Thermodynamics and Kinetics of Solids - Technische Fakultät · 2003-11-05 · Thermodynamics and...

Thermodynamics and Kinetics of Solids 1 ________________________________________________________________________________________________________________________

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Thermodynamics and Kinetics of Solids

Prof. Dr. W. Weppner Chair for Sensors and Solid State Ionics

Faculty of Engineering Christian-Albrechts-University, Kiel

Arrangements of atoms consist in equilibrium of a single or several homogeneous and physically different regions (phases). These are distinguishable in the case of differ-ent states of aggregation, structural arrangements or com-positions. We will investigate the question “Why does a specific phase or a phase change occur?” An answer is given by the formal theory of equilibrium (phenomenological thermodynamics) developed by Gibbs by using microscopic thermodynamic parameters. These may be determined in principle according to the laws of (quantum-) mechanics using the properties of individual atoms. Since these equations include approximately 1022 variables, drastic simplifications are necessary (statistical mechanics). In the case of solids, the calculation of the binding energy is a difficult problem. Transformation

heats of metals are, for exa mple , only of the order of 1 % of this binding energy. In spite of the not very realistic statistical models, the theory has contributed in a major way to the understanding of phenomena such as order–disorder transitions. Another fundamental question is „What is the mechanism of the transformation?“ The kinetic models make use of atomic processes for the description by phenomenological parameters. Fortu-nately, the kinetic properties are not very sensitive with regard to the nature of the binding forces. Deviations from the ideal structure are of large impor-tance since these influence the chemical activities and transport properties in a major way. .

I. Phenomenological Thermodynamics of Solids

1. Thermal Materials Properties

Knowledge of the thermal properties is essential for the understanding of the mechanical failure of materials, for heating processes or for the selection of materials with fast heat transfer, as e.g. in the case of microelectronics. 1.1. Heat Capacity and Specific Heat With the transfer of heat, atoms gain thermal energy and vibrate with an increased amplitude and frequency. This vibration is transferred to the neighbouring atoms and generates an elastic wave, a “phonon”. The energy of the phonon may be described by the wavelength or the fre-quency.

E = hcλ

= hν (1.1)

The number of phonons which is required to increase the temperature of a material by 1 K is expressed by the heat capacity or specific heat. The heat capacity Cp or CV is the necessary energy to increase the temperature of 1 mol of the material by 1 K

Fig. 1.1. Heat capacity as a function of temperature for metals

and ceramic materials

2 Thermodynamics and Kinetics of Solids ________________________________________________________________________________________________________________________

15.10.01

at constant pressure or constant volume, respectively. For sufficiently high temperatures: C p = 3R = 25 J/mol K (1.2)

For sufficiently high temperatures Dulong-Petit’s law holds: C p = 3R = 25 J/mol K

(R: general gas constant). This value of Cp is reached for metals already at room temperature, while that is only reached above 1000 °C for ceramic materials (Fig. 1.1). The specific heat c is the energy to heat a specific mass of a material by 1 K:

c = Heat Capacity

Atomic Weight

The specific heat depends significantly on the structure of the material; changes of the dislocation density, grain size or point defects have only a small influence. The largest influence comes from the lattice vibrations (phonons), but also other factores may be important, e.g. the transition from the normally directed magnetic moments in ferro-magnetic iron to disordered magnetic moments in paramagnetic iron at the Curie temperature (Fig. 1.2). 1.2. Thermal Expansion An increase in thermal energy increases the vibration of the atoms and generates a larger average distance be-

tween the atoms. The change in length ∆l per initial length lo at a temperature increase ∆T is the „linear thermal expansion coefficient“

α =∆l

l0∆T? (1.3)

Analogously, a volume coefficient of thermal expansion (αv) is defined. In the case of isotropic materials, αv = 3α. The thermal expansion coefficient depends on the strength of the atomic bond. Also for that reason, materials with low thermal ex-pansion coefficients have high melting points. Compared to metals, most ceramic materials have lower expansion coefficients because of strong ionic and covalent bonds. Certain glasses, e.g. quartz, have a lower packing factor which allows to add thermal energy with low volume changes. In spite of the fact that the bonds within polymer chains are

Fig. 1.2. Influence of the temperature of the specific heat of iron

Table 1.1. Linear expansion coefficient at room temperature


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covalent, the secondary bonds that keep the chains to-gether are weak which results in large expansion coeffi-cients. (Table 1.1).

Allotropic phase transformations show abrupt changes in the dimension. These abrupt changes result often in frac-tures. Linear expansion coefficients change continuously with temperature (Fig. 1.4). Interactions by electrical or mag-netic fields, which may be generated by magnetic do-mains, prevent the normal expansion up to temperatures above the Curie point. The expansion of Invar is for that reason very small at low temperatures because of the magnetic properties (cf. Table 1.1 and Fig. 1.5). If a material becomes restricted in the expansion during heating, a thermal pressure is generated which is related to the thermal expansion coefficient σ, the elasticity con-stant E and the temperature change ∆Τ: σ therm = αE∆T (1.4)

In the case of combinations of materials, e.g. in the case of covering layers (enamel layer of bathtubs, zirconia on top of turbine blades), temperature changes may result in contractions or expansions in different amounts. The thermal properties of the covering layer and the substrate material have to be carefully adopted. The same holds for composits; phases with lower expansion coefficients than that of the matrix may be expanded at temperature in-creases beyond the fracture point. Thermal pressures may be also generated if the tempera-ture is not uniform. By the annealing of glass, the faster cooling of the surface prevents a later contraction of the interior. A compression of the surface layer is generated.

1.3. Thermal Conductivity

The thermal conductivity K correlates the heat Q trans-ferred through a plane of size A with a temperature gradi-ent ∆T/∆x (Fig. 1.6):

QA

= Κ∆T∆x

(1.5)

Κ plays the same role as the diffusion coefficient D in the case of matter transport. Values for K are compiled in Table 1.2.

The transport of thermal energy occurs by two mecha-nisms: i) Motion of electrons

Fig. 1.3. Relationship between the linear expansion coefficient

of metals at 25 °C and the melting temperature

Fig. 1.5. Expansion coefficient as a function of temperature.

Fig. 1.4. Linear expansion coefficient of iron as a function of

temperature.


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Valence electrons gain energy, move into cooler zones of the material and transfer the energy to atoms in this re-gion; the amount of transferred energy depends on the number of excited electrons and their mobility. ii) Lattice vibrations The transfer of energy occurs by the thermally generated vibrations of the atoms. For the various classes of materials different mechanisms are predominant: Metals: The thermal conductivity is mainly caused by electrons. The thermal and electrical conductivity are correlated to each other:

Κ

σT= L = 2.3 x 10 -8 J Ω

s K2 (1.6)

σ is the electrical conductivity, L is the Lorentz constant. With increasing temerature, two factors influence the thermal conductivity: the higher energy of the electrons and the increasing contribution of lattice vibrations result in an increase; the larger scattering of the electrons in view of the larger lattice vibrations causes a decrease. This combination results in a quite different behaviour of different metals. The thermal conductivity of metals depends also on lat-tice defects, microstructure and the preparation method. Ceramic materials: In view of the small number of elec-trons, the thermal conductivity depends mainly on the phonons; the heat conduction of ceramic materials is mostly much smaller than in the case of metals. Glasses show lower thermal conductivity since the loosely packed amorphous structure offers only few points at which the silicate chains may have contact. With increasing temperature the heat conduction increases since higher energetic phonons are generated.

Crystalline ceramic materi-als show a higher thermal conductivity because of the lower scattering of the phonons. With increasing temperature, the scattering becomes larger and the thermal conductivity de-creases (see, e.g., Al2O3 and SiC in Fig. 1.7.).

Materials with dense packed structures and high elasticity constants form high energy phonons which results in high thermal conductivity. Lattice de-fects and porosity increase the scattering and reduce the heat conduction.

Some ceramic materials have similar thermal con-ductivities as metals; AlN und SiC are good thermal conductors, but neverthe-

Fig. 1.6. Heat transport in a temperature gradient

Table 1.2. Thermal conductivities of various materials


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less electrical isolators. These materials are therefore very useful for electronic applications.

Semiconductors: The thermal conductivity occurs both by phonons and by electrons. At low temperatures, phonons provide the main contribution. However, at higher tem-peratures electrons are sufficiently excited to increase the thermal conductivity in a major way. Polymers: The thermal conductivity is small since the energy is transferred by vibrations and motions of the molecular polymer chains. Increases in the degree of polymerisation, the crystallinity, minimization of branch-ing and extensive formation of connections result in higher thermal conductivities. An example is the applica-tion of polystyren and polyurethan for coffee cups. 1.4. Thermal Shock

Temperature gradients result in differently large expan-sions in different regions. In the case of such tensions, fractures may occur. The thermal shock is influenced by i) the thermal expansion coefficient: Smaller values

minimize the changes, ii) the thermal conductivity: Higher values favor the heat

transfer, iii) the elasticity constant: Lower values allow large de-

formations before critical tensions occur iv) the phase transformations: Structural changes may

result in additional changes in the dimension. The thermal shock sensitivity is determined by the maxi-mum temperature change that may be tolorated per sec-ond during quenching without changing the mechanical properties of the material. Some thermal shock resis-tances are Quartz glass: 3000 °C Sialon (Si3Al3O3N5): 950 °C PSZ, Si3N4: 500 °C SiC: 350 °C Al2O2, common glass: 200 °C

Thermal shock parameter: = σ f ΚEα

(1.7)

σf = fracture tension. 2. Equilibrium and Stability

A “state” is described by n (in simple cases 2) indepen-dent variables (e.g. V und T). If

dY x1 ,x 2, ...., x n( )= a1dx1 + a 2dx2 + .... + a n dxn (2.1)

is a total differential ("Pfaff’s differential”) of the quan-tity Y, i.e. Y is definded by a1(x1, x2, ...., xn), a2(x1, x2, ...., xn), ... , Y is called a quantity of state. In that case it is necessary that the integral

dY= a1dx1 + a2dx2 + ...a ndx n( )∫∫ = 0 (2.2)

Fig. 1.7. Temperature dependance of the thermal conductivity


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is independent of the path of integration, i.e. the inte-grability conditions

∂ a j

∂ x k

=∂ a k

∂ x j

(2.3)

hold (corresponding to the transformation of the line integral into an area integral according to Stoke’s rela-tion). Y = Y(x1, x2 , ..., xn) is called an equation of state. Example: dp =

∂p∂T

V

dT +∂p∂V

T

dV (2.4)

κ = −V

∂p∂V

T

: isothermal compression module

α =

1V

∂V∂T

p

: isobaric expansion coefficient

1p

∂P∂T

V

=ακp

= β : isochoric pressure coefficient

Other quantities are only given differentially, e.g. the amount of heat added to the system and the work per-formed:

δ Q ≠∫ 0 , δ W ≠ 0∫ (2.5)

However, the difference is given by a total differential: dU = δ Q- δ W (2.6)

(δA > 0, if work is performed by the system). Further contributions may be present, e.g.: i) Work of a galvanic element: δ Q = dU + pdV + ∅ dε (2.7)

According to (2.6) the work is: δ A = ∅ dε + pdV

ii) Magnetisation of a material

δ A = HdM ; A = HdM0

H '

∫ (2.8)

M: magnetic moment of the material H: magnetic field for the generation of the magnetic mo-ment (external field, which is necessary for the magneti-sation, is not considered to contribute to the inner energy U). Inner energy:

dU V, T( ) =∂U∂T

V

dT +∂U∂V

T

dV (2.9)

If dV = 0:

δ A = 0 ; dU = δ Q =∂U∂T

V

dT (2.10)

CV :=δ QdT

V

=∂U∂T

V

: heat capacity (2.11)

(CV

mass = specific heat;

CVmolar number

= molar heat)

Enthalpy: δ Q = dU + pdV = dU + d pV( ) − Vdp (2.12)

= dH - Vdp H = U + pV (2.13)

If dp = 0:

Cp :=δ QdT

p

=∂ H∂ T

p

(2.14)

+ ∅ - Fig. 2.1. Schematic representation of a galvanic cell which

performs electrical work


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=∂

∂ TδQdT

V

dT +∂U∂V

T

dV + pdV

p

= C V +∂U∂V

T

+ p

∂V∂T

p

As a result of the integrability conditions of the total differential of S(U, V) we have

∂U∂V

T

+ p = T∂p∂T

V

.

Accordingly Cp – CV is:

C p − CV = T∂p∂T

V

∂V∂T

p

= TVα2κ (2.15)

(c.f. the ideal gas for comparison: Cp - CV = R). It is al-ways Cp > CV (since κ > 0). Accordingly, it is more efficient to heat a material at constant volume instead at constant pressure since no work is performed at constant volume and all heat energy results in the increase of the inner energy. The difference between Cp und CV origi-nates according to (2.14) from: i) the volume dependence of the inner energy; ii) the work which is performed by the expansion. Comparison of the magnitudes of both effects:

∂U∂V( )

Tp

=Tp

∂p∂T

V−1

because∂U∂V

T+ p = T

∂p∂CT

V( see above)

=Tp

ακ −1 (2.16)

The right hand side is of the order of magnitude of 104 in

the case of solid materials, i.e. the difference Cp - CV is practically only determined by the volume dependence of the inner energy U(for comparison, only the work per-formed is essential in the case of gases). Determination of the equilibrium state of a material that consists of several components from the knowledge of macroscopic thermodynamic quantities. The change in the inner energy is in the case of the varia-

tion of the number of species Ni (atoms, molecules, …):

dU = TdS- pdV+ µii= 1

f

∑ dN i (2.17)

(starting equation of Gibbs systematics). U = U (S, V, Ni) : inner energy S, V, Ni : independent variables T, p, µi : conjugated variables. Change in the independent variables by Legendre trans-formation [dZ(x, y) = Xdx+Ydy ⇒ d(Z-xX) = -xdX+Ydy = dV(x, y)]. The quantities of state derived by Legendre transformation are called „thermodynamic potentials“ because they allow to derive the conjugated variables by differentiation with regard to the corresponding inde-pendent variables (similar to the force being derived from the force potential). With the choice of the potential also the variables are given. The “thermodynamic potentials” loose the property of the potential in the case of selection of other variables. F (T, V, Ni) = U-TS : (Helmholtz‘) free energy H (S, p, Ni) = U+pV : Enthalpy G (T, p, Ni) = H-TS = U+pV-TS=F+pV : Gibbs energy (also called free energy, exergy, Gibbs potential) Selection of the potential according to the choice of inde-pendent variables. Massieu’s potentials: Derivation of an analogue systematics starting from the entropy differen-tial. Considering µi as independent variable: pV potential, "identical vanishing (Guggenheim)" potential.

The eight Maxwell’s relationships are:

V =∂H∂p

S, N i

, V =∂G∂p

T, N i

, T =∂H∂S

p, N i

,

T =∂U∂S

V, N i

, p = - ∂U∂V

S, N i

, p = - ∂F∂V

T, N i

,

S = - ∂F∂T

V, N i

, S = - ∂G∂T

p, N i

In addition we have

µi =∂U∂Ni

S, V, N j

, µi =∂F∂N i

V, T, N j

,


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µi =∂H∂Ni

S, p, N j

, µi =∂G∂N i

T, p, N j

Multiplication of the system corresponds to the multipli-cation of all quantitative variables (Ni, V, S, ...) by the same factor α; intensive variables (p, T, µi, ...) remain unchanged. From

µ i =∂G∂Ni

T, p, N j

it follows that G p,T, αN i( )= α G p,T, N i( ).

Differentiation with regard to α provides

∂G p,T, αNi( )

∂ αNi( ) i

∑∂ αNi( )

∂α= G p,T, Ni( )

With α = 1, we have G = µi∑ Ni Duhem-Gibbs equation (2.18)

In the case of a single component system we have G=µN or

µ =GN

= g (2.19)

By Legendre transformation, the total differential of G results in:

d µiNi − Gi

∑

= SdT - Vdp + Ni∑ dµi := dΩ (2.20)

Ω T, p,µi( )= µ i

i∑ Ni − G : identical vanishing potential.

Because of Ω = 0 it follows that SdT - Vdp + N i

i∑ dµ i = 0 (2.21)

Duhem-Margules‘ equation According to this relationship, the µi’s are not independ-ent of each other: p,T = const: N1dµ1 + N2 dµ2 = 0 (2.22)

(2-component system)

or in molar fractions

γ 1∂µ 1

∂γ 1

+ 1 − γ 1( )∂µ2

∂γ 1

= 0 (2.23)

(2-component system) For a system of f components, the total differential of the quantity of state µi (p,T,γk) is:

dµ i =∂µ

i

∂γ k

dγ kk=1

f -1

∑ +∂µ

i

∂pdp +

∂µi

∂TdT (2.24)

In the case of p, T = const., the Duhem-Margules‘ rela-tion yields:

Ni∑ dµi = N ii

∑∂µ

i

∂γ kk=1

f -1

∑ dγ k = 0

Since the γk are independent variables, this equation has to hold for each dγk individually (Ni → Ni / ∑ Ni = γi ):

γ ii

∑ ∂µi

∂γ k

p, T, γ j≠ k

= 0 (2.25)

Duhem-Margules‘ conditions These conditions hold for all concentration changes of µi at constant p, T for a single phase.

Mixtures. i) Without chemical reactions: T, Ni = const.; i=1, ..., f. Because of G = ∑µiNi the total volume V of the system is

V =∂G∂p

T, N i

= Nii∑ ∂µ i

∂p

T, γ k

(2.26)

:= N i

i∑ vi

*

with v i* p,T, γ k( )=

∂µ i

∂p

T, γ k

: specific molar volume.

Before the mixture the total volume was

Vbefore = N ii

∑ vi = N ii

∑ ∂gi p,T( )∂p

since V =∂G∂p

T

(2.27)


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An ideal mi xture exists if the chemical potential may be presented by µ i = g i p,T( )+ RT ln γ i (2.28)

Then we have

v i* =

∂µ i

∂p

T, γ k

=∂gi p,T( )

∂p

T

= vi (2.29)

No volume contraction or expansion occurs in the case of an ideal mixture.

Generally, A is considered to be a quantitativ quantity, which may be presented by A = N i∑ ai

* in the case of a

mixture:

a i* =

∂A∂Ni

: specific molar quantity.

Because of G = ∑Niµi we have g i

* = µ i p,T, γ k( ) : chemical potential

Because of H = G + TS =G -T∂G∂T

= N i∑ µi − T N i∑ ∂µ i

∂T

we have

h i* p,T, γ k( )= µi − T

∂µi

∂T : specific molar enthalpy

Because of U = H - pV = N i∑ hi

* − p N i∑ v i* we have

u i* p,T, γ k( )= µi −T

∂µ i

∂T− p

∂µ i

∂p

: specific molar inner energy

Because of s = -∂G∂T

= − N i∑ ∂µ i

∂T we have

s i* p, T, γ k( )= −

∂µi

∂T : specific molar entropy

These specific molar quantities are related to the chemi-cal potential of the component i in the same way as the corresponding quantities of state with the Gibbs energy. The chemical potential of the i-th component of the mi x-ture of ideal gases is

µ i = g i p,T( )+ RT ln γ i (2.30)

gi (p,T) : Gibbs energy of the pure i-th component

Definition of an ideal mixuture (gas, liquid, solid state): The chemical potential of the i-th component of the mi x-ture may be described by (2.30). There is no volume contraction or volume expansion (only entropy of mi x-ture occurs). The general assumption for µi is: µ i = g i p,T( )+ RT ln a i (2.31)

ai is called activity of the component i. The entropy of mixture of an ideal mixture with any number of components is

S = -∂G∂T

p, Ni

= − N i∑ ∂µ i

∂T

p, γ k

= N i∑ ∂g i

∂T

p

+ R ln γ i

(2.32)

With s i = −∂g i

∂T

p

(a specific entropy of the pure component i) the entropy of the mixture is S = Ni∑ s i + Ni

i∑ R ln 1

γ i (2.33)

The second term is the entropy of mixture for any number of components. R ln 1

γ i: specific molar entropy of mixture.

ii) Mixtures with chemical reactions: We consider the following reaction: νA∑ A = νB∑ B

(νA,B : stoichiometric numbers). The reaction should take place ξ-times , i.e. ξνB moles of B are formed and ξνA moles of A are consumed:


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nA = nAo − ξν A (2.34)

dn A = −νA dξ

nB = nB

o + ξν B (2.35)

dn B = νBdξ

ξ: reaction number Determination of TdS: TdS = dU + pdV- µi∑ dN i

= dU +pdV + µA νA − µ BνBB∑

A∑

dξ (2.36)

Definition of the “affinity": A := µ A

A∑ νA − µ B

B∑ νB

Accordingly, we have dG = -SdT + Vdp + µ i∑ dNi

= - SdT + Vdp - Adξ (2.37)

⇒ A = -∂G∂ξ

p, T

(2.38)

In the case of a closed system (without exchange of mate-rial), equation (2.36) results in the case of dU=0, dV=0 (δQ=0) in

dS =δQT

+ δiS =AT

dξ (2.39)

δiS: internal generation of entropy by (irreversible) chemical reactions (ξ is an internal degree of freedom). Because of δQ=0 we have

δ iS=AT

dξ (2.40)

δiS=0 (reversible change of state), if dξ=0 (restricted equilibrium) or A=0 (unrestricted equilibrium) (mass action law)

1. Equilibrium of a closed system (no exchange of heat, no work): δQ=0, δA=0 (U=const, V=const).

The entropy may not decrease in the case of a thermally isolated system. As a consequence, the equilibrium state is the state of maximum entropy which is allowed by the external conditions, i.e. the state which has a maximum in entropy compared to all virtual changes of the system. For a small virtual change of the state ∆v we have under

the conditions dU=0, dV=0:

∆vS( )U,V

< 0 unrestricted equilibrium= 0 indifferent equilibrium > 0 restricted equilibrium

(2.41)

(Maximum of entropy) 2. Equilibrium of an open system without performing work (dV=0), but with exchange of heat which compen-sates internally the increase in entropy (dS=0). As a con-sequence of

dS =δQT

+ δiS

we have because of dS=0 δQ = -Tδ iS

and because of δQ=dU+δW=dU: dU = –Tδi S ≤ 0 (2.42) eva-82 An irreversible change results in a state of smaller inner energy (at constant entropy and volume)

∆vU( )S,V

> 0 unrestricted equilibrium= 0 indifferent equilibrium < 0 restricted equilibrium

(2.43)

(Minimum of the inner energy) 3. Equilibrium at T=const, V=const: F = U-TS


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∆F = ∆U-T∆S = ∆Q-∆A-T∆S = ∆Q-T∆S (2.44)

Because of ∆ S=∆QT

+ ∆i S we have

∆ F = -T∆ iS ≤ 0 (2.45)

∆vF( )V,T


(2.46)

(Minimum of the free energy) 4. Equilibrium at T=const, p=const: G = F+pV ∆G = ∆F+p∆V= ∆U-T∆S+p∆V= ∆Q-T∆S (2.47)

Because of ∆S =∆QT

+ ∆ iS we have

∆G = –∆iS ≤ 0 (2.48)

∆vG( )T ,p


(2.49)

(Minimum of the Gibbs energy) For the affinity of an ideal mixture we have with µ A = gA p,T( )+ kT ln γ A :

A = νAg A∑ p,T( )− νB gB∑ p,T( )+ kT ln ΠA

γ AνA

ΠB

γ BνB

= 0

(2.50) or

ΠA

γAν A

ΠB

γ BνB

= exp −νAg A p,T( )− νBgB p,T( )∑∑

kT

= K(p, T)

(2.51) In general we have because of µ A = µA

o + RT ln a A

ΠA

aAνA

ΠB

aBνB

=K(p,T) (2.52)

Pressure and temperature dependence of the mass equi-librium constant K(p,T):

i) Pressure:

∂ ln K

∂p

T

=1

kT− νA

∂gA

∂p

T

+ νB∂gB

∂p

T∑∑

=1

kTνB vB − νA vA∑∑( )

The equilibrium is shifted in favor of the side of smaller volume.

ii). Temperature:

∂ ln K

∂T

p

= −1R

νA

∂ gA / T( )∂T

p

∑

− νB

∂ gB / T( )∂T

p

∑

(2.53)

Because of

∂ g/T( )∂T

p

=1T

∂g∂T

−1

T2 g =-Ts -g

T2 =-Ts - h - Ts( )

T2 = −h

T2

(2.54) we have

∂ ln K∂p

p

=1

kT2 ν Ah A − ν BhB∑∑( )=r

kT2 (2.55)

r = νA∑ hA − νB∑ hB : heat generation

At an increase of temperature, the system changes toward the side of higher enthalpy.

Braun-Le Chatelier Principle: iii) Change of a quantitative variable: The other quantita-

tive variables are changed in such a way that their changes reduces the effect of the change of the original variable of the related intensive variable.

iv) Change of an intensive variable: The other quantita-tive variables are changed in such a way that their change increases the effect of the change of the original variable upon the variation of the re-lated quantitative quantity.


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Example: ∂ −p( )

∂V

T, A

<∂ −p( )

∂V

T, ξ

(2.56)

i.e.. κ T, A < κ T,ξ

(ξ: quantitative variable, A: intensive variable) Two-Phase-Systems with 1 component If p,T = const, the equilibrium condition is: δG=0 By neglecting surface effects, the Gibbs energy is: G = N ' g' + N' ' g' '

' , " : phases; N' , N" : numbers of mols of phases ', " ; g ‘ , g"

: molar Gibbs energies. Variation of the Gibbs energy: δG p, T = g‘ δN‘ + g " δN"

Since N‘ + N" = const. ⇒ δN‘ = - δN" ⇒ δG= (g‘ - g") δN‘ = 0 or: g‘ (p,T) = g" (p,T) (2.57) (equilibrium condition)

Because of ∂G∂N i

p, T

= µi the equilibrium condition may

be also written as:

µ ‘ = µ"

This condition results, e.g., in Nernst’s vapor pressure curve under the assumptions 1) vapor = ideal gas, 2) compressibility of the liquid = 0. We consider 2 adjacent points along the curve p(T): p,T and p+dp,T+dT. For these points we have g‘ (p,T) = g" (p,T) and g‘ (p + dp, T + dT) = g ‘"(p + dp, T + dT)

Taylor series development of the last equation:

g‘ p,T( )+∂g‘

∂p

T

dp +∂g‘

∂T

p

dT = g" p, T( )+∂g"∂p

T

dp +∂g"∂T

p

dT

↑ ↑ ↑ ↑ ↑ ↑

= g" p, T( ) v‘ s‘ g‘ p, T( ) v‘ s‘

From that results

dpdT

=s" -s‘

v" - v‘ (2.59)

Rewriting of s" - s ‘ : 0 = g‘ - g" = h‘ - Ts ‘ - h" + Ts" ⇒

s" − s‘ =h " -h‘

T=

rT

(2.60)

or

s" - s‘ =∆QT

=rT

r: latent transformation heat (= heat generated if the phase change is seen as a chemical process). Accordingly, the pressure dependence of a single component two phase system is:

dpdT

=r

T v" -v ‘( ) (2.61)

(Clausius-Clapeyron’s equation for any transition of 1st order).

p,T-dependence of a phase transition of 1. order: Fig. 2.2. Dependence of the molar Gibbs energy on the tempera-

ture and pressure. g‘ and g" stand for the molar Gibbs energies of the different phases

g‘

g‘

g" g"

T=const p=const

T p transition temperature

transition pressure

g g


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At a transition of 1st order, the first derivations of g‘ und g"

at the transition temperature and –pressure are different from each other. Phase transition of n th order (Ehrenfest): At the transition point we have:

∂ng ‘

∂Tn ≠∂n g"

∂T n and ∂ng ‘

∂pn ≠∂n g"

∂pn (2.62)

while all lower derivatives are equal. Phase transition of 2nd order:

∂g ‘

∂T≠

∂g"

∂T and

∂g‘

∂p≠

∂g"

∂p

i.e. -s ‘ = -s" and v ‘ = v" There is no heat of transition at a phase transition of 2nd order! Counters and denominators are = 0 in the Clausius-Clapeyron equation. Rule of Hôpital:

limdpdT

=

∂ s “ -s‘( )∂T

p

∂ v “ -v ‘( )∂T

p

or =

∂ s“ -s‘( )∂p

T∂ v“ -v‘( )

∂p

T

Rewriting the left hand side expression: ∂ v" − v ‘( )

∂T=

∂v"

∂T−

∂v‘

∂T= v‘ ∂ ln v"

∂T

p

− v‘ ∂ ln v' ∂T

p

= v ‘ α " − α ‘( )

↑

v‘ = v"

∂s∂T

p

=1T

δQdT

p

=cp

T

lim dpdT

=cp

" − cp‘

Tv ‘ α" − α‘( ) (2.63)

Rewriting the right hand side expression:

∂ v" − v‘( )∂p

= v‘ ∂ ln v "

∂p

T

- v‘ ∂ ln v ‘

∂p

T

= -v ‘ κ" − κ‘( )

∂s∂p

T

= −∂v∂T

p

= −vα

lim dpdT

=α" − α‘

κ" − κ ‘ (2.64)

(Maxwell’s straight line follows for the van-der-Waals gas from these stability conditions). Systems of several components and several phases p, T = const. Components: k = 1, ..., K Phases: p = 1, ..., P Equilibrium: minimum of G. The Gibbs energy of the entire system is the sum of the contributions of the individual phases and components:

G = µ kp( )

k =1

K

∑p=1

P

∑ nkp( ) (2.65)

(n kp( ): molar number of the k-th component in the p-th

phase) The sum of the molar numbers of the k-th component in all phases remains constant:

nkp( )

p=1

P∑ = const= c k k =1, ..., K

Necessary condition for the minimum of G is that the partial derivations of

G + λ kk= 1

K

∑ nkp( )

p=1

P

∑ − ck

(λk: Lagrange’s multiplicator) become equal to 0:

∂

∂n kp( ) G + λk

p=1

P

∑ n kp( ) −ck( )

k =1

K

∑

= 0

⇒ ∂

∂n kp( ) µk

p( )nkp( ) + λk n k

p( ) − λ kck( )p= 1

P

∑k=1

K

∑

= 0

⇒ µ kp( ) + λk = 0


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i.e. : µ kp( ) = const for all phases:

µ k1( ) = µ k

2( ) = ..... = µkP( )

k=1,.....,K (2.66)

The chemical potential of each component is the same in all phases. Gibbs phase rule Independent variables are: p,T, γ 1

1( ), γ 21( ), ..., γ K -1

1( ) , γ 12( ), γ 2

2( ), ..., γ K -12( ) ,..., γ1

P( ), γ 1P( ), ..., γ K -1

P( )

The γ kp( )' s are related to each other because of the condi-

tion γ kp( )

k=1

K

∑ = 1 .

The total number of available independent variables is accordingly: P (K-1) + 2 The number F of variables that may be changed arbitrar-ily (degree of freedom) is restricted however by the K(P-1) equations (2.66). The degrees of freedom F therefore adds up to: F = P (K-1) +2 - K (P-1) i.e. F = K +2 -P Gibbs phase rule (2.67) Because of F ≥ 0 we may also write: P ≤ K + 2 F = 0 : system is invariant F = 1 : system is monovariant F = 2 : bi-variant K = 1 , P = 3 : triple point K = 2 , P = 4 : quadruple point Example: 1 component (e.g. H2O) i) 1 phase (gaseous, liquid or solid) ⇒ F = 2; i.e. p and

T may be chosen arbitrarirly (within certain limits). ii) 2 phases co-exist ⇒ F = 1. In this case, only one

quantity of state may be chosen arbitrarily since the state is described by a point along the vapor pressure curve of the water, the vapor pressure curve of the

ice or along the curve of the dependence of the melt-ing point of the ice from the pressure;

iii) 3 phases co-exist ⇒ F = 0 (triple point) Regular mixtures Assumption (Hildebrand 1929): Simplest consideration of energetic interaction (next simple case besides ideal mixture):

µ1 = g1(T) + kT ln γ 1 + w γ 22 (2.68)

µ2 = g2 (T) + kT ln γ2 + w γ 1

2 (2.69)

w = constant, g1,2: molar Gibbs energies of the pure com-ponents 1,2.

This assumption for µ1 and µ2 fulfills the Duhem-Margules’ conditions

γ ii

∑ ∂µi

∂γ k

p, T, γ j ≠ k

= 0 :

γ 1∂µ 1

∂γ 1

+ γ2∂µ 2

∂γ 1

= γ 1kTγ1

− γ12wγ2 − γ 2kTγ2

+γ 2 2wγ1 = 0

w → 0 : Raoult' s law Heat of mixing (= enthalpy before the mixture – enthalpy after the mixture) for a regular mixture:

Because of g = h - Ts = h + T ∂g∂T

p

we have before the

mixture h before = γ1h1 + γ 2h 2 = γ 1g1 − γ1Tg1

‘ T( ) + γ 2g2 − γ 2Tg2‘ T( )

(2.70)

Because of h = g - T∂g∂T

p

and g = γ1µ1+γ2µ2 we have

according to eqs. (2.68) and (2.69) after the mixture: hafter = γ 1g1 T( )+ γ 1kT ln γ 1 + wγ 2

2 γ 1 + γ 2 g2 T( ) +γ 2 kT ln γ2 + wγ 1

2 γ2 − T γ 1g1‘ T( )+ γ 1k ln γ1

+γ 2 g2

‘ T( )+γ 2 k ln γ 2 (2.71)


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By subtraction of (2.71) from (2.70), the heat of mixing of a regular mixture is r = -wγ1 γ2

2 − wγ 2γ 12 = −wγ1γ 2

r = wγ1 γ1 −1( ) (2.72)

Metallic bond: “Electron gas”; stoichiometric proportions are not necessary; accordingly wide ranges of composi-tion exist. The thermodynamics differs widely from that of compounds with fixed stoichiometry (Fig. 2.3. and Fig. 2.4.).

Only few systems show Raoult’s relationship over a wide range of composition.

If the activity is proportional to the concentration (a i ≈ ci) we have Henry’s law. The activity constant may be repre-sented by the tangent (broken line) in Fig. 2.5.

Determination of the activity of a substance by measuring the vapor pressure over the solution and in the pure state: Example 1: Zn-Al-alloys. Separation: + Zn (l) = Zn (g) .....

− AlmZnn +1 l( )= AlmZnn l( )+ Zn g( )….

pZn , g0

pZn' , gZn

∗'

Al mZnn l( ) + Zn l( ) = Alm Znn +1 l( )K gZn∗

(∆G0 : Gibbs energy of the evaporation of 1 mole of Zn, -∆G1 and GZ n

are the specific molar Gibbs energies of

the solution of gaseous or liquid Zn in the alloy AlmZn). Because of

g 0 = −k T ln p Zn , g Zn∗ '

= −k T ln p Zn'

it follows:

gZn∗'

= g0 − gZn∗'

= k T ln pZn

'

pZn= k T ln a Zn (2.73)

Activities may be determined accordingly by measuring and a gas phase. Example 2: Fe-Ni-alloy. Measuring of the H2O/H2-ratio of the vapors in equilibrium with FeO/alloy and FeO/Fe − FeO s( ) + H2 g( )= Fe s( ) +H 2 O g( )

+ Zn g( )+ FemNin s( )+ FeO s( )+ H2 g( )= Fem +1Nin s( )+ H2O g( )

pH2 OpH 2

, g0

pH2O

'

p H 2' ,gFeO

∗

FemNin s( ) + Fe s( )= Fem+1Nin s( ) g Fe∗

Comparison: SnTe and Ag-Au

Fig.. 2.3. Phase diagrams of the systems: (a) Tin-Tellurium and

(b) Silver-Gold

Fig. 2.5. Activity of one of the components of binary mix-

tures (schematically)

Comparison: SnTe and Ag-Au

Fig. 2.4. Emfs of galvanic cells with alloy electrodes: (a) Tin-Tellurium and (b) Silver-Gold


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Abb. 2.7. Integral molar excess Gibbs energy of the system silver-gold

Fig. 2.6. Specific molar excess Gibbs energy as a function of the Ag-Au ratio

Fig. 2.8. Relationship between the specific and integral excess Gibbs energies (a) in a system that forms intermediate compounds and (b) in a system of complete mutual solubility.


15.10.01

gFe∗ = kT ln a Fe = kT ln

pH 2O / pH 2

pH 2O' / pH 2

' (2.74)

Determination of the activity of a component in solution by EMF measurement: gA

∗ = kT ln a A = −zqE

Calculation of integral values from specific and molar data. According to Duhem-Margules’ relationship, the varia-tion in the partial molar Gibbs energy of B (chemical potential) may be calculated according to

lnaB'

aB" = −

NANB

N B"

NB'

∫ d ln a A (2.75)

or

lnγ

B'

γ B" = −

NA

NBN B

"

N B'

∫ d ln γ A (2.76)

because of dNA = -dNB in the case of a binary system. For a diluted solution of A in B, γA has a constant value. For pure A, γA reaches the value 1. Therefore, the integral (2.76) may be determined precisely, while the integral (2.75) may be only estimated because aA→0 (ln aA →-∞).

From eqn- (2.75) the integral excess Gibbs energy may be determined by integration of the curve showed in Fig.2.6 and is plotted in Fig. 2.7. The value gA

∗ − kT ln N A = kT ln γ A is called excess

Gibbs energy g A∗ E( ).

Example: Ag-Au-alloy (see Figs. 2.3. – 2.7.). Instead of a solution, the system may have several homo-geneous phases which are separated by heterogeneous regimes. Comparison of the relation between the specific and the integral molar quantities for both limiting cases: Full lines in Fig. 2.8 show the integral excess Gibbs ener-gies (a) of a binary system with 2 compounds without signifi-cant solid solubility and (b) of a binary system with continuous solid solubility. The tangents of these lines intersect the ordinates (NA = 0, NA = 1) at the values of the specific Gibbs energies of B and A. A continuous solid solution results in a continuous vapor pressure curve (Fig. 2.9.a). In the other cases, the vapor pressure changes stepwise with the formation of a new compound with a higher amount of the volatile comp o-nent.

The magnitude of the pressure increase corresponds to the difference in the Gibbs energies. Considering a 2-

component system (Fig. 2.9.c) with A as volatile component and the inter-mediate solid phases AB3, AB, A3B, the Gibbs ener-gies are for each reaction step: Then, the integral Gibbs energies are

Fig.. 2.9. Schematic representation of the vapor pressure curves

of (a) a homogeneous system, (b) a three-phase system and (c)

a system with three compounds.

Fig. 2.8. Relationship between the specific and integral excess Gibbs energies (a) in a system that forms intermediate compounds and (b) in a system of complete mutual solubility.


15.10.01

3B+A=AB3 ... ∆G1 B+A=AB ... ∆G2= 1

3 ∆G1 + 23 g2

∗

3B+A=AB3 ... ∆G3=∆G2 + 2g 3∗

Then, the integral Gibbs energies are

3B+A=AB3 ... g1=kT ln p1/p4 12 AB3+A= 3

2 AB ... g 2∗ =kT ln p2/p4

12 AB+A= 1

2 A3B ... g3∗ =kT ln p3/p4

All integral Gibbs energies depend on the determination of the (under certain circumstances very small) pressure p1! Systems with heterogeneous and homogeneous ranges: Continuous (not necessarily linear) variation of the pres-sure in the homogeneous regimes; constant pressure in the heterogeneous regimes which is determined by the two limiting concentrations

Example 3: Cu-Zn (Fig. 2.12.)

Fig. 2.12. Specific molar Gibbs energies of zinc in the solid copper-zinc system at 500°C. The liquid is the standard state of zinc.

Fig.. 2.9. Schematic representation of the vapor pressure curves

of (a) a homogeneous system, (b) a three-phase system and (c)

a system with three compounds.

Example 1: Ce -Hg (Fig. 2.10.) The Hg -vapor pressure is presented:

Fig. 2.10. Mercury pressure of cer-mercury at 340°C.

Specific molar enthalpy of the solution of Zn in Cu-Zn-alloys (Fig. 2.13.):

Fig. 2.13. Specific molar heat of solution of liquid zinc in solid copper-zinc alloys.

Example 2: O in liquid and solid iron (Fig. 2.11.)

Fig. 2.11. Partial molar Gibbs energy of oxygen in liquid and solid ironliquid and solid iron


15.10.01

There is no longer a continuous curve. The relative positions of the ∆HZn-XZn lines are deter-mined by the slopes of the phase limits with the tempera-ture.

In the heterogeneous regions, the curve is a straight (bro-ken) line. Since the dG/dX-curve is not reversed, there is no benditure to the inside. Otherwise the phase would not be stable in that regime.

Because of the relationship ∆ G = 12 NTi gO 2

dx0

x

∫ between

the specific (partial) and integral Gibbs energy, ∆G corre-sponds to the integral under this curve. Richardson diagrams: Specific molar Gibbs energies of the equilibria metal-metal oxide as a function of tempera-ture. Dissoluted solutions i) Raoult: NA=aA (γA=1) (2.77) holds for many systems with high concentrations of the solvent (Fig. 2.16.).

Fig. 2.16 Dissoluted solution of B in A A: solvent, B: dissolved species B is completely surrounded by A. The forces A-B and B–B have no influence on the further addition or removal of A. The validity of Raoult’s law depends strongly on the system, but rarely exceeds a value larger than NA = 0.85. ii) Henry: aA=const. NA (2.78) (derived from the postulate of Henry that the solubility of a gas in a liquid is proportional to the pressure of the gas). (Eq. 2.78) is only valid for low concentrations. It also holds g i = hi + kT ln N i - si

E

(because of g i = kT ln γ i + kT ln N i )

Integral formation and Gibbs energies of the Cu-Zn-alloys:

Fig. 2.14. Integral enthalpies of formation at 500°C and Gibbs

energies of formation of solid alloys of the Cu-Zn-Systems at

500 °C. Reference states are solid copper and liquid zinc.

System Ti-O (Fig. 2.15.). Several phases exist, some with large range of homogeneity (stoichiometry):

Fig. 2.15. Specific Gibbs-Energies of dissotiation of 1 mole oxygen in the titanium-oxygen-system


15.10.01

If the dissolved species i is a di-atomic gas i2, the solubil-ity is not directly proportional to the gas pressure, but follows a square root dependence because of the dissocia-tion: Sievert’s law: g = h + 2 kT ln Ni − si 2

E T (2.79)

Regular solutions: In the case of a regular mixture with positive enthalpy of mixing, a critical temperature exists, above which the components are completely mixable and below which the system consists of a mixture of two solutions. Spinodal de-mixing occurs (Fig. 2.17.)

O - Q: ∂G M / ∂NA = 0

O' - Q' : ∂2G M / ∂N A2 = 0 ; spinodal point

(at this point, the two new phases occur first when the system is cooled down (and not at the equilibrium com-position O - Q) According to the Gibbs energy of a mixture of a binary regular solution, we have:

∂G

∂N A

= α 1− 2N A( )+ kT lnN A

1- N A

(2.80)

∂ 2∆G∂N A

2 = −2α + kT1

NAN B

= 0 (2.81)

Accordingly, we have along the spinodal composition

N A NB =kT2α

(2.82)

Order- Disorder-Transitions: Some binary systems with negative enthalpy of mixing show order-disorder transitions: At a given composition, the atoms of each component occupy well defined sites of a sublattice at low temperature; above a critical tempera-ture the atoms are randomly mixed in the lattice. Non-regular solutions In order to describe deviations from regular solutions, excess Gibbs energies are employed:

∆ GE = ∆G -kT NA ln NA + N B ln N B +K( ) (2.83)

∆ SE = ∆S- k NA ln N A + N B ln NB +K( ) (2.84)

∆ SE = 0 holds for regular solution for the entire regime of the solution. Fig. 2.18 shows the maximum or mini-mum excess entropies of mixture against the maximum or minimum enthalpies of mixtures for a selection of sys-tems with complete mutual solibility (but different types of chemical bonds). ∆ SE = 0 is rarely fulfilled (only for ideal solutions for

which ∆H = 0). There is a direct correlation between ∆H and ∆SE (empirically). The maxima of enthalpy and entropy (as a function of composition) are normally not reached at equi-atomic

Spinodal De -mixing:

Abb. 2.17 First and second derivation of the integral Gibbs

energy of a mixture of a binary A-B alloy with regard to the

molar number of A. The curve holds for a temperature below

Tc.

Abb. 2.18 Linear relationship between the maximum and mini-

mum excess entropies and enthalpies of mixtures for a variety

of binary alloy systems.


15.10.01

composition, as expected for regular solutions; both are shifted to higher or lower concentrations. There exists also an empirically observed relationship between the volume change and the entropy of mixture. The Table shows also a proportionality between volume changes and enthalpy of mixture. Covalent bonds exist between the meta- or semi metals, e.g. Bi, Sn, Pb, Tl, In, Cd and Cn. The more metal with metallic bond is added to the covalent metal, the more is the covalent bond bro-ken. This process consumes energy and increases the disorder. These alloys are commonly formed endother-mally and show positive excess entropies of mixing. Since the volume change is also positive, the covalent “cluster” is not completely destroyed, but bound too strongly in order to form a suitable dense packing. The covalent metal is listed as the second elemnent in Table 2.1. It may be realized that the entropy and en-thalpy maxima are nearly always below equi-atomic composition. That confirms the hypothesis that the cova-lent bonds are apparently more strongly affected if the corresponding metal has higher dissolution. I.e., if a co-valent metal is dissolved in a less covalent metal, more energy and order is consumed than vice versa. The mutual solubility of real metals should be nearly ideal as long as the components are only slightly diffe rent with regard to the atomic diameter (Ag-Cu in Fig. 2.19). The entropy for Zn-Sn is higher because of the destruc-tion of covalent bonds and the resulting vibration en-tropy. Changes in the site entropy may result in a minimum ? S = 0; changes in the vibration entropy may result in nega-

tive ?S values, as e.g. in the case of Mg-Bi or Sb. Solid Mg3Sb2 and Mg3Bi2 are partially stabilized by polar bonds. Fig. 2.19. shows this relationship in the melt which con-sists partially of Mg2+ and Sb3- or Bi3- ions. These forces show their influence mainly in the vibrational term of the entropy. If the alloy is formed by strong covalent bonds (Fig. 2.19., Fe-Si), the effect on the entropy is similar as in the case of polar bonds.

Tab. 2.1. Maximum or minimum values of the integral changes of volume, enthalpy and

entropy of mixt ures of liquid binary alloys

Abb. 2.19. Integral entropies of mixtures of the liquid systems

tin-cinc, silver-copper, iron-silicon and magnesium-antimony.

Thermodynamics and Kinetics of Solids - Technische Fakultät · 2003-11-05 · Thermodynamics and...

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Transcript of Thermodynamics and Kinetics of Solids - Technische Fakultät · 2003-11-05 · Thermodynamics and...