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258 Chemical equilibrium

5.6 Introduction to solution theory: ideal solutions

The remainder of this chapter is devoted to discussing the nature of the functions that

relate the concentration of a component in a solution to its activity. These are called

activity–composition relationships. The study of these relationships is a physically and

mathematically rich theory, called solution theory. We will cover some of the fundamental

concepts in these sections, and some specific applications to solids, liquids and gases in sub-

sequent chapters, but a thorough coverage of the theory would exceed the available space

and the intent of this book (see, for example, Guggenheim, 1952; Darken & Gurry, 1953;

Kerrick & Darken, 1975; Wood & Nicholls, 1978; Ganguly & Saxena, 1987; Navrotsky,

1987; Wood, 1987; Pitzer, 1995; Kress, 2003).

An important concept of solution theory is that of ideal solution. We define an ideal

solution as one in which the activity of every component, which we shall term its ideal

activity (sometimes also called thermodynamic mol fraction) follows a strict and simple rule,

which is specified in the next section. As with ideal gases, ideal solutions and ideal activities

are a simplification of the behavior of natural systems, which under some circumstances

may be approached more or less closely, but never perfectly, and often not even closely.

The importance of the concept of ideal solution rests, as for ideal gases, on the fact that

its mathematical definition is straightforward and it captures fundamental aspects of the

behavior of natural systems. For this reason the mathematical formalism of ideal solutions

provides a suitable starting point to describe the behavior of real, or also called non-ideal,

solutions.

5.6.1 Definition of ideal solution and ideal activity

An ideal solution is defined as one that complies strictly with two conditions. The first condi-

tion is that the enthalpy of mixing vanishes for all solution compositions, i.e. H

mixing

=0.

From (5.34) it follows that (h

−H

0,i

) is identically zero, and hence, from (5.44), that the

activity of any component in an ideal solution, called its ideal activity and symbolized

i, ideal

is:

ln a

i, ideal

=−



−S

0,i



. (5.61)

The second condition focuses on the definition of the quantity (s

−S

0,i

). From (5.33)we

find the following relationship:





−S

0,i



=−R



ln a

i, ideal

=S

mixing

. (5.62)

The entropy of mixing, S

mixing

, must be related in some way to the increase in configu-

rational entropy (= increase in the number of allowed microstates) that arises from mixing

of distinguishable particles. The particles could be, for example, ions in a crystal or in an

electrolyte liquid solution, molecules in a multicomponent molecular liquid, or molecules

in a mixed gas phase (the latter is the example discussed in Section 4.6.1). We can jus-

tify on physical grounds a relationship between S

mixing

and S

configurational

as defined in

Chapter 4 (equation (4.55)), but we have no reason to assume that the two quantities are in

general identical. This is so because end-member components may contain configurational

entropy that will also be present in the solution but that does not arise from mixing with

259 5.6 Introduction to solution theory

other components. Hence the second condition in the definition of an ideal solution, which

is that the entropy of mixing of an ideal solution equals the configurational entropy of the

solution minus a constant entropy contribution from each end-member component. We can

write this additional entropy term as R lnC

(Boltzmann’s postulate), and:

S

mixing

configurational

−



R ln C

. (5.63)

The constant ln C

may take different values for each solution component, and may possibly

vanish for some or all components, but it is important to emphasize that it is a constant in

terms of every thermodynamic variable, in particular, temperature, pressure and composition

of the solution. From (5.62) we write:





−S

0,i

+R ln C



configurational

(5.64)

and note that, because S

configurational

is a molar property of the solution, the quantity (s

−

0,i

+ RlnC

) behaves algebraically as a partial molar property (see (5.29)). Using the

definition of partial molar properties (5.28) and rearranging:



−S

0,i



=−R ln C



∂



configurational



∂n



P ,T ,n

j≡i

. (5.65)

The definition of ideal activity then becomes:

ln a

i, ideal

≡ln C

−



∂



configurational



∂n



P ,T ,n

j≡i

. (5.66)

Again, unorthodox but elegant.

5.6.2 Ideal activity and configurational entropy

We can approximate the partial derivative in (5.66) by calculating the change in the number

of microstates that arises when we add one molecule of component i to a solution. Say that

the solution contains m

molecules of component i. It may contain any number of molecules

of any number of additional components, the only requirement being that all of these other

quantities stay fixed. We will symbolize the number of microstates of this solution with O

The configurational entropy of this solution, S

configurational(m

)

, is given by equation (4.55):

configurational

(

)





, (5.67)

where O

is the number of microstates of the end-member configuration that corresponds

to the chosen standard state. Adding one molecule of component i changes the number of

mols of i by N

−1

(N =Avogadro’s number), which is an infinitesimally small increment

for all practical purposes. The configurational entropy changes to:

configurational

(

)





. (5.68)

260 Chemical equilibrium

The partial derivative in (5.66) can be approximated as follows:



∂



configurational



∂n



P ,T ,n

j≡i

≈

configurational

(

)

−S

configurational

(

)

−1

. (5.69)

Substituting (5.67) and (5.68)in(5.69), and then in (5.66), and simplifying we arrive at:

i, ideal

. (5.70)

Equation (5.70) is general, as we have made no assumptions regarding the nature of the

solution or its components, beyond the fact that it is an ideal solution. The reason why I

stated that (5.44) and (5.66) are “elegant” is because they lead naturally to (5.70). In order

to evaluate a

i, ideal

for a specific component in a specific solution, however, it is necessary

to know the rules under which mixing takes place, such as whether there is ordering over

different types of crystallographic sites and whether or not there are coupled substitutions.

We examine some specific cases in the next sections that will exemplify most of the common

calculation techniques that are involved in finding ideal activities. Given space constraints,

the coverage cannot possibly be exhaustive, but using (5.70) as a starting point it is always

possible to calculate an ideal activity in any solution that one may come across.

5.6.3 Ideal activity–composition relationships in fluids and in

simple crystalline solids

Consider a crystalline solid solution in which mixing takes place in only one type of crys-

tallographic site, that may however appear repeated in the standard formula unit. We call

the number of repetitions of the crystallographic site its site multiplicity, u. For example, a

simplistic view of Ca-free feldspar is that it is a solid solution of the components KAlSi

and NaAlSi

, in which the cations K and Na mix in only one site, so that in this case

u = 1. Olivine is a solid solution of Mg

SiO

and Fe

SiO

, so in this case Mg and Fe

mix in two crystallographic sites, and u = 2. If we disregard complications arising from

substitution of other cations in the Si and Al sites (we will get to these later), garnet is a

solid solution of the four components: Mg

,Fe

,Mn

and

, and in this case u = 3 for mixing of Mg, Fe, Mn and Ca.

We expand equation (4.1.1) and write the number of microstates O

as follows:



j=i

. (5.71)

In this equation N, typically ∼ Avogadro’s number, is the total number of atoms involved

in mixing (e.g. total number of Mg + Fe + Mn + Ca in garnet, Mg + Fe in olivine or

orthopyroxene, and so on), n

is the number of atoms of the component of interest, and the

j=i

s are the numbers of atoms of all other kinds (I have simply factored out n

! from the

product in the denominator of equation (4.1.1)). We now add one molecule of the component

of interest, which means that we add u atoms of i while leaving the amounts of every j =i

unchanged (e.g. if we want to calculate the activity of pyrope we add one molecule of

pyrope to garnet, which adds 3 Mg atoms). The number of microstates is now given by:

(

N +u

)

(

)



j=i

. (5.72)

261 5.6 Introduction to solution theory

Noting that for N ! u we can write (N +u)!≈N

N!, we get:

(N +u)!n

+u)!N!

≈

N!N





(5.73)

since n

/N = X

. Substituting in (5.70) we arrive at:

i, ideal

(

)

. (5.74)

In order to calculate the value of the constant C

we must specify the standard state. If

we choose as standard state the pure chemical species at the temperature and pressure of

interest then it follows from (5.45) that it must be a

i, ideal

=1 for X

=1, and hence C

=1.

With this choice of standard state, which is the norm, we have:

i, ideal

(

)

. (5.75)

You can see from (5.63) that in this case S

mixing

and S

configurational

are identical, which

recovers our discussion in Section 4.6.2. The condition S

mixing

= S

configurational

is not

an a priori assumption, but follows from a specific rule for mixing in the solution and a

specific choice of standard state. Equation (5.75) is deceptively simple, but must be applied

with care. In particular, one must give careful consideration to how the configurational

entropy of the entire solution is affected by compositional changes, as more than one

distinct crystallographic site may be involved. The following example, subsequent sections

and end-of-chapter problems will make this clear.

Worked Example 5.5 Chemical potential and activity in a mixture of ideal gases. Carbon–oxygen

equilibrium, part (ii)

Mixing of ideal gases is the simplest possible type of solution, as all locations in the gas

are equivalent (there is only one possible “site” on which mixing takes place). We seek an

expression for the chemical potential of an ideal gas in a gas phase composed of a mixture

of ideal gases. Let the gas phase occupy a volume V at pressure P and temperature T. The

total number of mols of gas in the mixture is N =



, where n

is the number of mols

of component i. Because ideal gases are made up of dimensionless and non-interacting

particles (Section 1.14) each component exerts a pressure, p

, equal to the pressure that the

same amount of gas would exert if it occupied by itself a volume V at temperature T . Thus:

V = n

RT . (5.76)

Dividing by the equation applied to the full mixture, P V = NRT :

(5.77)

or:

P . (5.78)

262 Chemical equilibrium

We call p

the partial pressure of component i, and it is obvious from (5.78) that P =



From (5.36) and (5.37) we see that:



∂µ

∂P





∂V

∂n



P ,T ,n

j≡i

∂

∂n



NRT



P ,T ,n

j≡i



∂



∂n



P ,T ,n

j≡i

0,i

, (5.79)

where V

0,i

is the molar volume of any ideal gas at P and T. Thus, the partial molar volume

of an ideal gas equals the molar volume of the pure component, v

= V

0,i

, and therefore

V

mixing

=0 for mixtures of ideal gases (see equation (5.35)).

In order to find the chemical potential of component i in the mixed gas phase at P and

T, µ

P ,T

we can integrate dµ at constant temperature T, from a standard state µ

0,i

P ,T

defined

as pure i at P and T, to a state in which the component is at partial pressure p

in the gas

mixture. Thus:



dµ

=µ

P ,T

−µ

0,i

P ,T

. (5.80)

We re-write this equation using (5.79), as follows:

P ,T

=µ

0,i

P ,T





∂µ

∂P



dP =µ

0,i

P ,T

+RT



, (5.81)

which, integrating and substituting (5.78), yields:

P ,T

=µ

0,i

P ,T

+RT ln X

. (5.82)

Comparing (5.82) with (5.45) shows that, if we define the standard state of an ideal gas as

the pure gas at the temperature and pressure of interest, then:

. (5.83)

From (5.75) it also follows that, with the standard state define in this way, a mixture of

ideal gases behaves as an ideal solution with site multiplicity u = 1. Note that we have not

assumed this to be the case – we have proved it.

For reasons that we will discuss in Chapter 9, it is more convenient to define the standard

state for gases as the pure gas at the temperature of interest and 1 bar. Calling the standard

state chemical potential in this case µ

0,i

1,T

, we can relate it to the standard state chemical

potential of pure i at P and T as follows:

0,i

P ,T

=µ

0,i

1,T





∂µ

0,i

∂P



dP =µ

0,i

1,T



0,i

dP =µ

0,i

1,T

+RT



(5.84)

or:

0,i

P ,T

=µ

0,i

1,T

+RT ln





. (5.85)

In equation (5.85) I have written the division by 1 explicitly, in order to make it clear that

the argument of the logarithm is not pressure, but rather a non-dimensional parameter that

263 5.6 Introduction to solution theory

equals the ratio between the pressures in one standard state, P bars, and the pressure in the

other standard state, 1 bar. The explicit division will generally be absent from equations

such as (5.85) if the denominator is 1, but it must always be remembered that this is an

abuse of notation, and that the argument of the logarithm function is a non-dimensional

variable, which in this case is numerically equal to pressure.

Substituting (5.85)in(5.82) and using (5.78) we get:

P ,T

=µ

0,i

1,T

+RT ln





, (5.86)

where again the division by 1 will be used only here, as a reminder that the argument of

the logarithm is a non-dimensional variable. Although we could call the ratio p

/1 the

activity of an ideal gas relative to a standard state of 1 bar and the temperature of interest, a

different terminology is used for gases, that we will examine in Chapter 9. Equations (5.82)

and (5.86) are equally valid representations of the chemical potential of an ideal gas in a

mixture of ideal gases, but (5.86) is preferred because it leads to a simpler generalization

to real gases. The standard state for gas species, real as well as ideal, is taken at 1 bar and

the temperature of interest.

We can now complete the speciation calculation for a C–O fluid at 1 bar that we began

in Worked Example 5.4. At 1 bar the ideal gas approximation is generally valid and there

is no ambiguity about the standard state, as this is simply the temperature of interest and 1

bar (which is also the pressure of interest). We use (5.83) and re-write (5.59) and (5.60)as

follows:





1/2

=exp



−



0,1

1,T



(5.87)

and:

=exp



−



0,2

1,T



. (5.88)

Together with the condition X

= 1 we have three equations that can be

solved for the three mol fractions at any specified temperature and 1 bar. An additional

simplification is possible in this case. The exponential functions in (5.87) and (5.88) (which

are the equilibrium constants K

and K

, compare with (5.59) and (5.60)) are very large

numbers. For example, we find that at 500

◦

C, K

and K

are of order 10

and 10

respectively. This means that the mol fraction of oxygen in the gas phase is vanishingly

small, which of course reflects the fact that oxygen is an extremely reactive chemical

species (more on this in Chapters 13 and 14). We can then make X

≈ 1 and

eliminate X

between (5.87) and (5.88) by squaring (5.87) and dividing by (5.88), leaving

us with two equations in X

and X

. We eliminate one of these variables between these

equations (say, X

) and with some manipulation end up with the following quadratic

equation:

(

)

−K

=0, (5.89)

where:

(

)

=exp







0,2

1,T

−2

0,1

1,T





. (5.90)

264 Chemical equilibrium

The mol fraction of CO

is simply 1 −X

, and X

(1) is calculated from either (5.87)

or (5.88). It is very straightforward to write a Maple

procedure to perform all of these

calculations, using previously written procedures that handle the thermodynamic calcula-

tions (Software Box 5.3). The results are plotted in Fig. 5.8 for temperatures between 25

◦

and 1000

◦

C. These are probably of limited applicability to real planetary environments, as

hydrogen-bearing species are unlikely to be absent from the gas phase. Recall that the cal-

culations assume that graphite is at equilibrium with the gas phase. They show that carbon

is a powerful reducing agent at high temperature, as the gas phase, which is almost pure

at low temperature, becomes almost pure carbon monoxide at T ∼ 1000

◦

C. Calcu-

lated oxygen mol fractions pose an interesting question: what is the physical meaning of

200 400 600 800 1000

0.2

0.4

0.6

0.8

roX

200 400 600 800 100

–70

–60

–50

–40

–30

–20

–10

Temperature (°C)

gol X

Excess graphite at 1 bar

Fig. 5.8 Speciation in a carbon–oxygen gas phase in equilibrium with graphite at P = 1 bar, assuming ideal gas behavior.

265 5.7 The geometric view of activity

numbers such as X

= 10

−40

(Fig. 5.8)? This represents a concentration of the order of

one molecule of oxygen per 10

mols of gas. The concept of fugacity, that we develop in

Chapter 9, allows us to circumvent this physically unappealing interpretation and attach to

numbers such as these a rigorous thermodynamic significance.

Software Box 5.3 Calculation of species distribution in a mixture of ideal gases

The Maple worksheet gas_mix_1.mw contains a procedure named COeq1 that

solves for species distribution in a C–O gas phase in equilibrium with graphite at 1

bar assuming ideal gas behavior (Worked Example 5.4 and 5.5). The procedure uses

dGPT (see Software Box 5.2) to calculate the values of equilibrium constants K

and

(equations (5.59) and (5.60)), and solves for the mol fraction of CO with equation

(5.89). Oxygen concentration is calculated from equation (5.88). Output for a specified

temperature range is sent to a file, name must be provided in the procedure call. The

data for CO–CO

–graphite equilibrium are stored in tab-delimited format in a file named

gasdata1. The procedure is easily modified to calculate other gas phase equilibria.

5.7 The geometric view of activity and Gibbs free energy of mixing

The concept of an ideal solution is that the only change in Gibbs free energy that takes place

when the solution forms arises from a change in the configurational entropy of the solution

(see equation (5.61) and associated discussion). The attentive reader may have noticed,

however, that we never examined whether an ideal solution in fact forms. The term



0,i

in equation (5.32) is the Gibbs free energy of a system that consists of a macroscopic

aggregate of components in the same relative proportions as they are present in the solution,

except that in the latter mixing occurs at a microscopic scale. A solution will form only if

G

mixing

< 0 (equation (5.32)). Starting from the relationship G

sol

= H

sol

−TS

sol

(this

is simply the definition of Gibbs free energy) it is straightforward to show using (5.32)to

(5.34) that:

G

mixing

=H

mixing

−TS

mixing

. (5.91)

From the definition of ideal solution (H

mixing

= 0) and the fact that mixing at the

microscopic scale always results in an increase in configurational entropy (a corollary

of Boltzmann’s postulate, see Section 4.6.2) we conclude that:

G

ideal

mixing

=−TS

ideal

mixing

< 0. (5.92)

An example of the behavior described by (5.92) is shown in Fig. 5.9, where we plot G

mixing

for a simple one-site ideal binary solution (equation (5.83)), using (5.62) to calculate

S

mixing

. From (5.62) it also follows that:

G

ideal

mixing



RT ln a

i, ideal

(5.93)

266 Chemical equilibrium

RT ln a

b, ideal

RT ln a

i, ideal

mixing, ideal

RT ln a

a, ideal

b, ideal

a, ideal

0, a

Fig. 5.9

Geometric representation of equations (5.93) and (5.94). Compare with Fig. 5.7. The curve represents the Gibbs free

energy of mixing, which vanishes for pure species. The intersections of the tangent line with the vertical axes give the

differences between chemical potentials in the solution and standard state chemical potentials = RT lna

i,ideal

. The

actual values of the standard state properties do not affect this relationship, so they can be set to zero, as in the ﬁgure.

and comparison with (5.29) shows that RT ln a

i, ideal

is a partial molar property. Using (5.45)

and (5.2.13) we find that, for an ideal binary solution such as that depicted in Fig. 5.9:

−µ

0,a

=RT ln a

a,ideal

=G

ideal

mixing

−X

∂



G

ideal

mixing



∂X

(5.94)

leading to the geometric interpretation of µ

−µ

0,a

(= RT ln a

ideal

) shown in Fig. 5.9.

There are two additional considerations. First, for solutions with more than two compo-

nents the function G

mixing ideal

is a higher-dimensional equivalent of the bowl-shaped

curve in Fig. 5.9, the tangent line becomes a tangent plane, and the geometric meaning of

−µ

0,a

stays the same. Second, (5.93) and (5.94) remain true for non-ideal solutions,

by simply removing the superscript “ideal” from the variables. We will expand on this

shortly.

5.8 More complex ideal activity–composition relationships

5.8.1 Crystalline solutions with multiple ionic sites

In order to apply equation (5.74) to crystalline solids it is necessary to know how ionic

substitution in the solid generates configurational entropy. Let us look first at a simple

case like olivine, in which substitution takes place only in two octahedral sites that can be

considered to be equivalent. Configurational entropy of olivine solid solutions arises only

267 5.8 More complex ideal activity–composition relationships

from mixing of cations in these two sites, so we have:

Fo,ideal





Fa,ideal

(

)

(5.95)

Consider now a mineral in which ionic substitution takes place in two distinct ionic sites,

A and B, such that the mineral’s structural formula is A

, and suppose that we are

interested in calculating the ideal activity of an end-member component consisting of ele-

ments α and β, α

. We need to consider two possibilities. Let us assume first that

substitution in the two sites is uncoupled, because only cations with the same charge mix in

each site. In this case each site makes its own independent contribution to configurational

entropy. One way of seeing why this is the case is to consider mixing on one site while

leaving the composition of the other one fixed at one of its end-members, and then perform

the converse operation. The total configurational entropy is the sum of the configurational

entropy contribution from each distinct crystallographic site, each of which has its own site

multiplicity. It follows from Boltzmann’s postulate that the ideal activity of the end-member

phase component is the product of activity contributions from each crystallographic site. In

our example we have:

,ideal

(

)





. (5.96)

Garnets within the compositional range (Mg, Fe, Mn, Ca)

(Al, Cr)

can be treated

in this way. For example:

Prp,ideal

grt





(

)

Alm,ideal

grt

(

)

(

)

Kno,ideal

grt





(

)

(5.97)

where Prp =pyrope, Alm =almandine and Kno =knorringite (Mg

). Conceiv-

ably, a small tetravalent cation could substitute for Si in the tetrahedral sites of garnet, in

which case the expressions in (5.97) would have to be multiplied by (X

)

to obtain the

correct ideal activities (but I am not aware of this ever being necessary).

A second possibility is that, owing to charge-balance constraints, solid solution forms

by coupled substitution. In this case a substitutes for α in site A only if b substitutes

simultaneously for β in site B. The ionic charges are such that [charge (a +b)] =[charge

(α +β)]. We shall assume for now that in a “perfect” crystal this substitution takes place

while preserving local charge balance, such that adjoining A and B sites are substituted

simultaneously. If this is the case then the “particles” that mix are not the individual ions

but the ionic pairs, ab and αβ . In virtually all minerals in which coupled substitution is

important the site multiplicity of the sites that undergo substitution is 1, even if the actual

multiplicities of A and B may be different. We then have, from (5.74) with u = 1:

αβ,ideal

αβ

. (5.98)

An important example of (5.98) is the calculation of activities of Tschermak’s components

in pyroxenes. The Tschermak’s substitution can be written in general as an exchange of the

form Al

−1

, where the superscripts O and T signify octahedral and tetrahedral

crystallographic sites, and R stands for a divalent cation. For example, in orthopyroxene