GROUP (8)
CHAPTER FIVE
AUXILIARY FUNCTIONS
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5.1 Introduction
 The power of thermodynamics lies in its provision of
criteria for equilibrium within a system and its ability to
facilitate determination of the effect, on the equilibrium
state, of change in the external influences which can be
brought to bear on the system.
 The practical usefulness of this power is consequently
determined by the practical of the equations of state of the
system, i.e. the relationships among the state functions
which can be established.
 The combination of the First and Second laws of
Thermodynamics leads to the following equation:
dU = TdS - pdV
or
U = f ( S, V )
 This equation of state gives the relationship between the
dependent variable U and the independent variable S and
V for a closed system of fixed composition which is in
states of equilibrium then undergoing a reversible process
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involving volume change against external pressure as the
only form of work performed on or by the system ( wp-v ) .
 The combination of the First and the Second Laws of
Thermodynamics provides also the following criteria for the
equilibrium :
1. for a closed system of constant energy and constant
volume, the entropy is maximum, and
2. for a closed system of constant energy and constant
volume, the internal energy is minimum.
 Since the entropy is an inconvenient choice as
independent variable from the point of view of
experimental measurements or control; it is desirable to
develop a simple equation similar to the previous one
which contains a more convenient choice of independent
variables.
 From experimental point of view, the most convenient pair
of independent variables would be temperature and
pressure because they are the most easily measured and
controlled parameters in a practical experiment.
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 From theoretical point of view, the most convenient pair of
independent variables would be volume and temperature
because when they are fixed for a closed system the ϵ(ί)’s
level values ; and hence the Boltzmann factor ( exp ( ϵ(ί)’s / kT ) and the partition function are fixed; this will
ease the theoritical calculations using statistical mechanics
rules.
 Thus , in this chapter, to meet the previously discussed
points the enthalpy function , H , the Helmholtz free energy
function ( the work function ) , A , the Gibbs free energy
function , G , and the chemical potential of species ί , μί ,
are introduced .
5.2 The Enthalpy H
 The enthalpy function H is defined as:
H = U + PV
thus:
dH = dU + PdV + VdP
= TdS – PdV + VdP + PdV
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therefore:
dH = TdS + VdP
ί. e,
H = ƒ ( S, P )
 The dependent variable, in this case, is enthalpy while the
pair of independent variables are the entropy and the
pressure.
 Since the enthalpy is a state function:
Δ H = H2 – H1
= ( U2 + P2V2 ) – ( U1 + P1V1 )
thus under constant pressure:
Δ H = ( U2 – U1 ) + P ( V2 – V1 )
= Δ U - wp-v= qp
 Therefore, the equation of statue, dH = TdS + VdP , gives
the relationship between the dependent variable H and the
independent variables S and P for a closed system of
fixed composition which is in state of equilibrium and is
undergoing a reversible process involving volume change
against external pressure as the only form of work
performed on , or by , the system ; the enthalpy change of
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the system equals the heat leaving or entering the system
at constant pressure.
5.3 Helmholtz Free Energy Function A
 Helmholtz Free Energy Function A is defined as :
A = U – TS
thus:
dA = dU – TdS – SdT
= TdS – PdV – TdS – Sdt
therefore
dA = – SdT – PdV
=ƒ(V, T)
 The dependent variable in this case is A and the pair of
independent variables are volume and temperature.
Since the Helmholtz free energy function is a state
function, thus:
Δ H = A2 – A1
= ( U2 – T2S2 ) – ( U1 – T1S1 )
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= ( U2 – U1 ) – ( T2S2 – T1S1 )
ί . e,
Δ A = q + w – ( T2S2 – T1S1 )
thus:
Δ A – w = q – ( T2S2 – T1S1 )
 In describing the work in the previous equation, a positive
sign is assigned to work done on the system and a
negative sign is assigned to work done by the system.
thus:
- w – ( - ΔA )= q – ( T2S2 – T1S1 )
 If the process is isothermal; that is T2 = T1 = T,
then:
-w – ( - Δ A ) = q – T ( S2 – S1 )
 From the second law of thermodynamics :
q ≤ T ( S2 – S1 ),
thus
- w ≤ –Δ A
 Therefore, for reversible isothermal process :
wmax = – Δ A
ί.e. the maximum amount of work done by system equal
the decrease in the work function .
since:
w =wmax–w deg
then:
w = –Δ A–TΔSirr
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= – ( ΔA + T ΔSirr )
Thus, for a process which occurs at constant V and T, we
can write:
Δ A + T Δ Sirr = 0
for an infinitesimal increment of such process:
δA + TdSirr = 0
 We know that for spontaneous processes dSirr is a positive
value , thus processes occur at constant T and V will be
spontaneous if dA is a negative value ; ί.e, for
spontaneous processes that occur at constant T and V :
dA<0
Since the condition for thermodynamic equilibrium is that
dSirr = 0 then with respect to the described process. ί. e , at
constant V and T equilibrium is defended by the condition
dA<0
Thus, for a closed system , held at constant T and V,
Helmholtz free energy can only decrease ; for a
spontaneous process, the attainment of equilibrium in the
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system coincides with the system having a minimum value
of A is corresponding to the fixed values of V and T
Consideration of A thus provides a criterion
of equilibrium for a closed system with fixed
composition at constant value V and T.
 Consider for example n atoms of some element occruing in
a crystalline phase and a vapor phase both contained in a
constant – volume vessel which is immersed in constant –
temperature heat reservoir. The point now is to determine
the equilibrium distribution of the n atoms between the sold
phase and the vapor phase.
At constant V and T, this distribution occurs at the
minimum value of A, and hence with low value of U and
high value of S since: A=U - TS
 The two extreme states of existence of this system are :
1. All n atoms are in the solid phase and none occur in
the vapor Phase.
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2. All n atoms are in the vapor phase and the solid
phase is absent.
 Starting with the system occurring in the first of these two
states, ί. e, the solid crystalline state ; the atom in such
case are held together by interatomic force ; thus , if an
atom to be removed from the crystal surface and placed in
the vapor phase ( the first atom is placed in vacuum ) ,
energy is absorbed as heat from the heat reservoir to the
system to break the interatomic bonds to increase the
internal energy , U , of the system, and its randomness ί.e,
the system entropy as shown in figure (5.1 ) which shows
the variation of internal energy and entropy with the
number of atoms in the vapor phase of the closed solid
vapor system at constant V and T .
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u
s
nv
Figure(5.1): the dependence of U and S on nv.
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 This figure shows that U increases linearly with nv while the
entropy increase is nonlinear.
 The saturated vapor pressure is calculated as :
P = [ nv ( eq , T ) kT ] / [ v-vs ]
Where V is the volume of the containing vessel, Vs is the
volume of the solid phase present, and nv ( eq , T ) is the
number of atoms in the vapor phase at the equilibrium
point which correspond to the minimum value of A as
shown in figure (5.2 ); this minimum value is obtained by
adding the values of U to the corresponding values of ( –
TS ) and thus having a curve that represents the variation
of A with nv.
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U
A=U-TS
-TS
Figure(5.2): the variation of U, TS and A.
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
As the magnitude of the entropy contribution to the value
of A, –TS, is temperature dependent and the internal
energy contribution is independent of temperature, the
entropy contribution becomes increasingly predominant as
temperature is increased and the compromise between U
and ( – TS ) which minimize A occurs at higher value of nv.
 This is illustrated in (5.3 ) which is drawn for T1 and T2
where T2 > T1
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U
_________________________________________
A=U-T1s
A=U-T2s
-T1s
-T2s
Figure(5.3): the variation of A with nv. at T1 and T2 where
T2>T1 .
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 The increase in temperature from T1 to T2 increases the
saturated vapor pressure:
from
to
p( T1 ) = nv ( eq,T1 ) kT1 / [ V–Vs( T1 ) ]
p( T2 )= nv ( eq,T2 ) k T2 [V–Vs(T2)]
The saturated vapor pressure increases exponentially with
increasing the temperature.
 For the constant – volume system, the maximum
temperature at which both solid and vapor phases are in
equilibrium occurs at nv ( eq,T ) = n; above this
temperature, the entropy contribution overwhelms the
internal energy contribution and hence all n atoms occur in
the gas phase.
 Conversely, as T decreases, then nv ( eq,T ) decreases
and, in the limit that T = 0K, the entropy contribution to A
vanishes and minimization of A coincides with minimization
of U , that is all n atoms occur in the solid phase .
 Now consider that the constant temperature heat reservoir
containing the constant volume system, is of constant
volume and is adiabatically contained, then the combined
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system, the particles containing system and the heat
reservoir, is one of constant U and V; accordingly the
combined system attains equilibrium at its maximum point
of entropy.
 If nv < nv ( eq ,T ) , the evaporating process occurs , this
process is accompanied by transfer of heat q , from the
heat reservoir to the particles containing system , thus the
entropy change of the combined system is given by :
Δ S combined system = ΔS heat reservoir +ΔS particles system
= – q/T + ( q/T + ΔSirr )
= ΔSirr
thus:
Δ A = – TΔSirr
Since minimization of A correspond to maximization of
entropy
 Also if nv > nv ( eq,T ) condensation will occur ; the entropy
change of the combined system , in this case , is given by :
Δ S combined system = ΔS heat reservoir + ΔS particles system
= +q/T + ( - q/T + ΔSirr )
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= ΔSirr
thus:
Δ A = – TΔSirr
Since at equilibrium:
thus:
ΔSirr = 0
ΔA = 0
 It should be noted that at, or near, equilibrium the
probability that nv deviates by even the smallest amount
from the value nv ( eq,T ) exceedingly small. This
probability is small enough that the practical terms it
corresponds
to
is
the
thermodynamic
statement
“spontaneous deviation of a system from its equilibrium
state is impossible".
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