Physical Chemistry of Solids
Concepts and formulae II
K.D. Becker,
unter Mitwirkung von M. Schrader, K. Talk
(WS 2007/08)
Contents
II
Thermodynamics of solids
II.1
Lattice energies of ionic crystals
II.2
Vibrational heat capacity of solids
II.3
Phase transformations
II.4
Stability of structures
II.5
Binary phase diagrams
1
Chapter II: Thermodynamics of solids
II.1 Lattice energy of an ionic crystal
Basic reaction to form an ionic crystal
H
M + ( g ) + X − ( g ) ⎯Δ⎯
→ MX ( s )
G
Analogous for other types of bonding
Ar ( g ) → Ar ( s ) (vdW )
C ( g ) → C ( s ) (cov alent )
Cu ( g ) → Cu ( s ) (metallic)
ΔHG can not be measured directly! Usually lattice energies are calculated using the Born–Haber cycle.
a) Born–Haber cycle
Example: M ( g ) + Cl ( g ) → MCl ( s )
+
−
ΔHG(MCl(s))
M+(g) + Cl-(g)
I1(M)
M(g)
- E(Cl)
Cl(g)
ΔHvap
½ D(Cl-Cl)
M(s) + ½ Cl2(g)
ΔH0f(MCl(s))
MCl(s)
Thus, the lattice enthalpy is given by
0
ΔH G = ΔH 0f − ΔH vap
−
1
D (Cl − Cl ) − I1 (M ) + E (Cl )
2
Example: NaCl and AgCl
Steps in the Born – Haber cycle
NaCl
AgCl
[kJ/mol]
[kJ/mol]
107,8
284,6
I1(M): M(g) → M (g) + e (g)
494
732
½ D(Cl-Cl): ½ Cl2(g) → Cl(g)
122
122
-349
-349
-411,1
-127,1
ΔHөvap(M): M(s) → M(g)
+
-
-
-
– E(Cl): Cl(g) + e (g) → Cl (g)
ΔH0f:
M(s) + ½ Cl2(g) → MCl(s)
2
At T=0, ΔHG is identical to the so-called lattice energy UG: UG = ΔHG(T=0)
However, ΔHG(T) differs little from UG and it is customary to use the approximation UG ≅ ΔHG(T=298K).
The two lattice energies/enthalpies in the above example are
U G ≅ ΔH G ( NaCl ( s )) = −786
kJ
mol
kJ
U G ≅ ΔH G ( AgCl ( s )) = −927
mol
Analogous calculations for: MO, MS, …
b) Calculation of the lattice energy of an ionic crystal
Coulomb interaction between positively and negatively charged ions I and j
EC ,ij = −
zi ⋅ z j ⋅ e 2
4 ⋅ π ⋅ ε 0 ⋅ rij
zi , z j > 0
i) Coulomb potential for a single cation interacting with the rest of the crystal (e.g. in NaCl)
EC , +
z+ ⋅ z− ⋅ e2 ⎛
12
8 6
⎞
⋅ ⎜6 −
+
− + ... ⎟
=−
4 ⋅π ⋅ ε0 ⋅ r ⎝
2
3 2
⎠
The term in brackets is the so-called Madelung constant A, which is characteristic for different crystal
structures (e.g. NaCl, A = 1,748), and r is the nearest neighbour distance in the crystal.
The total Coulomb energy of the arrangement of cations and anions on a lattice is given by
E C = −2 ⋅ N L ⋅
1
2
⎛ z ⋅ z ⋅ e2
⋅⎜ + −
⎜ 4 ⋅ π ⋅ ε0 ⋅ r
⎝
2
⎞
⎛
⎟ ⋅ A = −N L ⎜ z + ⋅ z − ⋅ e
⎟
⎜ 4πε 0 ⋅ r
⎠
⎝
⎞
⎟⋅A
⎟
⎠
Note that the total Coulomb energy of an ionic crystal is attractive.
ii) Repulsive potential
ER = B ⋅
1
(Born)
rn
or
⎛ r⎞
E R = b ⋅ exp⎜⎜ − ⎟⎟ (Born – Mayer)
⎝ ρ⎠
n and ρ are empirical constants, which have to be determined from experiment. The factors B and b
can be determined by consideration of the total energy of the crystal
U G = EC + E R
and the fact that
dU G
dr
r =r0
=0
where r0 now is the nearest neighbour distance in the crystal at equilibrium.
3
Using the Born repulsive potential, the Born–Landé equation is obtained:
UG = −
NL ⋅ z+ ⋅ z− ⋅ e 2 ⋅ A ⎛
1⎞
⋅ ⎜1 − ⎟
n⎠
4 ⋅ π ⋅ ε 0 ⋅ r0
⎝
The values for n range from 8 to 10; standard approximation is
n≈9
The Born–Mayer equation for the lattice energy can be found analogously by using the Born-Mayer
repulsive potential:
UG = −
N L ⋅ z+ ⋅ z− ⋅ e 2 ⋅ A ⎛ ρ ⎞
⋅ ⎜⎜1 − ⎟⎟
4 ⋅ π ⋅ ε 0 ⋅ r0
⎝ r0 ⎠
The values for
ρ
r0
range from 0,1 to 0,15; standard approximation is ρ = 34,5 pm.
Comment: n and ρ can be determined experimentally from the compressibility of the crystals.
Excursus
It was so far
Born–Lande:
UG = −
1,389 ⋅ 10 5 ⋅ z + ⋅ z − ⋅ A ⎛
1⎞
⋅ ⎜1 − ⎟
n⎠
r0
⎝
in kJ/mol with r0 in pm
Born–Mayer
UG = −
1,389 ⋅ 10 5 ⋅ z + ⋅ z − ⋅ A ⎛
ρ⎞
⎟
⋅ ⎜⎜1 −
r0
r0 ⎟⎠
⎝
in kJ/mol with r0 and ρ in pm
Kapustinskii found a useful approximate way to calculate the lattice energy of crystals. He noticed that
A
ν
is almost independent of the crystal structure, where ν is the number of atoms per formula unit.
Thus he used the approximation
⎛ A(NaCl ) ⎞
A=⎜
⎟ ⋅ν = 0,874 ⋅ν
2
⎝
⎠
and r0 is approximated by the sum of the ionic radii: r0 = r+ + r−
Example: NaCl
UG = −
1,389 ⋅ 10 5 ⋅ z1 ⋅ z 2 ⋅ 0 ,874 ⋅ ν ⋅ ⎛
34 ,5 ⎞
⎟⎟
⋅ ⎜⎜1 −
r+ + r−
r
+ + r− ⎠
⎝
How good are the results?
Example: NaCl, r0 = 281 pm, n = 9, ρ = 34,5 pm, r++r- = 283 pm
UG [kJ/mol]
Born–Lande
-768,0
Born–Mayer
-758,0
Kapustinskii
-753,3
4
This is to be compared to the result form the Born-Haber cycle, yielding -785,9 kJ/mol.
Remaining discrepancies between the (almost exact) lattice energies from the Born-Haber treatment
and the above indicated calculations are mainly due to the neglect of dispersion (London-) interactions
between the polarizable ions. However, it is to be noted that the simple model calculation using point
charges yields results usually only a few percent off the “correct” values.
II.2 Vibrational heat capacity of solids
The thermal energy of a solid consisting of N atoms and 2 degrees of freedom (kinetic and potential
energy) in 3 dimensions is
1
U = 3⋅ 2 ⋅ N ⋅ ⋅ k ⋅T
2
⇒ CV = 3 ⋅ N ⋅ k
⇒ CV = 3 ⋅ R ≈ 24,94
(Dulong – Petit)
kJ
mol
However, early experiments around 1890 already showed that the heat capacity decreases strongly
with decreasing temperatures.
Generally, the calculation of the heat capacity is possible in 3 steps
i) calculation of the average energy
ii) calculation of the total energy
iii)
ε per 1–dimensional linear oscillator
U = 3 ⋅ N ⋅ ε for a solid consisting of N atoms
⎛ ∂U ⎞
CV = ⎜
⎟
⎝ ∂T ⎠V
a) Einstein theory
All atoms are oscillating independently with the same frequency νE.
1⎞
⎛
2⎠
⎝
n = 0,1,2,...
ε n = ⎜ n + ⎟ ⋅ h ⋅ν E
step i)
∞
ε =
∑
n =0
∞
pn ⋅ ε n =
Nn
∑ 3N ⋅ ε
n
, where 3N is the total number of oscillators and where Nn represents
n =0
the number of oscillators at energy εn.
The Boltzmann distribution predicts that
⎛ ε ⎞
N n = const. ⋅ exp⎜ − n ⎟
⎝ k ⋅T ⎠
5
Making use of 3N =
∞
∑N
n =0
n
, we have
⎛ ε ⎞
exp⎜ − n ⎟
N
k ⋅T ⎠
pn = n = ∞ ⎝
3N
⎛ ε ⎞
∑0 exp⎜⎝ − k ⋅nT ⎟⎠
and obtain for the average oscillator energy
⎛
⎛ εn ⎞
exp
⋅
⎜
⎜−
⎟
_
1
k ⋅T ⎠
⎝
0
ε= ∞
= h ⋅ν E ⋅ ⎜
⎜
⎛ ε ⎞
⎛ h ⋅ν E
∑0 exp⎜⎝ − k ⋅nT ⎟⎠
⎜ exp⎜
⎝ k ⋅T
⎝
∞
∑ε
n
⎞
⎟
1⎟
+
2⎟
⎞
⎟ −1
⎟
⎠
⎠
step ii)
Total vibrational energy of Einstein solid:
⎞
⎛
⎟
⎜
1
1⎟
⎜
+ ⎟
U = 3 ⋅ N ⋅ ε = 3 ⋅ N ⋅ h ⋅ν E ⋅ ⎜
2
⎛ h ⋅ν E ⎞
⎟⎟
⎜⎜ exp⎜
⎟ −1
⎝ k ⋅T ⎠
⎠
⎝
step iii)
Differentiation
⎛
⎛ h ⋅ν E ⎞
⎜
exp⎜
⎟
2
k ⋅T ⎠
⎛ h ⋅ν E ⎞ ⎜
⎛ ∂U ⎞
⎝
CV = ⎜
⎟ ⋅⎜
⎟ = 3⋅ N ⋅ k ⋅⎜
2
⎝ ∂T ⎠V
⎝ k ⋅T ⎠ ⎜ ⎛
⎛ h ⋅ν E ⎞ ⎞
⎜ ⎜⎜ exp⎜ k ⋅ T ⎟ − 1⎟⎟
⎝
⎠ ⎠
⎝⎝
⎛
⎞
⎜ exp⎛⎜ h ⋅ν E ⎞⎟ ⎟
⎛ h ⋅ν E ⎞ ⎜
⎝ k ⋅T ⎠ ⎟
CV = 3 ⋅ R ⋅ ⎜
⎟ ⋅⎜
2 ⎟
⎝ k ⋅T ⎠ ⎜ ⎛
⎛ h ⋅ν E ⎞ ⎞ ⎟
⎜ ⎜⎜ exp⎜⎝ k ⋅ T ⎟⎠ − 1⎟⎟ ⎟
⎠ ⎠
⎝⎝
2
Introducing the Einstein temperature θE according to
h ⋅ν E = k ⋅ θ E → θ E =
h ⋅ν E
k
_
one obtains the following expression for C V
6
⎞
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎛
⎛θ ⎞
⎟
⎜
exp⎜ E ⎟
⎟
T ⎠
⎛ θE ⎞ ⎜
⎝
CV = 3 ⋅ R ⋅ ⎜ ⎟ ⋅ ⎜
2 ⎟
⎝ T ⎠ ⎜⎛
⎛ θE ⎞ ⎞ ⎟
⎜ ⎜ exp⎜⎝ T ⎟⎠ − 1⎟ ⎟
⎠ ⎠
⎝⎝
2
Limiting behaviour:
θE
i) T → ∞,
T
<< 1
1
⎛θ ⎞
CV ≅ 3 ⋅ R ⋅ ⎜ E ⎟ ⋅
2 ≈ 3 ⋅ R (Dulong–Petit)
⎝ T ⎠ ⎛ θE ⎞
⎜ ⎟
⎝T ⎠
2
θE
ii) T → 0,
T
>> 1
⎛θ ⎞
⎛ θ ⎞
C V = 3 ⋅ R ⋅ ⎜ E ⎟ ⋅ exp ⎜ − E ⎟
⎝T ⎠
⎝ T ⎠
2
lim
CV = 0
T →0
Energy at T = 0K, zero point energy
1
3
U (0 ) = 3 ⋅ N L ⋅ ⋅ h ⋅ν E = ⋅ R ⋅ θ E
2
2
b) Debye theory
Instead of isolated atomic oscillators, the Debye theory of heat capacity treats collective vibrations
(lattice vibrations) of the solids. In this model, the lattice vibrations show a distribution (spectrum) of
vibrational frequencies.
The minimum wavelength of a wave travelling in a solid is approximately given by the distance of the
atoms in the lattice
λmin ≤ λ ≤ ∞ ⇒ 0 ≤ ν ≤ ν max = ν D = Debye frequency
According to Debye, the spectrum of lattice vibrations can be approximated by
f (ν ) = α ⋅ν 2
⇒ f (ν ) =
9⋅ N
νD
3
⋅ν 2
0 ≤ν ≤ν D
The energy of the system of oscillators with this frequency distribution is
7
U=
νD
∫ f (ν ) ⋅ εν dν
0
ν
9
9⋅h⋅N D
= ⋅ N L ⋅ h ⋅ν D +
⋅∫
8
ν D3
0
dν
exp⎜
⎟ −1
⎝ k ⋅T ⎠
⎛ ∂U ⎞
⎜
⎟
⎝ ∂T ⎠V
Differentiating
CV =
ν3
⎛ h ⋅ν ⎞
9⋅N ⋅h
ν D3
⋅
h ⋅ν 4
⎛ h ⋅ν ⎞
⋅ exp⎜
⎟
2
k ⋅T
⎝ k ⋅T ⎠
νD
∫
⎛
⎛ h ⋅ν ⎞ ⎞
⎜ exp⎜
⎟ − 1⎟
⎝ k ⋅T ⎠ ⎠
⎝
0
2
dν
Changing integration variables such that x =
3
⎛T ⎞
CV = 9 ⋅ R ⋅ ⎜⎜ ⎟⎟ ⋅
⎝ θD ⎠
hν
, one obtains
kT
θD
x 4 ⋅ exp( x )
∫0 (exp(x ) − 1)2 dx
T
after introduction of the Debye temperature
θD =
h ⋅ν D
k
Limiting cases:
θD
i) T → ∞,
T
x << 1
<< 1
⇒ CV ≈ 3 ⋅ R
3
⎛T ⎞
12
⇒ CV = ⋅ π 4 ⋅ R ⋅ ⎜⎜ ⎟⎟ = a ⋅ T 3 “Debye’s T3 law”
5
⎝ θD ⎠
x→∞
ii) T → 0,
Zero point energy
U ( 0) =
9
⋅ R ⋅θD
8
Heat capacities according to Debye for different Debye temperatures
30
GD
300 K
20
CV
100 K
10
0
0
100
300
200
400
500
T
8
II.3 Phase transitions
Examples:
ZrO2: monoclinic
→
1170°C
tetragonal
→ cubic
2370°C
→ liquid
2680°C
BaTiO2: rhombohedral → orthorhombic → tetragonal → cubic
−90°C
0°C
1130°C
a) Gibbs energy and phase changes
T
H ( T ) = H ( T = 0 ) + ∫ C p dT
0
T
S=∫
0
Cp
T
dT
G = H −T ⋅S
Gα (T * ) = G β (T * )
H α (T * ) − T * ⋅ S α (T * ) = H β (T * ) − T * ⋅ S β (T * )
ΔH αβ = H α (T * ) − H β (T * ) = T * ⋅ ΔS αβ
b) Classification of phase transitions
⎛ ∂G ⎞
⎟ discontinuous at the transition.
⎝ ∂X ⎠
1st order if ⎜
2nd order if
⎛ ∂ 2G ⎞
⎜⎜ 2 ⎟⎟ discontinuous at the transition.
⎝ ∂X ⎠
Higher orders are more complicated.
Note:
dG = Vdp − SdT
⎛ ∂G ⎞
⎛ ∂G ⎞
⎟⎟ = V and ⎜
⎜⎜
⎟ = −S
⎝ ∂T ⎠ p
⎝ ∂p ⎠T
c) Transitions
p
α
dashed arrow: isobaric thermal transition (dp = 0)
solid arrow: isothermal pressure transition (dT = 0)
β
T
9
d) Melting (Lindemann theory)
xc
U = U pot + U kin = ∫ D ⋅ xdx =
0
4 ⋅ π ⋅ν ⋅
6⋅k
1
T
θD ~ m ⋅ 1
M
Vm3
Tm =
2
2
xc2
⋅m
1
⋅ D ⋅ x 2 = 3 ⋅ k ⋅ Tm
2
2
~ θ D2 ⋅ M ⋅ Vm3 ( s )
“At the melting point the crystal shakes itself into pieces.”
e) Phase diagram
p–T, pure substances
II.4 Stability of structures
Alternative attemps (non-thermodynamic) to determine and predict the stability of structures.
a) Different attempts to determine the radii of ions in crystals and to determine critical radius ratios
b) Mooser–Pearson–Plots
The “medium main quantum number” is
∑i ν i ⋅ ni
n=
∑i ν i
c) Ionicity
The bonding wave function of the molecule AB is
Ψb = a ⋅ ΨA + b ⋅ ΨB
a ionicity, f, with 0 ≤ f ≤ 1 can for example be defined as
f =
a 2 − b2
a 2 + b2
For solids, Phillips and van Vechten (PVV) define an ionicity fi according to f i =
where fi = 0 means fully covalent, fi = 1 means fully ionic.
10
Ei
Eg
2
The gap energy Eg is composed of an ionic and a covalent part: E g
= Ei2 + Ec2
There is a critical ionicity fi for the change from (ionic) six–fold coordination to (covalent) four–fold
*
coordination: f i = 0,785.
II.5 Binary phase diagrams
a) Binary mixtures
The Gibbs energy is
⎛ ∂G ⎞
⎟⎟
G = ∑ ni ⋅ μi where μi = ⎜⎜
= μi0 + R ⋅ T ⋅ ln ai with activity ai ( 0 ≤ ai )
∂
n
i
⎝ i ⎠ p ,T ,n j ≠i
ai = fi ⋅ xi
fi is the activity coefficient.
aB
Example: mixture A,B
Raoult
pos. deviation
Limiting cases:
Raoult:
xB → 1
μi =
μi0
aB → xB
fi → 1
Henry
+ R ⋅ T ⋅ ln xi
A
Henry:
xB → 0
a B = f i H ⋅ xB
μi = μi0 + R ⋅ T ⋅ ln fi H ⋅ xi
Gibbs energy of mixing (pure substances, nA, nB, n = nA + nB)
B
B
ΔGmix = G( end = mixture ) − G( start = unmixed pure A & B )
(
= n A ⋅ μ A + nB ⋅ μ B − n A ⋅ μ A0 + nB ⋅ μ B0
)
= R ⋅ T ⋅ (n A ⋅ ln a A + nB ⋅ ln aB )
ΔGmix = R ⋅ T ⋅ (x A ⋅ ln a A + xB ⋅ ln a B ) < 0
b) Ideal mixtures
fi ≡ 1
μi = μi0 + R ⋅ T ⋅ ln xi
11
neg. deviation
0
XB
1
B
ΔGmix = R ⋅ T ⋅ (x A ⋅ ln x A + xB ⋅ ln xB ) < 0 (spontaneous)
⎛ ∂ΔGmix ⎞
⎟⎟ = − R ⋅ (x A ⋅ ln x A + xB ⋅ ln xB ) > 0
ΔSmix = −⎜⎜
⎝ ∂T ⎠ p
ΔH mix = ΔGmix + T ⋅ ΔS mix = 0
⎛ ∂ΔGmix ⎞
⎟⎟ = 0
ΔVmix = ⎜⎜
⎝ ∂p ⎠T
c) Statistical models of mixtures: ideal mixtures
The entropy of mixing consists of a configurational and a vibrational part:
S mix = Sconfig + Svib
If only the configurational part is taken into account:
Smix = Sconfig = k ⋅ ln W
(Boltzmann equation)
W is the number of distinguishable arrangements (configurations) of the atoms on the lattice.
NA atoms of type A,
NB atoms of type B,
B
N (= NA + NB) particles on N sites.
B
W=
N!
N A! ⋅ N B !
So that
S mix = k ⋅ ln W = k ⋅ ln
N!
= k ⋅ (ln N ! − ln N A! − ln N B !)
N A! ⋅ N B !
Example: NA = 3, NB = 1, N = 4
B
→W=4
Stirling approximation:
ln N ! ≅ N ⋅ ln N − N
12
To see how good this approximation works compare the results in the table below:
N
N!
5
120
10
N ln N -N
Δ[%]
4,8
3,0
6
15,1
13
14
492
1134
1130
0,3
3,63·10
250
ln N!
3,2·10
1000
5912
5908
6
100000
0,07
6
1,05·10
1,05·10
~0
S mix = k ⋅ (N ⋅ ln N − N − N A ⋅ ln N A + N A − N B ⋅ ln N B + N B )
= −N ⋅ k ⋅ (x A ⋅ ln x A + x B ln x B )
ΔS mix = S mix − N A ⋅ S A0 − N B ⋅ S B0
W0 =1
S A0 = k ⋅ lnW 0
ΔS mix = − R ⋅ (x A ⋅ ln x A + xB ln xB )
ΔGmix = Gmix − n A ⋅ G A0 − nB ⋅ GB0
(
Gmix = ΔGmix + x A ⋅ G A0 + xB ⋅ GB0 = ΔGmix + GB0 + x A G A0 − GB0
)
Gmix = n A ⋅ G A0 + nB ⋅ GB0 + n ⋅ R ⋅ T ⋅ (x A ⋅ ln x A + xB ln xB )
(
)
(
= n A ⋅ G A0 + R ⋅ T ⋅ ln x A + nB ⋅ GB0 + R ⋅ T ⋅ ln xB
)
= n A ⋅ μ A + nB ⋅ μ B
with
μi = μi0 + R ⋅ T ⋅ ln xi , as expected.
d) Statistical models of mixtures: regular mixtures
ΔH mix ≠ 0
We consider the interactions (bonding energies) εij with the nearest neighbours.
The energy of mixing is
U mix = N AA ⋅ ε AA + N BB ⋅ ε BB + N AB ⋅ ε AB
Nij is the number of bonds between i and j.
Considering an atom of type A with neighbours of type A (Z is the coordination number, xi is the
probability to meet an atom of type i on a lattice site):
N AA = ( x A ⋅ Z + x A ⋅ Z + ... + x A ⋅ Z ) ⋅
1 1
1
= Z ⋅ N A ⋅ x A = Z ⋅ N ⋅ x 2A
2 2
2
13
N BB =
1
Z ⋅ N ⋅ x B2
2
N AB = (xB ⋅ Z + ... + xB ⋅ Z + x A ⋅ Z + ... + x A ⋅ Z ) ⋅
= (N A ⋅ x B ⋅ Z + N B ⋅ x A ⋅ Z ) ⋅
1
2
1
2
= (N ⋅ x A ⋅ x B ⋅ Z + N ⋅ x B ⋅ x A ⋅ Z ) ⋅
1
2
= Z ⋅ N ⋅ x A ⋅ xB
0
0
ΔU mix = U mix − N AA
⋅ ε AA − N BB
⋅ ε BB
with
0
N AA
=
1
Z ⋅ N ⋅ xA
2
and
0
N BB
=
1
Z ⋅ N ⋅ x B for the pure substances a and B, we have
2
1
1
1
⎧1
⎫
ΔU mix = Z ⋅ N ⋅ ⎨ ⋅ x A2 ⋅ ε AA + ⋅ xB2 ⋅ ε BB + x A ⋅ xB ⋅ ε AB − ⋅ x A ⋅ ε AA − ⋅ xB ⋅ ε BB ⎬
2
2
2
⎩2
⎭
1
1
⎧
⎫
= Z ⋅ N ⋅ ⎨ x A ⋅ xB ⋅ ε AB + ⋅ x A2 − x A ⋅ ε AA + ⋅ xB2 − xB ⋅ ε BB ⎬
2
2
⎩
⎭
(
)
(
)
1
⎧
⎫
ΔU mix = Z ⋅ N ⋅ x A ⋅ xB ⋅ ⎨ε AB − ⋅ (ε AA + ε BB )⎬ = Z ⋅ N ⋅ x A ⋅ xB ⋅ Δε = Z ⋅ n ⋅ N L ⋅ x A ⋅ xB ⋅ Δε
2
⎩
⎭
ΔU mix = Z ⋅ N L ⋅ Δε ⋅ x A ⋅ x B = x A ⋅ x B ⋅ Ω
with
Ω = Z ⋅ N L ⋅ Δε
In the model of the regular solution it is assumed that ΔVmix = 0 and that
ΔS mix = ΔS mix (ideal mixture)
Thus ΔH mix = ΔU mix and
ΔGmix = x A ⋅ x B ⋅ Ω + R ⋅ T ⋅ (x A ⋅ ln x A + x B ⋅ ln x B )
14
Activity coefficient
In the model of the regular mixture, the activity coefficients are given by:
2
⎛ (1 − x A )2 ⋅ Ω ⎞
⎞
⎛
⎟ and f = exp⎜ (1 − xB ) ⋅ Ω ⎟
f A = exp⎜
B
⎟
⎜
⎟
⎜
R ⋅T
R ⋅T
⎠
⎝
⎠
⎝
Limiting behaviour
xB → 1
fB → 1
xB → 0
⎛ Ω ⎞
f B → exp⎜
⎟
⎝ R ⋅T ⎠
deviation
deviation
Impurities
∂Gmix
x
= G A0 − GB0 + Ω ⋅ (1 − 2 ⋅ x ) + R ⋅ T ln
∂x
1− x
Limiting behaviour
i) x → 0,
∂Gmix
= lim (ΔG 0 + Ω ⋅ R ⋅ T ⋅ ln x ) → −∞
x → 0 ∂x
x→0
lim
15
ii) x → 1,
x ⎞
⎛
lim ⎜ ΔG 0 − Ω + R ⋅ T ⋅ ln
⎟ → +∞
x →1⎝
1− x ⎠
Conclusion: A pure crystal is unstable against contamination.
Critical upper temperature of demixing TC
For Ω>0, the phase diagram of a regular solution possesses a region of limited miscibility with an
upper critical point of demixing. The critical temperature of demixing is given by:
TC =
Ω
2⋅R
e) More complex binary phase diagrams
Solidus – liquidus regions
Systems with eutectics
Systems with formation of compounds
f) Solid – gas equilibria
Example: 2 A (s) + O2 (g) → 2 AO (s)
The equilibrium constant for this reaction is:
K=
⎛ ΔG R0
a 2 ( AO )
⎜−
exp
=
⎜ R ⋅T
a 2 ( A ) ⋅ [ p( O2 ) / p 0 ]
⎝
⎞
⎟ = p( O2 ) / p 0
⎟
⎠
(
)−1
Where p(O2) is the oxygen partial pressure in equilibrium with A and AO.
Because a(AO) = 1 and a(A) = 1 we have
K = ( p( O2 ) / p 0 )
−1
and
ΔG R0 = RT ln( p( O2 ) / p 0 )
Example:
4 Cu (s) + O2 (g) = 2 Cu2O (s)
ΔGR0(1000 K) = 190360 J/mol ± 1 kJ/mol
p(O2, 1000K ) = 1,1·10–10 bar
16
Literature
Chapter II Thermodynamics of Solids
Lehrbuch der Physikalischen Chemie
G. Wedler
VCH
Physikalische Chemie
P.W. Atkins
Wiley
Festkörperchemie
L. Smart, E. Moore
Vieweg, Braunschweig 2000…..
Thermodynamics in Materials Science
DeHoff
McGraw-Hill 1993
Introduction to the Thermodynamics of Materials
D.R. Gaskell
Taylor and Francis 1995
D.V. Ragone
Thermodynamics of Materials
Wiley 1995
D.A. Porter, K.E. Easterling
Phase Transformations in Metals and Alloys
Chapman & Hall 1997
Materials Thermochemistry
O. Kubaschewski, C.B. Alcock, P.J. Spencer
Pergamon, Oxford 1993
C.H.P. Lupis
Chemical Thermodynamics of Materials
Prentice Hall 1993
Physical Chemistry of Solids
Borg, Dines
Wiley 1992
Einführung in die Festkörperphysik
C. Kittel
Oldenbourg, München 2002…
H. Schmalzried, A. Navrotsky
Festkörperthermodynamik
Verlag Chemie 1982
Physikalische Metallkunde
P. Haasen
Springer, Berlin 1984
17
Thermodynamics of Solids
R.A. Swalin
Wiley 1972
18