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Solution
ThermodynamicsProf. Srinivas Krishnaswamy
Department of Chemical Engineering
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Chemical EngineeringThermodynamics (CHE C311)2011 - 2012
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Thermodynamic Potentials (U, H, A, G)
Fundamental Relations (4)
Partial Derivatives and associated relations
Maxwell relations Departure functions or Residual properties
General expressions for du, dh, ds
Volume Expansivity and Isothermal Compressibility
General expressions for heat capacities
The Clausius Clapeyron equation
Estimating departure functions
Looking Back!!!!!
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Revisiting Gibbs free energy
Gibbs free energy as a generating function
Chemical potential
Fugacity of pure species
Estimating Fugacity of pure species
Estimating Fugacity of component in a mixture
Mixing rules
Concept of partial molar property
Estimating partial molar properties Excess properties
Activity coefficient models
What next??
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Gibbs Free Energy as aGenerating function
An alternative form of a fundamental property relation asdefined in dimensionless terms:
The Gibbs energy when given as a function of T and Ptherefore serves as a generating function for the otherthermodynamic properties, and implicitly represents completeinformation.
P
T
P
RTGT
RT
H
P
RTG
RT
V
dTRT
HdP
RT
V
RT
Gd
2
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Fugacity and Fugacitycoefficient
Residual Gibbs energy:
G, Gig = the actual and ideal gas values of the Gibbs energy
at the same temperature and pressure
Residual volume:
igR GGG
1
ZP
RTV
PRTVVVV
R
igR
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Fugacity and Fugacitycoefficient (TB page 208 -209)
The fundamental property relation for residual properties
applies to fluids of constant composition
).(T
RT/GT
RT
H
).(P
RT/G
RT
V
).(dTRT
HdP
RT
V
RT
Gd
P
RR
T
RR
RRR
446
436
4262
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Fugacity and Fugacitycoefficient (TB page 208 -209)
*Pressure Explicit form
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Fugacity and Fugacitycoefficient
Volume Explicit form
Like departure functions, 4 ways to estimate fugacity
and fugacity coefficient
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Free Energy as Generating function
Beattie- Bridgman Equation
where Ao, Bo, C, a, b,c are constants
Fugacity and Fugacitycoefficient
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Fugacity from Generalized Equation of State
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Fugacity from Z factor Correlation
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ln = ln (f/P) = (Bo + B1)Pr/Tr
where Bo = 0.083 (0.422 / Tr1.6)
B1 = 0.139 (0.172 / Tr4.2)
Problem: Estimate the fugacity and fugacity coefficient for
ethylene using the Virial coefficient correlation
(0.9963 bar)
Fugacity coefficient throughvirial coefficient relation
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1. Estimate the fugacity of carbon monoxide at 50 bar and
200 bar if the following data is given:
P (in bar) 25 50 100 200 400
Z 0.9890 0.9792 0.9741 1.0196 1.2482
2. Determine the fugacity and fugacity coefficient of n-
octane at 427.85 K and 0.215 Mpa using the Lee Kesler
data (0.2368 MPa)
Fugacity problems
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Till date we have dealt with pure species (focus ongases)
But in most real situations, more than one species are
present resulting in a mixture
Methods discussed earlier need to be modified by
including an additional composition variable to predict
properties of real gas mixtures
Constants which appear in EOS which characterize purespecies behaviour can be averaged to get mixture
constants
Equations which express mixture constants in terms of
pure species constants are called MIXING RULES
Properties of real gas mixtures
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A mixing rule expresses a mixture constant am in terms ofcomposition expressed in mole fraction yi and the pure
component constant ai, i.e. am = yiyjaij
aii
= ai,
ajj
= aj,
(based on interaction of like pairs of
molecules) and aij is based on interaction of unlike pairs
of molecules
Equations which provide the interaction constant in terms of
pure component constants are called combining rulesaij = (aii + ajj) / 2 = (ai + aj) / 2 ----- sqrt(aij ajj) = sqrt(aiaj)
am = yiai am = [ yisqrt(ai)]2
Mixing rules
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Mixing Rules
For Cubic Equation of State (RK, SRK, PR)
For Beattie-Bridgman EOS
In the absence of
data, Kij can be set
equal to zero
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For Benedict-Webb-Rubin For Lee-Kesler Methodrecommended by Knapp et al.
Kij ~1
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For Virial EOS
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Pseudo-critical constant method
Kays Rule
Prausnitz-Gunn Rule Joffes Relation
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Rework this
problem using
the generalized
virial coefficient
correlation
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Prediction of Departure functions for real gas mixtures
Make use of
Kays rule/
Modified
Prausnitz Gunn/
Joffes relation to
evaluate Tcm, m
In terms of Virial EOS
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1. The Vander Waals constants for n-butane and n-octaneare
(1) n-butane a = 1.3874 Pa(m3/mol)2; b = 0.1163e- 3 m3/mol
(2) n-octane a = 3.7890 Pa(m3
/mol)2
; b = 0.237e- 3 m3
/molEstimate the Vander Waals constants for an equimolar mixture
of n-butane and n-octane
(am = 2.4405 Pa (m3/mol)2; bm = 0.1796e-3 m
3/mol
2. Estimate the enthalpy and entropy departures of an
equimolar mixture of n-butane and n-octane at 600K and 16
bar using the Vander Waals EOS
( 1.203 kJ/mol and 1.145 J/mol.K)
Practice problems
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Fugacity and Fugacity Coefficient for real gas
mixture
Determine the fugacity and fugacity coefficient of an equimolar mixtureof n- butane and n-octane at 600K and 16 bar using virial coefficient
correlation
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1. Determine the fugacity and fugacity coefficient for an
equimolar mixture of n-butane and n-octane at 600 K
and 16 bar using the van der Waals EOS (0.9018, 14.43
bar)
2. Rework the above problem using the pseudo-critical
constants method. Make use of Kays rule for pseudo-
critical constant and the Lee-Kesler data for the 3
parameter law of corresponding states (0.914, 14.62
bar)
3. Rework Problem 1 using the virial coefficient correlation
(1.104, 17.66 bar)
Estimation of fugacitycoefficient (Problems)
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For a pure substance, the fugacity at constant T is defined
by dg = vdP = RTdlnf
g go = RTln(f/P)
Similarly in mixtures the partial molar fugacity of a
component i is defined as
dgi = dP = RTdlnfi (put a bar and toodle ^ on fi)
= RTln = RTln
The partial fugacity is not a partial molar property
Fugacity of a component in amixture
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Fugacity of a component in a mixture
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Estimate the partial fugacities of the equimolar mixture of n-butane (1) and n-octane (2) at 600 K and 16 bar. Make use of
the RK equation of state
Tc and Pc values for n butane are 425.2 K and 37.97 bar
Tc and Pc values for n butane are 569.4 K and 24.97 bar
1. Estimate the pure component constants based on RK EOS
2. Use the mixing rules to find constants for mixture. If Kij
is
not known, set it equal to zero
3. Estimate Z (In this problem take it equal to 0.8688)
4. The calculate the partial fugacities
Problem
C ff
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By Virial Coefficient
Correlation
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Problem
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Fugacity of liquid and solid
If a liquid or solid is in equilibrium with its own vapor, then,
fv is
evaluated
and equatedto fl at T and
Psat and then
translated to
fl at required
T and P >
Psat
Poynting Pressure
Correction
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The saturation pressure of n-octane at 427.85 K is 0.215
MPa. Estimate the fugacity of liquid n-octane at 427.85 K
and 1.0 MPa.
(0.2475 MPa)
This approximation is not valid for substance like acetic
acid which forms dimers in the vapour phase.
Problem
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RECAP
(Finding real gas volume) / Departure
Functions / Fugacity
Equation of State
Compressibility factor correlations for EOS
Generalized factor correlations Virial coefficients
PURE
SUBSTANCE
(Finding real gas volume) / Departure
Functions / Fugacity
Equation of State
Compressibility factor correlations for EOS
Generalized factor correlations
Virial coefficients
MIXTURES(MIXING
RULES)
GAS / VAPOUR
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Partial Molar Property
The partial molar property is defined as, where M is an extensive
property and represents the
change in the property M of a
mixture per mol of componenti when temperature, pressure
and mole numbers of all other
constituents are held constant
= i (Chemical Potential)
Solution properties, M
Partial properties,
Pure-species properties,
We use smaller
case letters
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The following can be proven or
In general,
Partial Molar Property
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Differentiating G, we get
(Gibbs-Duhem relation)
At constant T and P,
or
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Method of Intercepts to evaluate partial molar
properties
From graph,
V1 = AB = AC BC
V2 = DF = DE + DF
P bl ti l l
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1. In order to prepare 2 m3 of alcohol water solution,
alcohol of mole fraction X1 = 0.40 is required to be
mixed with water at 25 oC. Determine the volumes of
alcohol and water needed to prepare the mixture
Partial molar volume of alcohol: 38.3 10 6 m3/mol
Partial molar volume of water: 17.2 10 6 m3/mol
Molar volume of alcohol = 39.21 10 6 m3/mol
Molar volume of water = 18 10 6 m3/mol
[1.223 m3, 0.8424 m3]
Problems on partial molarproperty
P bl ti l l
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In a binary liquid system, the enthalpy of species 1 and 2 at
constant temperature and pressure is represented by the
following equation
H = 400 x1 + 600 x2 + x1x2 (40x1 + 20 x2) where H is in J /mol
Determine expressions for partial molar enthalpies for
species 1 and 2 as function of x1 and the numerical values
of the pure species enthalpies H1 and H2
H1 (partial) = 420 60 x12 + 40 x1
3
Problems on partial molarproperty
P bl ti l l
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1. The partial molar volume of water (1) in methanol (2) at
25 oC and 1 atm is approximated by 18.1 + ax22 where a = -
3.2 cm3/mol. Determine an expression for the partial molar
volume of methanol at the same condition given that V2 =
40.7 cm3/mol at 25 oC and 1 atm. Determine the partialmolar volume of methanol at infinite dilution.
2. The molar volume of a binary solution at 25 oC is given a
Calculate partial molar volumes of components 1 and 2 at
X1 = 0.5 and 0,75
Problems on partial molarproperty
X1 0 0.2 0.4 0.6 0.8 1.0
V 106 (m3/mol) 20 21.5 24 27.4 32 40
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Useful intensive thermodynamic property
Partial derivative of H, A and G of the ith component in a
multicomponent system
Rate of increase in Gibbs free energy per mole ofcomponent i added
dGP,T = idni
GP,T = ini
The Chemical potential is thus seen to be the contribution
of that component to the Gibbs free energy of the
solution
Chemical Potential
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Consider 2 phases a and b in equilibrium
GP,T = ini
Role of in equilibrium
At the same temperature
and pressure chemicalpotential or partial molar
free energy of a component
in every phase must be the
same under equilibrium
conditions
Effect of pressure on chemical
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Effect of pressure on chemicalpotential
Effect of temperature on
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Effect of temperature onchemical potential
Do it yourself
Starting with G = H TS prove that
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Let us recap!!!
).(TRT/GT
RTH
).(P
RT/G
RT
V
).(dTRT
HdP
RT
V
RT
Gd
P
RR
T
RR
RRR
446
436
4262
Use lower case for g
P
R
T
R
T
fT
RT
H
P
f
RT
V
ln
ln
Mixture Fugacity
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Let us recap!!!
oi
i
i
i
f
f
RTg
fRTd
lng
lngd
0
i
i
Fugacity of component in a mixture P
ii
o
P
ii
o
T
fT
RT
HH
T
f
TRT
HH
ln
ln
T
ii
P
f
RT
VV ln
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Measure of the effective of a component
Results from interaction of molecules in non-ideal gas or
solution and is dimensionless
Depends on temperature, pressure and composition
The fugacity of component i as a pure substance atthe temperature and pressure is also the standard state
fugacity of component i
Activity and Activity coefficient
The numerator has
a toodle on top
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The activity coefficient is the mixture is defined as theratio of activity ai of a component to its mole fraction xi in
the liquid mixture
Activity and Activity coefficient
The numerator for f
has a toodle on top
and not a bar
Temperature dependence on
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Temperature dependence onactivity coefficient
Pressure dependency
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Gibbs Duhems equation
At constant T and P,
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1. The activity coefficient of component 1 in a binarysolution is represented by
Where a, b and c are independent of concentration. Obtain
an expression for 2 in terms of x1
2. For a binary system if the activity coefficient of
component is ln 1 = ax22, then derive the expression for
component 2
Problems
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1. Estimate the entropy change due to mixing when 2.8 l of oxygen and19.6 L of hydrogen at 1 atm and 25 oC are mixed to prepare a
gaseous mixture
0.754 cal/mol.K
2. At 25 oC, 0.7 moles of helium are mixed with 0.3 moles of argon.
Calculate the free energy and enthalpy change of mixing. Assume
ideal gas behavior (- 1511.79 J and 0)
3. A 20 L vessel is divided into 2 compartments with the help of a
removable partition. The first compartment contains 12 L of
hydrogen and the second compartment contains 10 L of nitrogen.
Now the partition is withdrawn and gases allowed to mix
isothermally at 1 atm and 298 K. Estimate G, H and S of mixing.
Problems
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Excess Property = Actual Property of mixture at given T
and P Property it would have as an ideal solution at
same temperature and mixture
ME = M Mideal
Excess Properties
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Properties of solution
Lewis & Randall Rule
For an ideal gas mixture (solution)
(Amagats Law)
(Daltons Law)
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Entropy change due to mixing
Entropy change due to mixing is given by
Phase equilibrium in Ideal
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Phase equilibrium in Idealsolutions
At equilibrium,
But
(Raoults Law)
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Raoults law: 2 unknowns 1 equation
Constraints
Antoine equation for
calculating Psat (t inCelsius and P in torr
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Ph ilib i bl
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Phase equilibrium problems
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Fl h l l ti
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Flash calculations
Rearranging Raoults law expression,
Ki is the equilibrium ratio or K
factor of component i
material balance around
the flash drum
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Objecti e assessment
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Revisiting Gibbs free energy
Gibbs free energy as a generating function
Chemical potential
Fugacity of pure species
Estimating Fugacity of pure species Estimating Fugacity of component in a mixture
Mixing rules
Concept of partial molar property
Estimating partial molar properties
Excess properties
Activity coefficient models
Objective assessment
Watch your thoughts they become yourwords
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Watch your words they become your
actions
Watch your actions they become your
habits
Watch your habits they become your
character
Watch your character it becomes your
destiny