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Chemical Engineering Department | University of Jordan | Amman 11942, Jordan
Tel. +962 6 535 5000 | 22888
1
Dr.-Eng. Zayed Al-Hamamre
Thermodynamics II
Solution Thermodynamics: Theory
Chemical Engineering Department | University of Jordan | Amman 11942, Jordan
Tel. +962 6 535 5000 | 22888
2
Content
Introduction
Thermodynamics of gas mixtures and liquid solutions
Chemical potential
Partial properties of mixtures
The Ideal-Gas Mixture Model
Chemical Engineering Department | University of Jordan | Amman 11942, Jordan
Tel. +962 6 535 5000 | 22888
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The purpose in this chapter is to develop the theoretical foundation for applications of
thermodynamics to gas mixtures and liquid solutions.
In the chemical, petroleum, and pharmaceutical industries multicomponent gases or liquids
commonly undergo composition changes as the result of mixing and separation processes, the
transfer of species from one phase to another, or chemical reaction.
The property of such systems depend strongly on composition as well as on temperature and
pressure
Introduction
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Fundamental property relation
What is the most important property ?
……G……….
For pure component; G = G (T, P)
For a homogeneous mixture e.g. containing i components mixture;
G = G (T, P, n1, n2, …, ni)
Also, for closed system: no mass transfer across boundary or in a single-phase fluid in a
closed system wherein no chemical reactions occur
dTndPnnd )()()( SVG
G = G (T, P) Since n is the total number of moles of the system (= constant);
dTdPd SVG
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Open system, single phase
,,,,,, 21 innnTPgnG G
ni is the number of moles of species i
i
i
nPTinPnT
dnn
ndT
T
ndP
P
nnd
j,,,,
)()()()(
GGGG
all mole numbers held constant all mole numbers except ni held constant
The fundamental property relation for single phase fluid systems of variable mass and
composition
ii
i dndTndPnnd )()()( SVG
Fundamental property relation
SG
VG
nPnT TP ,,
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nPnT TP ,,
G
SG
V
The chemical potential of species i
jnTPii n
n
,,
)(
G
Fundamental property relation
This equation forms the basis for the definition of partial properties
For pure species
Gi molar Gibbs energy,
inn
0
,
i
ii
TPi
ii nG
n
GnG
n
Gn
G is intensive property independent of the system size ni
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The Chemical Potential and Phase Equilibria
Consider the following:
o A closed system and multicomponent
o Containing two phases in equilibrium.
o Mass transfer occurs if the equilibrium is disturbed
o Each individual phase is an open system, free to transfer
mass to the other
ii
i dndTndPnnd )()()( SVG
ii
i dndTndPnnd )()()( SVG
The total Gibbs energy of the two-phase system )()( GGG nnn
ii
iii
i dndndTndPnnd )()()( SVG
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compare with for the whole closed system
The Chemical Potential and Phase Equilibria
dTndPnnd )()()( SVG
0i
iii
ii dndn
Mass conservation requires
ii nn constant 0 ii dndn
0i
iii dn
Quantities idn are independent and arbitrary (never be zero).
),,2,1( Niii Hence
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The Chemical Potential and Phase Equilibria
Chemical Potential
Is an extensive property,
Provides a measure of the work of a system is capable when a change in mole numbers occurs
e.g. chemical reaction or a transfer of mass.
)(
For π phases at equilibrium, and N is the number of species, generalization to multiple phases
in equilibrium
Niiii ,,2,1 N ( – 1) equations
In addition to thermal and mechanical equilibrium criteria
TTT PPP
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Partial (molar) properties
Partial molar properties are defined as partial derivatives with respect to moles
jnTPii n
M,,
M
M denotes for any extensive properties
It is a response function, i.e., a measure of the response of total property nMto the addition at constant T and P of a differential amount of species I to a finite amount of solution.
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GASHUVM ,,,,,
iiiiiii GASHUVM ,,,,,
iiiiiii GASHUVM ,,,,,
Solution properties Pure-species properties
Partial molar properties
NotationsPartial (molar) properties
jnTPiii n
G,,
G Re-call
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Partial (molar) properties
),....x,xM(T,P,xnM i21
In general, for a homogenous mixture
The total differential of M is
ii
nTPiP,nT,n
dnn
MdT
T
(nM)dP
P
(nM)d(nM)
j,,
Could also be written as
(11.9) dnMdTT
(M)ndP
P
(M)nd(nM) ii
P,xT,x
differentiation at constant composition
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Partial (molar) properties
Since ni = xin
And
Substitute these terms to Eq. (11.9), and then rearrange:
Rearrange,
iii ndxdnxdn
MdnndMd(nM)
=0.0 =0.0
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Partial (molar) properties
(11.12)
(11.11)
0
(11.10)
0
iii
iii
iii
iii
P,xT,x
iii
P,xT,x
MnnM
MxM
MxM
dxMdTT
MdP
P
MdM
dxMdT-T
MdP-
P
MdM-
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Partial (molar) properties
Differentiating Eq 11.11 i
iii
ii dxMMdxdM
Comparison of this equation with Eq. (11.10) (Subtraction gives) yields
0,,
iii
xPxT
MdxdTT
MdP
P
M
Gibbs-Duhem equation at constant T and P
As a special case at constant T and P: 0i
ii Mdx
This equation must be satisfied for all changes in P, T, and the Mi caused by changes of state in a homogeneous phase
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Partial Properties in Binary Solutions
2211 MxMxM
22221111 dxMMdxdxMMdxdM
For a binary solution
AB
Gibbs-Duhem equation is
02211 MdxMdx C
i
ii MxM
0i
ii Mdx
For a binary solution
Dividing by dx1, we have the Gibbs-Duhem equation in derivative forms
01
22
1
11
dx
Mdx
dx
Mdx
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121 xx
21 dxdx 21
1
MMdx
dM
SinceDEq. B becomes
From Eq A and D
121 dx
dMxMM
112 dx
dMxMM
These equations can be used to obtain partial
molar properties from solution property.
Partial Properties in Binary Solutions
iixx
MMMii
11
limlim
As a solution becomes pure in species i, both properties approach pure species
property
In the limit of infinite dilution iiix
MMMi
0lim
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Partial Properties in Binary Solutions
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Partial Properties in Binary Solutions
1
2222 ,,1,,1,,1,,1
2211
2211
2211
2211
nVTnPTnSVnPS
T,VT,V
T,PT,P
V,SV,S
S,PS,P
n
A
n
G
n
U
n
H
dn) (μdn)(μPdV -SdT dA
dn) (μdn)(μVdP -SdT dG
dn) (μdn)(μPdVTdS dU
dn) (μdn)(μVdPTdS dH
For a binary (For two components) solution
Also,
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The need arise in a laboratory for 2000 cm3 of an antifreeze solution consisting of 30mol % methanol in water. What volumes of pure methanol and of pure water at 25 Cmust be mixed to form the of antifreeze, also at 25 C ? Partial molar volumes formethanol and water in a 30 mol % methanol solution and their pure-species molarvolume, both at 25 C , are:Methanol (1) and water (2):
Example
132
132
131
131
molcm 068.81 molcm 765.17
molcm 40.727 molcm 632.38
VV
VV
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mol 272.58) 246.83)(7.0(
mol 974.24) 246.83)(3.0(
246.83 025.24
2000
025.24
)765.17)(7.0()632.38)(3.0(
22
11
313
3
13
2211
nxn
nxn
cmmolcm
cm
V
Vn
molcmV
VxVxV
t
Example Cont.
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SolutionThe line drawn tangent to the V-x1 curve at x1=0.30, illustrates the values of V1=40.272 cm3 mol-1 and V2=18.068 cm3 mol-1.
32
132
31
131
1053
) 068.18)( 272.58(
1017
) 727.40)( 497.24(
cmV
molcmmolV
cmV
molcmmolV
t
t
t
t
Example Cont.
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We show now how partial properties are related to one another. By Eq. (11.8), μi ≡ Gi, and Eq. (11.20 may be written:
) . (dnG(nS)dT(nV)dPd(nG) ii 1711
Application of the criterion of exactness, Eq. (6.12) , yields the Maxwell relation,
). ( P
S -
T
V
T,nP,n
166
Relations among Partial Properties
jj P,T,niT,n
i
P,T,niP,n
i
n
(nV)
P
G and
n
(nS) -
T
G
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One can write the RHS in the form of partial molar, and change the composition from n to x.
Relations among Partial Properties
T,x
ii
T,x
i
P,x
ii
P,x
i
PV
P
G
T S
T
G
Every equation that provides a linear relation among thermodynamic properties of a
constant-composition solution has as its counterpart an equation connecting the
corresponding partial properties of each species in the solution.
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25
P(nV) nU , nH moles n For
PVU H
dTSdPVGd
dTT
GdP
P
GGd
P
nVnUnH
iii
xP
i
xT
ii
iii
,,
iii VUH
)
P))
jjj nT,P,nT,P,nT,P,n
(
n
(
n
(
Relations among Partial Properties
This may be compared with Eq. (6.10). These examples illustrate the parallelism that exists
between equations for a constant composition solution and the corresponding equations for
the partial properties of the species in solution. We can therefore write simply by analogy
many equations that related partial properties.
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Example
212121 2040600400 xxxxxxH
12 1 xx 3
11 20180600 xxH
21
1
60180 xdx
dH
121 dx
dHxHH
31
211 4060420 xxH
112 dx
dHxHH
312 40600 xH
4201 H 6402 H
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The Ideal-Gas Mixture Model
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Dalton Law: Every gas has the same V and T.
iit
t
iitii
PP
n
nyPyP where
The Ideal-Gas Mixture Model
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Application of partial properties to molar volume
jji
niP,T,niP,T,ni
igiig
i
nnn
n
n
P
RT
n
(nRT)/P
n
)(nVV
jjj
P
RTVVV igig
iig
i
partial molar volume = pure species molar volume= mixture molar volume
Note: Partial pressure of species i (It is not partial molar property)
Molar Volume and Partial Molar Volume
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Properties of each component species are independent of the presence of other species.
A partial molar property (other than volume) of a constituent species in an ideal-gas mixture is equal to the corresponding molar property of the species as a pure ideal gas at the mixture temperature but at a pressure equal to its partial pressure in the mixture.
Partial molar properties
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Partial molar properties
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Partial molar entropy
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Partial molar Gibbs energy
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Property Change of Mixing
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Property Change of Mixing
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Property Change of Mixing
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Property Change of Mixing
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The Ideal-Gas Mixture Model
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y
yR SyS
S
/P p y yy -RSyS
HyH
UyU
HyH
) (T,pM(T,P)M
iii
igi
ii
ig
igi
iiii
iigi
ii
ig
igi
ii
ig
igi
ii
ig
igi
ii
ig
iigi
igi
1ln
is mixing gas idealan of changeentropy The
P. and T mixture at the valuespecies-pure theis
(11.25))(where ln
0
(11.23)
(11.23)
(11.21)
Summary
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).( P y yRT (T) ΓyG
). ( P yRT (T) Γμ
(T) Γ
). ( P RT (T) ΓG
PRTd dPP
RTdPV dG
).( yRT GGμ
ii
iii
iig
iiigi
i
iigi
igi
igi
iigi
igi
igi
2911ln
2811ln
constant n integratio theis
2711ln
ln From
2611ln
Summary
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What is the change in entropy when 0.7 m3 of CO2 and 0.3 m3 of N2 each at 1 bar and 25 C blend to form a gas mixture at the same condition? Assume ideal gases.
Example
1-1-
1-1-
iii
igi
ii
igmixing
K mol) (g J
K mol) (g J 8.314
y
yR SySS
is mixing gas ideal an of changeentropy The
079.5
)7.0
1ln7.0
3.0
1ln3.0(
1ln