Gibbs' gen. funct,residual - Copy ht.ppt

Gibbs' gen. Funct, residual

2. Useful forms for liquids or solids:
 dHCdTV TdP
dSC
dT
T
VdP
P
P
 
 
1

Eq 6.28
Eq 6.29

4.  dHCdTV TdP
dSC
dT
T
VdP
P
P
 
 
1


5.  
molJH /3400
10
)11000)(204.18()15.323)(10513(1
)15.29815.323(310.75
6




 
molJS /13.5
10
)11000)(204.18(10513
15.298
15.323
ln310.75
6





6. SdTVdPdG 
Gibbs energy G = G (P,T)
dT
RT
G
dG
RTRT
G
d
2
1






Thermodynamic property of great potential utility
as it can be directly measured and controlled
G = G (P,T). Both P and T can be directly measured
and controlled. G a property of interest and important.
G/RT = g (P,T)
The Gibbs energy serves as a generating function
for the other thermodynamic properties.
Eq 6.10
d(G/RT) = V/RT dP – H/RT
2

dT
It is more convenient to deal with a dimensionless for of 6.10.
Advantages of this equation: All terms are dimensionless. Moreover, in
contrast to 6.10, enthalpy rather than entropy appears on the right side.

7. dT
RT
H
dP
RT
V
RT
G
d
2






dT
T
RTG
dP
P
RTG
RT
G
d
PT





















 )/()/(
TP
RTG
RT
V









)/(
PT
RTG
T
RT
H









)/(
Eq 6.38
Eq 6.37
Eq 6.39
At contst T
At const P

8. Gibbs energy as generating function
The Gibbs energy when given as a function of P and T serves as
a generating function for the other thermodynamic properties.
G/RT = g (P,T)

9. In
 thermodynamics a residual property is defined as the
difference between a
 real gas property and an ideal
gas
 property, both considered at the
same
 pressure, temperature, and 
composition.
In
 thermodynamics, a 
departure function 
is defined for any
thermodynamic property as the difference between the property
as computed for an
 ideal gas and the property of the species as
it exists in the real world, for a specified temperature
 
T 
and
pressure
 
P. Common departure functions include those
for
 enthalpy, entropy, and internal energy.

10. Residual properties
•The definition for the generic residual property:
–M and M
ig
are the actual and ideal-gas properties,
respectively at the same T & P
–M is the molar value of any extensive thermodynamic
properties, e.g., V, U, H, S, or G.
•The residual Gibbs energy serves as a generating function for
the other residual properties:
igR
MMM 
dT
RT
H
dP
RT
V
RT
G
d
RRR
2










11. dT
RT
H
dP
RT
V
RT
G
d
RRR
2








 const T
dP
RT
V
RT
G
d
RR











P
RR
dP
RT
V
RT
G
0
P
RT
P
ZRT
VVV
igR



P
R
P
dP
Z
RT
G
0
)1(
PT
RTG
T
RT
H









)/(










P
P
R
P
dP
T
Z
T
RT
H
0










PP
P
R
P
dP
Z
P
dP
T
Z
T
R
S
00
)1(
RT
G
RT
H
R
S
RRR

const T
const T
Z = PV/RT: experimental measurement .Given PVT data or an appropriate equation
of state, we can evaluate H
R
and S
R
and hence all other residual properties.
Eqn 6.39
Eqn 6.49
Eqn 6.48
Eqn 6.3
Eqn 6.46

14. These equations provide a convenient base for real gas calculations

15.  
 
  
  
































































P
P
ig
P
P
H
ig
P
ig
P
P
ig
P
P
T
T
ig
P
igRig
P
dP
T
Z
RTTTDCBATTMCPHRH
P
dP
T
Z
RTTTCH
P
dP
T
Z
RTDCBATTICPHRH
P
dP
T
Z
RTdTCHHHH
0
2
000
0
2
00
0
2
00
0
2
0
),,,;,(
),,,;,(
0




























































































PP
P
ig
PP
P
S
ig
P
ig
PP
P
ig
PP
P
T
T
ig
P
igRig
P
dP
Z
P
dP
T
Z
TR
P
P
R
T
T
DCBATTMCPSRS
P
dP
Z
P
dP
T
Z
TR
P
P
R
T
T
CS
P
dP
Z
P
dP
T
Z
TR
P
P
RDCBATTICPSRS
P
dP
Z
P
dP
T
Z
TR
P
P
R
T
dT
CSSSS
00
00
00
00
00
0
00
0
00
00
0
0
)1(lnln),,,;,(
)1(lnln
)1(ln),,,;,(
)1(ln
0

16. Calculate the enthalpy and entropy of saturated isobutane vapor at 360 K from the
following information: (1) compressibility-factor for isobutane vapor; (2) the vapor
pressure of isobutane at 360 K is 15.41 bar; (3) at 300K and 1 bar, the ideal-gas heat
capacity of isobutane vapor:
molJH
ig
/18115
0

KmolJS
ig
 /976.295
0 TRC
ig
P
3
10037.337765.1/












P
P
R
P
dP
T
Z
T
RT
H
0 









PP
P
R
P
dP
Z
P
dP
T
Z
T
R
S
00
)1(
Graphical integration requires plots of and (Z-1)/P vs. P.P
T
Z
P
/







9493.0)1037.26)(360(
4
0











P
P
R
P
dP
T
Z
T
RT
H
6897.0)02596.0(9493.0)1(
00








 
PP
P
R
P
dP
Z
P
dP
T
Z
T
R
S
molJH
R
/3.2841
KmolJS
R
 /734.5
 
mol
J
HEICPHRHH
Rig
5.21598)0.0,0.0,3037.33,7765.1;360,300(
0 
 
Kmol
J
HREICPSRSS
Rig

 676.286
1
41.15
ln)0.0,0.0,3037.33,7765.1;360,300(
0

20. Residual properties by equation of state:
an alternative method•The calculation of residual properties for gases and vapors
through use of the virial equations and cubic equation of state.
•Z-1 = BP/RT
G
R
/RT = BP/RT
•If Z = f (P,T):










P
P
R
P
dP
T
Z
T
RT
H
0 









PP
P
R
P
dP
Z
P
dP
T
Z
T
R
S
00
)1(


P
R
P
dP
Z
RT
G
0
)1(
Eq 3-38
Eq 6-49
Eq 6-54
Eq 6-46
Eq 6-48

21. Equation of State
•Equation of state is a relation between state
variables. More specifically, an equation of state
is a thermodynamic equation describing the state
of matter under a given set of physical
conditions.
•A mathematical expression yielding PVT
relations.
•Equations of state are useful in describing the
properties of fluids, mixtures of fluids and solids.

22. Different kinds of Equations of State
• Major equation of state
–Classical Ideal Law
•Cubic equations of state
–Van der Waals equation of state
–Redlich-Kwong equation of state
–Soave modification of Redlich-Kwong
–Pent-Robinson equation of state
–Elliot, Suresh, Donohue equation of state
•Non-cubic equations of state
–Dieterici equation of state
•Virial equation of state

24. 1
( )( )
Z
Z q
Z Z

 
 

 
 
Redlich-Kwong equation of state

25. Residual Properties from the Virial
Equation of State
RT
BP
Z1


P
R
P
dP
Z
RT
G
0
)1(
two-term virial equation
RT
BP
RT
G
R











P
P
R
P
dP
T
Z
T
RT
H
0 










P
P
R
P
dP
RT
BP
T
T
RT
H
0
)1(
 




















P
P
R
P
dP
T
B
dT
dB
TR
P
T
RT
H
0
2
1
RT
BP
Z1







dT
dB
T
B
R
P
RT
H
R
RT
G
RT
H
R
S
RRR

R
S P dB
R R dT

Eq 6.49 Eq 6.54
Eq 6.46
Eq 6.56
Eq 6.55

26. 2
1 CBZ 


P
R
P
dP
Z
RT
G
0
)1(
Three-term virial equation
ZCB
RT
G
R
ln
2
3
2
2
 










P
P
R
P
dP
T
Z
T
RT
H
0



















2
2
1

dT
dC
T
C
dT
dB
T
B
T
RT
H
R
RT
G
RT
H
R
S
RRR

RTZP
densityoftermsIn
ZRTPV


,
Application up to moderate pressure and required data for second
and third virial coefficients
Eq. 6.49
Eq.3.40
Eq. 6.61
Eq. 6.62
Eq. 6.46

27. Using the cubic equation of state
•The generic cubic
equation of state:










P
P
R
P
dP
T
Z
T
RT
H
0
))((
)(
bVbV
Ta
bV
RT
P




)1)(1(1
1
bb
b
q
b
Z


 



bRT
Ta
q
)(

qIb
d
Z 
)1ln()1(
0



I
dT
dqd
T
Z












0
Tconst
bb
bd
I





0 )1)(1(
)(
qI
Td
Td
Z
RT
H
r
r
R






 1
ln
)(ln
1

 qI
Td
Td
Z
R
S
r
r
R






 1
ln
)(ln
ln

 RT
bP

in terms of
density










PP
P
R
P
dP
Z
P
dP
T
Z
T
R
S
00
)1(
Textbook – Refer to page 218
Eq 3.42
Eq 3.51Eq 6.46
Eq 6.48

28. Find values for the residual enthalpy H
R
and the residual entropy S
R

for n-butane gas at 500 K and 50 bar as given by Redlich/Kwong
equation.
176.1
1.425
500

rT 317.1
96.37
50

rP 09703.0
r
r
T
P
 8689.3
)(




r
r
T
T
q

))((
1






ZZ
Z
qZ
0
1
685.0Z
qI
Td
Td
Z
RT
H
r
r
R






 1
ln
)(ln
1

 qI
Td
Td
Z
R
S
r
r
R






 1
ln
)(ln
ln


Tconst
bb
bd
I





0 )1)(1(
)(
13247.0ln
1















 Z
Z
I
mol
J
H
R
4505
Kmol
J
S
R

546.6
Use Table 3.1 for data
Eq. 3.53 Eq. 3.54
Eq. 3.52
Eq. 6.65
Eq.6.67
Eq. 6.68

30. Sample Problem
6.14 page 242. Calculate the Z, H
R
, and S
R
by the
Redlich/Kwong equation for one of the following
and compare results with values from suitable
generalized correlations :

31. Two-phase systems
•Whenever a phase transition at constant temperature and
pressure occurs,
–The molar or specific volume, internal energy, enthalpy, and
entropy changes abruptly. The exception is the molar or
specific Gibbs energy, which for a pure species does not
change during a phase transition.
–For two phases α and β of a pure species coexisting at
equilibrium:

GG
where G
α
and G
β
are the molar or specific Gibbs energies of the individual phases
SdTVdPdG 

32. 
GG

dGdG
SdTVdPdG 
dTSdPVdTSdPV
satsat 





V
S
VV
SS
dT
dP
sat






VdPTdSdH 

STH 
The latent heat of phase transition


VT
H
dT
dP
sat



The Clapeyron equation
Ideal gas, and V
l
<< V
v
lv
sat
v
l
Z
P
RT
VV 

)/1(
ln
Td
Pd
RH
sat
lv

The Clausius/Clapeyron equation

33. 

VT
H
dT
dP
sat



The Clapeyron equation: a vital connection between the properties of different phases.
)(TfP
T
B
AP
sat
ln
For the entire temperature range from the triple point
to the critical point
CT
B
AP
sat

ln The Antoine equation, for a specific temperature range





1
ln
635.1
DCBA
P
sat
r
The Wagner equation, over a wide
temperature range.
rT1
Temperature dependence on vapour pressure of liquids