Douce A.P. Thermodynamics of the Earth and Planets

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478 Melting in planetary bodies

positions of the atoms, that yield the minimum free energy of the crystal. This is a

structure with perfect long-range order (Section 7.4). In fact, this description of a crys-

tal of an element is identical to the example that we considered in Section 7.4: there

is a total of N sites, of which N/2 are occupied lattice points, and N/2 are vacant

interstitial sites.

As temperature increases the atoms vibrate with increasing amplitude about their equi-

librium positions, but remain “anchored” to their corresponding lattice points, so that the

configurational entropy of the crystal remains constant. The vibrational, or thermal, entropy

of course increases, as additional vibrational energy levels become accessible (Section

4.6.2). At some temperature the amplitude of the vibrations reaches a critical value relative

to the lattice spacing, and the atoms become detached from their lattice positions. One

particularly fruitful way of modeling this, known as the Lennard-Jones and Devonshire

theory of melting (Lennard-Jones & Devonshire, 1939a,b), is to postulate that the lattice

points and interstitial sites still exist when such vibrational instability sets in, but now the

atoms are distributed at random over the two types of sites. This corresponds to the sudden

loss of the long range order of the material. There is an entropy discontinuity and hence a

first-order phase transition (Section 7.6.3). The loss of long-range order causes the crystal to

lose its resistance to shear, because there is no longer an “organized” system of interatomic

potentials that generates a restorative force when interatomic bonds are stretched in a given

direction. From a mechanical point of view the solid becomes a liquid at the first order phase

transition. These are complementary microscopic descriptions of melting or, equivalently,

alternative ways of defining the difference between a solid and a liquid, i.e.: (i) as condensed

phases that either have long-range order (solids) or not (liquids); (ii) as condensed phases

that either have shear strength or not; or (iii) as condensed phases in which the amplitudes

and modes of atomic vibrations stay within certain bounds or not. Each of these approaches,

and others, have been pursued in order to construct fundamental theories of melting. Excel-

lent discussions can be found in Poirier (1991, most recommended), Ubbelohde (1978),

Cotterill (1980), Mulargia (1986) and the remarkably clear and succinct paper by Oriani

(1951). Here I will focus only on the point of view of melting as a sudden loss of long range

order, as it is the one that is most helpful in understanding the chemical–thermodynamic

aspects of melting.

There are several contributions to the increase in entropy during melting. One of them

is the configurational entropy that arises from the loss of long-range order, as we dis-

cussed in the previous paragraph. In systems more complex than elements there are

additional contributions to configurational entropy, arising from chemical mixing. For

example, in a silicate crystal there may be ordering between cations occupying different

kinds of octahedral sites (say, Ca and Mg in clinopyroxene) that persists to the melting

point but not in the melt. There is then an increase in configurational entropy arising

from chemical mixing, in addition to the lattice point–interstitial site disorder. For most

substances melting is also accompanied by an increase in volume (the low pressure poly-

morph of H

O ice is an important exception, that we discuss later). Expansion arises

from repulsion among atoms with the same type of charge, that in the ordered crystal

are shielded by oppositely charge atoms, and entropy increases in concert with expansion,

as additional vibrational energy modes become available. Additional entropy contribu-

tions may arise if particles in the melt can acquire rotational degrees of freedom, which

do not exist in the solid, or if changes in electron occupation levels become possible

(for example, silicon becomes metallic on melting, and this contributes to its entropy

of melting).

479 10.1 Principles of melting

We can write the entropy of melting of a substance, 

S, as a sum of various

contributions:



S = S

expansion

+S

lattice disorder

+S

chemical mixing

+S

rotation

+S

electronic

+···. (10.1)

For a simple substance only the first two terms are non-zero, and we write them as follows:



S =



∂S

∂V





V +S

lattice disorder

(10.2)

where the expansion term equals the rate of increase in entropy (number of vibrational

frequencies) with volume at constant temperature, times the increase in volume during

melting. Now, from the definition of Helmholtz free energy we have:



∂S

∂V





∂

∂V



−

∂F

∂T







∂

∂T



−

∂F

∂V







∂P

∂T



. (10.3)

Recall that this is one of Maxwell’s relations (Section 4.9.1) – you can memorize them,

look them up or, better, as this usually brings out their physical meaning, derive them as

needed from the cross second derivatives of the appropriate thermodynamic potential. We

recall equations (8.59) and (8.60):



∂P

∂T



=αK

γC

(10.4)

and substituting in (10.2) we get:



S = αK



V +S

lattice disorder

. (10.5)

In Section 8.4 we saw that above the Debye temperature the product αK

is nearly constant.

This is a good approximation for equation (10.5). If the configurational entropy associated

with loss of long-range order is also a constant then the entropy of melting should be a

linear function of the volume of melting.

We can test this hypothesis by using the second identity in (10.4) to write (10.5)as

follows:



S = γC



+S

lattice disorder

. (10.6)

It was pointed out by Stishov et al. (1973) and Lasocka (1975) that if values of 

for many elements are plotted against the corresponding values of (

V/V) the points

scatter about a straight line with y intercept equal to R ln2. Figure 10.1a shows this for

a number of metals. The volume of melting is divided by the volume of the solid phase.

I have also divided equation (10.6)byR ln 2, so that a y intercept equal to 1 corresponds

to 

S = S

lattice disorder

= R ln2. The straight line is not a calculated correlation line,

but rather an arbitrary line drawn through the point {0,1}, intended as a guide for the eye.

There is clearly quite a bit of scatter, possibly arising to some extent from experimental

uncertainties, but the pattern appears to be robust: for all of these metals the configura-

tional entropy associated with loss of the crystal’s long range order is approximately R ln 2.

Now, in the previous section we noted that, according to the Lennard-Jones and Devonshire

480 Melting in planetary bodies

0 0.02 0.04 0.06

1.2

1.4

1.6

1.8

V/(V

R ln2

)

S / (R ln2)

0 0.1 0.2 0.3

V/(V

Rln2)

S / (R ln2)

LiF

LiCl

LiBr

NaF

NaCl

NaBr

NaI

KCl

KBr

RbCl

RbBr

CsCl

0 0.05 0.1 0.15 0.2 0.25 0.

V/(V

R ln2 )

(R ln2)

Forsterite

Fayalite

Pyrope

Enstatite

Diopside

Na, K

Quartz

a b c

Fig. 10.1

Entropy of melting versus volume of melting for metals (stars), alkali halides (circles) and silicates (triangles). The

same group of metals are shown in all three panels, and the same halides in panels (b) and (c). The lines are eyeballed

but forced to intersect the vertical axis at 

S(Rln2)=1.DatafromUbbelohde(1978),Bottinga(1985)andRichet

andBottinga(1986).

model of melting, for a simple substance the configurational entropy of the melt corresponds

to mixing N/2 occupied lattice points and N/2 vacant interstitial sites over a total of N

sites, and that this is exactly the same problem that we considered in Section 7.4. From

equation (7.26) we see that, since the solid has no configurational entropy, the entropy

increase that we should expect in this case is, indeed, equal to R ln 2. The importance of

this result is that it validates the interpretation of the melting phase transition as a sud-

den loss of long-range order. Note that this is different from a critical phase transition.

In a critical phase transition the order parameter approaches zero continuously, so that

there are no discontinuities in entropy, or enthalpy, at the phase transition. In the case

of melting there is a discontinuous jump in the order parameter (see Fig. 7.10). It has

been argued that this discontinuity may vanish at high pressure, leading to a solid–melt

critical point analogous to the vapor–liquid critical point. Unequivocal experimental evi-

dence for this is lacking, however, at least for conditions such as those in the interiors of

terrestrial planets.

The melting behavior of alkali halides (Fig. 10.1b) appears to follow that of elements

fairly closely. This suggests that the alternation of anions and cations is largely preserved in

the melt, so that there is no contribution to configurational entropy beyond the redistribution

of both types of atoms among lattice points and interstitial sites, as happens in pure elements.

Observations such as this one have given rise to the quasicrystalline model of melts. In this

481 10.2 Melting point depression

model the melt is thought of as a crystal in which the number of defects is of the same order

as the total number of crystallographic sites (see Ubbelohde, 1978, Chapter 5).

In contrast to halides, the melting behavior of silicates is clearly different from that

of metals (Fig. 10.1c). In this case there is a very large entropy component in excess

of the R ln 2 term that arises from lattice point–interstitial site mixing, with the largest

contribution likely arising from chemical mixing. Quartz is anomalous, however, in that

its configurational entropy of melting is significantly lower than that of simple substances.

This suggests that SiO

melt is strongly polymerized, and preserves much of the long range

order of the crystal.

10.1.2 Melting in complex natural systems

Melting in multicomponent systems in which a multi-phase assemblage is stable at the

solidus is much more complex. To begin with, if melt–solid equilibrium is multivariant

then we must distinguish a solidus and a liquidus (Section 6.6.1). The microscopic picture

of melting becomes fuzzy. For example, is the structure of the melt analogous to one of

the solid phases in particular, or is it a composite of the various solid phases at equilibrium

at the solidus? From a purely thermodynamic point of view the picture may seem clearer,

as calculating the composition of a melt at equilibrium with a given solid assemblage as

a function of intensive variables is no different from calculating equilibrium among solid

phases (Chapters 5 and 8) or between solids and a gas phase (Chapter 9). In fact, in Chapter 6

we calculated some simple melt–solid phase diagrams (Section 6.6). Those calculations,

and the resulting phase diagrams, are, however, no more than crude approximations. They

are only intended to illustrate the qualitative behavior of the systems that we modeled, for

instance, the effect of enthalpy of fusion on the width of a melting loop, or on the magnitude

of the melting point depression. Implementing quantitatively rigorous calculations that

allow us, for example, to predict basaltic melt compositions as a function of mantle source

composition, pressure of melting and oxygen fugacity is much more complex than what we

did in Chapter 6. Fundamentally, this is so because it is never acceptable to assume that

mixing in the melt phase is ideal, and it is not always obvious what the basis for a non-

ideal model ought to be. The problem begins by defining what chemical species to use in

a multicomponent melt (such as what we did in fluid speciation calculations, Section 9.6).

Once the species have been defined, it is not always straightforward to determine their

standard state thermodynamic properties and their excess mixing properties. As of this

writing, and in the humble opinion of yours truly, only one comprehensive calculation engine

cum data base exists that generates reliable thermodynamic models of igneous systems over

a wide range of bulk compositions and intensive variables. This is MELTS and its extensions,

developed by Ghiorso, Sack, and collaborators (see, for example, Ghiorso & Sack, 1995;

Ghiorso, 1997; Ghiorso et al., 1983, 2002). Discussing these models is beyond the scope

of this book, as well as the available space. I will rather focus on some of the constraints

on the conditions of formation of planetary magmas.

10.2 Melting point depression. Eutectics, cotectics and peritectics

Eutectics are a fundamental aspect of planetary magmatism, for two reasons. First, the fact

that the solidus temperature of an assemblage of immiscible phases may be hundreds of

482 Melting in planetary bodies

degrees lower than the melting point of each individual phase (e.g. Figure 6.21) makes

igneous activity possible at conditions that otherwise it might not be. Second, the invariant

(or nearly so) nature of eutectic melting equilibria restricts the range of common magmatic

compositions. Consider the case of silicate planetary bodies. They all have peridotitic man-

tles. Over a wide range of conditions the initial melting behavior of peridotite is more or

less close to eutectic, and the composition of the resulting low-temperature melts is broadly

speaking basaltic. Because of this, all silicate planetary bodies have basaltic crusts. Before

you get incandescently upset at this sweeping generalization, let me emphasize that this is

a book on thermodynamics, not igneous petrology. My purpose is to use simple thermody-

namic reasoning to understand many of the first-order characteristics of planetary bodies.

Along the way, and because space and time are both finite, we will be forced to ignore

important details. Some of these details are very dear to this writer, such as the fact that

about one third of the Earth’s solid surface is not basaltic, because it is not a simple product

of mantle melting.

The fundamental thermodynamic reason for eutectic behavior is the fact that the chem-

ical potential of a component in a stable solution is lower than its standard state chemical

potential (e.g. Fig. 5.9). If solid phases do not form a solution, and therefore remain in their

standard states, but the components mix stably in the melt, then solid–melt equilibrium is

only possible at a temperature lower than the melting temperature of the isolated phases.

You can see this graphically by constructing a diagram similar to Fig. 5.3, in which the

low entropy curve corresponds to one of the pure solids, and the high-entropy curve to

the corresponding component in the melt. If the chemical potential of the melt compo-

nent decreases equilibrium must shift to a lower temperature. As we saw in Sections 6.5

and 6.6, the magnitude of this effect can be calculated by using the melting point depression

equation, (6.49). This equation is important not only for the study of eutectics, but also to

study other controls on magma generation. There are reasons to revise equation (6.49)as

applied specifically to melting reactions, which we do next.

10.2.1 A revision of the melting point depression equation

Although (6.49) is the traditional way of integrating (6.48), our microscopic understanding

of melting, Section 10.1.1, suggests that a better approximation may be to consider the

entropy of melting, rather than the enthalpy of melting, to be constant. Using the identity



H =T ·

S, we re-write equation (6.48) for the depression of the melting point relative

to that of a pure substance as follows:

ln K





dT (10.7)

where T

is the melting point of a pure one-component substance, and K

is the ratio of

the activity of this component in the melt to its activity in the solid (see equation (6.44)).

Equation (10.7) integrates to:







(10.8)

which is equivalent to (6.49) (but not identical, see Exercise 10.1). We shall use this ver-

sion of the melting point depression equation throughout this chapter. One advantage of

this equation relative to (6.49) is that it is more compact. More importantly, experimental

483 10.2 Melting point depression

studies of natural rock compositions (see Kojitani & Akaogi, 1997) support the theoretical

expectation (Section 10.1.1) that entropy of melting varies less strongly with temperature

than enthalpy of melting.

10.2.2 Multicomponent eutectics and cotectics

The conclusion that the eutectic temperature must be lower than the melting point of the

isolated phases generalizes immediately to systems of any number of components. Consider

a system of c components, in which there are c mutually immiscible solid phases. An

assemblage consisting of the c solid phases plus a melt is univariant, or pseudoinvariant at

constant pressure. Call the equilibrium temperature T

. At equilibrium there is one equation

like (10.8) for each of the c components. Let i be the component that has the lowest melting

temperature, T

0,i

. We then have, because the solid remains pure:

i, melt



0,i





. (10.9)

But if the melt is a stable phase then a

i, melt

< 1 (Fig. 5.9). Because the exponent in (10.9)is

always positive for a melting reaction, it follows that T

0,i

, i.e. the eutectic temperature

is lower than the melting point of the least refractory phase.

What happens if one of the phases disappears, but the corresponding component is still

present in the melt? We now have two degrees of freedom, so the system becomes pseu-

dounivariant at constant pressure. The c–1 solid phases can exist at equilibrium with melt

over a temperature range. The melt composition will necessarily change along this tempera-

ture range, because the Gibbs free energy of each of the solid phases varies with temperature

at a different rate (in order for this not to be the case all the solid phases would have to have

the same entropy and heat capacity – why?). The resulting temperature vs. melt composi-

tion curve is called a cotectic, with which you are almost certainly familiar from igneous

petrology. In general, c cotectics radiate away from a eutectic. For our purposes the impor-

tant point is to recall that temperature along each of the cotectics increases away from the

eutectic, and to understand why.

Suppose that the solid phase consisting of component j disappears from the eutectic

assemblage. This does not have to be the lowest melting point phase, it can be any of the

solid phases. The chemical potential of j in the melt decreases relative to its value when the

melt was at equilibrium with the solid phase, because the melt is no longer saturated in this

phase. The chemical potentials of the other melt components must increase (equation (5.31))

so that preserving solid–melt equilibrium requires that temperature increase relative to that

of the eutectic. Alternatively, you can pretend that temperature does not increase. In that

case the solid assemblage has a lower Gibbs free energy than the melt, and the latter will

crystallize, unless temperature rises.

The same line of argument can be applied to show that each time that a solid phase

disappears, and a degree of freedom is gained, melting temperature must keep increas-

ing (exercise left to the reader). The sequence in which this happens depends on the bulk

composition of the system relative to the thermodynamically determined locations of the

eutectic and cotectics. It may be different for different bulk compositions, something that

is very important in the detailed study of igneous phase relations, but that will not concern

us here. The important point for many of our subsequent discussions is that, in general,

peridotites consisting of several major components and a comparable number of phases

484 Melting in planetary bodies

(e.g. lherzolites) are less refractory than simpler ultramafic rocks such as dunites or

pyroxenites. Similarly, the mantle of an icy satellite consisting of a mixture of H

O, NH

and CH

ices melts at a lower temperature than one consisting of pure H

10.2.3 Incongruent melting and peritectics

The phase rule requires that an assemblage of c solid phases plus a melt phase in a c-

component system be pseudoinvariant at constant pressure, but it is mute with respect to

the chemography of the melting reaction. At a eutectic the only phase on the high entropy

side of the reaction is melt. Such a reaction is known as a congruent melting reaction.

However, as we saw in Worked Example 6.3, there are also incongruent melting reactions

in which the high entropy assemblage consists of melt plus one or more solid phases. If

the total number of phases at an incongruent melting reaction is c +1 then the variance is

still 1. The isobaric pseudoinvariant point is in this case known as peritectic. In contrast

to eutectics, peritectics are not minimum melting points. It can be shown that requiring

a peritectic to be a minimum melting point leads to a violation of Schreinemakers rule

(Exercise 10.2).

Our simplified calculation of the temperature and composition of eutectics (e.g.

equations (10.9)or(6.70) and (6.71)) assumes that the melt is an ideal mixture of the

same chemical species that make up the solid phases. Incongruent melting is impossible

with this assumption. In order to see why, consider a binary system. The congruent melting

reaction, A +B → liquid, allows for the liquid to be a combination of species A and B.

The incongruent melting reaction, A →liquid +B, in contrast, makes it impossible for the

liquid to be a linear combination of non-negative amounts of species A and B. It follows that

calculation of the temperature and composition of a peritectic requires that one specify melt

species that are not present in the solid phases, and that may interact non-ideally in the melt

phase. How to do this is beyond the scope of this book, but is at the core of thermodynamic

models such as MELTS.

10.3 Partitioning of trace components between solids and melts

Virtually all minerals incorporate in their crystalline structures trace amounts of components

that are not present in their nominal formulas. For example, end-member forsterite is never

SiO

, as a small fraction of the Mg cations are replaced by elements such as Ni and

Co. When the mineral melts these components also enter the melt and, under the right

circumstances, equilibrium may be established between trace components in the crystal and

in the melt, for instance: Ni

SiO

(crystal)

→

←

SiO

(melt)

. The usual convention is to write

the equilibrium of a trace element between crystal and melt with the crystal as product (this

convention, which I have always found counterintuitive, probably arises from an excessive

preoccupation with magmatic differentiation, as opposed to magma generation):

melt

 z

crystal

. (10.10)

In this equation z stands for the trace component of interest. It could be an element or

some convenient species such as Ni

SiO

. If the concentrations of the trace component in

the crystal and the melt are low enough that Henry’s law is valid (Section 5.9.2), then the

485 10.3 Partitioning of trace components

thermodynamic equilibrium condition for (10.10) can be written as follows:

−

P ,T

+RT ln



z,s

z,m



=0, (10.11)

where s and m stand for solid and melt, K

are the respective Henry law constants, X is

mol fraction and 

P ,T

is the standard state Gibbs free energy of melting of the trace

component at the pressure and temperature of interest. We can re-write (10.11) as follows:

z,s

z,m

z,s

exp





P ,T



. (10.12)

As a first approximation it is often assumed that the exponential function is a constant, or at

least that it varies very slowly, over a restricted pressure–temperature range. Furthermore,

for restricted ranges in the compositions of solid and melt the mol fraction of z can be con-

sidered to be related by a constant factor to its mass concentration, expressed, for example,

in weight percent or ppm. Using C

for the mass concentration, we simplify (10.12) to:

z,s

z,m

s/m

, (10.13)

where D

s/m

is called the solid–melt partition coefficient. This equation is sometimes called

Nernst’s distribution law, and D

s/m

is Nernst’s distribution coefficient (Chapter 11). It

incorporates the right-hand side term of equation (10.12), plus the ratio of constant factors

that convert mol fraction to mass fraction in the solid and melt. Partition coefficients are

determined empirically rather than calculated from thermodynamic properties, so that these

different contributions are lumped together and in general cannot be isolated. If enough

experimental measurements exist, however, it may be possible to determine the pressure

and temperature dependencies of the partition coefficient.

Trace elements are classified as compatible vs. incompatible, depending on whether the

value of D is greater or smaller than one, respectively. A compatible trace element prefers

the crystal, and an incompatible one the melt. From the point of view of thermodynamics this

distinction arises from two independent causes, which can be identified in equation (10.12).

The first is the melting point of the trace component relative to that of the crystalline host, or,

in other words, whether the trace component is more or less refractory than the host. If it

is more refractory then 

P ,T

> 0, because at the temperature at which the host crystal

melts the stable phase for the trace component is the solid. The trace element in this case

will tend to behave compatibly. Conversely, it will tend to be incompatible if the trace

component is more fusible than the host and 

P ,T

< 0. The second factor is the relative

value of the Henry’s law constant in the crystal to that in the melt. Given two trace cations

in the same crystalline host, the one closer in size and charge to the essential cation in the

crystal will generally have a smaller Henry law constant and be more compatible. It follows

that a given trace component can be compatible in some systems and incompatible in others.

Whether a trace element is compatible or incompatible is important. Suppose that we have

a crystal that melts progressively, such that melt fraction, ϕ, varies continuously between

0 and 1. We shall come back to the “melts progressively” statement and analyze it from

a rigorous thermodynamic point of view, but for now we note that, if C

z,i

is the initial

concentration of component z in the crystal before melting starts, and melt and crystal

486 Melting in planetary bodies

remain at equilibrium throughout the melting process, then for any melt fraction we have,

by mass balance:

z,s

(

1 −ϕ

)

z,m

ϕ = C

z,i

(10.14)

or:

z,m

z,i

s/m

(

1 −ϕ

)

+ϕ

. (10.15)

This equation is known as the batch melting equation, meaning that crystal and melt remain

at equilibrium from the beginning of melting until (at least) melt fraction ϕ is attained.

Other possibilities exist, and may be more common in nature (Maaløe, 1985; Winter, 2001;

Philpotts & Ague, 2009), but (10.15) is sufficient for our purposes. Variation in C

z,m

with

ϕ for various values of D

s/m

is shown in Fig. 10.2. We begin by noting that for ϕ = 0

we get C

z,m

= C

z,i

s/m

. This is the concentration in the first infinitesimal fraction of

melt; it is very high for a strongly incompatible element (e.g. D

s/m

=0.001, Fig. 10.2) and

virtually zero for a strongly compatible one (D

s/m

=100). For ϕ = 1, C

z,m

z,i

,asnow

the composition of the melt is identical with the starting composition.

An incompatible component (Fig. 10.2) is strongly enriched in the first melts and its con-

centration initially decreases very rapidly. The rate of decrease slows down with increasing

melt fraction. For an incompatible trace component: (i) at sufficiently large melt fractions

the concentration is more or less independent of the value of D and depends only on melt

fraction, and (ii) the same is true for sufficiently small values of D. The reason is that, in

both cases, C

z,m

≈ C

z,i

/ϕ (see equation (10.11)). A strongly compatible trace element, in

contrast, may be virtually absent from the melt until the melt fraction is almost 1. These

concepts, and elaborations based on the nature of the melting process (e.g. whether melt

and crystal remain at equilibrium or are separated as soon as a new melt increment forms)

constitute much of the basis of trace element geochemistry. Application to multicomponent

0 0.2 0.4 0.6 0.8 1

0.01

0.1

100

Melt fraction

z,m

z,i

100

0.1

0.01

0.001

Fig. 10.2

Concentration of trace component z in melt, C

z,m

, relative to bulk initial concentration, C

z,i

. Numbers in boxes are

solid–melt partition coefﬁcients, D

s/m

487 10.4 The effect of “impurities”

systems (i.e. rocks) differs chiefly in the fact that the partition coefficient that is used, called

the bulk partition coefficient, is a weighted average of the partition coefficients for each of

the minerals in the assemblage. We will not pursue these topics here, but in Section 10.4.2

we will examine the conditions under which trace element partitioning may have important

effects on melt generation in planetary mantles.

10.4 The effect of “impurities” on melting temperature

10.4.1 Melting point depression revisited (again!)

Let T

be the melting point of a pure one-component crystal, and T the melting point of

the crystal in the presence of a component, z, that dissolves in the melt but not in the solid.

We shall refer to this component as an “impurity”, and to the component that makes up the

crystal as the “major” component. The impurity could be an abundant component that is

altogether excluded from the solid, or a strongly incompatible trace component – all that

matters is that its concentration in the solid is vanishingly small relative to its concentration

in the melt and can thus be ignored. The difference with our discussion of eutectics is that

now we do not require that this additional component make up a phase of its own. Let the

mol fraction of the impurity in the melt be X

z, m

. Because we ignore the concentration of the

impurity in the solid the activity of the major component in the solid species is 1, and K

in the melting point depression equation (10.8) equals the activity of the major component

in the melt, just as in the case of a eutectic. Let γ

r,m

be the activity coefficient of the major

component in the melt. Then K

=γ

r,m

(1 −X

z, m

) and (10.8) becomes:

r,m



1 −X

z, m









(10.16)

or:

T = T



r,m







1 −X

z, m





. (10.17)

This equation yields the crystal–melt equilibrium temperature as a function of the mol

fraction of the impurity in the melt. Calculating an accurate value for T , however, depends

on knowing accurate values for (i) X

z,m

and (ii) γ

r, m

. This is the crux of the problem, as

it requires that we know the distribution of chemical species in the melt and the possible

non-ideal interactions among them, something that is generally far from simple.

We can derive an important generalization, however. If the concentration of the trace

component in the melt remains low then γ

r, m

∼ 1, and we can ignore it. Because the

exponent R/

S is always a positive quantity, T<T

. Say that the molecular weight of

the impurity is m

, and that of the major component m

. If we express concentrations in

mass percent (what we normally and incorrectly call weight percent), we have:

z,m

100−C

z,m

. (10.18)