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Chapter 3

Thermodynamics of Heterogenous Systems

Thermodynamics plays a very important role in heterogenous kinetics both by providing the existence ranges of the studied transformations and by its use in the most important approximate methods of modeling.

3.1. Introduction: aims of thermodynamics

The thermodynamic study of a system can have four objectives classified in the order in which we want to deepen our understanding of the studied system:

– identifying a transformation by characterizing the initial and final states under the aspects of both chemical species and phases, which, in turn, help express the reaction of the studied transformation;

– identifying the parameters that govern the conditions of the transformation. We thus define a space with n variables specifying the field in which the transformation is possible (field of stability of the products resulting from the transformation). These variables can be temperature, total pressure, compositions of the phases (concentrations, partial pressures), surface tensions of the solids, etc., and the dimension of space thus defined is given by the variance of the system;

– getting the thermodynamic quantities, in particular equilibrium constants and their variations with the variables to specify the course of the transformation, associated enthalpies, necessary to calculate heat equilibria; and

– modeling solid and liquid phases and in particular solid solutions within the framework of the stoichiometry and structure elements, which will allow representations of mechanisms during the further kinetic studies.

60 Handbook of Heterogenous Kinetics

A thermodynamic study thus will be more or less sophisticated according to our deep understanding of the system. However, it is certain that kinetics and mechanism studies require a thorough thermodynamic knowledge of the system going until the fourth step.

3.2. General survey of thermodynamics of equilibrium

Consider a reaction represented by means of the following equation:

i i j ji j

A Aβ β=∑ ∑

Coefficients β are the arithmetic stoichiometric numbers (all positive). This equation can also be written in the following form:

1 1 2 20 k kA A Aν ν ν= + + ⋅⋅⋅ + + ⋅⋅ ⋅

By placing all the components in the second member, the algebraic stoichiometric numbers, νk, are equal in absolute value to the corresponding arithmetic stoichiometric numbers but are positive for the components produced by the reaction and negative for reactants.

3.2.1. Chemical potential of a component in a phase

First, remember that a component is defined by both the knowledge of the chemical species and the phase in which it is present. The same chemical species in another phase is another component.

The chemical potential of component i in its phase is defined as its partial molar Gibbs energy, that is, by the partial derivative of Gibbs energy with respect to the amount of the component:

, ,

j

ii P T n

Gn

μ⎛ ⎞∂

= ⎜ ⎟∂⎝ ⎠ [3.1]

The total Gibbs energy of the phase is as follows:

i ii

G n μ= ∑

this sum being extended to all the components present in the phase.

Thermodynamics of Heterogenous Systems 61

Chemical potential is a function of temperature, pressure, and phase composition (definite, e.g., by the mole fractions xi of its components). This function can always be written in the following form:

[ ]0( , , ) ( , ) ln ( , , )i i i i i iT P x T P RT x T P xμ μ γ= +

On the condition of supplementing this equation (which defines two quantities: standard chemical potential 0

iμ and coefficient of activity iγ of the component) by the condition under which the coefficient of activity becomes equal to 1, this is what we define in the convention of reference. The product i ixγ is the activity of the component.

3.2.1.1. Some conventions of definition of activity

Several conventions are commonly used, but we will retain the two more frequently used conventions.

3.2.1.1.1. Convention I or the pure component reference

The state of reference is a hypothetic state in which any component i is pure (xi = 1) at temperature T under pressure P and in the same physical state as the phase (even if this physical state is hypothetical for this pure component under these conditions). Then:

0 ( , ) ( , )i iT P g T Pμ =

where ig is the molar Gibbs energy of pure component i. The chemical potential is as follows:

(I)( , , ) ( , ) ln ( , , )i i i i i iT P x g T P RT x T P xμ γ⎡ ⎤= + ⎣ ⎦

where (I)iγ is the coefficient of activity.

3.2.1.1.2. Convention II or reference infinitely diluted solution

The state of reference is now the infinitely diluted solution in one (or several) of the components called solvent(s). The other components are called “the solutes” (usually the less abundant ones).

Consequently, for a solvent, this convention is the same as the preceding one, the pure state being almost identical to the infinitely diluted solution for the solvent. On the other hand, for a solute, chemical potential will take the form:

(II)( , , ) ( , ) ln ( , , )i i i i i iT P x T P RT x T P xμ μ γ∞ ⎡ ⎤= + ⎣ ⎦


iμ∞ represents the chemical potential of solute i in the infinitely diluted solution because (II)

iγ tends toward 1 if the mole fractions of all the solutes tend toward zero.

To go further in the expression of chemical potential with composition, it is necessary to clarify the variation in the coefficient of activities with this composition by means of models of solutions. Let us examine some such models.

3.2.1.2. Some models of solutions

We will briefly describe three major models of solutions.

3.2.1.2.1. The phase has a single component i

In fact, this is a pure phase and the chemical potential is its molar Gibbs energy and variations in this property with pressure and temperature give those of chemical potential immediately, that is to say:

i ii

gs

T Tμ∂ ∂

= = −∂ ∂

(molar entropy of the pure phase i)

i ii

gv

P Pμ∂ ∂

= =∂ ∂

(molar volume of the pure phase i)

3.2.1.2.2. Component i belongs to a gas phase with several components

In this case, the chemical potential of i can be written in the following form:

(0)(0)( , ) ln i

i ifg T P RT

Pμ

⎡ ⎤= + ⎢ ⎥

⎣ ⎦

(0)( , )ig T P represents the Gibbs energy of a pure gas i under the pressure of the standard state, P(0) is in general equal to 1 atm (or 1 bar), and fi is the fugacity of i in the mixture. For conditions that are far from condensation (low pressures), this fugacity merges with the partial pressure.

At higher pressures, fugacity can often be calculated by the Lewis rule, which expresses that the fugacity of a gas in a mixture under total pressure P is given by the product of its mole fraction by the fugacity of the same gas at the pure state under the same total pressure; hence:

0i i if f x=


The chemical potential thus can be written as

0(0)( , ) ln ( , ) ln xi

i i i ifg T P RT g T P RT

Pμ

⎡ ⎤= + = +⎢ ⎥

⎣ ⎦

3.2.1.2.3. Component i belongs to a condensed phase with several components

Two models of condensed solutions are commonly used:

– perfect solution, for which the coefficient of activity (in any convention) is 1 for all the components and whatever the values of the variables temperature, pressure, or composition;

– ideal diluted solution, for which the coefficient of activity is 1 in convention II.

In general, on examining the various expressions of chemical potential, we can obtain only one form:

0 0( , ) lni i iT P RT aμ μ= + [3.2]

Table 3.1 provides the various significances of 0iμ and ia in the usual cases.

Phase containing component i

Nature of solution Reference state

0 ( , )i T Pμ ia

Solid or liquid Pure component Pure at T and P ( , )ig T P 1

Perfect mixture Pure at T and P(0) (0)( , )ig T P Pi /P(0)

Non-perfect (Lewis) Pure at T and P ( , )ig T P xi Gas

Non-perfect (general) Pure at T and P(0) (0)( , )ig T P fi /P(0)

Perfect solution Pure at T and P ( , )ig T P xi

Diluted ideal solution (solute)

Infinitely diluted at T and P

0 ( , )i T Pμ xi Condensed solution (liquid or solid) Diluted ideal

solution (solvent)

Pure at T and P ( , )ig T P xi

Table 3.1. Usual expressions for chemical potentials


A third model, given by Debye and Hückel [GOK 96a], is also used for diluted solutions of charged components (ions, structure elements).

3.2.2. Variance of a system at equilibrium

The variance [SOU 90] of a system is the unrestricted number of variables of state, selected among the p external intensive variables (temperature, pressure, electric and magnetic fields, etc.) and mole fractions (or other intensive variables of composition, or chemical potentials) that the experimenter must fix to reach any state of equilibrium of this system and so that all the other physicochemical variables are given.

This variance is calculated by Gibbs theorem (or phase rule). If the system contains n constituents connected to each other by r linearly independent chemical reactions and if it consists of ϕ phases, the variance is as follows:

( )v n r p ϕ= − + − [3.3]

The difference n − r is called the number of independent components.

In traditional physicochemical systems, we choose p = 2 because the external intensive constraints are generally limited to temperature and pressure. For systems that involve only condensed phases, pressure is a constraint that practically does not have any influence (if it remains in a range of usual values) and thus we take p = 1.

We define the degree of freedom, L, such as the number of variables of state, selected among the p external intensive variables (temperature, pressure, electric field, magnetic field, etc.) and mole fractions (or other intensive variables of composition, or chemical potentials) that the experimenter must fix to reach the states of equilibrium compatible with a whole of constraints κ already imposed on the system, that is to say:

L v κ= −

3.2.3. Associated extensive properties of a transformation, partial molar properties

If we look at an unspecified extensive property X, it corresponds with each component in its phase by a partial molar property iX defined by:

, , j

ii P T n

XXn

⎛ ⎞∂= ⎜ ⎟∂⎝ ⎠

[3.4]


The partial derivation was carried out by keeping temperature, pressure, and the amount of matters jn of all the other components j i≠ in the studied phase constant.

If iν is the algebraic stoichiometric number of i in reaction R, the result of the following operation:

R ( ) i ii

X XνΔ = ∑ [3.5]

is called the value of X associated with reaction R, or the change of X for the reaction.

We thus define the Gibbs energy associated with reaction R by:

R ( ) i i i ii i

G Gν ν μΔ = =∑ ∑ [3.6]

Similarly, the enthalpy associated with R is:

R ( ) i ii

H HνΔ = ∑ [3.7]

The values of partial molar properties involved in operator [3.5] can be considered under any conditions; thus, we can define a standard Gibbs energy associated with R for which the reactants and the products are selected in their state of reference:

0 0R ( ) i i

iG ν μΔ = ∑ [3.8]

Similarly, we can calculate this property for the components considered under the equilibrium conditions:

equil equilR ( ) i i

iG ν μΔ = ∑ [3.9]

We call affinity of the transformation, the property defined by (ξ being the extent of the reaction):

,P T

Gξ

⎛ ⎞∂ℵ = −⎜ ⎟∂⎝ ⎠

[3.10]


and it is shown that:

R ( )i i Gν μℵ = − = −Δ∑ [3.11]

3.2.4. Chemical potential of an ion or a structure element

For ions in a solution and structure elements in a solid, the derivation of relation [3.4] presents a difficulty. Indeed, in the case of ions, it is impossible to keep the amounts of ions of all the components in an ionic solution constant while varying the amount of only one of them because then the electric neutrality is not respected. The same condition applies to structure elements for which we cannot carry out derivation because we would modify the ratio of sites related to the crystal.

We will bypass this difficulty by remembering that in the case of ions, we can always write a reaction by using either ions or the molecular notation; thus, for example, the precipitation of silver chloride by the action of hydrogen chloride on silver nitrate can be represented by one of the following two notations:

– The ionic notation:

Ag Cl AgCl+ −+ =

– The molecular notation:

3 3AgNO HCl AgCl HNO+ = +

Indeed, there is no difficulty in using derivation [3.4] for molecules. This means that any property attached to this reaction by operator RΔ defined by relation [3.5] is real; therefore, the same property attached to the same reaction but written in the form of ions (e.g., Gibbs energy) will be quite real. Thus, if this derivation is applied, we obtain a property that presents a real part and an imaginary part, the latter being such that it is cancelled in the application of operator [3.5]. For example, for chemical potentials, we can write for an ion:

0 0( , ) lni i i iT P RT a Iμ μ= + +

where Ii represents the imaginary part, and we will have:

0i iIν =∑


and thus we will be able to preserve, for the application of operator R ,Δ the following form of chemical potential:

0 0( , ) lni i iT P RT aμ μ= +

Conversely, we can use the same relation for structure elements because quasi-chemical reactions can also be written with building units (see section 2.1.4) that respect the ratios of sites. This shows that the properties associated with the quasi-chemical reactions are real.

Structure elements are as follows:

– either points defects in very minor amounts for which we can apply the approximation of the very diluted solutions or, at least, if these elements are ions, the Debye and Hückel model;

– or normal elements of the crystal lattice that will be regarded as solvents, and, if the defects are much diluted, their activity will be 1.

Actually, we can apply this to structure elements and ions in all the models of solutions available in thermodynamics and in particular those deduced from statistical thermodynamics such as the model of strictly regular solution. The basic assumptions of these models apply without reservation to structure elements and even, in this case, the assumption of a pseudo-network, which it is necessary to admit in the case of liquid phases, does not obviously present any difficulty for crystallized solids.

3.2.5. Feasibility of chemical reactions: De Donder inequality

It can be shown, starting from the second principle of thermodynamics, that the affinity of a transformation ℵ and its velocity r must satisfy, so that the transformation is possible, the following De Donder inequality:

0rℵ ≥

From this, we deduce that to make the reaction possible from left to right (positive velocity), its affinity should be positive, and so we can write:

R0 or ( ) 0i ii

Gν μℵ= ≥ Δ ≤∑ [3.12]


the equilibrium condition resulting in the following equality:

equil equili R0 or 0 or ( ) 0iv Gν μℵ = = Δ =∑ [3.13]

Now try to represent all the equilibrium states of a transformation. For that, we start from, in particular, an equilibrium state:

equil 0 i ii

ν μ =∑

On modifying the variables with an infinitesimal amount, the chemical potential of an unspecified component i undergoes an infinitesimal change, d .iμ If we want that the new state thus obtained were still an equilibrium state, it must satisfy [3.13] and thus:

i( d ) 0 or d 0 or d 0 i i i i

iν μ μ μ+ = = ℵ =∑ ∑ [3.14]

We can write the differential of the chemical potential in the following form:

1d d d

Ni

i i ik k

S T V Pnμ

μ=

∂= − + +

∂∑ [3.15]

and substituting into [3.14], we obtain all the equilibrium states in the system of selected variables.

3.2.6. Law of mass action for equilibriums

3.2.6.1. Equilibrium constant

Let us again examine the equilibrium condition [3.13] in the form:

equil 0i iν μ =∑

On substituting the chemical potential from expression [3.2] into this expression, we obtain:

0 ln 0ii i i

i iRT aνν μ + =∑ ∑


which can be written as:

iia Kν =∏ [3.16]

defining the equilibrium constant K by:

0 0( )( , ) exp expi ii R G

K P TRT RTν μ ⎡ ⎤Δ

= = −⎢ ⎥⎣ ⎦

∑ [3.17]

Relation [3.16] is the law of mass action between the values of the activities ai of the involved components at equilibrium.

3.2.6.2. Variations in equilibrium constant with temperature

It can be shown [SOU 90b] that equilibrium constant varies with temperature according to the Vant Hoff law:

0R

2

( )d lnd

HKT RT

Δ= [3.18]

Therefore,

0 00 0R R( ) ( )

exp , with expH S

K K KRT R

⎛ ⎞ ⎛ ⎞Δ Δ= − =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ [3.19]

Usually, for relatively narrow temperature ranges, we can consider constant standard enthalpy and entropy with temperature.

3.3. Phenomena leading to solid-gas equilibriums

Phenomena that occur during heating of a solid can lead to solid-gas equilibriums. These are as follows:

– sublimation of solid;

– gas desorption from solid surface (and its reverse adsorption with cooling);

– dissolved gas departure from solid;

– evolution of solid stoichiometry;

– decomposition of solid solution with gas departure; and

– stoichiometric reactions using gases as reactants and (/or) products.


We will study these phenomena from the thermodynamic point of view, but for that, it is convenient to provide a classification from their thermodynamic characteristics and in particular the variance of the system.

The systems considered include ϕ phases and a gas phase (gases are miscible in all proportions). The number of solid phases sϕ thus will be:

s 1ϕ ϕ= −

Most commonly, transformations involve one of the following three classes.

3.3.1. Systems with variance p − 1

If the variance is p − 1, then, according to phase rule [3.3], the number of phases is c + 1 and thus the number of solid phases is s .n rϕ = − It will be, for example, the case of a univariant system if p = 2. There will be only one variable to fix, for example, temperature. Pressure and compositions of all phases, at equilibrium, are functions of this temperature (Figure 3.1a).

3.3.2. Systems with variance p

We deduce by the same reasoning as above that s 1.n rϕ = − − This will be, for example, the case of a divariant system if p = 2. If the system involves two independent components, the transformation will occur within the same solid phase. There will be two variables to fix, for example, temperature and pressure. In a P versus T diagram, there will be as many equilibrium curves as there are compositions xi (Figure 3.1b).

Figure 3.1. Representation of a univariant system (a) and a divariant system (b)

T T

P P x1 x2 x3

x4 (a) (b)


3.3.3. Systems with variance p + 1

We deduce that s 2.n rϕ = − − This will be, for example, the case of a trivariant system if p = 2. We need at least three independent components, with at least a solid phase. There will be three variables to fix, for example, temperature, pressure, and the composition of one phase. The composition of each phase thus will be a function of these three variables.

One of the primary aims of the thermodynamic study thus will be to determine the variance by means of experiments from which we will deduct the number of solid phases, an important information to model systems. The practice shows that this step, when it is possible, is much easier than the one that consists of determining the number of solid phases from direct experiments and then deducing the variance.

3.4. Thermodynamic approach of solid-gas systems

We will now combine the preceding thermodynamic classification with that of the phenomena and their effects. For that we will, except contrary specification, take into account only two external physical variables, pressure and temperature, and thus p = 2.

3.4.1. Univariant systems

3.4.1.1. Sublimation of solid

Take as an example the direct passage of magnesium from solid state into vapor state:

(solid) (gas)Mg Mg=

The number of independent components n − r = 1, and the variance v is 1; therefore, if temperature is fixed, the vapor pressure of magnesium is also fixed, as is shown by the law of mass action:

Mg ( )P K T=

3.4.1.2. Decomposition of solid solution with gas departure

For example, in the case of plaster, the transformation of soluble anhydrite (also called anhydrite γ), which is a solid solution of calcium sulfate and water, into


anhydrite δ, which is anhydrous calcium sulfate, occurs with the loss of water, and we can write:

( ) ( )4 2 4 2CaSO H O γ CaSO δ H Ox x⋅ = +

As in section 2.5.1, this representation is incorrect because of the intervention of variable x. Also, it is more accurate to write the double equilibrium of phase transformation and transfer of water in the form:

( ) ( )4 4CaSO γ CaSO δ=

2 2H O(inside the solid) H O(gas)=

It is easier to assume that n − r = 2 (calcium sulfate and water) and therefore 1,v = and, for example, if the vapor pressure of water is fixed, temperature and the

value of x, that is, the water content of the solid solution, are also fixed.

3.4.1.3. Stoichiometric reaction involving a gas and two solids

There are numerous reactions such as the decomposition of calcium carbonate into carbon dioxide and lime or dehydration of a hydrated salt into a less hydrated form and water vapor or dissociation of an oxide into metal and oxygen. In all these cases, we can see that n − r = 2 and therefore 1.v = According to the law of mass action, gas pressure will be, at equilibrium, a function of temperature only.

3.4.1.4. Stoichiometric reaction involving a gas and three solids

Take the example of the following reaction between barium carbonate and titanium oxide:

3 2 3 2BaCO TiO BaTiO CO+ = +

Here n − r = 3 and therefore 1v = , and at equilibrium, the pressure of carbon dioxide is a function of temperature only.

3.4.2. Divariant systems

3.4.2.1. Adsorption and desorption of gases

Usually, two types of adsorptions are distinguished:


– physical adsorption, often considered as the condensation of gas in an adsorbed phase onto the surface of a solid, which can be written in the form:

G(gas) G(adsorbed)=

Experiment shows that the system is divariant, which can be interpreted in the following manner: there is only one independent component (gas G) under two phases, but p = 3, because with temperature and pressure, it is necessary to add the field of force of the solid, which constitutes the third external physical variable. We can also preserve p = 2 and state that there are two independent components: the gas and the solid;

– chemical adsorption, by the formation of a true chemical bond between gas and some surface sites s of solid, which can be written as:

G(gas) Gs s+ = −

Or, if this adsorption is a dissociative one:

2G (gas) 2 2Gs s+ = −

We still have v = 2 with p = 2 and n − r = 2 and therefore sϕ = 1.

Thus, variance does not allow us to distinguish physical adsorption from chemical adsorption. In both cases, the amount of gas fixed per unit of area is a function of both pressure and temperature.

3.4.2.2. Dissolution of a gas into solid

Take again the example of plaster, a solid solution of water and calcium and sulfate, and examine the equilibrium of the variation in composition with water loss, which can be symbolized by:

2 2H O(solid) H O(gas)=

In this case, n − r = 2 (water and calcium sulfate) and therefore 2.v = The water content of the solution is a function of temperature and water vapor pressure above the solid (see section 3.7.3).

3.4.2.3. Stoichiometric reaction with two gases and a solid

We will take the example of the Boudouard equilibrium:

2CO C 2CO+ =


In this case, 2v = , n − r = 2, and therefore s 1ϕ = . Thus, by fixing the total pressure and temperature, equilibrium is reached and, in particular, the composition of the gas phase is fixed, for example, by partial pressures, which are functions of both temperature and total pressure.

3.4.2.4. Equilibrium between non-stoichiometric solid and gas

We will quote as an example the variation in the stoichiometry of zinc oxide with oxygen pressure. Zinc oxide is an oxide of Wagner with interstitial cations. The equilibrium is represented as (see section 2.5.3.1.3):

i 2 Zn O1Zn 2e O Zn O2

°° ′+ + = +

The number of independent components is 2 (the oxide and the gas), the number of solid phase is 1, and the variance is therefore 2. The content of interstitial ions is a function of oxygen pressure and temperature. We will reconsider quasi-chemical equilibriums in the solids in the following text (see section 3.6).

3.4.2.5. Stoichiometric reaction with two gases and two solids

We will take the example of decomposition of a metallic oxalate into a metal oxide:

2 2M(COO) MO CO CO= + +

In this case, 2v = , n − r = 3, and therefore s 2;ϕ = thus, by fixing the total pressure and temperature, the composition of the gas phase is fixed because, at equilibrium, the partial pressures of two gases are functions of both T and P.

3.4.3. Trivariant systems

3.4.3.1. Evolution of the stoichiometry of a solid by means of a gas

We examine the equilibrium between a non-stoichiometric solid and a gas likely to react with an element of the solid. We will take the example of hydrogen with an oxide, for example, the ceria, from the following quasi-chemical reaction:

Ce O 2 O Ce 22Ce O H V 2Ce H O°° ′+ + = + +


Actually, this equilibrium can be regarded as the superposition of two simpler equilibriums:

The first one (from reaction [2.R.10]):

Ce O O Ce 212Ce O V 2Ce O2

°° ′+ = + +

and, the second one:

2 2 21H O H O2

+ =

In our case, 3v = , n − r = 3, and s 2.ϕ = If we fix temperature and the partial pressures of hydrogen and water, then the total pressure, the partial pressure of oxygen, and the concentrations of the various point defects are fixed.

3.4.3.2. Stoichiometric reaction with three gases and two solids

We will take as an example the decomposition of cerium(III) hydroxy-carbonate according to the following reaction:

3 2 2 22Ce(OH)CO 2CeO H O CO CO= + + +

In this case, 3v = , n − r = 4, and therefore s 2.ϕ = To determine the equilibrium, it is necessary to fix temperature, the total pressure, and the partial pressure of a gas (water, for example), and then the partial pressures of the other two gases are also fixed as functions of the three variables: temperature, the total pressure, and the partial pressure of water.

3.4.3.3. Stoichiometric decomposition of a solid into three gases

We will illustrate this case by the decomposition of ammonium carbonate according to the following reaction:

4 2 3 2 2 3(NH ) CO CO H O 2NH= + +

We still have 3v = , with n − r = 3 and therefore s 1.ϕ = As discussed previously, equilibrium is reached if we fix the temperature, total pressure, and partial pressure of a gas, for example, water, and the partial pressures of the other two gases are functions of these three variables.


Variance Phenomena

1 Sublimation of a solid Decomposition of a solid solution with gas departure Stoichiometric reaction with a gas and two solids Stoichiometric reaction of three solids and a gas

2 Adsorption or desorption Dissolution of a gas by a solid Evolution of the stoichiometry of a solid Stoichiometric reaction with two gases and a solid Stoichiometric reaction with two gases and two solids

3 Variation in stoichiometry by a foreign gas with the solid Stoichiometric reaction with three gases and two solids Stoichiometric decomposition of a solid in three gases

Table 3.2. Classification of solid-gas systems according to the variance

Table 3.2 recapitulates the various enumerated cases and can be used as a guide for the identification of a transformation starting from the variance of the system.

3.5. Thermodynamics of systems containing solid phases only

In these systems, only condensed phases are involved and thus practically pressure is not an equilibrium factor, the gas phase above the solid is inert, and thus p = 1.

3.5.1. Non-variant systems

In this case, we find polymorphic transformations of solids such as α-sulfur into β-sulfur. The presence of only one independent component with two phases leads to a non-variant system when temperature at the equilibrium between the two forms is fixed.

We also come across variance zero with stoichiometric transformations between two solids, for example, the synthesis of tricalcium silicate, which involves two independent components with three phases:

2 3 5SiO 3CaO Ca SiO+ =


3.5.2. Univariant systems

We will put in this category the precipitation of a phase starting from a foreign element dissolved in a solid, for example, the precipitation of chromium oxide resulting from chromium ion substitution in nickel oxide:

Ni O 2 32Cr 3O 2e Cr O° ′+ + =

From the thermodynamic point of view, this case is similar to the precipitation of a solid from a liquid phase; there are two independent components, two phases, and therefore the variance is 1. We can define the solubility product, which is in this case:

[ ]2 2s NiCr eK ° ′⎡ ⎤= ⎣ ⎦

because the concentration of the normal oxygen ions in the network, in large excess, can be kept constant.

3.6. Specific study of quasi-chemical equilibriums

We will thoroughly study the equilibrium corresponding to quasi-chemical reactions. From section 3.2.4, we can define the associated properties of these reactions and therefore equilibrium constants. However, frequently, the number of superimposed equilibriums is important in that it leads to very complex calculations. We can simplify these with the help of certain justified approximations. These approximations are of two orders:

– approximation of the number of defects taken into account (the approximation of the prevalent defect) in the studied solid, as is the case of the Wagner approximation in non-stoichiometric binary solids (see section 2.3.2);

– approximation of a larger amount of charged defects (the approximation of the majority defects) due to the Brouwer approximation, when the preceding approximation of the prevalent defect cannot be retained.

We will thus see (section 3.6.3) how thermodynamics makes it possible to discuss the effect of a foreign element in an ionic solid quantitatively.


3.6.1. Equilibrium between an oxide and oxygen: the Wagner prevalent defect approximation

3.6.1.1. Example of a n-semiconductor

In the Wagner classification, such as we have established in section 2.3.2, zinc oxide is non-stoichiometric and has zinc ions in interstitial positions and free electrons. In the presence of oxygen, the oxide tends to approach stoichiometry:

i 2 Zn O1Zn 2e O Zn O2

°° ′+ + = +

There are two equations for this problem: the law of mass action is applied to the preceding equilibrium, on the assumption that there are very diluted defects:

[ ]2

1/ 2i OZn e P K°° ′⎡ ⎤ =⎣ ⎦

and the electric neutrality thus can be written in a phase with constant volume:

[ ]i2 Zn e'°°⎡ ⎤ =⎣ ⎦

The resolution of this system of two equations leads to the expression of the concentration of the two defects according to oxygen pressure at a given temperature:

[ ]2

1/ 3 1/ 3

i 1/ 6O

22 Zn e KP

°° ′⎡ ⎤ = =⎣ ⎦

This result can be verified from experiments, because as we will see in Chapter 5 (see section 5.5.2.4), electric conductivity is directly proportional to the concentration of interstitial zinc ions. Thus, at equilibrium, the logarithm of conductivity must lead to a line versus the logarithm of oxygen pressure with a slope of −1/6 (Figure 3.2).

3.6.1.2. Generalization with all the Wagner solids

We can again consider the preceding assumption for equilibrium between any Wagner oxide and a gas corresponding to the anion. We obtain laws of the same type as the preceding ones for the concentration of point defects or electric conductivity.


Figure 3.2. Variation in the electric conductivity of zinc oxide with oxygen pressure

Point defect Semiconductor Power of pressure

Interstitial cation n −1/2m Interstitial anion p 1/2m

Cation vacancy n −1/2m Anion vacancy p 1/2m

Table 3.3. Influence of oxygen pressure on the concentration of a Wagner oxide

Table 3.3 recapitulates the various cases, acknowledging that the power of the pressure is positive or negative, according to whether we are dealing with p- or n-type semiconductor, and is worth in absolute value 1/nm, where n indicates the atomicity of the gas molecule and m the number of entities that constitute the whole of the disorder, for example, 3 (one ion and two electrons) for zinc oxide but 2 (one oxygen ion and one hexavalent uranium ion trapped in two electron holes) for uranium dioxide.

3.6.2. General equilibrium of an oxide with oxygen in the Brouwer approximation of majority defects

We now assume a case in which the approximation of the prevalent defect is not acceptable any more. So, we will consider the equilibrium of cerium oxide (CeO2) with oxygen at a given temperature. This oxide presents, beside cerium(III) ions in cation positions, neutral, singly, and doubly ionized oxygen vacancies (within the meaning of the effective charge), which are in equilibrium with each other. Thus, three equilibrium conditions are involved in the system:

log P

log σ

Slope −1/6


The equilibrium of oxide with oxygen:

Ce O O Ce 212Ce O V 2Ce O2

°° ′+ = + +

The equilibrium leading to the two other forms of oxygen vacancies:

O Ce O CeV Ce V Ce°° °′+ = +

OO Ce CeV Ce V Ce° ′+ = +

Each equilibrium corresponds to the law of mass action, from which a system of three equations can be derived:

[ ]2

2 1/ 2O Ce O 1V Ce P K°°

′⎡ ⎤ =⎣ ⎦

[ ]O

2

O Ce

V

V CeK

°

°°

⎡ ⎤⎣ ⎦ =′⎡ ⎤⎣ ⎦

[ ][ ]O

3

O Ce

V

V CeK

°=

′⎡ ⎤⎣ ⎦

The electric neutrality is added:

[ ]O O CeV 2 V Ce° °° ′⎡ ⎤ ⎡ ⎤+ =⎣ ⎦ ⎣ ⎦

Solving this system of four equations with four unknown factors makes it possible to deduce the expressions of the concentrations of the four point defects as a function of oxygen pressure and the three equilibrium constants (i.e., at a given temperature).

We realize that the more the number of defects taken into account in the solid, the more the number of quasi-chemical equilibriums and the more complex the system becomes. Brouwer recommended a method of usually very sufficient approximation. It consists of considering the expression of electric neutrality in only the two majority entities (one on each side of the equality).


Case 1 Case 2

Concentrations O O2V V° °°⎡ ⎤ ⎡ ⎤<⎣ ⎦ ⎣ ⎦ O O2V V° °°⎡ ⎤ ⎡ ⎤>⎣ ⎦ ⎣ ⎦

OV°°⎡ ⎤⎣ ⎦ 1/ 31

2 / 3 1/ 62K

P 21/ K

[ ]CeCe′ 1/ 3 1/ 3

11/ 6

2 KP

1 21/ 4

K KP

OV°⎡ ⎤⎣ ⎦ 2 / 3

2 11/ 3 1/ 32

K KP

1 21/ 4

K KP

[ ]OV 1 2 3K K KP

1 2 3K K KP

Pressure ranges

2 61 2

42K K

P >>

Range 1

2 61 2

42K K

P <<

Range 2

Table 3.4. Concentrations of defect points in cerium oxide in both cases of the Brouwer approximation

Thus, in the case of cerium oxide, we can consider two Brouwer cases defined by:

Case 1: [ ]O Ce2 V Ce°° ′⎡ ⎤ =⎣ ⎦ or O2 V°°⎡ ⎤⎣ ⎦ >> OV°⎡ ⎤⎣ ⎦

Case 2: [ ]O CeV Ce° ′⎡ ⎤ =⎣ ⎦ or O2 V°°⎡ ⎤⎣ ⎦ << OV°⎡ ⎤⎣ ⎦

In each case, we replace the electric neutrality by its approached form. We solve the new systems and obtain the results given in Table 3.4. Of course, the concentrations of neutral particles are not affected by the approximation. The limiting pressure is calculated by clarifying in each zone the inequalities that result from the approximation.

Figure 3.3 shows plots, in each pressure ranges, of the variations in the concentrations of point defects versus oxygen pressure. This kind of diagram is referred to as a Kröger-Vink diagram [KRO 73].

REMARK.– Such models make it possible to explain the property of some semiconductor oxides, which switch from the p-type to the n-type with oxygen pressure or temperature.


Figure 3.3. The Kröger-Vink diagram for cerium oxide

3.6.3. Doping a solid with foreign elements: quantitative aspect

We will now approach the quantitative aspect of doping, that is, the effect of the introduction of a foreign element into a solid on the concentrations of its various structure elements. As in Chapter 2 (section 2.6), we will distinguish between stoichiometric solids and non-stoichiometric solids. We will discuss this through an example, in both cases, the effects of the introduction of a cation.

3.6.3.1. Doping of a Schottky solid with a cation of lower valence

We will discuss the concrete case of α-alumina (Al2O3), which can, at first approximation, be considered as the Schottky solid, that is, it contains aluminum vacancies and oxygen vacancies. Let us consider the effect of introduction of magnesium cations in substitution (valence 2) at the concentration C. We assume that the solid is closed compared with magnesium and thus C is a constant.

The heated solid is the site of the Schottky equilibrium:

Al O0 2V 3V°°′′′= +

The corresponding equilibrium condition is:

2 3

Al O SV V K°°⎡ ⎤′′′ ⎡ ⎤ =⎣ ⎦⎣ ⎦

The electric neutrality gives:

[ ]O Al Al2 V 3 V Mg°° ′′′ ′⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦ , with AlMg C′⎡ ⎤ =⎣ ⎦


In this expression of the electric neutrality, we can consider two Brouwer cases following the relative values of C compared with the aluminum vacancy concentration. We will consider the case in which C is sufficiently large; indeed, on the contrary, we can easily check that the defect concentrations are the same as those for pure (without dopant) alumina, that is, independent of the magnesium concentration. We will thus write:

O Al2 V Mg°° ′⎡ ⎤ ⎡ ⎤=⎣ ⎦ ⎣ ⎦ [ ]3/ 2

1/ 2Al s

2V KC

⎛ ⎞′′′ = ⎜ ⎟⎝ ⎠

associated with the equilibrium condition. This relation leads to the concentration of oxygen vacancies as:

[ ]3/ 2

1/ 2Al

2V sKC

⎛ ⎞′′′ = ⎜ ⎟⎝ ⎠

and OCV2

°°⎡ ⎤ =⎣ ⎦

These relations show that the concentration of oxygen vacancies increases with the introduction of magnesium, whereas that of aluminum vacancies decreases.

These expressions are valid if the concentration C is sufficiently large, or:

[ ]Al3 VC ′′′>> or 2 / 5 3 / 5 1/ 5

0 3 2C C K>> =

Figure 3.4 provides the curves of variations in the logarithms of defect concentrations versus the amounts of magnesium added.

A similar calculation with a cation of higher valence than the valence of aluminum would lead to variations in opposite directions.

Figure 3.4. Defect concentrations in magnesium-doped alumina


REMARK.– Doping by substitution with cations of the same valence as that of the basic cation occurs without affecting defect concentrations because the electric neutrality is not affected by the presence of a doping agent.

3.6.3.2. Doping with controlled electronic imperfection

To quantitatively calculate the effect of doping with cations of a non-stoichiometric solid, we examined the case of potassium-doped zinc oxide. This oxide is at equilibrium with oxygen:

i 2 Zn O1Zn 2e' O Zn O2

°° + + = +

The potassium concentration is fixed at C; therefore:

[ ]ZnK C′ =

The preceding equilibrium leads to:

[ ] 2 1/ 2i

1 eZn

KP°°

=′⎡ ⎤⎣ ⎦

The electric neutrality takes the form:

[ ] [ ]i Zn2 Zn eK°° ′ ′⎡ ⎤ = +⎣ ⎦

We can, therefore, have two Brouwer ranges:

– range 1 characterized by equality i2 Zn c°°⎡ ⎤ =⎣ ⎦ or [ ]e c′ <<

– range 2 characterized by equality [ ]i2 Zn e°° ′⎡ ⎤ =⎣ ⎦ or [ ]e c′ >>

Table 3.5 provides the expressions of the concentrations for the two Brouwer ranges. The border between the two ranges obeys the condition:

62

4PCK

=

The effect of the doping agent is felt only within range 1. Figure 3.5a shows the variations in defect concentrations versus potassium concentration at a given oxygen pressure. It is noted, especially within range 1, that the electron concentration is inversely proportional to the square root of the potassium concentration.


Element Range 1 Range 2

6PC >> 2

4K

PC << 2

4K

[ ]e′ 1/ 4

2 1KC P

⋅ 1/ 3

1/ 62 PK

−⎛ ⎞⎜ ⎟⎝ ⎠

iZn°°⎡ ⎤⎣ ⎦ 2C

1/ 31/ 61 2

2P

K−⎛ ⎞

⎜ ⎟⎝ ⎠

Table 3.5. Defect concentrations in zinc oxide doped with potassium

Figure 3.5b shows the variations in defect concentrations versus oxygen pressure at a given potassium concentration, C.

If potassium ions had been replaced by a cation with valence 3+, we would have ended with a variation in opposite directions.

3.7. Thermodynamics of systems: water vapor-hydrated salts

3.7.1. Experimental approach of equilibriums between water vapor and hydrated salts

Equilibrium between two hydrated forms of a salt and water vapor can be represented by the reaction (n and q being two positive numbers, n can be null):

2 2 2S H O H O S ( )H On q n q⋅ + = ⋅ + [3.E.1]

When equilibrium states of such a system are studied, we generally proceed by isothermal and isobaric thermogravimetry under controlled water vapor pressure.

If we indicate by ε the total amount of water contained in the solid per mole of salt, two kinds of curve are obtained at equilibrium.

The first one represented in Figure 3.6a is characterized by the presence of two steps, which correspond to the compositions of both hydrates with n and n + q moles of water, separated by a vertical line at a given pressure. If we change temperature, the shape of the curve remains the same, only the vertical line is now placed at another value of pressure.


Figure 3.5. Variations in defect concentrations of zinc oxide doped with potassium

The second form of curve illustrated in Figure 3.6b is characterized by a continuous variation in the content of water in the solid phase when the pressure varies. At another temperature, a new curve presenting the same character is obtained.

The first experimental result thus obtained is in fact the variance of the system. Indeed, in case (a), we see that if temperature is fixed, we have equilibrium between two hydrated forms for only one determined pressure; therefore, the variance of the system is 1. On the other hand, in case (b), the water content of the solid varies continuously with pressure and each point depends on temperature; thus, the variance of the system is 2.

Figure 3.6. Isotherm curves for equilibriums between water and hydrated salts: (a) univalent, (b) divalent


Let us again consider the Gibbs theorem for these systems. In both cases, the number of independent components is 2 (salt and water) and the number of external physical parameters taken into account is also 2 (pressure, temperature). In both systems, there is a gas phase and if sϕ is the number of solid phases, the Gibbs theorem is written:

s2 2 1v ϕ= + − − ; therefore, s 3 vϕ = −

We then deduce in both cases the number of solid phases (Table 3.6). If variance is 1, case (a), there are two solid phases at equilibrium: two hydrates with n and n + q moles of water. If variance is 2, then both hydrated forms have only the limiting compositions of a hydrated phase whose composition varies continuously with pressure and temperature. In fact, we have a solid solution or a non-stoichiometric hydrate. The water contained in the solid is sometimes described as zeolithic because zeolites constitute a family of solids having this property. Other hydrates have also been reported in literature [MUT 68], [GER 68], [NOV 71], [SOU 70a], [SOU 70b], [SOU 70c].

Curve of Figure 3.6 Variance Phase

number Solid phase number

(a) 1 3 2 (b) 2 2 1

Table 3.6. The number of solid phases at the equilibrium between a hydrated salt and water vapor

3.7.2. Study of the equilibriums with variance 1

In equilibrium (3.E.1), both hydrates constitute two solid phases. The law of mass action applied to this equilibrium is as follows:

00 1

1 1( ) exp HK P KRT

⎛ ⎞Δ= = − ⎜ ⎟

⎝ ⎠

which means that for a given temperature, at the equilibrium conditions, the vapor water pressure is fixed.

Figure 3.7 gives the curve of this equilibrium as the pressure-temperature curve. The left side of the curve represents the field of stability of the higher hydrate (n + q) and the right side represents the field of stability of the lower hydrate (n). The enthalpy associated with the reaction is usually positive (endothermic


dehydration). The preceding curve could have been traced in the system of axis ln P versus 1/T, which would have given a line (by neglecting variations in enthalpy and entropy with temperature) with slope:

01( )

Slope HR

Δ= −

Figure 3.7. Pressure-temperature curve for a system with variance 1

3.7.3. Study of hydrates with variance 2

3.7.3.1. Presentation of models

Since we have only one non-stoichiometric solid phase, we will approach the study of this system by quasi-chemistry of structure elements. However, this approach presents some difficulties. Indeed, hydrated salts are relatively complex solids with at least three principal components: the anion (itself often complex), the cation, and water. If salt admits several limiting hydrates, water molecules are not all equivalent. All these complexities require a simplification of the representation of solid. With this intention, we consider hydrated solids as pseudo-binary (see section 2.4.1) of which one of the components is the water concerned with dehydration and the other component is the skeleton of anhydrous salt or incorporates possible n molecules of water not implicated in the equilibrium under study. We will disregard specific defects related to the skeleton and thus take into account the following structure elements:

– molecules of skeleton in normal positions;

– water molecules in normal positions: 22 H OH O ;

– water molecules in interstitial positions: 2 iH O ;

– water vacancies: 2H OV ;

T

P

S · (n + q) H2O

S · nH2O


– empty interstitial positions: iV ; and

– non-located and mobile water molecules in the lattice: <<H2O>>.

Water vapor is noted as [H2O].

We make the assumption of the prevalent defect, which leads to three types of hydrates:

– non-stoichiometric hydrates with excess of non-located water molecules;

– non-stoichiometric hydrates with water vacancies; and

– non-stoichiometric hydrates with excess of interstitial located water molecules.

We assume perfect solutions, although the defects are not necessarily very diluted; however, they are electrically neutral.

3.7.3.2. Hydrates with mobile water molecules

The equilibrium that involves interstitial mobile water molecules can be written as follows:

[ ]2H O = <<H2O>> [3.E.2]

We consider index 1 as the property relating to the skeleton and index 2 as those relating to water in the solid. We will take the quantity ε of water contained per mole of solid as a variable, but this quantity includes n moles related to the skeleton and the true variable will be the difference ε − n. Then, the mole fraction of water in the solid is as follows:

2Amount of water

Amount of water Amount of skeleton 1 ε nxε n−

= =+ + −

If we indicate by KII the equilibrium constant of reaction [3.E.2], it becomes:

002 II

II II( )

expx H

K KP RT

⎛ ⎞Δ= = −⎜ ⎟

⎝ ⎠

Hence,

0 0II II

II

( ) ( ), or exp

1K n HPn P

K P n RTε

εε

⎛ ⎞− Δ− = = −⎜ ⎟− + − ⎝ ⎠

[3.20]


Figure 3.8a is the resultant isotherm. The concavity of this curve is turned upward (except in the vicinity of saturation, which is introduced a priori and not included in the model).

Figure 3.8. Shapes of an isotherm (a) and iso-composition curves (b) for hydrates with mobile water molecule

Figure 3.8b provides some calculated curves, each one at a constant composition in a system of axis ln P versus 1/T, which linearizes these curves.

3.7.3.3. Non-stoichiometric hydrates with water vacancies

The equilibrium is now written as:

[ ]2 22 H O 2 H OH O H O V= + [3.E.3]

On indicating by x1 and x2 the water site fractions occupied by vacancies and molecules, respectively, the law of mass action gives:

1III

2

x PK

x=

There exist q sites likely to accommodate water per mole of skeleton, and thus the site fractions are:

1( )q nxqε− −

= and ( )

2

nx

qε −

=


We deduce:

III

qPnK P

ε − =+

[3.21]

with

0 00III III

III III( ) ( )exp expG HK K

RTRT⎛ ⎞ ⎛ ⎞Δ Δ

= − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

[3.22]

The curve in Figure 3.9 represents an isotherm. We note there that, contrary to the preceding case, concavity is turned downward.

Figure 3.9. Isotherm curve for the equilibrium of a water hydrate (with vacancies) with water

3.7.3.4. Non-stoichiometric hydrates with interstitial water molecules

We write the equilibrium as:

[ ]2 i 2 iH O + V H O= [3.E.4]

The calculation of the water content shows that we are led to the same result as obtained previously (Figure 3.9). This means that such a solution with localized water molecules can be regarded either as a sub-stoichiometry of the higher hydrate or as on over-stoichiometry of the lower hydrate. In fact, it is normal to obtain in both cases the same result because structure elements are defined compared with one or the other hydrate, which does not modify obviously the solution but simply the base for the description of the structure.

In the case of the located water molecules, these curves become lines in a system ln P versus 1/T (Figure 3.10b). The slope of these lines makes it possible to calculate the standard enthalpy associated with equilibrium [3.E.4], which is, in general,


positive. All the curves are thus parallel and result from each other by simple transformations.

By comparing Figures 3.8a and 3.9, the curvature of the isotherm makes it possible to conclude the localization or absence of the water molecules in the skeleton.

Figure 3.10. Curves of iso-compositions of a non-stoichiometric hydrate

3.7.3.5. Limits of the divariant area in the diagram P(T)

In the pressure-temperature space, the divariant area is not unlimited. In the majority of the cases, a phenomenon represented by equilibrium comes to limit this area. We can distinguish two categories of such a phenomenon:

– those that involve water vapor such as the liquefaction of water

[ ]2 2 liquidH O H O=

– those that do not involve water, such as the precipitation of the lower hydrate in a new crystallographic structure, that is, under a new phase. We can represent this precipitation as the stoichiometric phase, indicated by phase 1, and that which does not accept the non-stoichiometry as the stoichiometric phase, indicated by phase 2:

2 phase1 2 phase2S H O S H On n⋅ = ⋅

As an example, we will discuss this last case. By assuming again the notations of section 3.7.3.3, this equilibrium can be written with structure elements such as that of a simple disappearance of vacancies:

2H O 0V = [3.E.5]


To study the limit of divariant area in the diagram P, T, we superimpose equilibriums [3.E.3] and [3.E.5]. By application of the law of mass action at equilibrium [E 3.5], it becomes:

0

1( )1 exp V

V

Gx

K RT⎛ ⎞Δ

= = ⎜ ⎟⎝ ⎠

And, on substituting into equation [3.22] and by taking into account the sum of the site fractions as 1, we obtain:

00III ( )( )

exp exp 1V GGP

RT RT⎡ ⎤⎛ ⎞⎛ ⎞ ΔΔ

= − − −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

This limit is represented in Figure 3.11 and separates the area of phase 1 and the divariant area of phase 2. The introduction of the new equilibrium decreases the variance down to 1.

Figure 3.11. Divariant area limited by the precipitation of the lower hydrate

REMARK.– Phase 1 can, itself, become another divariant area if the lower hydrate (with n molecules of water) is itself non-stoichiometric. In such a case, the two areas are separated by the curve representing the superposition of the two phases, which can be described as two phases representing the hydrate with n moles of water, one sub-stoichiometric and the other over-stoichiometric, compared with n.

3.8. Sequence of transformations, juxtaposition of stability area

If we change an intensive property such as temperature in a wide range, successive transformations of a system can occur. This will result in a succession of equilibriums. Between these equilibriums, we will define temperature ranges in which no reaction occurs. These intervals will define stability areas of the phases.


Figure 3.12. Shape of a pressure-temperature diagram of the hydrated forms of copper sulfate

We will illustrate this by an example: the dehydration of pentahydrate copper sulfate.

This salt gives in a certain temperature range a series of univariant dehydrations whose curves P versus T are represented in Figure 3.12. Thus, the phenomenon depends on the vapor pressure P imposed on the system.

For pressures such as P < P1, only a single equilibrium is possible between the pentahydrate and anhydrous salts, and we have:

4 2 4 2CuSO 5H O CuSO 5H O⋅ = +

If we plot, for such a pressure, the curve giving the water content in the solid at equilibrium versus temperature, we obtain the representation as in Figure 3.13a. In this range of pressure, we can thus say that there are both a stability area for the pentahydrate form on the left-hand side of the curve of the transformation (Figure 3.12) and a stability area for the anhydrous salt on the right-hand side.

Figure 3.13. Dehydrations of pentahydrate copper sulfate (a) shows two steps, this means that we obtain one stable phases (b) shows three steps, this means we obtain two stable

phases (c) shows four steps, this means we obtain three stable phases


For pressures such as P1 < P < P2, the thermogravimetric study leads to the curve of Figure 3.13b, which shows that the system is the site of two successive univariant transformations, from the pentahydrate to the monohydrate and then from the monohydrate to the anhydrous salt, following equilibriums:

4 2 4 2 2CuSO 5H O CuSO H O 4H O⋅ = ⋅ +

4 2 4 2CuSO H O CuSO H O⋅ = +

When we vary temperature, this leads to three stability areas, as shown in Figure 3.12, inside which the three forms – pentahydrate sulfate, the monohydrated sulfate, and the anhydrous one – are stable.

We notice that at each triple point, for the pressure P1 and P2, two equilibriums occur simultaneously and the system becomes non-variant, both pressure and temperature are commanded.

For pressures such as P > P2, thermogravimetric study leads to the curve of Figure 3.13c, which shows that the system is the site of three successive univariant transformations, from the pentahydrate into trihydrate, then into monohydrate, and finally into the anhydrous, following equilibriums:

4 2 4 2 2CuSO 5H O CuSO 3H O 2H O⋅ = ⋅ +

4 2 4 2 2CuSO 3H O CuSO H O 2H O⋅ = ⋅ +

4 2 4 2CuSO H O CuSO H O⋅ = +

When we vary temperature, this leads to four stability areas, as shown in Figure 3.12, inside which the four forms are stable. We still obtain a non-variant triple point. We note that it is impossible to consider a quadruple point where four curves would be merged; indeed, at this point, the superposition of three equilibriums would involve a negative variance (−1), which is impossible.

We recall the reaction path way in the pressure-temperature diagram, the whole of the successive transformations met during the pathway, starting from an initial point and arriving at a final one. For example, it is seen that the reaction pathway at constant pressure, starting from the pentahydrate copper sulfate and leading to anhydrous salt, can be, according to the pressure range, more or less complex.


3.9. Equilibrium of the formation of a solid from a solution

Usually, a solid is generated starting from a solution that can be either fluid (mixture of gas or liquid solution) or solid (real solid with point defects and thus regarded as a solution of structure elements). The thermodynamics of such heterogenous systems is not basically different from what we have already considered, but the concept of supersaturation is usually used.

3.9.1. Solubility product and supersaturation

3.9.1.1. Thermodynamic definitions

A precipitation is a transformation that leads to a pure component S in a condensed phase (liquid or, more often, solid) starting from components belonging to the same solution (usually, condensed liquid or solid), which we will represent by the equation:

sA Ai i j ji jβ β S β= +∑ ∑ [3.23]

the sum being extended to all the components of the solution that take part in precipitation.

The Gibbs energy associated with this reaction is:

0 0 0c s s( ) ( ) ( ) ln ( ) lnj j j j i i i i

j j i iG g T T RT a T RT aβ β μ β β μ βΔ = + + − −∑ ∑ ∑ ∑

At equilibrium, this Gibbs energy is null and therefore:

0 0 0 equils

s s s

( ) ( ) ln ( ) lnj j i ij j i i

j j i i s

g T T RT a T RT aβ β β β

μ μβ β β β

= − − + +∑ ∑ ∑ ∑

which can be written as:

s

s

0 0 0equils

s ss

equil

( ) ( ) ( )( ) ( ) exp

( )

i

j

jii ji

i ji

jj

g T T TaK T

RTa

ββ

β

β

ββμ μ

β β⎛ ⎞

− +⎜ ⎟⎜ ⎟= = ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

∑ ∑∏

∏ [3.24]

The preceding constant, s ( )K T , is referred to as the solubility product of S in the initial solution. It depends on temperature.


REMARK.– When the solvent, which is present in large amount, is one of the reactants of precipitation, its activity can be taken equal to 1 and thus omitted in the preceding product.

According to condition [3.12], we will distinguish three cases:

– If s

0s ,j j i i

j igβ β μ β μ+ >∑ ∑ the solution, which is the stable phase, and the

addition of a minor amount of the pure phase will involve its disappearance by dissolution.

– If 0s s ,j j i i

j igβ β μ β μ+ <∑ ∑ the precipitate is stable and the addition of a bit

of one of the reactive component of the solution will result in an increase in the amount of the precipitated pure phase.

– If 0s s ,j j i i

j igβ β μ β μ+ =∑ ∑ the affinity of precipitation is null, and there is

equilibrium between the solution and the precipitated pure phase.

If we assume the stability conditions of the solution and modify the conditions to bring the system under the equilibrium conditions, the pure condensed phase should appear. Frequently, this phase does not appear, and we can then possibly be temporarily out of equilibrium with a solution that would meet the condition:

0s s j j i i

j igβ β μ β μ+ <∑ ∑ . We can practically reach this situation by increasing the

pressure if the solution is a gas, and by decreasing the temperature or the concentrations in all the other cases. Such a system is known as supersaturated. Supersaturation S is a measurement of the Gibbs energy excess compared with equilibrium, that is, of the distance from equilibrium conditions. In practice, we can often characterize it by the difference in either temperature (we speak then about supercooling) or pressure, or compared with the activities by the following expression:

s

ss

1( )

i

j

ii

jj

aS

K Ta

ββ

ββ

= ∏

∏ [3.25]

In the particular case in which we have 0,jβ = for any given value of j and s 1β = , the preceding relations become:

s

0 0s

equils

( ) ( )( ) ( ) exp

i i ii

ii

g T Ta K T

RT

ββ

β μ⎛ ⎞−⎜ ⎟= = ⎜ ⎟⎜ ⎟⎝ ⎠

∑∏ [3.26]


and

s ( )

ii

i

aS

K T

β

=∏

[3.27]

REMARK.– In the case of supersaturation, the Gibbs energy associated with precipitation is:

0R R

s s

( ) ( ) ln ln lnjii j

j

G G a a RT Sββ

β βΔ = Δ − + = −∑ ∑ [3.28]

3.9.1.2. Other definitions of supersaturation

There exist other definitions of supersaturation that can create some confusion, and some of them are without physical significance (remain in the simplified case

0,jβ = for any value of j, and s 1β = ).

Some authors prefer the following expression, which has the advantage of being invariant in a change of the definition of the building unit:

1

s ( )

i iiiia

SK T

β β⎛ ⎞∑= ⎜ ⎟⎜ ⎟

⎝ ⎠

∏ [3.29]

We sometime also encounter:

( )equil i

i ii

S c cβ

= −∏ [3.30]

This latest expression is without physical significance, but it can find its utility in certain empirical formulae, for example, to choose a particular set of values for the equilibrium concentrations equil

ic .

For any chosen definition, it is obvious, in principle but often forgotten in practice, that species Ai must be the real species that build the solid; in particular, in solution, if they are complexes, their concentrations govern the equilibrium and not the total concentration of an even majority species.


3.9.2. Extension to formation of a real solid

Real solid differs from the ideal solid by points defects whose existence modifies the Gibbs energy of formation; fortunately, it is rare that we cannot avoid taking into account these defects because the variation in Gibbs energy due to the formation of the defects is in general small compared with the one that is associated with the reaction of precipitation, which makes it possible to calculate the properties of the ideal solid. Possibly, we can take into account the defects, as we have already done in sections 3.6.1 and 3.6.2, if these defects are important for a property used later on.

3.9.3. Extension to the transformation of a solid into another solid

These are the cases that really interest us, so we must necessarily understand the equation of supersaturation.

3.9.3.1. Case in which both solids are perfect

The reaction has the following form:

s sA' i i j ji j

β S β β S β A′ + = +∑ ∑

In this case, supersaturation is expressed in a manner similar to the formation of a solid starting from a liquid phase replacing 0

sg by 0 0s s s ' s ' s( ) /g gβ β β′ ′− in relations

[3.26] and [3.27].

3.9.3.2. Case in which the solids have point defects

We will see later that solid phases are created by quasi-chemical reactions. The reactants are thus structure elements (i.e. point defects) of the initial phase added, possibly of components present in a surrounding liquid phase. Similarly, the products of the reaction will be, in addition to the final solid, structure elements of the initial solid and/or components of the fluid phase.

Actually, just as for the crystallization starting from a fluid solution, it is also necessary to carry out an analysis in order to determine the true reactants, which is, in turn, complicated by the difficulties inherent in the solids: minor defects, heterogeneity of composition, and difficulty in performing the online chemical analysis.

However, once this study is carried out, the problem will be formulated in a manner similar to [3.23], the liquid phase being replaced by the initial solid phase,


classically discussed in the calculation of quasi-chemistry of a solid in a manner identical to solving a problem of solution, the solutes being the point defects of this solid. The application of the law of mass action to the reaction of precipitation of the new solid will give the solubility product, formally identical to [3.26], and we will be able to define supersaturation by [3.27].

It is seen that the transformation of one solid into another is not really different from precipitation starting from a liquid solution; the application of the methods of quasi-chemistry allows the use of the same formalism. It will however be necessary to be careful with the connection of the two modes of description: in particular, the energy of formation of the composed gas phase or pure liquid is expressed starting from the elements, whereas the energy of formation of a point defect in a solid is expressed compared with the ideal solid.

We will illustrate the methodology of the reaction of precipitation of γ-alumina starting from bœhmite (see section 17.4).

3.10. Variations in the equilibrium conditions with sizes of solid phases

Until now, we considered that thermodynamic properties, in particular chemical potentials and Gibbs energy, depend on temperature and pressure. In any rigor, these properties also depend on the area of the interfaces between the phases. This dependence is in fact significant for phases of very small sizes only, such as grains with radius less than 100 nm. In this granular range, low sizes will result in a variation in the standard Gibbs energies associated and thus of equilibrium constants, defined by [3.17], with sizes of solid phases.

3.10.1. Variation in equilibrium constant with curvature radii

If we take into account the variable area σ and if we indicate by γ the surface tension of a phase, the infinitesimal change in internal energy in a reversible transformation becomes [SOU 90c]:

dd d dd

U T S P Vnσγ= − + [3.31]

We can thus write the infinitesimal change in Gibbs energy:

chimddd

G dGnσγ= +


indicating by Gchim the Gibbs energy for the large-sized phases (or infinite curvature radii, i.e., plane surfaces).

For the standard properties of a reaction, we thus write:

0 0chim

ddi

i

G Gnσγ ν ⎛ ⎞Δ = Δ + ⎜ ⎟

⎝ ⎠∑ [3.32]

However, if there is only a single solid with several components and if imV is the

partial molar volume (compared with molar volume) of this component, we have:

d d d d. d d d dim

i i

V Vn V n Vσ σ σ⎛ ⎞ = =⎜ ⎟

⎝ ⎠

It can be shown that if R1 and R2 are the two principal curvature radii of the surface, we have:

1 2

d 1 1dV R Rσ

= +

radii being taken positively for a convex surface.

Then, switching to equilibrium constants, indicated by K∞ , the “chemical” equilibrium constant, that is, without curvature:

dln d ii m

i

KRT VVKσγ ν

∞

= − ∑

Hence,

1 2

1 1lnii m

i

KRT VR RK

γ ν∞

⎛ ⎞= − +⎜ ⎟

⎝ ⎠∑ [3.33]

In the particular case in which only a spherical interface of radius r exists, this gives:

1 2

1 1 2R R r

+ = [3.34]


and the relation between the equilibrium constants becomes:

2lni

i mi

KRT VrKγ ν

∞

⎛ ⎞= − ⎜ ⎟

⎝ ⎠∑ [3.35]

Table 3.7 gives the orders of magnitude of surface energies of various types of interfaces.

Kind of interface γ (J/m2)

Liquid-gas 0.1 Liquid-liquid 0.01--0.1 Liquid-solid 1

Solid-gas 1 Solid-solid 1

Grain boundaries 0.01–1

Table 3.7. Orders of magnitude of surface energies

Take the example of the reaction of gaseous nickel carbonyl formation starting from nickel grains [DEF 51].

4Ni 4CO Ni(CO)+ =

The equilibrium constants are:

2ln mr VK

RTK r

γ

∞

= − [3.36]

In these expressions, rK and K∞ are the values of the equilibrium constants for radius R and an infinite radius (or sufficiently large), respectively.

This influence of grain sizes in fact increases the variance of the system by 1, by increasing the term p of the Gibbs theorem, with the introduction of surface tension.

Thus, for example, for a univariant system with large grains, we obtain a divariant system for fine grains, and the single curve of the diagram (P, T) in Figure 3.1 is transformed into an area of the plane in which we can draw a series of curves, each one characterized by a value of the radius of the grains (Figure 3.14).


If the condensed sample would be limited by a curved surface with principal curvature radii R1 and R2, relation [3.35] becomes:

1 2

1 1ln imii

KRT VK R R

γ ν∞

⎛ ⎞⎛ ⎞= − +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠∑ [3.37]

Figure 3.14. Representation of a divariant system due to the introduction of curvature

This modification of thermodynamics of the systems can lead to new results and, under certain conditions, a solid phase that would not be stable in a massive state can result in very fine grains. For example, [SUC 74] at 44°C, under a water pressure of 4.1 hPa, the larger grains (diameter >225 μm) of pentahydrate copper sulfate are transformed into the monohydrate salt, whereas if the grains are smaller, the trihydrate salt is formed.

3.10.2. Influence of curvature radii on tension of vapor

Consider a monoatomic compound composed of small spherical grains of radius r and examine the tension of vapor P in equilibrium with these grains. Indicate by P∞ the tension of vapor above a plane and by VM the molar volume of the solid. The vaporization equilibrium is written:

solid vaporM M=

And the equilibrium constant is K = P. Using relation [3.35] on the convex side, we obtain:

2ln mVPRT

P rγ

∞

= [3.38]


And with a surface with two curvature radii:

1 2

1 1ln mPRT VP R R

γ∞

⎛ ⎞= +⎜ ⎟

⎝ ⎠ [3.39]

3.10.3. Influence of curvature radii on point defect concentrations

Examine the influence of a curvature radius on the oxygen vacancy concentration in cerium dioxide in the presence of oxygen. The reaction is:

O Ce Ce O 21O 2Ce 2Ce V O2

°°′+ = + +

Equilibrium associated with the electric neutrality gives:

2

3 1/ 2O O4 VK P°°⎡ ⎤= ⎣ ⎦

The term ii mVν∑ is equal to the difference

Lm mV V− between the molar volume of cerium oxide in which all the oxygen ions would be replaced by vacancies and the molar volume of stoichiometric cerium oxide. This difference is positive because the vacancies expand the oxide, as the repulsive interactions between cations are no longer compensated by a shielding effect of negative ions. Equation [3.35] thus gives:

( )1 2

1 1 ln Lm m

KRT V VK R R

γ∞

⎛ ⎞= − − +⎜ ⎟

⎝ ⎠ [3.40]

We now calculate the vacancy concentration by observing the electric neutrality, which becomes:

( )O O

1 2

1 1V V exp3

Lm mV V

RT R R

γ°° °°

∞

⎡ ⎤⎛ ⎞− ⎛ ⎞⎢ ⎥⎜ ⎟⎡ ⎤ ⎡ ⎤= − +⎜ ⎟⎣ ⎦ ⎣ ⎦ ⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦ [3.41]

These surface effects will be taken into account for the phenomenon of nucleation (Chapter 8), which necessarily leads to low-size solid phases such as in the case of coalescence (Chapter 12).

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