Chapter 8Multicomponent Homogeneous Nonreacting Systems: Solutions
Notes onThermodynamics in Materials Science
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-1
Thermodynamics in Materials Scienceby
Robert T. DeHoff(McGraw-Hill, 1993).
Extensive Quantities of theState Functions: F/, G/, H/, S/, U/, V/
Using G/ as example:G G T P n n n nk c( , , , ... ... )1 2
A differential form of G/:
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-2
k
c
k nPTknTnP
dnnGdP
PGdT
TGdG
kjkk1 ,,
/
,
/
,
//
c
kk
nPTkPT dn
nGdG
kj1 ,,
/
,/
and at constant T & P:
A differential form of G/:
Partial Molal Quantities of theState Functions: .
Using as example:
kj nnPTkk n
GG,,
/
A differential form of G/ (use definition of ):
KG
kkkkkk VUSHGF ,,,,,
G
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-3
c
kkk
nTnP
dnGdPPGdT
TGdG
kk1,
/
,
//
c
kkkPT dnGdG
1,
/
and at constant T & P:
A differential form of G/ (use definition of ):kG
Gibbs-Duhem EquationThe contributions of the components sum to the
whole: c
kkknGG
1
/
c c
Differentiating the products on the right yields:
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-4
c
k
c
kkkkk GdndnGdG
1 1
/
n dGk kk
c
10
12
12 Gd
nnGd
For a binary system:
Inspection yields the Gibbs-Duhem equation:
Reference States:F/O, G/O, H/O, S/O, U/O, V/O
• T, S, V, P have absolute values.• F, G, H, U have relative values.• The difference in values between states is unique.• To compare values, use the same reference state.
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-5
• To compare values, use the same reference state.• Superscript O refers to the reference state.
c
kk
Ok
O nGG1
/
• Preferably, for a solution use the pure components in the same phase as the solution as the reference state.
Rule of mixtures.
Rule of Mixtures2211
2
1
/ nGnGnGG OO
kk
Ok
O
21
22
21
11
21
/
nnnG
nnnG
nnG OO
O
2211 XGXGG OOO
OG2X
For binary
For binary
Extensive
Molar
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-6
OG1 X
G2XOG
2X1 2
Mixing Values for SolutionsF/
mix, G/mix, H/
mix, S/mix, U/
mix, V/mix
For solution: Gibbs free energy of mixing.O
somix GGG //ln
/
For component k: Change experienced when 1 mole of k is transferred from its reference state to the given solution.
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-7
Okkk GGG
k
c
k
Okk
c
kkmix nGnGG
11
/
c
kkkmix nGG
1
/and
Contributions of the components add to the whole:given solution.
Mixing Values
OG2XG G
X X00
Osomix GGG //
ln/
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-8
OG1 X
XlnsoG
2X1 2
mixG
2X1 2
Mixing values for SolutionsF/
mix, G/mix, H/
mix, S/mix, U/
mix, V/mix
Differential form:
c
kkmix dnGGd /
c
k
Okkk
Okkkkkmix dGndnGGdndnGGd
1
/
0 0Gibbs-Duhem
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-9
kkkmix dnGGd
1c
kkkkkmix GdndnGGd
1
/
n d Gk kk
c
01
Gibbs-Duhem for mixing:
Total derivative:
Graphical Evaluation of Partial Molal Values
Consider a binary system (alloy):G G X G Xmix 1 1 2 2
121 XX dX dX1 2
d G G dX G dXmix 1 1 2 2
GSubstitute & rearrange:
Note:
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-10
222 1
dXGdXGG mix
mix
111 1
dXGdXGG mix
mix
d GdX
d GdX
G Gmix mix
2 12 1
G1
G2
and
Substitute & rearrange:
X2
Gmix
0 1X1 01
Derivation: Graphical Evaluationof Partial Molal Values
2211 XGXGGmix 1
d G G dX G dXmix 1 1 2 2 2
121 XX dX dX1 2 b3a3GdGG mix 4Rearrange (2)
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-11
212 dX
GdGG mix 4Rearrange (2)
1
22
11 X
XGXGG mix 5Rearrange (1)
211
212 dX
GdXG
XXXG mixmixInsert
(5) in (4)
222 1
dXGdXGG mix
mixYielding
Integration of the Gibbs-Duhem Equation(s)
For a binary system (alloy):X d G X d G1 1 2 2 0
d GXX
d G12
12and
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-12
1
d G G X G X G XX
X
10
1 2 1 1 1 22
2
10
2
2 02
2
2
1
21
X
X
dXdX
GdXXG
Now integrate right side:
Integrating the left side from X2 = 0 to X2:
Molar Values of the State FunctionsNote: c
kk
kk
n
nX
1
dG G dXk kk
c
1d G G dXmix k k
k
c
1
Then,
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-13
k 1
G G Xk kk
c
1
X dGk kk
c
10
k 1
G G Xmix k kk
c
1
X d Gk kk
c
10
Chemical Potential of (Open) Multicomponent Systems
U U T P n n n nk c( , , , . . . . . . )1 2
dU TdS PdV dnk kk
c
1
U c
termsc 2
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-14
kk S V n n
Un
j k, ,
W dnk kk 1
kk S V n k S P n k T V n k T P n
Un
Hn
Fn
Gn
j j j j, , , , , , , ,
Chemical Potential of (Open) Multicomponent Systems
kj nnPTkkk n
GG,,
kkk TT
GS kkkkk T
TSTGH
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-15
kk nPnPk TT ,,
kk nT
k
nT
kk PP
GV,,
knPkkkk T ,
kk nP
k
nT
kkkkk T
TP
PVPHU,,
knT
kkkkk P
PSTUF,
Activities and Activity Coefficients
Definition of activity, ak (dimensionless):
kkOkk aRTln
O XRTlnXaDefinition of activity coefficient, k (dimensionless):
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-16
kkkOkk XRTlnkkk Xa
If k<1 ak<Xk k is less apparent than its mole fraction.k=1 ak=Xk k is as apparent as its mole fraction.k>1 ak>Xk k is more apparent than its mole
fraction.
Ideal SolutionNo heat of mixing. 0mixH0kH
0mixV0kV
0mixU0kUc
Entropy increases.
No change in internal energy.No volume change.
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-17
kk XRS ln k
c
kkmix XXRS ln
1
kk XRTF ln k
c
kkmix XXRTF ln
1
kk XRTG ln k
c
kkmix XXRTG ln
1
Gibbs free energy decreases.
Helmholtz free energy decreases.
Ideal Solution
mixGmixS
mixH
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-18
T2X 2X
mix00
Ideal Solution
• All plots (e.g. Gmix vs. Xk) are symmetrical with composition.
• Slopes of plots of Smix, Fmix, Gmix are infinite at Xk =0 & Xk =1.
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-19
infinite at Xk =0 & Xk =1.• Entropy of mixing is independent of
temperature.
Ideal SolutionActivity is the same as mole fraction.
Activity coefficient is one.1
kk Xa
1
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-20
a2 a1
X20
1k
0 1
Slope = 1
Dilute Solutions:Raoult & Henry’s Laws
Raoult’s Law for the solvent in dilute solutions:1111
lim XaX
lim Xa O
Henry’s Law for the solute in dilute solutions:1slope 1slope
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-21
2202
lim Xa O
X
20 XsmallforOOslope 2
2a
2X
1a
2X
Oslope 1
1slope 1slope
Real Solutions: Relation of Activity Coefficient to Free Energy
kkk Xa
kkkkkk XRTRTXRTG lnlnlnID
kXS
kkk GGG
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-22
Ideal partial molal free energy of mixing:kkkk GGG
kID
k XRTG ln
kXS
k RTG lnExcess partial molal free energy of mixing:
Regular SolutionHeat of mixing is a function of composition, only.
ckmixmix XXXXHH ,...,..., 21
kk XRS ln k
c
kmix XXRS ln
PTHH mixmix ,Entropy is the same as for ideal solution.
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-23
kk XRS ln kk
kmix XXRS ln1
kkk XRTUF ln k
c
kkmixmix XXRTUF ln
1
kkk XRTHG ln k
c
kkmixmix XXRTHG ln
1
Gibbs free energy decreases.
Helmholtz free energy decreases.
Regular SolutionmixS
mixGX 2X00
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-24
mixH
mixG2X 2X
T
Regular SolutionmixSmixH
mixGX 2X00
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-25
mixG2X 2X
T
Regular SolutionmixSmixH
mixGX 2X00
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-26
mixG2X 2X
Regular SolutionmixSmixH
mixGX 2X00
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-27
mixG2X 2X
T
Regular SolutionmixSmixH
mixGX 2X00
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-28
mixG2X 2X
T
Problem 8.6 DeHoffmoleJXXH CnPnPn
2500,12)(XfHmix
1222 XaXH
Find: Given:Rewrite in general form:
2211222 2 dXXaXdXaXHd
22 2 XaXaXHdDifferentiate:
Substitute dX1= dX2:
09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) 8-29
2122
2 2 XaXaXdX
121222
2 312 XaXXXaXdX
Hd
Substitute dX1= dX2:
Substitute X1+X2=1:
202
2
1
21
2
2
dXdX
HdXXH
X
X
20 121
21
2
2
31( dXXaXXXH
X
X
Gibbs-Duhem: