Thermodynamics of Multi-component Systems
Consider a binary solid solution of A and B atoms:
Let nA= # of moles of A
nB= # of moles of B
def:
mole(or atom) fraction
nA
xA 
nB  nA
;
xB 
nB
nB  nA
Consider nA moles of A and nB moles of B. Before they are mixed:
G1  xAGA  xBGB
GB
G1
GA
0
XB
1
Variation of the free energy
before mixing with alloy composition.
XA moles of A
XB moles of B
G1  X AGA  X BGB
The Free Energy of the system changes on mixing
MIX
fixed T
XA moles of A
XB moles of B
1 mole of solid solution
G2 G1 Gmix
G1  X AGA  X BGB
G2  G1  Gmix
Since
Defining
G1  H1  TS1
G2  H 2  TS 2
H mix  H 2  H1
Gmix  H mix  T S mix
Smix  S2  S1
Lets consider each of the terms
Hmix -
recall that for condensed systems ΔH ≈ ΔU and so
Δ Hmix= heat of solution ≈ change in internal
energy before and after mixing.
Smix -
change in entropy due to mixing. Take the mixing to be
“perfect”i.e., random solid solution
Smix = (molar) Configurational entropy
Smix  k ln W
Boltzmann’s Eqn.
W = # of distinguishable ways of arranging the atoms → “randomness”
The number of A atoms and B atoms in the mixture is:
NA = nANa
NB = nBNa
Na , Avogadro’s number
From combinatorial mathematics: W 
 N A  NB 
N A NB !
!
ln W can be approximated using Stirling’s approximation, ln N !  N ln N  N
and using, Nak = R, where R is the gas constant, we obtain,
S mix   R  X A ln X A  X B ln X B 
Let’s Re-examine the Hmix term:
2 models
(a) Ideal solution model
(b) Regular solution model
Ideal Solutions
Hmix = 0
Physically this means the A atoms interact with the B atoms as if
they are A and vice versa.
,
The only contribution to alterations in the Gibbs potential is in the
configurational entropy i.e.,
Gmix  T Smix  RT ( X A ln X A  X B ln X B )
Examples:
(a) Solution of two isotopes of the same element
(b) Low pressure gas mixtures
(c) Many dilute ( xA << xB or xB << xA) condensed phase
solutions.
Recall that the total free energy of the solution
G2  G  G1  Gmix  X AGA  X BGB  RT  X A ln X A  X B ln X B 
G
GB
G1
Low T
GA
0
XB
0
1
XB
-TSmix
High T
Gmix
Gmix
1
Regular Solutions
Assume a random solid solution and consider how the A&B
atoms interact.
MIX
fixed T
XA moles of A
XB moles of B
G1  X AGA  X BGB
1 mole of solid solution
G2 G1 Gmix
In a general the interaction of an A atom with another A atom
or a B atom depends upon
(i) interatomic distance
(ii) atomic identity
(iii) 2nd, 3rd, … next –near-neighbor identity and distances.
The Regular Solution model assumes only nearest-neighbor
interactions, pairwise.
“Quasichemical” model
Assume the interatomic distance set by the lattice sites.
Let :
VAA  A  A
Bond energy
VBB  B  B
Bond energy
VAB  A  B
Bond energy
Note that all the V’s are < 0.
Consider a lattice of N sites with Z nearest-neighbors per site. :
Each of the N atoms has Z bonds so that there are
NZ
2
bonds in the lattice.
Division by 2 is for double counting.
Let PAA be the probability that any bond in the lattice is an A-A bond:
1
 NZPAA
2
then
N AA
and
PAA  x A2
so
N AA
N BB
1
 NZx A2
2
1
 NZxB2
2
N AB
1
1
 NZx A xB  NZxB x A
2
2
N AB  NZxA xB
The energy for the mixed solution is
H mix 
1
NZ  x A2VAA  xB2VBB  2 x A X BVAB 
2
prior to mixing
1
H1  NZ  xAVAA  xBVBB 
2
and
H mix
1
 NZ  xA2VAA  xB2VBB  2 x A xBVAB  x AVAA  xBVBB 
2
combining terms & using xA + xB = 1
Hmix  ZxA xB VAB  1 (VAA  VBB )

2

Hmix  xA xB where
  Z[VAB 1/ 2(VAA  VBB )]
Notice the ΔHmix can be either positive or negative.
ΔHmix > 0,  > 0 and VAB > 1/2 (VAA+ VBB)
from ΔGmix = ΔHmix – TΔSmix
At low temps clustering of As and Bs result, i.e., “phase separation”.
for ΔHmix < 0,  < 0 and VAB < 1/2 (VAA+ VBB) The A atoms are
happier with B atoms as nearest-neighbors.
Short range ordering i.e., PAB is
increases over the random value.
For a Regular Solution
Gmix  xA xB  RTxA ln xA  xB ln xB 
 H mix  TSmix
Variation of ΔGmix with composition
 < 0
0
ΔGmix
xB →
–TΔSmix
1
ΔHm
ΔGm
> 0
T < TC
Note
ΔHmix
0
ΔGmix
ΔGmix
xB →
–TΔSmix
1
2 minima
change in curvature
2 pts of inflection.
 2Gmix
0
 inf pts.
2
 xB
Gmix
0
 extrem um pts.
 xB
As T increases, the –TΔS term begins to dominate.
0
XB →
ΔGmix
1
T < Tc
T = Tc
the critical temperature:
Gmix extremum  Gmix inflection.pts.
The inflection pt.
& extremum merge