Download Chapter 5 auxiliary functions

Chapter 5
auxiliary functions
5.1
Introduction
•The power of thermodynamics lies in its provision of the 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 practicality of the equations of state of the
system ί.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:
Or
du = Tds - pdVˇ
u = f ( s , Vˇ )
This equation of state gives the relationship between
the dependent variables u and the independent
variables s and v for a closed system of fixed
composition which is in states of equilibrium and is
undergoing a process involving volume change
against external pressure as the only form of work
performed in or by the system (p – v work)
* the combination of the first and the second
laws of thermodynamic provides also the following
criteria for the equilibrium :
1. For closed system of constant energy and
constant volume the entropy is a maximum.
2. for closed system of constant entropy and
volume the energy a minimum
* Since the entropy is an inconvenient choice of
independent variable from the point of view of
experimental measurement of control; it is desirable to
develop a simple equation similar to the previous one
which contains a more convenient choice of
independent variable.
* 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.
* from theoretical point of view the most convenient
pair of independent variable would be volume and
temperature when they are fixed fore closed system
the Eίs level values ; and hence the Boltzman , factor (
exp (- Eί / kT ) and the partition function are fixed this
will ease the theoritical calculations using the
methods of statistical mechanism
* 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
he enthalpy function H is defined as:
H = υ + pv
Thus :
dH = dυ + pdv+ vdp
= Tds – pdv + vdp + pdv
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 state function, we have:
Δ H = H2 – H1 = ( υ2 + p2v2 ) – (u1+ p1v1 )
Thus :under constant pressure:
Δ H = ( υ2 – υ1 ) + p ( v2 – v1 )
= Δ υ + Wp-v = qp
* Therefore, the equation of state, dH = Tds +vdp ,
gives the relationship between the dependent
variable H and independent variables S and P for a
closed system of fixed composition whish is in
states of equilibrium and is undergoing a process
involving volume change against external pressure
as the only form of work performed on , or by , the
system ; the enthalpy change of the system equals
the heat leaving or entering the system
.
5.3
Helmholtz Free Energy Function A
* Helmholtz Free Energy Function A is defined as :
A = υ – Ts
Thuse :
dA = dυ – 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 A = A2 – A1 = (U2 – T2S2) – (U1 – T1S1)
= (U2 – U1) – (T2S2 –T1S1)
. ί . e,
Thus
Δ A = q + w – ( T2S2 – T1S1)
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 . if the apposite convention is used . ί . e, the
work done by the system is assigned a positive value ,
the previous equation become 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 tha maximum amount of work done by system
equal the decrease in the work function
* Since =wmax–w deg; thus w = –Δ A–TΔsirr= – (ΔA + T
Δsirr ) Thus for process which occurs at constant V and
T , we can write :
Δ A + T Δ sirr = 0
For an infinitesrnal increment of such on process : SA +
TdS = 0
* For spontaneous processes dsirr is a positive value
, thus processes occur at constant T and V will be
spontaneous if d A is negative value ; ί . e, for
spontaneous processes that occur at constant T and
V : dA < 0
* since the condition for thermodynamic equilibrium
is that dsisr = 0 then with respect to the described
process . ί . e , at constant V and T equilibrium is
defended by the condition that
dA = 0
* Thus in closed system , held at constant T and V
Helmholtz free energy can only decrease , for
spontaneous process or remain constant , at
equilibrium ; the attainment of equilibrium in the
system coincides with the system having a minimum
value of A constant with 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 m n atoms of seme element
accusing in both asides crystalline phase and a
vapor phase contained in a constant – volume
vessel which is immersed in constant – temperature
heat reservoir . the point now is to determine the
equilibrium 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 values with
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 sold phase and none occur
in the vapor phase .
2- all n atoms are in the vapor phase and the solid
phase is absent .
* starting with 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 vapor phase ( the first
atom is placed in vacuum) , energy is a boarded as
heat from the heat reservoir to the system to break
the interatomic bonds to increase the internal
energy , υ , of the system and its randomness ί. e ,
the system entropy as shown in figure ( figure 5.1
page 97 ) which chins the variation of internal
energy and entropy with the number of atom in in
the vapor phase of the closed solid vapor system at
constant V and T .
*this figure shows that U increase linearly with nυ
while the entropy increase is nonlinear figure (1a)
the note of deincrease of S with nυ increase .
* The saturated vapor pressure is calculated as :
P = [ nv( q,T) kT ] ÷ [ V – Vs ]
Where V is the volume ob the containing vessel, Vs is
the volume of solid phase present, and nυ ( 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 (2) (figure 5.2 page 98 ) 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 nυ
.
* As the magnitude of the entropy contribution the
value of A , –Ts , is temperature dependent and the
internal energy contribution is independent of
temperature , the entropy contribution becomes
incuriously predominate as temperature is increased
and the compromise between U and (– Ts) which
minimize A occurs at layer of nυ .
* This is illustrated in figure ( figure 5.3 page 99 )
which is draw for T1 and T2 where T1 < T2
* As increase in the temperature from T1 to T2
increase the saturated vapor pressure from p(T1) =
nv (eq.T1) le T1 [v–vs(T1)] to p(T2) = nv (eq.T2) le T2
[v– vs(T2)] the saturated vapor pressure increase
exponentially with increasing the temperature .
* for the constant – volume system , the maximum
temperature at which both solid vapor phase occur in
the temperature at which minimization of A occurs at
nυ (eq.T) = n, above this temperature , the entropy
contribution overwhelms the enternal energy
contribution and hence all n atomsoccur in the solid
phase
* Conversely, as T decrease, then nυ (eq.T) decrease
and, in the limit that T = ok, the entropy contribution to
AS vanishes and minimization of A coincides with
minimization o U , that is all n atoms occur in the solid
phase .
* Now considers that the constant temperature heat
reservoir containing the constant-volume system, is of
constant-volume and is adiabatically continued , then
the cornbraid system , the particle containing system
and heat reservoir , is one o constant U and V ;
accordingly the combined system attains equilibrium at
its maximum point of entropy .
* if nυ > nυ (eq.T) , the evaporating process stimulating
occur , this process is accompanied by transfer of heat
q , from the heat reservoir to the particles containing
system , thus the entropy chang of the combined
system is giving by :
Δ s combined system = Δs heat reservoir +Δs particles system
= – q/T + ( q/T + Δsirr )
= Δsirr
Thus : Δ A = – TΔsirr
Hence minimization of A correspond to maximization of
entropy
* also if nυ > nυ (eq.T) condensation will occur ; the
entropy change the combined system , in this case , is
given by :
Δ s combined system = Δs heat reservoir +Δs particles system
= q/T + (– q/T + Δsirr )
= Δsirr
Thus : Δ A = – TΔsirr
AT equilibrium Δsirr = 0 thus ΔA = 0
* It should be point that at , or near , equilibrium the
probability that nυ deviates bu even the smallest
amount from the value nυ (eq.T) exceedingly small .
this probability is small enough that the practical
terms it corresponds to the thermodynamic
statement that spontaneous deviation of asystem
from its equilibrium state is impossible .
5.4
The Gibbs Free Energy G
The Gibbs Free Energy G is defined as :
G = H – TS
Thus G = U + PV – TS
And dG = du + pdv + vdp – Tds – sdT
= Tds – pdv + pdv + vdp – Tds – sdT
= – sdT = Vdp =ƒ ( p , T )
* the dependent variable in this case G and the
pair of independent variable are the pressure
and temperature .
*since the Gibbs Free energy function is state
property :
Thus ΔG = G2 – G1 = ( H2 – T2S2 ) – ( H1 –
T1S1 )
= ( υ 2 + P2V2 – T2S2 ) – ( υ1 + P1V1 – T1S1 )
= ( υ2 – υ1 ) + ( P2V2 – P1V1 ) – ( T2S2 – T1S1 )
If the process is carried out under constant •
temperature and pressure .
Thus ΔG = Δ υ + P(V2 –V1) – T(S2 –S1)
= q + w + P(V2 –V1) – T(S2 –S1)
Since w = wp –v +wَ where wp –v is the work
carried out due to volume change and wَ is the
sum of all forms of work other than the p – v work :
ΔG = q + wp –v + wَ + P(V2 –V1) – T(S2 –S1)
= q – P(V2 –V1) + wَ + P(V2 –V1) – T(S2 –S1)
= q + wَ – T(S2 –S1)
* in describing the work in the pervious equation ,
appositive sign is assigned to work done on the
system and negative sign is assigned to work done by
the system IF the opposite convention is used ί . e. the
work done by the system is assigned appositive value ,
the pervious equation because :
ΔG = q + wَ – T(S2 –S1)
* From the second law of thermodynamics :
q ≤ T(S2 –S1)
≤ – ΔGَw
therefore for reversible processes that occur at
constant temperature and pressure ; the maximum
amount of work , other than the p – v work is given by
equation :
max = – ΔGَw
* again the pervious inequality can b written as ;
= – (ΔG + T Δsirr )َw
* in the case of an isothermal , isobaric process during
which no work other thann the p – v work is done , that
= 0 , َis w
then :
ΔG + TΔsirr = 0
For infinitesimal changes , the pervious equation
because :
dG + Tdsirr = 0
such process can only spontaneously If The
Gibbs Free Energy decrease since in any
spontaneously process dsirr > 0
* As the condition for the thermodynamic
equilibrium is that d sirr = 0 then with respect
to isothermal and isobanic processes ,
equilibrium is defined by the coordination that :
dG=0
* thus for a closed system under going process at
constant T and P The Gibbs Free Energy can only
decrease or remain constant
, and hence the
attainment of equilibrium in the system coincides
with the system having the minimum value of G
constant with the fixed value of P and T .
5.5
Function Thermodynamic
equation For a closed system
* The Function Thermodynamic equation are foni
Thermodynamic equation of states that related
thermodynamic state function as dependent variable
to pair of independent thermodynamic state function
for closed system of fixed composition which is in
statue of equilibrium and the under
going a process involving a change in the two
independent variable . these equation are
:
du = Tds – pdv
thus , U = ƒ ( S , V )
dH = Tds + vdp
thus , H = ƒ ( S , P )
dA = – sdT – pdv thus , A = ƒ ( T , V )
dG = – sdT + vdp thus , G = ƒ ( T , V )
* applying the partial differentiation principles , the
following relationship are obtained :
1- T = ( ∂ u \ ∂ s )υ = ( ∂ H \ ∂ s )p
p = ( ∂ u \ ∂ v )s = ( ∂ A \ ∂ v )T
v = ( ∂ H \ ∂ p )s = ( ∂ G \ ∂ p )T
T = ( ∂ A \ ∂ T )v = ( ∂ G \ ∂ T )p
2 - ( ∂ u \ ∂ s )υ ( ∂ s \ ∂ v )u ( ∂ v \ ∂ u )s = –1
( ∂ H \ ∂ s )p ( ∂ s \ ∂ p) H ( ∂ p \ ∂ H )s = –1
( ∂ A \ ∂ T )υ ( ∂ T \ ∂ A )v ( ∂ T \ ∂ v )A = –1
( ∂ G \ ∂ T )p ( ∂ T \ ∂ p )G ( ∂ p \ ∂ G )T = –1
3- Maxwell's Relations:
( ∂ T \ ∂ v )s = ( ∂ p \ ∂ s )υ
( ∂ T \ ∂ P )s = ( ∂ v \ ∂ s )p
( ∂ s \ ∂ v )T = ( ∂ p \ ∂ T )υ
( ∂ s \ ∂ p )T = ( ∂ v \ ∂ T )p
5.6
examples of the use of
thermodynamic relations
1- Equation of State Relating the Eternal Energy of a
closed one-component system to the experimentally
measurable Quantities T , P and v .
Since
du = Tds – pdv
Thus
( ∂ u \ ∂ v )T = T ( ∂ s \ ∂ v )T – P
Use the Maxwell's relation : ( ∂ s \ ∂ v )T = ( ∂ p \ ∂ T ) v
( ∂ u \ ∂ T )t = T( ∂ p \ ∂ T ) v – P
It can be shown that ideal gases which obeys the
equation of state pv = nRt , ( ∂ u \ ∂ v )T = 0 ; ί.e the
enternal energy of the as volume .
2- Equation of State Relating the Eternal Energy of a
closed one-component system to the experimentally
measurable Quantities T , P and .
since : dH = Tds + Vdp
thus : ( ∂ H \ ∂ P )T = T ( ∂ s \ ∂ p )T + V
Use the Maxwell's relation : ( ∂ s \ ∂ p )T = – ( ∂ v \ ∂ T )p
yields
( ∂ H \ ∂ P ) T = – T ( ∂ v \ ∂ T )p + V
If again the system is a fixed quantity o an ideal gas ,
the equation of statue indicates that the enthalpy o an
ideal as is independent of its pressure .
3. The Gribs-Helmholz Equation :
Since : G = H – Ts
And : ( ∂ G \ ∂ T )p = – S
Thus : G = H + T ( ∂ G \ ∂ T )p
Therefore for closed system of fixed composition
undergoing processes at constant pressure :
G = H + T ( dG / dT )
Thus : GdT = HdT + TdG
Or
: TdG – HdT = – HdT
Then : (TdG – HdT) / T2 = – HdT / T2
Or
: d ( G / T ) = – ( H / T2 ) dT
Thus : [ d ( G / T ) ]/dT = – H / T2
Also we can write :
[ d ( ΔG / T ) ]/dT = –Δ H / T2
Similarly : [ d ( A / T ) ]/dT = – U / T2
And
: [ d (ΔA / T ) ]/dT = – Δ U / T2
The preview two equation are applicable to closed
system of fixed composition under going process at
constant volume
4. The relationship between cp and cv
The difference between cp and cv is given by :
cp – cv = ( ∂ H \ ∂ T )p – ( ∂ u \ ∂ T ) υ
= [ ∂ ( u + pv ) \ ∂ T ]p – ( ∂ u \ ∂ T ) υ
=(∂u\∂T)p+p(∂v\∂T)p–(∂u\
∂T)υ
But
: U=ƒ( υ,T)
Thus
: du = ( ∂ u \ ∂ v )T dv + ( ∂ u \ ∂ T )υ
dT
There : ( ∂ u \ ∂ T ) p = ( ∂ u \ ∂ v )T ( ∂ v \ ∂ T )
p + ( ∂ u \ ∂ T )υ
Hence : cp – cv = ( ∂ u \ ∂ T ) υ + ( ∂ u \ ∂ v )T (
∂v\∂T)p
+ p ( ∂ v \ ∂ T ) p – ( ∂ u \ ∂ T )υ
= ( ∂ v \ ∂ T ) p [ p + ( ∂ u \ ∂ v )T ]
= ( ∂ v \ ∂ T ) p [( ∂ u \ ∂ v )T + ( ∂ u \ ∂ v )T ]
= T( ∂ v \ ∂ T ) p ( ∂ s \ ∂ v )T
The T Maxwell's relation: ( ∂ s \ ∂ v )T = ( ∂ p \ ∂ T ) υ
leads to :
Cp – Cυ = ( ∂ v \ ∂ T ) p ( ∂ p \ ∂ T ) υ
Since
: ( ∂ p \ ∂ T )υ ( ∂ T \ ∂ v ) p ( ∂ p \ ∂ T ) υ = – 1
Then
: ( ∂ p \ ∂ T ) υ = – ( ∂ v \ ∂ T ) p ( ∂ p \ ∂ v )T
Thus
: cp – cv = – T ( ∂ v \ ∂ T ) p ( ∂ v \ ∂ T ) p ( ∂ p
\ ∂ v )T
Since
:
α = 1/v ( ∂ v \ ∂ T ) p
And
:
β = – 1/v ( ∂ v \ ∂ p ) T
Thus
:
cp – cv = ( V T α2 ) / β ί.e. cp / cv > 1
Where α is the coefficient of thermal expensive at
constant pressure , and β is the compreslility factor at
constant temperature .
5.7
The Gibbs Free Energy and the
composition of the system
* thus for the discussion has been restricted to
closed system of fixed composition ; as an example
, the system containing fixed number of moles of
component , etc . in such cases the system has to
independent variable which when fixed uniquely fix
the state of the problem .
* How ever , if the composition of the system varies
during the process for example , if the system
contains the gaseous co, co2 and O2 , then at
constant T and P minimization of G would occur
when the equilibrium o the reaction Co + 1/2 O2 is
established .
* Thus , G is a function of T, P and the number of
moles of all species present in the system i,e .
G = G ( T , P , and n1, n2 , n3 , …. , nj , …. nr )
When n1, n2 , …. Are the number of moles o species
1, 2 , …..
* Differentiation of the pervious equation yields :
dG = (∂G /∂T)p dT + (∂G /∂p)T dp
+ ∑ ί =1 ί=r (∂G /∂nί) t.p all nj=1.2. dni
= – SdT + Vdp +
∑ ί =1 ί=k ί (∂G /∂nί) t.p all nj=1.2. dni
Where
∑ ί =1 ί=r (∂G /∂nί) t.p all nj=1.2. dni
Is the sum of r terms , one of the ί species .
5.8
The chemical potential
* the term (∂G /∂nί) t.p all nj=1.2...nr dni is called
chemical potential of spices ί and is designed as μί
that is :
μί = (∂G /∂nί) t.p all nj=1.2.r..exp..nr dni
* thus μί is defined as the increase in Geibbs free
energy u a homogeneous phase per an infinitesimal
addition of the species ί with the addition being mole
at constant T, p and the system doesn't change
measurably in composition .
* alternatively , if the system is very large that the
addition of one mole of species ί at constant T and P and
the system doesn't change measurably in composition ,
then μί increase in G for the system accompanying the
adition of mole spices ί
* Thus , μί is the amount by which the capacity of the
system for closing work , other than the p-v work , is
increased due to the addition of infinitesimal addition of
species ί at constant T and P per one male of ί since dG
= – swَ
* The pervious fundamental equation can be written as :
dG = – SdT + Vdp + ∑ ί =1 ί=r μί , dni
similarly :
* μί = (∂U /∂nί) s,v, all nj=1.2.r..exp..ni
= (∂H /∂nί) s,p, all nj=1.2.r..exp..ni
= (∂U /∂nί) s,v, all nj=1.2.r..exp..ni
therefore :
du = Tds – pdV + ∑ ί =1 ί=r μί dni
dH = Tds + Vdp + ∑ ί =1 ί=r μί dni
dA = – Tds – pdV + ∑ ί =1 ί=r μί dni
dG = – Tds + Vdp + ∑ ί =1 ί=r μί dni
* the process fundamental equation can thus be
applied to opened systems which exchange matter
as well as heat with there surrounding and to closed
system which undergoes composition changes
we can also conclude that , for aclosed system •
which undergoes a process involving reversible
change of composition e,g a reversible chemical
reaction .
Sq = Tds
Sw = – pdυ + ∑ μί dni
Thus : the term ∑ μί dni in the chemical work done by
, and the total َthe system which was denoted w
work w is the sum of the p-v work and the chemical
work .