So far we have discussed the thermodynamic properties of materials, which have been considered as pure and to consist of only a single component. We will now continue with systems containing two or more components and thereby solutions.
Solutions are thermodynamic phases with variable composition, and are common in chemical processes, in materials and in daily life. Alloys – solutions of metallic elements – have played a key role in the development of human civilisation from the Bronze Age until today. Many new advanced materials are also solutions.
Examples are tetragonal or cubic ZrO2, stabilized by CaO or Y2O3, with high toughness or high ionic conductivity, and piezoelectric and dielectric materials based on BaTiO3or PbZrO3. In all these cases the mechanical or functional proper-ties are tailored by controlling the chemical composition of the solid solution. The chemical and thermal stability of these complex materials can only be understood if we know their thermodynamic properties.
The understanding of how the chemical potential of a component is changed by mixing with other components in a solution is an old and fascinating problem. The aim of this chapter is to introduce the formalism of solution thermodynamics.
Models in which the solution is described in terms of the end members of the solu-tion, solution models, are given special attention. While the properties of the end members must be described following the methods outlined in the previous chapter, the present chapter is devoted to the changes that occur on formation of the solutions. In principle one could describe the Gibbs energy of a mixture without knowing the properties of the end members, but since it is often of interest to apply a solution model in thermodynamic calculations involving other phases, the solu-tion model often is combined with descripsolu-tions of the Gibbs energies of the end members to give a complete thermodynamic description of the system.
57 Chemical Thermodynamics of Materials by Svein Stølen and Tor Grande
© 2004 John Wiley & Sons, Ltd ISBN 0 471 492320 2
3.1 Fundamental definitions
Measures of composition
The most important characteristic of a solution is its composition, i.e. the concen-tration of the different components of the phase. The composition of a solution is best expressed by the ratio of the number of moles of each component to the total number of moles. This measure of the composition is the mole fraction of a com-ponent. In the case of a binary solution consisting of the components A and B, the mole fractions of the two components are defined as
x n
n n
A A
A B
= + and x n
n n
B B
A B
= + (3.1)
and it is evident that
xA +xB=1 (3.2)
For an infinitesimal change in composition of a binary solution the differentials of the two mole fractions are related as
dxA = -dxB (3.3)
In dealing with dilute solutions it is convenient to speak of the component present in the largest amount as the solvent, while the diluted component is called the solute.
While the mole fraction is a natural measure of composition for solutions of metallic elements or alloys, the mole fraction of each molecule is chosen as the measure of composition in the case of solid or liquid mixtures of molecules.1In ionic solutions cations and anions are not randomly mixed but occupy different sub-lattices. The mole fractions of the atoms are thus an inconvenient measure of composition for ionic substances. Since cations are mixed with cations and anions are mixed with anions, it is convenient for such materials to define composition in terms of ionic fractions rather than mole fractions. In a mixture of the salts AB and AC, where A is a cation and B and C are anions, the ionic fractions of B and C are defined through
X n
n n X
B B
B C
= C
+ = -1 (3.4)
1 Note that volume fraction rather than mole fraction is recommended in mixtures of molecules with significant different molecular mass. This will be discussed in Chapter 9.
In a binary solution AB–AC, the ionic fractions of B and C are identical to the mole fractions of AB and AC. It may therefore seem unnecessary to use the ionic fractions. However, in the case of multi-component systems the advantage of ionic fractions is evident, as will be shown in Chapter 9.
Mixtures of gases
The simplest solution one can imagine is a mixture of ideal gases. Let us simplify the case by assuming only two types of ideal gas molecules, A and B, in the mix-ture. The total pressure in this case is the sum of the partial pressures of the two components (this is termed Dalton’s law). Thus,
ptot =pA +pB (3.5)
where pAand pBare the partial pressures of the two gases and ptotis the total pres-sure. By applying the ideal gas law (eq. 2.23), the volume of the gas mixture is
Vtot =n VA m,A +n VB m,B (3.6)
where nAand nBare the number of moles of A and B in the mixture and Vm,Aand Vm,Bare the molar volumes of pure A(g) and B(g). In this case, where both A and B are ideal gases, Vm,A= Vm,B. It follows that, for a mixture of ideal gases
pA =x pA tot (3.7)
The chemical potential of an ideal gas A is given by eq. (2.26) as
mA A m m
wheremAo is the standard chemical potential of the pure ideal gas A at p
A
o = 1 bar at a given temperature T. For a mixture of the ideal gases A and B at constant pressure ( ptot p
A
o bar
= =1 ) the chemical potential of A for a given composition of the solu-tion, xA, is, by using eq. (3.7)
The difference between the chemical potential of a pure and diluted ideal gas is simply given in terms of the logarithm of the mole fraction of the gas component.
As we will see in the following sections this relationship between the chemical potential and composition is also valid for ideal solid and liquid solutions.
In mixtures of real gases the ideal gas law does not hold. The chemical potential of A of a mixture of real gases is defined in terms of the fugacity of the gas, ƒA. The fugacity is, as discussed in Chapter 2, the thermodynamic term used to relate the chemical potential of the real gas to that of the (hypothetical) standard state of the gas at 1 bar where the gas is ideal:
mA A m m
A
o A
A
o A
o A
(x ) RT ln f ln( )
p
RT f
= + æ
è çç
ö ø
÷÷ = + (3.10)
Solid and liquid solutions – the definition of activity
In the solid or liquid state the activity, a, is introduced to express the chemical potential of the components of a solution. It is defined by
mA =mA* +RT lnaA (3.11)
wherem*Ais the chemical potential of A in the reference state. For p = 1 barmA m
A o
* = .
One of the most important tasks of solution thermodynamics is the choice of an appro-priate reference state, and this is the topic of one of the following sections.
3.2 Thermodynamics of solutions
Definition of mixing properties
The volume of an ideal gas mixture is given by eq. (3.6). Let us now consider only solid or liquid mixtures. Our starting point is an arbitrary mixture of nA mole of pure A and nBmole of pure B. The mixing process is illustrated in Figure 3.1. We
p, T constant
mixing Vm,B nB Vm,A
nA
A A B B
n V +n V
Figure 3.1 Mixing of nAmoles of A and nBmoles of B at constant p and T. The molar vol-umes of pure A and B are VAand VB. The partial molar volumes of A and B in the solution are VAand VB, respectively.
will first derive the expressions for the volume of the system before and after the mixing. The volume before mixing is
V(before)=n VA m,A +n VB m,B (3.12)
where Vm,A and Vm,Bare the molar volumes of pure A and B. We now mix A and B at constant pressure p and temperature T and form the solution as illustrated in Figure 3.1. The expression for the volume of the solution is then
V(after)=n VA A +n VB B (3.13)
where VA and VB represent the partial molar volumes of A and B (defined by eq.
1.87) in the solution. These partial molar volumes may be seen as apparent vol-umes that when weighted with the number of A and B atoms give the observed total volume of the solution. The difference in the volume of the solution before and after mixing, the volume of mixing, is designatedDmixV:
DmixV =V(after)-V(before)=nA(VA -VA)+nB(VB -VB) (3.14) The volume of mixing for one mole of solution is termed the molar volume of mixing,Dmix mV , and is derived by dividing eq. (3.14) by the total number of moles (nA +nB) in the system
D D
D D
mix m mix
A B
A mix A B mix B
V V
n n x V x V
= + = +
( ) (3.15)
The molar volume of mixing of two binary systems is shown in Figure 3.2.
Pb–Sn shows positive deviation from the ideal behaviour at 1040 K [1] while the volume of mixing of Pb–Sb at 907 K is negative, with a minimum at xPb¹ 0.5 and asymmetric with respect to the composition [2].
0.0 0.2 0.4 0.6 0.8 1.0
–0.16 –0.12 –0.08 –0.04 0.00 0.04
Pb–Sb Pb–Sn
xPb
3 mixm/mmolV- D61 10
Figure 3.2 Molar volume of mixing of molten Pb–Sn at 1040 K [1] and Pb–Sb at 907 K [2]
as a function of composition.
The phenomenology described above can be applied to any thermodynamic extensive function, Yi, for a solution. The integral molar enthalpy, entropy and Gibbs energy of mixing are thus
DmixHm =xADmixHA +xBDmixHB (3.16)
DmixSm =xADmixSA +xBDmixSB (3.17)
DmixGm =xADmixGA +xBDmixGB (3.18)
The three functions are interrelated by
DmixGm =DmixHm -TDmixSm (3.19)
SinceDmixGA =mA -m , the integral molar Gibbs energy of mixing can alterna-Ao tively be expressed in terms of the chemical potentials as
DmixGm =xA(mA -mAo)+xB(mB -mBo)=RT x( A lnaA +xBlnaB) (3.20) wheremAo andmBoare the chemical potentials of pure A and B, whereasmAandmB are the chemical potentials of A and B in the given solution. Using G = H – TS, the partial molar Gibbs energy of mixing is given as
Dmix A Dmix A Dmix A A A
o A
G = H -T S =m -m =RT lna (3.21)
The partial molar entropy, enthalpy or volume of mixing can be derived from eq.
(3.21) and are given by the relations
D D
Corresponding equations can be derived for the partial molar properties of B.
Ideal solutions
In Section 3.1 we showed that the chemical potential of an ideal gas in a mixture with other ideal gases is simply given in terms of a logarithmic function of the mole fraction. By comparing eqs. (3.9) and (3.10) we see that the fugacity/activity of the ideal gas is equal to the mole fraction. A solution (gas, liquid or solid) is in general called ideal if there are no extra interactions between the different species in addi-tion to those present in the pure components. Thermodynamically this implies that the chemical activity is equal to the mole fraction, ai = , over the entire composi-xi tion range. The molar Gibbs energy of mixing for an ideal solution then becomes DidmixGm =RT x( A lnxA +xBlnxB) (3.25)
The Gibbs energy of mixing of an ideal solution is negative due to the positive entropy of mixing obtained by differentiation of DidmixGm with respect to temperature:
In the absence of additional chemical interactions between the different species that are mixed the solution is stabilized entropically; the solution is more disor-dered than a mechanical mixture of the components. The origin of the entropy con-tribution is most easily understood by considering the discon-tribution of two species on a crystalline lattice where the number of lattice sites is equal to the sum of the number of the two species A and B. For an ideal solution, a specific number of A and B atoms can be distributed randomly at the available sites, i.e. in a large number of different ways. This gives rise to a large number of different structural configurations with the same enthalpy and thus to the configurational entropy given by eq. (3.26). This will be discussed further in Chapter 9.
Two other characteristic properties of ideal solutions are
DidmixHm =DidmixGm +TDidmixSm = 0 (3.27)
Or in words: in the absence of additional chemical interactions between the two types of atom, the enthalpy and volume of mixing are both zero.
The partial molar properties of a component i of an ideal solution are readily obtained:
D D
The thermodynamic properties of an ideal binary solution at 1000 K are shown in Figure 3.3. The integral enthalpy, entropy and Gibbs energy are given in Figure 3.3(a), while the integral entropy of mixing and the partial entropy of mixing of component A are given in Figure 3.3(b). Corresponding Gibbs energies are given in Figure 3.3(c). The largest entropic stabilization corresponds to the minimum Gibbs energy of mixing, which for an ideal solution is RT ln( )1
2 or –RT ln 2, or about 0.7 times the thermal energy (RT) at 1000 K.
Excess functions and deviation from ideality
Most real solutions cannot be described in the ideal solution approximation and it is convenient to describe the behaviour of real systems in terms of deviations from the ideal behaviour. Molar excess functions are defined as
DexcmixYm =Dmix mY -DidmixYm (3.31)
The excess molar Gibbs energy of mixing is thus
Dexcmix m Dmix m A A B B
The activity coefficient of component i, gi, is now defined as a measure of the deviation from the ideal solution behaviour as the ratio between the chemical activity and the mole fraction of i in a solution.
gi i
i
a
=x or ai = gixi (3.33)
For an ideal solutiongi =1.
The partial molar Gibbs energy of mixing of a component i in a non-ideal mix-ture can in general be expressed in terms of activity coefficients as
DmixGi =RT lnai =RT lnxi +RT lngi (3.34)
Using eq. (3.34) the excess Gibbs energy of mixing is given in terms of the mole fractions and the activity coefficients as
Dmix m A A B B Dmix D
kJmol-1 kJmol-1 JKmol --11
JKmol--11
(a)
(b)
(c)
Figure 3.3 Thermodynamic properties of an arbitrary ideal solution A–B at 1000 K. (a) The Gibbs energy, enthalpy and entropy. (b) The entropy of mixing and the partial entropy of mixing of component A. (c) The Gibbs energy of mixing and the partial Gibbs energy of mixing of component A.
Implicitly:
DexcmixGm =RT x( A lngA +xBlngB) (3.36)
SinceDidmix m Dmix id m
H = V = 0 (eqs. 3.27 and 3.28), the excess molar enthalpy and volume of mixing are simply
DexcmixVm =Dmix mV (3.37)
DexcmixHm =DmixHm (3.38)
The excess molar entropy of mixing is the real entropy of mixing minus the ideal entropy of mixing. Using a binary A–B solution as an example,DexcmixSm is
DexcmixSm =DmixSm -R x( AlnxA +xBlnxB) (3.39)
For a large number of the more commonly used microscopic solution models it is assumed, as we will see in Chapter 9, that the entropy of mixing is ideal. The dif-ferent atoms are assumed to be randomly distributed in the solution. This means that the excess Gibbs energy is most often assumed to be purely enthalpic in nature.
However, in systems with large interactions, the excess entropy may be large and negative.
As shown above, the activity coefficients express the departure from ideality and thus define the excess Gibbs energy of the solution. Deviation from ideality is said to be positive wheng >1(ln g is positive) and negative when g < 1(ln g is negative).
A negative deviation implies a negative contribution to the Gibbs energy relatively to an ideal solution and hence a stabilization of the solution relative to ideal solu-tion behaviour. Similar arguments imply that positive deviasolu-tions from ideality result in destabilization relative to ideal solution behaviour.
The activities of Fe and Ni in the binary system Fe–Ni [3] and the corresponding Gibbs energy and excess Gibbs energy of mixing are shown in Figures 3.4 and 3.5, respectively. The Fe–Ni system shows negative deviation from ideality and is thus stabilized relative to an ideal solution. This is reflected in the negative excess Gibbs energy of mixing. The activity coefficientsgi, defined by eq. (3.33) as a xi/ i, are readily determined from Figure 3.4.gNi for the selected composition xNi= 0.4 is given by the ratio MQ/PQ. At the point of infinite dilution, xi= 0, the activity coefficient takes the value gi¥. gi¥ is termed the activity coefficient at infinite dilution and is, as will be discussed in Chapter 4, an important thermodynamic characteristic of a solution. The activity coefficient of a given solute at infinite dilution will generally depend on the nature of the solvent, since the solute atoms at infinite dilution are surrounded on average by solvent atoms only. This determines the properties of the solute in the solution and thusgi¥.
The formalism shown above is in general easily extended to multi-component systems. All thermodynamic mixing properties may be derived from the integral Gibbs energy of mixing, which in general is expressed as
Dmix m Dmix D
In solution thermodynamics the standard or reference states of the components of the solution are important. Although the standard state in principle can be chosen freely, the standard state is in practice not taken by chance, but does in most cases reflect the type of model one wants to fit to experimental data. The choice of
0.0 0.2 0.4 0.6 0.8 1.0
Figure 3.5 The molar Gibbs energy of mixing and the molar excess Gibbs energy of mixing of molten Fe–Ni at 1850 K. Data are taken from reference [3].
0.0 0.2 0.4 0.6 0.8 1.0
Figure 3.4 The activity of Fe and Ni of molten Fe–Ni at 1850 K [3]. At xNi= 0.4 the activity coefficient of Ni is given by MQ/PQ.
standard state is naturally influenced by the data available. In some cases the vapour pressure of one of the components is known in the whole compositional interval. In other cases the activity of the solute is known for dilute solutions only.
In the following, the Raoultian and Henrian standard states will be presented.
These two are the far most frequent standard states applied in solution thermody-namics. Before discussing these standard states we need to consider Raoult’s and Henry’s laws, on which the Raoultian and Henrian standard states are based, in some detail.
Henry’s and Raoult’s laws
In the development of physical chemistry, investigations of dilute solutions have been very important. A dilute solution consists of the main constituent, the solvent, and one or more solutes, which are the diluted species. As early as in 1803 William Henry showed empirically that the vapour pressure of a solute i is proportional to the concentration of solute i:
pi =x ki H,i (3.41)
where xiis the mole fraction solute and kH,i is known as the Henry’s law constant.
Here we have used mole fraction as the measure of the concentration (alternatively the mass fraction or other measures may be used).
More than 80 years later François Raoult demonstrated that at low concentra-tions of a solute, the vapour pressure of the solvent is simply
pi =x pi i* (3.42)
where xiis the mole fraction solvent and p*i is the vapour pressure of the pure solvent.
Raoult’s and Henry’s laws are often termed ‘limiting laws’. This use reflects that real solutions often follow these laws at infinite dilution only. The vapour pressure above molten Ge–Si at 1723 K [4] is shown in Figure 3.6 as an example. It is evi-dent that at dilute solution of Ge or Si, the vapour pressure of the dominant compo-nent follows Raoult’s law. Raoult’s law is expressing that a real non-ideal solution approaches an ideal solution when the concentration of the solvent approaches unity. In the corresponding concentration region Henry’s law is valid for the solute.
The Ge–Si system shows positive deviation from ideality and the activity coeffi-cients of the two components, given as a function of xiin Figure 3.6(b), are thus positive for all compositions (using Si and Ge as standard states).
Raoult’s law is obeyed for a solvent at infinite dilution of a solute. Mathemati-cally this implies
(d A/d A)
a x xA®1=1 (3.43)
In terms of activity coefficients eq. (3.43) can be transformed to
SincegA ®1when xA ®1the expression for Raoult’s law becomes d
This is a necessary and sufficient condition for Raoult’s law.
A solute B obeys Henry’s law at infinite dilution if the slope of the activity curve aBversus xBhas a nonzero finite value when xB ® 0:
Figure 3.6 (a) The vapour pressure above molten Si–Ge at 1723 K [4]. (b) The corre-sponding activity coefficients of the two components.
The finite value of the slope when xB® 0, gB¥, is the activity coefficient at infi-nite dilution defined earlier. In terms of activity coefficients eq. (3.46) becomes
The finite value of the slope when xB® 0, gB¥, is the activity coefficient at infi-nite dilution defined earlier. In terms of activity coefficients eq. (3.46) becomes