ₯ππ¦
and hence
ππΏ
ππ
=
ππ¦ π₯
ππ₯ π¦
(5.30)
• Application of equation (5.30) to Eqns. (3.12) and (5.10)-(5.12) gives a set
of relations which are referred to as the Maxwell’s Equations:
ππ
ππ
ππ
=−
ππ
π
π
(5.31)
ππ
ππ
ππ
=
ππ
π
π
(5.32)
ππ
ππ
ππ
=
ππ
π
π
(5.33)
ππ
ππ
ππ
=−
ππ
π
π
(5.34)
• Consider the dependence of entropy of an ideal gas on π and π:
π = π π, π
and by differentiation
ππ =
ππ
ππ
ππ +
π
ππ
ππ
ππ
π
(π)
Combination of the definition of the constant volume heat capacity (Eqn.
πΏπ
(2.6), ππ = πΆπ£ ππ and Eqn. (3.8), ππ = πππ£ applied to a reversible constant
π
volume process gives,
πππ = πΏππ£ = ππ = πππ£ ππ
the first P.D on the right-hand side of eqn. (i) is obtained as
ππ
πππ£
=
ππ π
π
and the second P.D in eqn. (i) is obtained from Maxwell’s equation (5.33).
• Thus eqn. (i) can be written as,
πππ£
ππ
ππ =
ππ +
π
ππ
ππ
π
(ii)
From the ideal gas law
ππ
ππ
π
ππ
=
π
(iii)
and thus eqn. (ii) can be written as
πππ£
ππ
ππ =
ππ +
ππ
π
π
(iv)
integration of which between two states gives
π2
π2
π2 − π1 = πππ£ πΌπ
+ πππΌπ
π1
π1
•
Considering a closed system of fixed composition Eqn. (3.12) gives,
ππ = πππ − πππ
and thus
ππ
ππ
=π
−π
ππ π
ππ π
Using Maxwell’s eqn. (5.33) allows this to be written as
ππ
ππ
=π
−π
ππ π
ππ π
(v)
which is an equation of state relating the internal energy of a closed system of
fixed composition to measurable quantities π, π and π.
If the system is 1 mole of an ideal gas, substitution of eqn. (iii) into eqn. (v) gives
ππ
=0
ππ π
Therefore the internal energy of an ideal gas is independent of the volume of the
gas.
• Similarly for a closed system of fixed composition, eqn. (5.10) gives,
ππ» = πππ + πππ
in which case
ππ»
ππ
ππ
=π
ππ
π
π
+π
Substituting Maxwell’s eqn. (5.34) gives
ππ»
ππ
= −π
+π
ππ π
ππ π
which is an equation of state which gives the dependence of enthalpy on
π, π and π. If the system is an ideal gas, this equation of state shows that the
enthalpy of an ideal gas is independent of its pressure.
The Gibbs-Helmholtz
Eqn.(5.2) gives
πΊ = π» − ππ
Eqn. (5.12) gives
ππΊ = −πππ + πππ
and therefore,
ππΊ
= −π
ππ π
Thus at constant pressure and composition,
ππΊ
πΊ =π»+π
ππ
or
πΊππ = π»ππ + πππΊ
Dividing throughout by π2 and rearranging gives,
πππΊ − πΊππ
π»ππ
=− 2
π2
π
which on comparison with the identity
π₯
π¦ππ₯ − π₯ππ¦
π
=
π¦
π¦2
shows that
π πΊ π
π»
=− 2
ππ
π
(5.36)
The above equation is known as the Gibbs-Helmholtz equation and is
applicable to a closed system of fixed composition undergoing processes at
constant composition.
• For an isobaric change of state of a closed system of fixed composition,
eqn. (5.36) gives the relation of the change in πΊ to the change in π» as
π(βπΊ π)
βπ»
=− 2
ππ
π
(5.36a)
• This equation is used in experimental thermodynamics since it allows βπ»,
the heat of reaction, to be obtained from a measurement of βπΊ, the free
energy change for the reaction, with temperature, or, conversely βπΊ to be
obtained from a measurement of βπ».
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