chemical engineering tongji university · partial molar property •partial molar property (q )...
TRANSCRIPT
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Solution thermodynamics
Min HuangChemical Engineering
Tongji University
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Mixture/solution
• Chemical Potential• Partial Molar Property• Partial Pressure• Ideal-Gas Mixtures– enthalpy of an ideal gas– entropy of an ideal gas
• Gibbs energy of an ideal-gas mixture Gig = Hig – TSig,
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Mixture/solution
• Because the properties of systems in chemical engineering depend strongly on composition as well as on temperature and pressure,
• Need to develop the theoretical foundation for applications of thermodynamics to gas mixtures and liquid solutions
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Mixture/solution
• definition of a fundamental new property called the chemical potential, upon which the principles of phase and chemical-reaction equilibrium depend.
• This leads to the introduction of a new class of thermodynamic properties known as partial properties.
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Chemical potential
• Gibbs free energy of a multicomponent mixture is a function of T, P and each species mole number, therefore, the total differential of the Gibbs free energy,
, , ,
i i i
i
iP N P N i P N
i i
i
G G GdG dT dP dN
T P N
SdT VdP G dN
æ ö¶ ¶ ¶æ ö æ ö= + + ç ÷ç ÷ ç ÷
¶ ¶ ¶è ø è ø è ø
= - + +
å
å
C
C
(CP1)
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Chemical potential
• Recall
• in extensive function, therefore we have
i
r
i
i
i
r
i
i
dn
pVTSdnpVTSdG
å
å
=
=
=
--÷ø
öçè
æ+-=
1
1
µ
µ
( )
jinTpi
in
nG
¹
úû
ùêë
é
¶
¶º
,,
µ
-
Chemical Potential and Phase Equilibrium
• Consider a closed system consisting of two phases in equilibrium.
• Within this closed system, each individual phase is an open system, free to transfer mass to the other, that is
( ) ( ) ( )
( ) ( ) ( ) bbbbb
aaaaa
µ
µ
i
i
i
i
i
i
dndpnVdTnSnGd
dndpnVdTnSnGd
å
å
=
=
++-=
++-=
1
1
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Chemical Potential and Phase Equilibrium
• The change in the total Gibbs energy of the two-phase system is the sum of these equations.
( ) ( ) ( )
011
11
=+
+++-=
åå
åå
==
==
bbaa
bbaa
µµ
µµ
i
i
ii
i
i
i
i
ii
i
i
dndn
dndndpnVdTnSnGd
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Chemical Potential and Phase Equilibrium
• And• Therefore,
• Thus, multiple phases at the same T and P are in equilibrium when the chemical potential of each species is the same in all phases.
ba
ii dndn -=
( )
),,2,1(
01
Ni
dn
ii
i
i
ii
!==
=-å=
ba
aba
µµ
µµ
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Partial Molar Property
• Partial Molar Property (q )
• It is a response function, representing the change of total property q due to addition at constant T and P of a differential amount of species i to a finite amount of mixture/solution.
( )
jinTpi
in
nG
¹
úû
ùêë
é
¶
¶º
,,
//
qqµ
-
Partial Molar Property
• Let q be any molar property (molar volume, molar enthalpy, etc.) of a mixture consisting of Nimoles of species i,
• And partial molar thermodynamic property
• Therefore,
( )( )
, ,
, ,
j i
i i
i T P N
NT P x
N
qq q
¹
¶= =
¶
iiN N=å
C
( )1
, ,i ii
x T P xq q=
=åC
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Partial Molar Property
• Or
So that, by the product rule of differentiation,
• substitute G with Nq into Eq. CP 1
1
i i
i
N Nq q=
=åC
( ) i i id N N d dNq q q= +å å PM 1
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Partial Molar Property
• substitute G with Nq into Eq. CP 1
, , ,
, ,
i i i
i i
i
iP N T N i P N
i i
iP N T N
N N NdN dT dP dN
T P N
N dT N dP dNT P
q q qq
q qq
æ ö¶ ¶ ¶æ ö æ ö= + + ç ÷ç ÷ ç ÷
¶ ¶ ¶è ø è ø è ø
¶ ¶æ ö æ ö= + +ç ÷ ç ÷
¶ ¶è ø è ø
å
å
C
C
PM 2
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Partial Molar Property
• Subtracting PM 2 from PM 1,
• Or,, ,
0
i i
i i
iP N T N
N dT N dP N dT P
q qq
¶ ¶æ ö æ ö- - + =ç ÷ ç ÷
¶ ¶è ø è øåC
, ,
0
i i
i i
iP N T N
dT dP x dT P
q qq
¶ ¶æ ö æ ö- - + =ç ÷ ç ÷
¶ ¶è ø è øåC
This is the generalized Gibbs-Duhem Equation
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Partial Molar Property
• For constant temperature and pressure,
• Substitute q with G
,1
,1
0
0
i i T Pi
i i T Pi
N d
x d
q
q
=
=
=
=
å
å
C
C
1
1
0
0
i i
i
i i
i
SdT VdP N dG
SdT VdP x dG
=
=
- + =
- + =
å
å
C
C
-
Partial Molar Property
• At constant temperature and pressure,
• Recall Gibbs-Duhem Equation
1
1
0
0
i i
i
i i
i
N dG
x dG
=
=
=
=
å
å
C
C
, ,
0
i i
i i
iP N T N
dT dP x dT P
q qq
¶ ¶æ ö æ ö- - + =ç ÷ ç ÷
¶ ¶è ø è øåC
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Partial Molar Property
• Rearrange and times n on both sides,
• Derivative with respect to Ni
å-
=++÷÷ø
öççè
涶
-÷÷ø
öççè
涶
-1c
jjjii
NT,Np,
0θdNθdNdppθndT
Tθn
ii
å-
=++÷÷ø
öççè
涶
-÷÷ø
öççè
涶
-1c
j i
jji
NT,
i
Np,
i 0dNθd
dNθddppθdT
Tθ
ii
å-
=¶¶
++÷÷ø
öççè
涶
-÷÷ø
öççè
涶
-1c
j j
iji
NT,
i
Np,
i 0NθdNθddp
pθdT
Tθ
ii
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Partial Molar Property
• Rearrange,
• Substitute back to the Gibbs-Duhem Equation
å-
=÷÷
ø
ö
çç
è
æ
¶
¶+÷÷
ø
öççè
æ
¶
¶+÷÷
ø
öççè
æ
¶
¶=
1
1,,,
C
j
j
pTj
i
xT
i
xp
ii xd
xdp
pdT
Td
qqqq
å å=
-
= úú
û
ù
êê
ë
é
÷÷
ø
ö
çç
è
æ
¶
¶+÷÷
ø
öççè
æ
¶
¶+÷÷
ø
öççè
æ
¶
¶+
÷÷ø
öççè
æ
¶
¶-÷÷
ø
öççè
æ
¶
¶-=
C
j
C
j
j
pTj
i
xT
i
xp
ii
NT
i
Np
i
xdx
dpp
dTT
x
dpp
dTT
ii
1
1
1,,,
,,
0
qqq
qq
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Partial Molar Property
• Since
• Finally
• For Binary
1
1 1,
0ii ji j j T P
x dxx
q-
= =
æ ö¶=ç ÷
ç ÷¶è øå åC C
å å= =
÷ø
öçè
æ
¶
¶=÷
ø
öçè
æ
¶
¶=÷÷
ø
öççè
æ
¶
¶C
i xp
C
i
ii
xpxp
ii dT
TdTx
TdT
Tx
1 ,1,,
qq
q
00
,1
22
,1
11
2
1
1
1
1
,1
=÷÷
ø
ö
çç
è
æ
¶
¶+÷
÷
ø
ö
çç
è
æ
¶
¶=÷
÷
ø
ö
çç
è
æ
¶
¶å å=
-
= pTpTi
C
j pT
ii
xx
xx dx
xx
qqq
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Ideal-Gas Mixtures
• Ideal gas
• pi is known as the partial pressure of species i
t
ii
V
RTnp =
tV
nRTp =
),,2,1( Ni pyp or yn
n
p
piii
ii!====
( ) ( )
P
RT
n
n
P
RT
n
pnRT
n
nVV
j
ii
ni
npTinpTi
igig
i
=÷÷
ø
ö
çç
è
æ
¶
¶
úû
ùêë
é
¶
¶=ú
û
ùêë
é
¶
¶=
,,,,
/
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Ideal-Gas Mixtures
• 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.
ig
i
ig
ii
ig
i
ig
i VM whenpTMPTM ¹= ),(),(
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Ideal-Gas Mixtures
• Since the enthalpy of an ideal gas is independent of pressure,
• For ideal gases, this enthalpy change of mixing is zero.
ig
i
i
i
ig
ig
i
ig
i
ig
i
ig
i
ig
ii
ig
i
HyHp T, mixture at value pureHH
pTHpTH pTHpTH
å==
=
=
)(),(),(),(),(
-
Ideal-Gas Mixtures
• The entropy of an ideal gas does depend on pressure, is given as
• Therefore, integration from pi to p
T const at pRddS igi ln=
i
i
i
ig
i
ig
i yRp
pRpTSpTS lnln),(),( =-=-
1
2
1
2p p
pnR lnTTlnncS -=D
-
Ideal-Gas Mixtures
• Recall• Therefore,
• Where Siig is the pure-species value at the mixture T and P.
ig
i
ig
ii
ig
i
ig
i VM whenpTMPTM ¹= ),(),(
i
ig
i
ig
i
i
ig
i
ig
i
yRSS
yRpTSpTS
ln
ln),(),(
-=
-=
-
Ideal-Gas Mixtures
• By the summability relation
• the left side of the second equation is the entropy change of mixing for ideal gases. Since l/yi > 1, this quantity is always positive, in agreement with the second law.
ii
i
ig
i
i
i
ig
i
i
i
i
ig
i
i
i
ig
i
yyRSyS
yyRSyS
1ln
ln
åå
åå
=-
-=
-
Ideal-Gas Mixtures
• For the Gibbs energy of an ideal-gas mixture Gig = Hig – TSig,
• for partial properties is
i
ig
i
ig
i
ig
i
i
ig
i
ig
i
ig
i
ig
i
ig
i
ig
i
yRTGG
yRTTSHG
STHG
ln
ln
+=º
+-=
-=
µ
-
Recap Mixture/solution• Gibbs free energy of a multicomponent mixture• Chemical Potential• Partial Molar Property• Partial Pressure of Ideal-Gas• Ideal-Gas Mixtures– enthalpy of an ideal gas– entropy of an ideal gas
• Gibbs energy of an ideal-gas mixture Gig = Hig –TSig,