the basic relation connecting the gibbs energy to the...
TRANSCRIPT
The basic relation connecting the Gibbs energy to the temperature and pressure in any closed system:
(nG) (nG)d(nG) dP dT (nV)dP (nS)dTP T
and pressure in any closed system:
T,n P,nP T
applied to a single-phase fluid in a closed system wherein no h i l tichemical reactions occur.
Consider a single-phase, open system:
Chemical potential
i(nG) (nG) (nG)d(nG) dP dT dn
P T
jT,n P,n i P,T,niP T n
Define the chemical potential:
j
ii P,T,n
(nG)n
The fundamental property relation for single-phase fluid systems of constant or variable composition:
i ii
d(nG) (nV)dP (nS)dT dn
When n = 1,
dG VdP SdT dT,x
GVP
i i
idG VdP SdT dx
GST
P,xT
C id l d t i ti Consider a closed system consisting of two phases in equilibrium:
i ii
d(nG) (nV) dP (nS) dT dn
i ii
d(nG) (nV) dP (nS) dT dn
nM (nM) (nM)
i i i ii i
d(nG) (nV)dP (nS)dT dn dn
i i i ii i
d(nG) (nV)dP (nS)dT dn dn
The two phase system is closed thus:
d(nG) (nV)dP (nS)dT
i i i ii i
dn dn 0
i idn dn
ii
M lti l h t th T d P i ilib i h
i i ii
( ) dn 0
Multiple phases at the same T and P are in equilibrium when chemical potential of each species is the same in all phases.
Define the partial molar property of species i:
j
ii P,T,n
(nM)Mn
The chemical potential and the particle molar Gibbs energy are
i iG
The chemical potential and the particle molar Gibbs energy are identical:
1 2 inM M(P,T, n , n ,..., n ,...)
For thermodynamic property M:
M M i i
iT,n P,n
M Md(nM) n dP n dT M dnP T
i iiT ,n P ,n
M Md(nM) n dP n dT M dnP T
M M
i i id n x dn n dx d nM n dM M dn
i i iiT ,n P ,n
M MndM Mdn n dP n dT M (x dn ndx )P T
i i i ii iT n P n
M MdM dP dT M dx n M x M dn 0P T
i iT,n P,nP T
0
dxMdTMdPMdM 0
,,
i
iinPnT
dxMdTT
dPP
dM
0 iiMxM0 iiMnnMSummability i
ii
Calculation of mixture properties from partial properties
i
iiyrelations
i iiT P
M MdM dP dT M dx 0P T
iT,n P,nP T
i i i ii i
dM x dM M dx
i iiT,n P,n
M MdP dT x dM 0P T
The Gibbs/Duhem equation
This equation must be satisfied for all changes in P, T and the caused by changes of state in a homogeneous phase
iMchanges of state in a homogeneous phase.
For the important special case of changes at constant T and P:and P
i ii
x dM 0
2211 MxMxM For binary systems 2211 MxMxM
Const. P and T, using Gibbs/Duhem equation
y y
22221111 dxMMdxdxMMdxdM
dMdMdM dM2211 dxMdxMdM
121 xx
211
MMdxdM
21 ddMxMM 12 dx
dMxMM 1 12
M x MM (A) (B)
121 dx 1dx
2x (A)
The three kinds of properties used in solution thermodynamicsthermodynamics
Solution properties M, for example G,S, H, Up p p , , ,
G ,S , H , Ui i i iPartial properties, , for exampleiM
G ,S , H , Ui i i iPure-species properties Mi, for example
, , ,i i i ii
, , ,i i i ii
EXAMPLEDescribe a graphical interpretation of equations (A) and (B)Describe a graphical interpretation of equations (A) and (B)
2dM M I
1 1d x x
1 21 2
dM I I I I 1 2
1I I
d x 1 0
dMI M x2 1
1I M x
d x
1 11
dMI M 1 x
d x
1
1 1I M 2 2I M
The need arises in a laboratory for 2000 cm3 of an antifreeze solution consisting of 30 mol-% methanol in water. What volumes of pure methanol and of pure water at 25°C must be mixed to form the 2000 cm3 of antifreeze at 25°C? The partial and pure molarmust be mixed to form the 2000 cm3 of antifreeze at 25 C? The partial and pure molar volumes are given.
3 -1 3 -11 1V 38.632 cm mol V 40.727Methan cm ool : m l 1 1
3 -1 3 -12 2V 17.765 cm mol V 18.06Water 8 cm m: ol
2211 VxVxV molcmV /025.24)765.17)(7.0()632.38)(3.0( 3
molVVn
t
246.83025.24
2000
moln 974.24)246.83)(3.0(1
moln 272.58)246.83)(7.0(2
3111 1017)727.40)(974.24( cmVnV t
31053)06818)(27258( cmVnV t222 1053)068.18)(272.58( cmVnV
The enthalpy of a binary liquid system of species 1 and 2 at fixed T and P is:
)2040(600400 212121 xxxxxxH
Determine expressions for and as functions of x1, numerical values for the pure-species enthalpies H1 and H2, and numerical values for the partial enthalpies at infinite dilution and
1H 2H
1H
2Hinfinite dilution and1H 2H
)2040(600400 212121 xxxxxxH
12 1x 1 x
311 20180600 xxH
21dHxHH
31
211 4060420 xxH 3
12 40600 xH
121 dx
xHH
01 x 11 x01x
molJH 4201
11x
molJH 6402
mol mol
i
iidnGdTnSdPnVnGd )()()(
iG (nS) iG (nV)
jP ,n i P,T,nT n
jT ,n i P,T,nP n
i
xP
i STG
, i
iT ,x
GV
P
dTSdPVGd iii
H U PV
nH nU P (nV)
(nH) (n U) (n V)Pn n ni i iP,T,n P,T,n P,T,nj j j
, , , , , ,j j j
H U P Vi i i
كه رابطه اي خطي بين خواص ترموديناميكي يك محلول با تركيب ثابت هر معادلهاز ك اظ ا خ ك ا ث ال ل ا ك ا ا ا ن ا ق
For every equation providing a linear relation among the thermodynamic properties of a constant-composition برقرار نمايد، داراي يك معادله المثني است كه خواص جزيي متناظر هر يك از
.هم مربوط مي نمايد اجزاء مخلوط را بهthermodynamic properties of a constant composition solution there exists a corresponding equation connecting the corresponding partial properties of each species in the p g p p p psolution.
nRTPt
p ni i yi tVyiP n
n RTipi tV p y P i 1,2,... , Ni i
V
Gibbs’s theoremA partial molar property (other than volume) of a constituent آل دهنده مخلوط گاز ايده يك جزء تشكيل) غير از حجم ( هر خاصيت جزيي مولي
آل در دماي مخلوط ولي صورت گاز ايده برابر همان خاصيت مولي جزء خالص بهم مخلوط در آن جزي فشار معادل فشاري باشددر
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 mixtureباشد در فشاري معادل فشار جزيي آن در مخلوط مي.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
iigi VM ),(),( i
igi
igi pTMPTM
ig ig(nV ) (nRT P) RT n RTigVi n n P n Pi i iT,P,n nT,P,n j jj
j jjig igV Vi i
Since the enthalpy of an ideal gas is independent of pressure:
ig ig igH (T P) H (T ) H (T P) ig igH Hig ig igi i i iH (T, P) H (T, p ) H (T, P) ig ig
i ii
H y H
i iM y M i ii
yig ig ig
i ii
H H y H 0 M M y M i i
iM M y M
Property change of mixingEnthalpy change of mixing
ig igU y U Similarly
Property change of mixing
g gi i
iU y U Similarly
ig ig igU U y U 0 i ii
U U y U 0 Internal energy change of mixing
The entropy of an ideal gas does depend on pressure
dT dPig igP
dT dPdS C RT P
igidS Rd ln P (T const.)
ig igi i i i
i i
P PS (T, P) S (T, p ) R ln R ln R ln yp y P
ig igi i iM (T, P) M (T, p )
ig igi i iS (T, P) S (T, P) R ln y
ig ig ig igi i i i
i iS y S R y ln y
ig ig igi i i i
i iS S y S R y ln y
ig ig igi i iG H TS
ig ig igi i i iG H TS RT ln y ig ig
i iH (T, P) H (T, P)
ig igS (T P) S (T P) R ln y
ig ig igi i i iG G RT ln y
i i iS (T, P) S (T, P) R ln y
RTig igdG V dP dP RT d ln P (const. T)i i P
igi i i(T) RT ln y P ig
i iG (T) RT ln P
igi i i i
i iG y (T) RT y ln y P
Chemical potential:Chemical potential:
id f d t l it i f h ilib i– provides fundamental criterion for phase equilibria
– however the Gibbs energy hence μi is defined in relation tohowever, the Gibbs energy, hence μi, is defined in relation to the internal energy and entropy - (absolute values are unknown).
igi iG (T) RT ln P i i ( )
Fugacity: a quantity that takes the place of μi
G (T) RT l fi i iG (T) RT ln f
With units of pressureWith units of pressure
i i iG (T) RT ln f ig i
i ifG G RT lnP
igi iG (T) RT ln P
i i P
ii
fP
RG RT ln
PFugacity coefficient
i iG RT ln
ig
Ri
iGlnRT
Ri 0G For ideal gases ig
i 1
P dPl (Z 1) ( t T) i i0ln (Z 1) (const. T)
P
When the compressibility factor is given by two terms virial equation
iii
B PZ 1
RT
P Biiln dPi 0ln dPi RT
P
i i0
dPln (Z 1) (const. T)P
B Piiln i RT
v vi i iG (T) RT ln f Saturated vapor:
l li i iG (T) RT ln f Saturated liquid:
vv l ii i l
i
fG G RT ln
f v l
i iG G
vili
fRT ln 0
f
v l satf f fi v l sati i if f f
v l sati i i= =
For a pure species coexisting liquid and vapor phases are in equilibrium p p g q p p qwhen they have the same temperature, pressure, fugacity and fugacity coefficient.
f
The fugacity of pure species i as a compressed liquid:
i i iG (T) RT ln f sat i
i i sati
fG G RT ln
f
sat sati i iG (T) RT ln f
i i iG (T) RT ln f
Psati i iPsat
iG G V dP (const. T) dG = V dP – S dT
Pi
isat
f 1ln V dPRTf
Since Vi is a weak function of P
satPi sati
RTf l satf V (P-P ) sat sat sat
i i if = P l satV (P P )i i isati
f V (P-P )ln =f RT
i i if P l satsat sat i i
i i iV (P-P )f = P exp
RT
For H2O at a temperature of 300°C and for pressures up to 10,000 kPa (100 bar) calculate values of fi and i from data in the steam tables and plot them vs. P.
For a state at P: iii fRTTG ln)(
For a low pressure reference state: ** ln)( iii fRTTG
)(1ln *i GGf
iii TSHG
)(1ln **iii SSHHf
)(ln * iii
GGRTf
)(ln * iii
SSTRf
kPaPfi 1**
The low pressure (say 1 kPa) at 300°C: gJHi 8.3076*
gKJSi 3450.10* gK
i if H 3076.818.015ln (S 10 345)
iln (S 10.345)1 8.314 573.15
For different values of P up to the saturated pressure at 300°C, one obtains the values of fi ,and hence . Note, values of fi and at 8592.7 kPa are obtainedi i
P = 4000 kPaT = 300 ºC
Hi = 2962 J/gS 6 3642 J/gKT = 300 C Si = 6.3642 J/gK
if 18.015 2962 3076.8ln (6.3642 10.345)1 8.314 573.15
if 3611 kPa
i 3611/ 4000 0.9028
T = 300 ºC Psat = 8592.7 kPa
Hi = 2751 J/gSi = 5.7081 J/gKVi = 1.403 cm3/g
satif 18 015 2751 3076 8 i 18.015 2751 3076.8ln (5.7081 10.345)1 8.314 573.15
satf 6738 9 kPaif 6738.9 kPa
sati 6738.9 / 8592.7 0.7843 i
Values of fi and i at higher pressure
l satsat sat i i
i i i
V (P P )f P exp
RT
l 3iV (1.403)(18.015) 25.28 cm / mol
i25.28(P 8592.7)f 6738.9exp
8314 573.15
P = 10000 kPa
i25.28(10000 8592.7)f 6738.9exp 6789.4 kPa
if 6738.9exp 6789.4 kPa8314 573.15
i 6789.4 /10000 0.6789 i
Fugacity and fugacity coefficient: species in solution
igi i i(T) RTln y P
For species i in an ideal gas mixture:
For species i in a mixture of real gases or in a solution of liquids:
i i i(T) RTln y P
i i iˆ(T) RT ln f
Fugacity of species i in solution (replacing the partial pressure)
Multiple phases at the same T and P are in equilibrium when the fugacity of each constituent species is the same in all phases:
ˆ ˆ ˆi i i
ˆ ˆ ˆf f ... f
R igM M M The residual property:
R igi i iM M M The partial residual property:
R igR igi i iG G G
ig ii i
i
f̂G G RT ln
y P
i i i iˆG (T) RT ln f
ig igi i i iG (T) RT ln y P i i i i( ) y
ii
i
f̂ˆy P
The fugacity coefficient of species i in solution
Ri i
ˆG RT ln
For ideal gas RiG 0 ig
ig if̂ˆ 1 Ri i
ˆG RT ln i
i1
y P
igi if̂ y P
Fundamental residual-property relationrelation
2nG 1 nGd d(nG) dTRT RT RT
i ii
d(nG) (nV)dP (nS)dT dn G H TS
ii
i dnRTGdT
RTnHdP
RTnV
RTnGd 2 ),,( inTPf
RTnG
G/RT as a function of its canonical variables allows evaluation of all other thermodynamic properties, and implicitly contains complete property information.
ii2
i
GnG nV nHd dP dT dnRT RT RTRT
igig ig ig GG V H
RR R Ri
i2i
GnG nV nHd dP dT dnRT RT RTRT
igig ig igi
i2i
GnG nV nHd dP dT dnRT RT RTRT
R R R
i i2i
nG nV nH ˆd dP dT ln dnRT RT RT
or
R R R
i i2i
nG nV nH ˆd dP dT ln dnRT RT RT
R R
T x
V (G / RT)RT P
T,x
R RH (G / RT)TRT T
P,xRT T
R
i(nG / RT)ˆln
n
j
ii P,T,n
n
Develop a general equation for calculation of values form compressibility-factor data.
iˆln
j
R
ii P,T,n
(nG RT)ˆlnn
P (nZ n) dPˆl
j
PR
PdPnnZ
RTnG
0)( j
i0 i P,T,n
( ) dlnn P
(nZ)i
i
(nZ) Zn
n 1
i1
n
d
P
ii PdPZ
0)1(ˆln Integration at constant
temperature and composition
Generalized correlations for the fugacity coefficient
rP dP )()1(l r
rr
rii Tconst
PdPZ
0).()1(ln
10 10 ZZZ
r rP P
TdPZdPZ 10 )()1(l r r
rr
r
r
r TconstP
dPZP
dPZ0 0
10 ).()1(ln
rP dP00 r
r
r
PdPZ
0
00 )1(ln
rP rdPZ 11ln
10 lnlnln
rP
Z0
ln
For pure gas
Table E13-E16
p g0 1( )
Estimate a value for the fugacity of 1-butene vapor at 200°C and 70 bar.
1271T 0127.1rT
731.1P
0 0.627
1 1.096
Table E15 and E16
731.1rP
0.191
1.096
0 1( )
0.638
f P (0.638)(70) 44.7 bar
Generalized correlations for the fugacity coefficient
BPP0 1rPZ 1 (B B )
cr
r c
BPPZ 1T RT
P dP
r
( )T
0 1cBP B BRT
rP
ri r0
r
dPln (Z 1) (const. T )P
cRT
0 1rPln = (B +ωB )0
1.6r
0.422B 0.083T
r
ln (B +ωB )T
1
4.2r
0.172B 0.139T
For pure gasr
Fugacity coefficient from the virial E.O.SThe virial equation
BPRTBPZ 1
The mixture second virial coefficient B
i j
ijji ByyB
For a binary mixture
2222211212211111 ByyByyByyByyB
Fugacity coefficient from the virial E.O.S
nBPZRT
nnZ
22 ,1,,11
)(1)(
nTnTP nnB
RTP
nnZZ
22 ,,, nTnTP
)()(1ˆlP nBPdnB
22 ,10
,11
)()(1lnnTnT n
nBRTPdP
nnB
RT
1 1 11 1 2 12 2 1 21 2 2 22B y y B y y B y y B y y B
1 2 11 1 2 12 2 1 21 2 1 22B y (1 y )B y y B y y B y (1 y )B
1 11 2 22 1 2 12 11 22B y B y B y y (2B B B )
12 12 11 222B B B
1 11 2 22 1 2 12 11 22B y B y B y y (2B B B )
nny /
1 11 2 22 1 2 12B y B y B y y 1 2 1 2
11 22 12n n n nB B Bn n n n
nny ii /
1 21 11 2 22 12
n nnB n B n B
2 1 2n n n n(nB) B
1 11 2 22 12nB n B n Bn
2
2 1 211 122
1 T,n
( ) Bn n
2
2 1 211 122 2
1 T,n
n n n n(nB) B ( )n n n
2,
2 1 211 12
1
n n n(nB) B ( )n n n n
21 T,n
n n n n
11 2 1 2 12(nB) B (y y y )n
11 2 1 12(nB) B y (1 y )n
21 T,n
n 21 T,nn
2(nB) B
P (nB)ˆ 2
1 11 2 12Pˆln = B +y δ
RT2
211 2 12
1 T,n
( ) B yn
2
11 T,n
P (nB)ˆlnRT n
RT
PˆSimilarly: 22 22 1 12
Pˆln B yRT
For multi-component systems
k kk i j ik ijP 1ˆln B y y (2 )
RT 2
i jRT 2
Where: 2B B B Where: ik ik ii kk2B B B
cij 0 1ij ij
RTB = (B +ω B ) i jω +ω
ω =ij ijcij
B (B ω B )P ijω
2
cij ci cj ijT = (T T )(1-k )
E i i l i t ti t
cij cijZ RTP ci cjZ +Z
Z
Empirical interaction parameter
cij cijcij
cij
P =V
c cjcijZ =
2
31/3 1/3ci cj
cij
V +VV =
2 cij 2
Determine the fugacity coefficients for nitrogen and methane in N2(1)/CH4(2) mixture at 200K and 30 bar if the mixture contains 40 mol-% N2.
molcmBBB
3
22111212 6.200.1052.35)8.59(22
30P 0501.0)6.20()6.0(2.35)200)(14.83(
30ˆln 212
22111 yB
RTP
ˆ
18350)620()40(010530ˆl 22 BP
9511.01
1835.0)6.20()4.0(0.105)200)(14.83(
ln 212
21222 yB
RT
83240̂ 8324.02
Estimate and for an equimolar mixture of methyl ethyl ketone (1) / toluene (2) at 50°C and 25 kPa. Set all kij = 0.
1̂ 2̂
2ji
ij
cij
cijcijcij V
RTZP
2cjci
cij
ZZZ
33/13/1
2
cjci
cij
VVV
)1()( ijcjcicij kTTT
)( 10 BBP
RTB ij
cij
cijij 22111212 2 BBB
0128.0ˆln 1222111 yB
RTP
987.01̂
0172.0ˆln 1221222 yB
RTP
983.02̂
The ideal solution
• Serves as a standard to be compared:
idii
id MxM
iiid
i xRTGG ln
cf igig yRTGG ln
i
iii
iiid xxRTGxG ln
i
iiMxM
cf. iii yRTGG ln
iP
i
xP
idiid
i xRTG
TGS ln
ii
idi xRSS ln i
iii
ii
id xxRSxS ln PxP ,
iid
iidi P
GP
GV
iid
i VV ii
iid VxV
TxT PP ,
iiiiid
iid
iidi xRTTSxRTGSTGH lnln i
idi HH
i
ii
iid HxH
The Lewis/Randall Rule
iii fRTT ˆln)(
• For a special case of species i in an ideal solution:id
iiid
iidi fRTTG ˆln)(
iiid
i xRTGG ln
iii fRTTG ln)( iii fRTTG ln)(
iiid
i fxf ˆThe Lewis/Randall rule
iidi ˆ
The fugacity coefficient of species i in an ideal solution is equal to the fugacity coefficient of pure species i in the same physical state as the solution and at the same T and P.
Excess properties
• The mathematical formalism of excess properties is analogous to that of the residual properties:
– where M represents the molar (or unit-mass) value of i h d i (
idE MMM
any extensive thermodynamic property (e.g., V, U, H, S, G, etc.)
– Similarly we have:Similarly, we have:
ii
Ei
EEE
dnRTGdT
RTnHdP
RTnV
RTnGd 2
The fundamental excess-property relation
The excess Gibbs energy and theThe excess Gibbs energy and the activity coefficient
• The excess Gibbs energy is of particular interest:idE GGG
iii fRTTG ˆln)(
iiiid
i fxRTTG ln)(
ii
iEi fx
fRTGˆ
ln
ii
ii fx
f̂
The activity coefficient of species i in solution.A factor introduced into Raoult’s law to account for liquid-phase non-idealities.
iE
i RTG lnq p
For ideal solution,
c.f. iR
i RTG ̂ln
1,0 iE
iG
ii
Ei
EEE
dnRTGdT
RTnHdP
RTnV
RTnGd 2
iii
EEE
dndTRTnHdP
RTnV
RTnGd ln2
EE
PRTG
RTV )/(
xTPRT ,
xP
EE
TRTGT
RTH )/(
Experimental accessible values:activity coefficients from VLE data,VE and HE values come from mixing experimentsxP ,
jnTPi
E
i nRTnG
,,
)/(ln
V and H values come from mixing experiments.
i
ii
E
xRTG ln
Important application in phase-equilibrium thermod namics),.(0ln PTconstdx
iii thermodynamics.