CT-CH 1-Lecture Slides For Academic Version
CT-CH 1-Lecture Slides For Academic Version
CT-CH 1-Lecture Slides For Academic Version
Basic Thermodynamics
1.1 First law of thermodynamics
dU = dQ − PdV
dQ dQ
Total change of S: dS = + d ip S >
T T
3 Computerized Thermodynamics for Materials Scientists and Engineers
Hillert and Selleby, 2018
1.2 Second law of thermodynamics
∆Q
Tb Ta
∆Q ∆Q ∆Q(Tb − Ta )
∆Stot =− + = > 0 where Tb > Ta
Tb Ta TaTb
Entropy produced!
Introduce G = U + PV − TS
Rewrite dG = VdP − SdT − Td ip S < VdP − SdT
At constant P and T: dG = −Td ip S < 0
since dipS always
or − dG = Td ip S > 0 positive
D = −(∂G / ∂ξ )T , P = Td ip S / dξ 0 ξeq ξ
FIG.1.1.
Identify D with TdipS/dξ i.e.
TdipS=Ddξ in previous eqs.
D = −(∂G / ∂ξ )T , P = Td ip S / dξ
For a phase transformation between two states,
α −> β, one obtains by integration
The process:
liquid solid ∆G = Gsol-Gliq
Gm(kJ)
Solidification: liq fcc -100
D=Gm(liq)-Gm(fcc)
100
-100
-200
-300
1700 1725 1750
T
A
nA’ nA’’
V1 V2
P’ = ξRT/V1 = P’’=(nA-ξ)RT/V2
dG = VdP − SdT + ∑ µi dN i
and if P, T and Nj are constant
µ k = (∂G / ∂N k ) P ,T , N j
GB ≡ µ B = Gm + x A
dGm Gm .G o
B
dxB
.µ
G A ≡ µ A = Gm − xB
dGm oG . . .G (x )
B
(1 − xB )
dGm
µ .
A
dxB m B
dxB
A
xB 1-xB= xA
A xB B
FIG.1.2.
µB − µ A = Gm .G o
B
∂G ∂G ∂G ∂G
Gm + x A m − m − Gm − xB m − m = .µ
∂xB ∂x A ∂x A ∂xB oG . . .G (x )
B
(1 − xB )
dGm
µ .
A
dxB
∂Gm ∂Gm dGm m B
− = A
∂xB ∂x A dxB xB 1-xB= xA
A xB B
FIG.1.2
α+ β
B
A xαB / β xBβ / α B
FIG.1.3.
A xϕB B
FIG.1.4.
∑ xi dµi − Vm dP + S m dT = 0
Phases
• are often denoted by a Greek letter e.g. α, β, γ.
• are identified by their structure (not composition)
• may be
• stoichiometric i.e. fixed composition e.g. Al2O3, CaO
• line compounds e.g. Al2O3 – Fe2O3
• solution phases e.g. fcc, liquid
A1 B1 L10 L12
A2 B2 D03 L21
In Gibbs-Duhem relation
∑ xi dµi − Vm dP + S m dT = 0
there are c+2 terms (c =# of independent components),
each term consisting of one extensive quantity and one
potential. The two quantitites in a pair are conjugate
variables.
In a one phase system we may vary the potentials in
c+2-1 ways since G-D gives a relation between the
potentials.
In a system with p phases we get the variance, v, (or
degrees of freedom): v = c + 2 - p
FIG 1.5.
or expressed differently
µ H O = µ H + 0.5µO
2 2 2
D = µ H 2 + 0.5µO2 − µ H 2O
FIG 1.6.
FIG 1.7a.
D = x ϕA µ αο
A + x ϕ αο
µ
B B − G ϕ
m
(a + b) D = a ( µ αο
A − o α
G A ) + b ( µ αο o β
B − G B ) − ∆ o ϕ
f G Aa Bb
FIG 1.7b.
∂Q ∂H
CP ≡ =
∂T P ∂T P
Gα ( P, T , N iα , ξ1 , ξ 2 ...)
Such an analytical expression is regarded as a
thermodynamic model from which all thermodynamic
information may be obtained.