Contents for previous class
• 상, 상태도, 상변태에대한이해
• 열역학계와열역학법칙에대한이해
• Gibbs 와 Helmholtz Free Energy의유도
• 평형의개념
• 단일성분계의이해와변태거동
• 1차변태, 2차변태와변태시압력효과에관한이해
• 응고구동력과과냉의역할에대한이해
Contents for today’s class
• Ideal solution과 regular solution의이해
• 이상용액과규칙용액에서 Gibbs Free Energy
• Chemical potential과 Activity 의이해
• Real solution의이해
* Binary System (two component) A, B
- Mixture ; A – A, B – B ; 각각의성질유지, boundary는존재, 섞이지않고기계적혼합
A B
- Solution ; A – A – A ; atomic scale로섞여있다. Random distribution
A – B – A Solid solution : substitutional or interstitial
- compound ; A – B – A – B ; A, B의위치가정해짐, Ordered state
B – A – B – A
* Single component system One element (Al, Fe), One type of molecule (H2O): 평형상태압력과온도에의해결정됨
: 평형상태온도와압력이외에도조성의변화를고려
- Solution ; A – A – A ; atomic scale로섞여있다. Random distribution
A – B – A Solid solution : substitutional or interstitial
• Solid solution:– Crystalline solid – Multicomponent yet homogeneous – Impurities are randomly distributed throughout the lattice
• Factors favoring solubility of B in A– Similar atomic size: ∆r/r ≤ 15%– Same crystal structure for A and B – Similar electronegativities: |χA – χB| ≤ 0.6 (preferably ≤ 0.4)– Similar valency
• If all four criteria are met: complete solid solution• If any criterion is not met: limited solid solution
: 합금에대한열역학기본개념도입위해 2원고용체, 1기압의고정된압력조건고려
* Composition in mole fraction XA, XB XA + XB = 1
1. bring together XA mole of pure A and XB mole of pure B
2. allow the A and B atoms to mix together to make a homogeneous solid solution.
Gibbs Free Energy of Binary Solutions
Gibbs Free Energy of The SystemIn Step 1
- The molar free energies of pure A and pure Bpure A; pure B;
;XA, XB (mole fraction)
),( PTGA
),( PTGB G1 = XAGA + XBGB J/mol
Free energy of mixture
Gibbs Free Energy of The System
In Step 2 G2 = G1 + ΔGmix J/mol
Since G1 = H1 – TS1 and G2 = H2 – TS2And putting ∆Hmix = H2 - H1 ∆Smix = S2 - S1
∆Gmix = ∆Hmix - T∆Smix
∆Hmix : Heat of Solution i.e. heat absorbed or evolved during step 2∆Smix : difference in entropy between the mixed and unmixed state.
How can you estimate ΔHmix and ΔSmix ?
Mixing free energy ΔGmix
가정1 ; ∆Hmix=0 :; A와 B가 complete solid solution
( A,B ; same crystal structure); no volume change
- Ideal solution
wkS ln= w : degree of randomness, k: Boltzman constant
thermal; vibration ( no volume change )Configuration; atom 의배열방법수 ( distinguishable )
ΔGmix = -TΔSmix J/mol
∆Gmix = ∆Hmix - T∆Smix
Entropy can be computed from randomness by Boltzmann equation, i.e.,
= +th configS S S
)]ln()ln()ln[(
, ,
1ln!!)!(ln
000
oBoBoBoAoAoAooo
BABBAA
BA
BAbeforeaftermix
NXNXNXNXNXNXNNNk
NNNNXNNXN
kNNNNkSSS
−−−−−=
=+==→
−+
=−=Δ
이므로
NNNN −≈ ln!ln
Excess mixing Entropy
),_(_ !!)!(
)__(_ 1
BABA
BAconfig
config
NNsolutionafterNNNNw
pureBpureAsolutionbeforew
→+
=
→=If there is no volume change or heat change,
Number of distinguishable way of atomic arrangement
0kNR =using Stirling’s approximation
and
Chemical potential
= μA AdG' dn
⎛ ⎞∂μ = ⎜ ⎟∂⎝ ⎠ B
AA T, P, n
G'n
The increase of the total free energy of the system by the increase of very small quantity of A, dnA, will be proportional to dnA.
(T, P, nB: constant )
μA : partial molar free energy of A or chemical potential of A
⎛ ⎞∂μ = ⎜ ⎟∂⎝ ⎠ A
BB T, P, n
G'n
Chemical potential
= μ + μA A B BdG' dn dn
= − + + μ + μA A B BdG' SdT VdP dn dn
for A-B binary solution, dG’
for variable T and P dG = -SdT + VdP
Chemical potential
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂= μ = = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠j j j j
ii i i iT,P,n S,V,n S,P,n T,V,n
G U H Fn n n n
• The chemical potential is the change in a characteristic thermodynamic state function (depending on the experimental conditions), the characteristic thermodynamic state function is either: internal energy (U), enthalpy (H), Gibbs free energy (G), or Helmholtz free energy (F)) per change in the number of molecules.
From Wikipedia, the free encyclopedia
= − + μ
= + + μ
= − − + μ
= − + + μ
∑∑∑∑
i i
i i
i i
i i
dU TdS PdV dn
dH TdS VdP dn
dF SdT PdV dn
dG SdT VdP dn
Ideal solution : ΔHmix= 0
Quasi-chemical model assumes that heat of mixing, ΔHmix,is only due to the bond energies between adjacent atoms.
Structure model of a binary solution
Regular Solutions
∆Gmix = ∆Hmix - T∆Smix
Bond energy Number of bond
A-A εAA PAAB-B εBB PBBA-B εAB PAB
= ε + ε + εAA AA BB BB AB ABE P P P
Δ = ε ε = ε − ε + εmix AB AB AA BB1H P where ( )2
Internal energy of the solution
Regular Solutions
Δ =mixH 0
Completely random arrangement
ideal solution
=AB a A B
a
P N zX X bonds per moleN : Avogadro's numberz : number of bonds per atom
∆Hmix = ΩXAXB where Ω = Nazε
Δ = εmix ABH P
0
Regular Solutions
=ε )(21
BBAAAB εεε +=
↑→< ABP0ε ↓→> ABP0ε
0≈ε
Ω >0 인경우
Regular Solutions
Reference state
Pure metal 000 == BA GG
)lnln( BBAABABBAA XXXXRTXXGXGXG ++Ω++=
G2 = G1 + ΔGmix ∆Gmix = ∆Hmix - T∆Smix
Regular Solutions
)lnln( BBAABABBAA XXXXRTXXGXGXG ++Ω++=
BBBB
AAAA
XRTXG
XRTXG
ln)1(
ln)1(2
2
+−Ω+=
+−Ω+=
μ
μ
Free Energy Change on Mixing
ABBABA XXXXXX 22 +=
1−= μ + μA A B BG X X Jmol
Activity, aideal solution regular solution
• μA = GA + RTlnaA μB = GB + RTlnaB
⎛ ⎞ Ω= −⎜ ⎟
⎝ ⎠B
BB
aln (1 X )X RT
γ = BB
B
aX
μ = +μ = +
A A A
B B B
G RTln XG RTln X
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