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Chapter 4
FUGACITY
Fundamental equations for closed system consisting of n moles:
nVdPnSdTnUd (1)
dPnVnSdTnHd (2)
nVdPdTnSnAd (3)
dPnVdTnSnGd (4)
HOMOGENEOUS OPEN SYSTEM
An open system can exchange matter as well as energy with its surroundings.
For a closed homogeneous system, we consider U to be a function only of S and V:
U = U(S, V) (5)
In an open system, there are additional independent variables, i.e., the mole numbers of the various components present.
nU = nU(S,V, n1, n2, ....., nm) (6)
where m is the number of components.
The total differential of eq. (6) is
ii
n,V,Sin,Sn,V
dnnnU
dVV
nUdS
SnU
nUdijii
(7)
Where subscript ni refers to all mole numbers and subscript nj to all mole numbers other than the ith. Chemical potential is defined as:
ijn,V,Sii n
nU
(8)
We may rewrite eq. (7) as
i
iidnnVdPnSdTnUd (9)
For a system comprising of 1 mole, n = 1 and ni = xi
i
iidxdVPdSTdU (10)
Eqs. (9) and (10) are the fundamental equations for an open system corresponding to eq. (1) for a closed system.
Using similar derivations, we can get the following relations:
i
iidndPnVnSdTnHd (11)
(12)
(13)
i
iidnnVdPdTnSnAd
i
idndPnVdTnSnGd
It follows that:
jjjj n,P,Tin,V,Tin,P,Sin,V,Sii n
nGnnA
nnH
nnU
(14)
(15)
(9)
(16)
(17)
(18)
(21)
(19)
(20)
Equations (19 – 21) can be written as
nSdnSdnSdnSd 321 (22)
nVdnVdnVdnVd 321 (23)
i3
i2
i1
i dndndndn (24)
Substituting eqs. (22 – 24) into eq. (18) gives:
nSdnSdnSdTnUd 321
nVdnVdnVdP 321
i
i3
i2
i1
i dndndn
i
2i
2i
2222 dnnVdPnSdT
i
ii dnnVdPnSdT
i
3i
3i
3333 dnnVdPnSdT
• All variations d(nS)(2), d(nV)(2), dn1(2), dn2
(2), etc., are truly independent.
• Therefore, at equilibrium in the closed system where d(nU) = 0, it follows that
0nSnU
2 12 TT
0nSnU
3 13 TT
0nSnU
1TT
TTT 21
0nVnU
2 12 PP
0nVnU
3 13 PP
0nV
nU
1PP
PPP 21
0
nnU
21
1
12
1
0
nnU
1
0
nnU
31
1
13
1
111
12
11
1
0
nnU
22
1
22
2
0
nnU
2
0
nnU
32
1
23
2
122
22
21
2
0
nnU
2m
1
m2
m
0
nnU
m
0
nnU
3m
1
m3
m
1mm
m2
m1
m
(25)
(26)
(27)
(28)
(29)
Thus, at equilibrium
TTT 21
PPP 21
12
11
1
22
21
2
m2
m1
m
(30)
jn,P,Tii n
nG
jn,P,Tii n
nMM
Eq. (14):
Eq. (30):
ii G (31)
Important relations for partial molar properties are:
i
iiMxM (32)
and Gibbs-Duhem equation:
0MxdTTM
dPPM
iii
x,Px,T
(33)
(34)
(35)
(36)
GIBB’S THEOREM
(37)
(38)
(39)
(32):
Equation (20) of Chapter 3:
dPTV
TdT
CdSP
P
(3.20)
For ideal gas:
dPT
VT
dTCdS
P
igiig
Pigi i
(40)P
dPR
TdT
CdS igP
igi i
For a constant T process
TdT
CdS igP
igi i (constant T)
P
p
P,TS
p,TS
igi
i
igi
iigi
TdT
RdS (constant T)
iii
iigi
igi ylnR
PyP
lnRpP
lnRp,TSP,TS
iigii
igi ylnRP,TSp,TS
According to eq. (37):
iigi
igi p,TSP,TS
whence
iigi
igi ylnRP,TSP,TS
iigi
igi ylnRSS
By the summability relation, eq. (32):
i
iigii
i
igii
igi ylnRSySyS
Or:
i
iii
igii
igi ylnyRSyS
(41)
This equation is rearranged as
i
iii
igii
igi ylnyRSyS
i i
ii
igii
igi y
1lnyRSyS
the left side is the entropy change of mixing for ideal gases.
Since 1/yi >1, this quantity is always positive, in agree-ment with the second law.
The mixing process is inherently irreversible, and for ideal gases mixing at constant T and P is not accompanied by heat transfer.
(42)
Gibbs energy for an ideal gas mixture: igigig TSHG
Partial Gibbs energy :
igi
igi
igi STHG
In combination with eqs. (38) and (41) this becomes
iigii
igi
igi
igi ylnRTGylnRTTSHG
iigi
igi
igi ylnRTGG
or:
(43)
An alternative expression for the chemical potential can be derived from eq. (2.4):
dPVdTSdG igi
igi
igi
At constant temperature:
(2.4)
PdP
RTdPVdG igi
igi (constant T)
Integration gives:
PlnRTTG iigi (44)
Combining eqs. (43) and (44) results in:
PylnRTT iiigi (45)
Fugacity for Pure SpeciesThe origin of the fugacity concept resides in eq. (44), valid only for pure species i in the ideal-gas state.
For a real fluid, we write an analogous equation:
iii flnRTTG (46)
where fi is fugacity of pure species i.
Subtraction of eq. (44) from Eq. (46), both written for the same T and P, gives:
Pf
lnRTGG iigii (47)
Combining eqs. (3.41) with (47) gives:
Pf
lnRTG iRi
The dimensionless ratio fi/P is another new property, the fugacity coefficient, given the symbol i:
Pfi
i
(48)
(49)
Equation (48) can be written as
iRi lnRTG (50)
The definition of fugacity is completed by setting the ideal-gas-state fugacity of pure species i equal to its pressure:
Pf igi (51)
Equation (3.50):
P
0i
Ri
PdP
1ZRTG
(constant T) (3.50)
Combining eqs. (50) and (3.50) results in:
P
0ii P
dP1Zln (constant T) (51)
Fugacity coefficients (and therefore fugacities) for pure gases are evaluated by this equation from PVT data or from a volume-explicit equation of state.
An example of volume-explicit equation of state is the 2-term virial equation:
RTPB
1Z ii
RTPB
1Z ii
P
0
ii dP
RTB
ln (constant T)
Because the second virial coefficient Bi is a function of temperature only for a pure species,
RTPB
ln ii (constant T) (52)
FUGACITY COEFFICIENT DERIVED FROM VOLUME-EXPLICIT EQUATION OF STATE
Use equation (3.63):
ii
ii
i
i
i
iiii
Ri
bVbV
lnRTb
aV
bVZln1Z
RTG
Combining eqs. (3.63) and (50) gives:
ii
ii
i
i
i
iiiii bV
bVln
RTba
VbV
Zln1Zln
(53)
VAPOR/LIQUID EQULIBRIUM FOR PURE SPECIES
Eq. (46) for species i as a saturated vapor:
Vii
Vi flnRTTG
(54)
For saturated liquid:
Lii
Li flnRTTG (55)
By difference:
Li
ViL
iVi f
flnRTGG
Phase transition from vapor to liquid phase occurs at constant T dan P (Pi
sat). According to eq. (4):
d(nG) = 0
Since the number of moles n is constant, dG = 0, therefore :
0GG Li
Vi
Therefore:
sati
Li
Vi fff
(56)
(57)
For a pure species, coexisting liquid and vapor phases are in equilibrium when they have the same temperature,
pressure, and fugacity
An alternative formulation is based on the corresponding fugacity coefficients
sati
satisat
i Pf
whence:
(58)
sati
Li
Vi (59)
FUGACITY OF PURE LIQUID
The fugacity of pure species i as a compressed liquid is calculated in two steps:
1. The fugacity coefficient of saturated vapor is determined from Eq. (53), evaluated at P = Pi
sat and Vi = Vi
sat. The fugacity is calculated using eq. (49).
ii
ii
i
i
i
iiiii bV
bVln
RTba
VbV
Zln1Zln
(53)
Pf ii (49)
2. the calculation of the fugacity change resulting from the pressure increase, Pi
sat to P, that changes the state from saturated liquid to compressed liquid.
An isothermal change of pressure, eq. (3.4) is integrated to give:
P
Pi
satii
sati
dPVGG (60)
iii flnRTTG
According to eq. (46):
satii
sati flnRTTG
( – )
sati
isatii f
flnRTGG (61)
Eq. (60) = Eq. (61):
P
Pisat
i
i
sati
dPVff
ln
Since Vi, the liquid-phase molar volume, is a very weak function of P at T << Tc, an excellent approximation is often obtained when Vi is assumed constant at the value for saturated liquid, Vi
L:
satiisat
i
i PPVff
ln
RT
PPVexpfactorPoynting
ff sat
iisat
i
i (62)
Remembering that:
sati
sati
sati Pf
The fugacity of a pure liquid is:
RT
PPVexpPf
satiisat
isatii (63)
Phase RuleFirst law of thermodynamics:
WQdU
thermodynamic properties (U, T, P, V) reflect the internal state or the thermodynamic state of the system.
Heat and work quantities, are not properties; they account for the energy changes that occur in the surroundings and appear only when changes occur in a system.
They depend on the nature of the process causing the change, and are associated with areas rather than points on a graph.
P1
P2
P
V V2V1
(P1, V1)
(P2, V2)
W
• The intensive state of a PVT system containing N chemical species and phases in equilibrium is characterized by the intensive variables, temperature T, pressure P, and N – 1 mole fractions for each phase.
• These are the phase-rule variables, and their number is 2 + (N – 1)().
• The masses of the phases are not phase-rule variables, because they have no influence on the intensive state of the system.
• An independent phase-equilibrium equation may be written connecting intensive variables for each of the N species for each pair of phases present.
• Thus, the number of independent phase-equilibrium equations is ( – 1)(N).
• The difference between the number of phase-rule variables and the number of independent equations connecting them is the number of variables that may be independently fixed.
Called the degrees of freedom of the system F:
F = 2 + (N – 1)() – ( – 1)(N)
F = 2 – + N (60)
• The intensive state of a system at equilibrium is established when its temperature, pressure, and the compositions of all phases are fixed. • These are therefore phase-rule variables, but they are
not all independent. • The phase rule gives the number of variables from this
set which must be arbitrarily specified to fix all remaining phase-rule variables number of degrees of freedom (F).
PURE HOMOGENEOUS FLUID
T, P, V
N = 1
= 1
F = 2 – + N
= 2 – 1 + 1 = 2
• The state of a pure homogeneous fluid is fixed whenever two intensive thermodynamic properties are set at definite values
N = 1 = 2F = 2 – + N = 2 – 2 + 1 = 1
VAPOR/LIQUID EQULIBRIUM FOR PURE SPECIES
TV, PV, VV
TL, PL, VL
The state of the system is fixed when only a single property is specified. For example, a mixture of steam and liquid water in equilibrium at 101.325 kPa can exist only at 373.15 K (100°C). It is impossible to change the temperature without also changing the pressure if vaporand liquid are to continue to exist in equilibrium
Intensive variables: TV, PV, VV, V, TL, PL, VL, L = 8
Independent equations:
TV = TL = T
PV = PL = P
V = L
VV = f(T, P) ……. biggest root of the eos
V = f(T, P, VV)
VL = f(T, P) ………. smallest root of the eos
L = f(T, P, VL)
Here we have 7 equations with 8 unkowns. It means that we must define one intensive variable (T or P)
Algorithm:
1. Input: T
2. Assume P
3. Calculate ZV and ZL (cubic equation)
4. Calculate VV and ZL
5. Calculate V (eq. 53 with V = VV)
6. Calculate L (eq. 53 with V = VL)
7. Calculate Ratio = V/ L
8. If Ratio 1, assume new value of P
9. Go to step no. 3