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Chapter 2. Thermodynamics
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
Applications of Thermodynamics to Light-Water Reactor Materials ................................ 2
Basic Thermodynamic Properties and Laws....................................................................... 2
Criterion of Chemical and Phase Equilibrium .................................................................... 7
Single-component phase equilibria ..................................................................................... 7
The Phase Rule ................................................................................................................. 11
Solution Thermodynamics ................................................................................................ 12
Binary Phase Equilibria .................................................................................................... 17
Chemical Equilibrium ....................................................................................................... 24
Aqueous Electrochemistry and Ionic Reactions ............................................................... 28
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2.1 Applications of Thermodynamics to Light-Water Reactor
Materials
It is generally recognized that materials failure limits both the maximum burnup attainable by
LWR fuel and the lifetime of the structural alloys in and around the core. The processes that
affect the behavior of these materials are both equilibrium and nonequilibrium; or, the processes
exhibit both thermodynamic and kinetic aspects. The nonequilibrium, kinetic features are
dominant in some phenomena, such as cladding corrosion by the coolant and production of
defects in metals by irradiation. In other processes, thermodynamic properties either completely
dictate the nature of the response of the materials to reactor conditions or control the driving
force for kinetic steps. Thermodynamics plays a particularly important role in the following:









Pressure-temperature-volume properties (equations of state); water, UO2
Vapor pressures: fuel, coolant, fission products
Thermal properties: the specific heat; coefficient of thermal expansion
Phase diagrams; single component and binary
Chemical and physical state of fission products in fuel
Tendency of hydrogen to form hydrides in cladding and carbon to form carbides in steel
Susceptibility of metals and alloys to aqueous corrosion
Response of uranium dioxide stoichiometry to the oxygen potential of the environment
Thermal stability of point defects in solids
This chapter is devoted to a brief review of materials thermodynamics, with emphasis on
chemical equilibrium. It is intended to serve as the basis of the many applications of
thermodynamics that will be encountered in subsequent chapters.
2.2
Basic Thermodynamic Properties and Laws
Thermodynamic properties depend only on the state or condition of the system but not on
the process or the path by which the particular state was achieved. Pure substances have only
five fundamental thermodynamic properties, which are those that cannot be derived from other
thermodynamic properties. These are, with their common symbols:
T = temperature
p = pressure
V = volume
U = internal energy
S = entropy
In addition, there are three derived thermodynamic properties that are combinations of
the primitive properties. These are:
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H = U + pV = enthalpy
F = U - TS = Helmholz free energy
(2.1)
G = H - TS = Gibbs free energy
The particular combination of the natural variables that these three functions represent
describe commonly encountered processes. For example, when pressure and temperature are
fixed, chemical equilibrium is achieved when the Gibbs free energy is a minimum. With the
same restraints, the heat absorbed or evolved in a chemical reaction is the change in enthalpy.
The eight basic thermodynamic properties can be classified as intensive or extensive. An
intensive property is independent of the quantity of substance. Temperature and pressure are
intensive properties. All of the others are extensive but can be made intensive by dividing by the
quantity of the substance. Taking the number of moles, n, as the measure of quantity, the
intensive counterparts of V,…..G are v = V/n, …. g = G/n. The lower-case designations are
reserved for intensive properties, which are also called specific values of the property. The
specific volume v is the reciprocal of the molar density.
Other thermodynamic properties are defined as partial derivatives of one of the eight
properties listed above. The heat capacities (also called specific heats)
 u 
CV   
 T  V
 h 
CP   
 T  p
(2.2)
represent the increases in internal energy and enthalpy, respectively, per degree of temperature
increase. They are written as partial derivatives because of the restraints indicated by the
subscripts on the derivatives. For CV, the increase in temperature is required to occur at a fixed
volume. For Cp, on the other hand, the pressure is maintained constant during the increase in
temperature.
The coefficient of thermal expansion  and the coefficient of compressibility  involve
the fractional changes in volume as temperature or pressure is increased:
1  v 

v  T  p
 
1  v 

v  p  T
 
(2.3)
Because the specific volume (or density) of a substance depends on both temperature and
pressure,  and  are defined as partial derivatives in order to indicate the property that is to be
held constant during the increase of the other property. Both  and  are positive numbers,
which accounts for the negative sign in the definition of .
Four of the basic properties have absolute values: T, p, V, and S. Assignment of zero
entropy to crystalline solids at 0 K is a consequence of the 3rd law of thermodynamics. The
energy-like properties U, H, F, and G are relative. Any one of them may be given an arbitrary
value at a chosen state (e.g., specified p and T).
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To a good approximation, the internal energy and the enthalpy of both gases and
condensed phases (liquids and solids) are nearly independent of pressure (or volume). This
simplification permits Eq (2.2) to be integrated:
T
u (T)  u o   C V (T)dT  C V (T  To )
To
(2.4)
T
h (T)  h o   C P (T )dT   C P (T  To )
To
where To is an arbitrary reference temperature where the internal energy or the enthalpy is set
equal to uo or to ho. Either of these could be chosen as zero, but the two must be related by
ho = uo + pvo. The second equalities in Eq (2.4) follow from the commonly employed
approximation of temperature-independent specific heats.
Other important properties of pure, or single-component, substances are the enthalpy
changes that accompany changes of phase. For the solid-liquid transition, the heat or melting, or
fusion, is:
hM = hL – hS
(2.5)
where hL and hS are the molar enthalpies of the liquid and solid phases, respectively. Conversion
of a liquid to its vapor is characterized by the heat of vaporization:
hvap = hg - hL
(2.6)
The enthalpy changes represent the heat absorbed as the phase with the lower enthalpy is
converted to the phase with the higher enthalpy. Consequently, hM and hvap are positive
quantities.
Relations similar to Eqs (2.5) and (2.6) can be written for all types of phase transitions.
Of particular importance are the transformations of crystalline solids from one type of crystal
structure to another type.
The concepts of heat and work are fundamentally different from the properties of a
material. Heat, in particular, is often confused with the thermodynamic properties temperature
and internal energy. It is neither. To say that a body (or system) contains a certain quantity of
heat is incorrect; the body or system possesses internal energy. Heat appears as this energy
crosses the system’s boundary in the form of conduction, convection, or radiation.
Work is a catch-all term for forms of energy transfer that have in common that they are
not heat but are in principle completely interconvertible among themselves. The most common
form of work is that produced by a force F acting over a distance X, which represents
displacement of the system boundary. This action involves a quantity of work given by W =
FX. If we multiply X by A, the area over which the force acts, and divide F by A, the work
equation becomes F = (F/A)(AX). Since F/A defines pressure p, and since the product AX is
the volume change V, the work involved can also be written as W = pV. This form of
mechanical work done on (or by) the system is called “pV” work.
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Another common form of work is shaft work, by which a system can exchange work with
its surroundings by means of rotational motion rather than expansion or contraction of the system
boundaries, as in the pV form. An example of shaft work is that performed as high-pressure
steam spins a turbine in an electric power plant. A third form of work is electrical work, which is
best exemplified by a battery that runs a motor by means of the electrical current generated by a
chemical reaction
The First law of thermodynamics is an empirical observation, never refuted, that the
change in the internal energy of a closed system resulting from addition of heat and performance
of work is given by:
U = Q - W
(2.7)
where
U = U(final) - U(initial) = change in system internal energy
Q = heat added to the system
W = work done by the system
Equation (2.7) applies to a system, which is a region of space whose boundaries enclose
the substance characterized by the properties T, p,… (note that heat and work are not properties
of a system). The material outside of the system boundaries is called the surroundings. The law
of energy conservation states that the sum of the energies of the system and surroundings is a
constant:
U + Usurr = 0
(2.8)
Equations (2.7) and (2.8) actually apply to the total energy, which is the sum of the internal
energy and the potential and kinetic energies of the system. For simplicity, the last two have been
neglected.
Without realizing it, most people have an intuitive feeling for the Second Law of
thermodynamics. We know, for example, that without any external intervention, heat will never
flow from a cold body to a hot one,and steam will not spontaneously decompose into H2 and O2.
The opposite of these processes, which we know to be correct, cause the entropy of the universe
to increase, which is precisely what the Second Law requires of spontaneous changes.
In the microscopic view of thermodynamics, entropy characterizes the state of disorder of
a system. Consequently, entropy changes are closely related to heat, but are not at all associated
with work. The work that raises a weight by a frictionless pulley does not affect the state of order
or organization of the system or the surroundings, and so work is entropy-neutral. Heat added to
a body, on the other hand, increases disorder by spreading the molecules over a larger number of
quantum states than they occupied prior to addition of thermal energy. This qualitative notion of
the connection of heat and entropy is embodied in the following quantitative statement of the
Second Law:
δ Q 
ΔS  
(2.9)

 T 
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where S and the integral represent changes from an initial state to a final state. The equality in
Eq (2.9) applies if the process is reversible, not just in heat transfer but in the mechanical aspects
as well*.
In common with enthalpy changes, entropy changes are associated with changes in phase
of a pure substance. For melting and vaporization, these are:
sM =sL – sS
(2.10)
where sL and sS are the molar entropies of the liquid and solid phases, respectively. Conversion
of a liquid to its vapor is characterized by the entropy of vaporization:
svap = sg - sL
(2.11)
Both sM and svap are inherently positive quantities, since the degree of order of a liquid is
greater than that of a solid but less than that of a gas or vapor.
The preceding discussion of the Second Law has dealt exclusively with changes in a
system, without regard to the entropy changes in the surroundings. Considering the system and
surroundings (the “universe”), the total entropy change is:
S+Ssurr  0
(2.12)
where the equality applies to reversible processes. Contrary to the energy analog given by Eq
(2.8), entropy is not conserved in irreversible processes.
The First law (Eq (2.7)) written in differential form for a unit quantity of substance is
du = q - w. For reversible changes, the second law (Eq (2.9)) is q = Tds and if only
expansion/contraction work is permitted, w = pdv. The combined First and Second laws takes
the form:
du = Tds - pdv
(2.13)
Although Eq (2.13) was derived by requiring total reversibility in the heat and work
terms, it nevertheless applies to irreversible processes as well. The reason for this powerful
generalization is that each term involves only state functions (i.e., thermodynamic properties) or
differentials of state functions. The changes in state represented by the differentials in Eq (2.13)
are independent of the path by which they occur, or, of whether the process is reversible or
irreversible.
Analogous differential forms of the combined First and Second laws can be derived for
the other three energy-like properties. For the enthalpy function, h = u + pv. The differential of
this property is dh = du + pdv + vdp, which, when combined with Eq (2.13), gives:
*
A reversible process is one that can be made to go backward without any change in the system or the surroundings.
To possess this feature, the reversible process must proceed through a series of infinitesimal stages in which internal
equilibrium and external equilibrium are maintained. Such processes are sometimes called quasistatic.
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dh = Tds + vdp
(2.14)
For the Gibbs function, g = h – Ts, the differential is dg = dh –Tds – sdT. Substituting Eq (2.14)
gives:
dg = -sdT + vdp
(2.15)
2.3
Criterion of Chemical and Phase Equilibrium
The combined First and Second laws, expressed by Eq (2.13), allow only for work due to
expansion or contraction. If the system performs other types of work, such as shaft work or work
derived from a chemical reaction, this work must be subtracted from the right hand side of Eq
(2.13). The same term appears in the right hand sides of Eqs (2.14) and (2.15). With the
constraints of constant T and p, Eq (2.15) reduces to:
dGT,p = -Wext
(2.16)
where Wext is the external, or non-pv, work performed by the system. The physical meaning of
this equation is the following: the maximum external work that a system can perform in any
process occurring at constant temperature and pressure is equal to the decrease in the system’s
Gibbs free energy during the process.
A useful definition of a system in equilibrium is one that cannot perform useful (nonexpansion) work. Applying this notion to Eq (2.16), the criterion of equilibrium for closed
systems constrained by fixed T and p is:
dGT,p = 0
(2.17)
Or, at equilibrium, the Gibbs free energy of a system with constant T and p is a minimum.
This criterion of equilibrium is useful chiefly for systems with more than one phase or more than
one component. For a single-phase, single-component system, fixing any two properties
determines the others, and Eq (2.17) is a trivial example of this fact. However, for heterogeneous
(multiphase) systems or homogeneous (single-phase) systems containing chemically reacting
species, Eq (2.17) provides the essential starting point for determining the state of equilibrium.
2.4
Single-component phase equilibria
A pure, or single-component, substance can exist in one, two or three phases. Important
two-phase mixtures, and the process by which one phase is transformed to the other, include:
liquid-gas (vaporization); solid-gas (sublimation); solid-liquid (melting, or fusion); solidI solidII (allotropy).
In a two-phase mixture at fixed pressure and temperature, let the system contain nI moles of
phase I and nII moles of phase II, with molar Gibbs free energies of gI and gII, respectively. The
total Gibbs free energy of the two-phase mixture is G = nIgI + nIIgII. The requirement of
equilibrium is that G remain unchanged (at its minimum value) for any variations in the state of
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the system that do not change its temperature or pressure. Because p and T are fixed, so are the
molar Gibbs free energies gI and gII. The only possible change is the conversion of some of one
phase to the other. Since the system is closed, an increment dnI of phase I implies an equal and
opposite change dnII = -dnI in phase II. The equilibrium criterion of Eq (2.17) yields dGT,p = gIdnI
+ gIIdnII = (gI - gII)dnI = 0, or:
gI =gII
(2.18)
An important application of Eq (2.18) is to the liquid-gas (vaporization) transformation, where
I = L(liquid) and II = g(gas). At equilibrium, gL = gg. In general, the molar Gibbs free energies
depend of both T and p. For the liquid phase, the dependence of gL of pressure is small. The
dependence of gg on pressure, however, is significant. At constant temperature, this effect is
given by Eq (2.15) as dgg = vgdp = RTdp/p, where the ideal gas law (pvg = RT) has been used to
eliminate vg. Integrating at constant temperature T from the hypothetical gas state at p = 1 atm
where gg = g og to the actual vapor at p = psat, the saturation pressure at temperature T, yields
gg = g og + RTlnpsat. The superscript o denotes the reference or standard state at 1 atm pressure.
Equating gL and gg yields:



RT ln p sat   g og  g L  Δ g ovap   Δ h ovap  TΔ s ovap

(2.18a)
Assuming that the heat of vaporization is independent of pressure (and hence of temperature),
the superscript o can be dropped from this term. However, the entropy of vaporization is pressure
dependent, and the entropy change on vaporization in the above equation is the difference
between the liquid and the gas at 1 atm pressure. The entropy difference between the liquid and
the 1-atm vapor is approximately temperature-independent. With these approximations, the
variation of the saturation pressure with temperature is:
p sat
 Δ s ovap 
 Δ h vap 
 exp  
 exp 
 RT 
 R 




(2.19a)
o
A formula of the same mathematical form but with different coefficients ( Δ s sub
instead of
Δ s ovap and hsub in place of hvap); this is the sublimation pressure curve of the substance.
Δ so 
 Δh 
p sat  exp  sub  exp   sub 
 RT 
 R 
(2.19b)
The heats and entropies of sublimation and vaporization are related to the corresponding
quantities for melting by:
o
hsub = hvap + hM
(2.20)
Δ s sub
 Δ s ovap  Δ s M
The superscript o is omitted from the entropy of melting because neither the entropies of the
solid or liquid are pressure-sensitive.
9
0
2
M
n
C
a
logp sat
4
Z
n
C
d
S
n
6
T
i
C
r
8
1
0
P
b
W
P
t
1
2
0
.
4 0
.
8 1
.
2 1
.
6 2
.
0 2
.
4
3
1
0
/
T
Figure 2.1 shows the saturation pressures of selected metals. Sublimation pressures are
shown as solid lines and vaporization pressures as dashed lines.
Fig. 2.1 Vapor Pressure of Selected Metals – solid lines: sublimation; dashed lines:
vaporization. Pressure in atm and temperature in Kelvins. (from G. V. Samsonov,
Handbook of the Physicochemical Properties of the Elements, Plenum (1968))
In each region, the lines are straight because the enthalpy and entropy changes in Eqs
(2.19a) and (2.19b) are nearly temperature-independent. The slopes of the lines are -hsub/R for
the solid-gas transition and -hvap/R for the liquid-gas phase change. The changes in slope at the
melting temperature for Ca, Zn, and Cd in Fig. 2.1 are due to the replacement of hsub by hvap.
According to Eq (2.20), the difference between these two enthalpy changes is the heat of fusion.
The changes in slope are barely discernible because the heats of fusion are only ~ 4 – 5% of the
heats of sublimation for all metals.
In addition to the vapor pressure formulas of Eqs (2.19a) and (2.19b), the phase
relations of a pure substance include the solid-liquid equilibrium. These three equilibria are
shown graphically on a phase diagram. The most familiar graphical representation of the phase
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relationships of a pure substance is the p-T diagram such as that shown in Fig. 2.2. These
representations contain three lines that intersect at the triple point. In the areas labeled “solid”,
“liquid”, and “vapor/gas”, a single phase is stable over a range of temperature and pressure. The
line separating the liquid and vapor regions is the liquid vapor pressure equation given by Eq
(2.19a). Equation (2.19b) separates the solid and vapor zones. The slope of the melting line
represents the effect of pressure on the melting temperature. The three 2-phase equilibrium lines
intersect at a common point called the triple point. At this unique combination of pressure and
temperature, all three phases coexist at equilibrium.
The melting line is derived as follows: The Gibbs free energy of the solid and liquid
phases are related to the corresponding enthalpies and entropies by g = h – Ts. At the melting
temperature TM, gL = gS, or:
gL – gS = gM = hM - TMsM = 0
(2.21a)
If the pressure is increased by dp, the melting temperature changes by dTM. Because the new
state is still in equilibrium, the change is the free energy of melting, d(gM), must be zero.
Obtaining the difference in the liquid and solid free energy changes from the fundamental
differential of Eq (2.15) yields:
d(gM) = - sMdTM + vMdp = 0
(2.21b)
Eliminating sM between Eqs (2.21a) and (2.21b) yields
dTM TMΔ v M

dp
ΔhM
(2.22)
here hM is the enthalpy of fusion and vM = vL - vs is the volume change on melting. For most
materials, vM is positive - the solid is denser than the liquid. Water is the notable exception to
this general rule because the liquid is denser than the solid.
pressure
meltin g
SOLID
LIQUID
critical point
a ti
o riz
p
a
v
triple point
sub
lim
n
atio
VAPOR/GAS
temperature
on
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Fig. 2.2 Generic phase diagram of a pure substance
A related aspect of the melting process is the variation of gM with temperature at
constant pressure. Under these conditions, the solid and liquid are no longer in equilibrium.
Although this situation is not encountered in pure substances, analysis of binary (twocomponent) phase diagrams requires gM for each species at temperatures other than their
melting points. In Eq (2.21b), setting dp = 0 for the constant-pressure requirement and replacing
TM by the variable temperature T yields:
Δh
 Δ g M 

  Δ s M   M
T  p
TM

where Eq (2.21a) has been used to eliminate sM. Assuming hM to be independent of
temperature, this equation can be integrated to give:

T 

Δ g M (T)  Δ h M 1 
T
M 

(2.23)
If T > TM, gM is negative, implying that the liquid has a lower free energy than the solid, and
hence in a single-component system, is the stable phase. Conversely, if T < TM, only the solid
exists at equilibrium. This sharp demarcation of phase stability breaks down in multicomponent
systems; a component can exist in a liquid solution at temperatures well below its melting point
when pure.
2.5
The Phase Rule
Analysis of systems other than pure substances requires understanding of the notions of
phase, components and degrees of freedom. Components are distinct chemical species whose
quantities can be independently varied. The relative amounts of the components are designated
by compositions, for which mole fraction is a common measure. Phases are regions of a system
in which all properties are uniform and are distinct from other regions in the same system.
Degrees of freedom are the number of system variables (e.g., properties, composition) that can be
independently specified without changing the phase(s). the numbers of components (C), phases
(P), and degrees of freedom (F) are related by the Gibbs Phase Rule:
F=C+2–P
(2.24)
For pure substances (C = 1), Eq (2.24) reduces to F = 3 – P. This relation can be understood
using Fig. 2.2. The areas labeled SOLID, LIQUID, and VAPOR/GAS permit both p and T to be
varied. This corresponds to P = 1, and F = 3 – 1 = 2. Or, these single-phase regions possess two
degrees of freedom. Two phases are present for p – T combinations that fall on the sublimation,
vaporization, and melting lines in Fig. 2.2. These correspond to P = 2 and F = 1. The single
degree of freedom can be either p or T; specification of one fixes the other according to Eqs
12
(2.19a), (2.19b) or (2.22). When three phases coexist at equilibrium, F = 0. This means that the
three-phase mixture occurs at a unique combination of p and T called the triple point.
Application of the phase rule to multicomponent systems is not as straightforward as it is
for pure substances. For example, take the problem of identifying the number of components in a
gas containing H2, O2 and H2O. At low temperature and in the absence of an ignition source, the
hydrogen does not burn and the mixture is a true three-component system. Temperature, pressure
and two mole fractions can be independently specified, which in Eq (2.24) corresponds to C = 3
and F = 3 + 2 – 1 = 4. At high temperatures, on the other hand, the chemical reaction 2H2(g) +
O2(g) = 2H2O(g) determines a relation between the concentrations of the three molecular
species. This restraint effectively reduces the number of components from three to two. The two
components are the elements H and O, irrespective of their molecular forms. The sole
composition variable is the H/O element mole ratio. With C = 2, the phase rule gives F = 3,
which corresponds to the variables p, T and H/O needed to fix the equilibrium composition of
the molecular species.
If a mixture does not change composition in a process, a binary system can be treated as a
pseudo single-component substance. Thus, analysis of air flowing through an orifice need not
consider N2 and O2 as distinct components as long as the properly averaged properties are used.
2.6
Solution Thermodynamics
The objective of this section is to understand the thermodynamics of single-phase, twocomponent (binary) systems. These include mixtures of ideal gases and the simplest models of
nonideal binary solid or liquid solutions. The terms “mixture” and “solution” are nearly, but not
quite, synonymous. A solution unequivocally refers to a homogeneous system of two or more
components. This term is applied to liquids an solids, but not to gases. Salt dissolved in water is
an aqueous solution of NaCl; a gold-silver alloy is a solid solution of these two elements.
However, air is a mixture of oxygen and nitrogen (plus minor species), not a solution of O2 in N2.
A multiphase system of a single component is referred to as a mixture of phases. These semantic
distinctions between mixtures and solutions are usually clear from the context in which the
words are used. Whether a single-phase mixture or a solution, the composition is denoted by the
mole fractions of the components present:
xi  ni / n
where
n  ni
(2.25)
n is the total moles in the system and ni is the number of moles of component i. By definition, the
sum of the mole fractions is unity.
To good approximation, gas mixtures and many condensed-phase (i.e., solid or liquid)
solutions can be considered to be ideal. In an ideal gas mixture, there are no intermolecular
interactions in either the pure components or in the mixture. Pure solids and liquids must exhibit
strong intermolecular attractions simply to exist as condensed phases. A binary solution of A and
B is ideal if the average of the A-A and B-B intermolecular forces is just equal to the strength of
the A-B interaction.
13
With some exceptions, the properties of an ideal solution are mole-fraction-weighted
averages of the properties of the pure components. If Q denotes an extrinsic property (i.e., for n
moles of mixture or solution), and qA and qB are the properties per mole of the two pure
components, the average is:
Q  n A q A  n Bq B
q  Q / n  x A q A  x Bq B
or
(2.26)
q is the intensive value of the property, or the value of the property per mole of mixture or
solution. Equation (2.26) applies to the volume V, the internal energy U, the enthalpy H, and the
heat capacities CP and CV. The entropy, however, contains an additional term that arises from the
increased randomness afforded by mixing. When two pure substances are mixed at constant
temperature and pressure, the entropy change, or the entropy of mixing is given by:
smix = -R(xAlnxA + xBlnxB)
(2.27)
where R = 8.314 J/mole-K (or 1.987 cal/mole-K) is the gas constant. Since the definitions of the
free energies include the entropy, the right hand side of Eq (2.27) appears in the expressions for
g and f analogous to Eq (2.26) for the generic property q.
Nonideality is a common characteristic of condensed phase (liquid or solid) solutions.
The very existence of solids and liquids requires strong interactions (bonding) between the
molecules. Departure from ideal behavior in a binary solution occurs when the mean of the A-A
and B-B bond strengths differs from the A-B bond strength. Divergence from ideal behavior can
be positive or negative. Quantitative treatment of nonideality in a binary system is achieved by
modification of Eq (2.26). The entropy of mixing, Eq (2.27) is unchanged and is handled just as
for ideal solutions. Because chemical and materials equilibrium analyses are based on
minimization of the Gibbs free energy, the nonideal version of Eq (2.26) will be presented in
terms of this property.
The total Gibbs free energy of a binary solution at a specified temperature and pressure
and its value per mole of solution are given by:
G  n A gA  n B gB
or
g  x A gA  x BgB
(2.28)
where g A and g B are the partial molar Gibbs free energies of the two components in the
solution. Nonideality causes these properties to differ from their pure-component counterparts,
the molar free energies gA and gB. g A depends on the nature of the component B with which it
shares the solution. In addition to depending on temperature (and to a much smaller extent, on
presssure), g A depends on the solution composition. The same characteristics apply to g B .
The partial molar Gibbs free energy of a component in solution is called the chemical
potential of the component, or i = g i . With this replacement, Eq (2.28) becomes:
G  n Aμ A  n Bμ B
or
g  x Aμ A  x Bμ B
(2.29)
14
The physical meaning of the chemical potential is best appreciated by recognizing that
the total Gibbs free energy of a binary solution depends on the number of moles of A and B as
well as on temperature and pressure. Taking the differential of G(T,p,nA,nB) holding T and p
constant gives:
 G
dG  
 n A

 G 


dn A  
dn B
 T ,p,n B
 n B  T ,p ,n A
dG  μ A dn A  μ B dn B
or
(2.30)
The partial derivatives in the first form of dG define the chemical potentials in the second form:
 G
μ A  
 n A


 T ,p ,n B
 G 

μ B  
 n B  T ,p ,n A
(2.31)
This equation shows that A represents the change in the Gibbs free energy of the solution when
a small quantity of A is added while the amount of B is held constant.
Equation (2.30) can be “integrated” in a physical sense by simultaneously adding the pure
components to a vessel at rates proportional to their concentrations in the final solution. This
procedure maintains all concentrations constant during the process, so that the integral of Eq
(2.30) is identical to the first equality in Eq (2.29). This procedure demonstrates that A and B
in Eq (2.30) are identical to those in Eq (2.29).
Another important relation involving the chemical potentials can be derived from the
total differential of Eq (2.29), dG = AdnA + nAdA + BdnB + nBdB. Eliminating dG using Eq
(2.30) and dividing by n to convert mole numbers to mole fractions yields:
x A dμ A  x B dμ B  0
(2.32)
This relation is known as the Gibbs-Duhem equation. As will be seen later, it is very important in
analyzing nonideal solutions.
An alternative to partial molar properties as a means of characterizing nonideality in
solutions is the concept of excess properties*. Instead of Eq (2.28), the Gibbs free energy of a
binary solution can be expressed by:
g = xAgA + xBgB + gex – Tsmix = xAgA + xBgB + hex – Tsex – Tsmix
(2.33)
Because gA and gB refer to the pure components, these terms and Tsmix represent the free energy
of the solution if it were ideal. The nonideal features are contained in the gex term, which has
been broken into enthalpy and entropy contributions according to the definition the Gibbs free
*
A partial molar property and the corresponding excess property are not independent quantities; one can be obtained
from the other, although these connections are not given here.
15
energy in Eq (2.1). The motivation of this last step arises from the possibility of attaching
physical meaning to hex and sex.
Formulas analogous to Eq (2.33) apply to other thermodynamic properties. For example,
the volume and enthalpy of a binary solution are given by:
v = xAvA + xBvB + vex and
h = xAhA + xBhB + hex
(2.34)
The excess volume and enthalpy are directly measurable. For example, if 50 cm3 of roomtemperature water is mixed with 50 cm3 of room-temperature sulfuric acid in an insulated vessel,
the temperature rises to 123oC, and after cooling, the final volume is 90 cm3 rather than 100 cm3.
These changes are direct measures of vex and hex. In this example, water and sulfuric acid interact
strongly and produce highly negative deviations from ideality.
For ideal solutions, both hex and sex are zero. The behavior of a fair number of nonideal
binary solid or liquid solutions can adequately represented by the regular solution model. In this
theory, the molecules mix randomly as they do in ideal solutions, so that sex = 0 and the excess
Gibbs free energy reduces to the excess enthalpy. The analytical formulation of hex in terms of
composition is restricted by the limiting behavior as the solution approaches pure A and pure B.
In these limits, hex must be zero at xA = 0 and at xB = 0. The simplest function that obeys these
restraints is the symmetric expression:
hex = xAxB
(2.35)
where  is a temperature-independent property of the A-B binary pair called the interaction
energy. The form of Eq (2.35) is supported by molecular modeling, which suggests that  is
equal to the difference between the energy of attraction (bond energy) of the A-B pair and the
mean of the bond energies of the A-A and B-B interactions.
Although the thermodynamic behavior of species in solution is ultimately tied to their
chemical potentials, a connection between this property and the concentration of the component
is needed. This relationship is made via a quantity called the activity of a solution species. The
activity is a measure of the thermodynamic “strength” of a component in a solution compared to
that of the pure substance; the purer, the stronger. The chemical potential and activity are related
by the definition of the latter:
i = gi + Rtlnai
(2.36)
The activity tends to unity for pure i, where i = gi, the molar free energy of pure i. When
component i becomes infinitely dilute in the solution, ai  0 and its logarithm approaches -.
This is also the limit of the chemical potential of i at infinite dilution. This inconvenient behavior
of the chemical potential at zero concentration is avoided by using the activity in practical
thermodynamic calculations.
For nonaqueous or solid solutions, the activity coefficient is the ratio of the activity to the mole
fraction:
i = ai/xi
(2.37)
16
A useful connection between the activity coefficients of species in a solution is obtained
by eliminating ai between Eqs (2.36) and (2.37) and substituting the resulting equation into the
Gibbs-Duhem equation, Eq (2.32). For two component (A-B) solutions, this procedure yields:
xAdlnA + xBdlnB = 0.
(2.38)
The activity coefficient is the all-important characterization of nonideality in solid or liquid
solutions. The significance of Eq (2.38) is in the need to measure the activity coefficient of one
species as a function of composition, from which the activity coefficient of the other species
follows by integration.
For regular solutions (i.e., solutions that obey Eq (2.35)), the activity coefficients can be
shown to be:
(2.39)
RT ln γ A  Ωx 2B and RT ln γ B  Ωx 2A
These activity coefficients satisfy the Gibbs-Duhem equation, Eq(2.38).
Contrary to condensed-phase solutions, gas mixtures are generally nearly ideal. When
two pure ideal gases A and B at the same p and T are combined, the mixture occupies a volume
V equal to the sum of the initial volumes. Each component obeys the ideal gas law, but because
its molecules are spread over a larger volume, its pressure is reduced from p to a smaller value
termed the partial pressure. Denoting these by pA and pB, each component obeys the ideal gas
law:
pAv = xART
pBv = xBRT
(2.40)
In Eqs (2.40), a convenient value R is 82 cm3-atm/mole-K. The total pressure of the mixture (as
would be measured by a gauge) is the sum of the partial pressures:
pA + pB = p
(2.41)
The mixture also obeys the ideal gas law, as can be seen by adding Eqs (2.40) and using Eq
(2.41) to eliminate the partial pressures:
pV = nRT
(2.42)
Dividing each of Eqs (2.40) by Eq (2.42) relates the partial pressures to the mole fractions
pA n A

 xA
p
n
pB n B

 xB
p
n
(2.43)
Equations (2.40) and (2.40) are known as Dalton’s law.
The mixing rules for ideal gases for the volume, internal energy, and enthalpy follow the
generic form of Eq (2.26). The entropy and free energies of the mixture must include Eq (2.27)
in addition to Eq (2.26).
17
An important feature of ideal gas mixtures, given here without proof, is that the chemical
potential of a species in the mixture at a given partial pressure is the same as the molar Gibbs
free energy of the pure gas at the same pressure, or  i (in mixture at pi) = gi (pure, at pi).
Contrary to condensed phases, the Gibbs free energy of a pure gas is pressure-dependent.
In order to provide a common pressure reference for all pure gases (arbitrarily chosen at 1 atm),
gi in the above equation is expressed in terms of gio , which is the molar Gibbs free energy of
species i at temperature T and 1 atm pressure. The effect on gi of the difference in pressure
between pi and 1 atm is obtained by combining Eq (2.15) with the ideal gas law. For constant
temperature, this gives:
dgi RT

dp
p
Integrating this equation from pi to 1 atm and identifying gi at pi with the chemical potential
yields:
i = gio + RTlnpi.
(2.44)
The 1-atm reference condition of the pure gas is denoted by the superscript zero. With this
standard state, pi must be expressed numerically in units of atmospheres.
2.7
Binary Phase Equilibria
An important application of the thermodynamics of solutions summarized in the preceding
section is to the analysis of equilibrium of two (or more) components distributed between two (or
more) phases. The most common combinations are a gas phase and a condensed phase* or two
condensed phases. The latter includes liquid-solid, liquid-liquid, and solid-solid pairs. The two
coexisting phases are denoted by I and II, but the number of components is restricted to two,
labeled A and B. The total Gibbs free energy of this two-phase mixture is G = GI + GII. A change
in the state of the system at constant temperature and pressure is provoked by moving dnAI moles
of component A from phase I to phase II. If the system is at equilibrium, this movement does not
change the system’s Gibbs free energy, and Eq (2.17) results in dG = dGI + dGII = 0. The free
energy changes of each phase are related to the chemical potentials according to Eq (2.30),
which yields:
 AIdnAI +  BIdnBI +  AIIdnAII +  BIIdnBII = 0
where nAI …..nBII are the numbers of moles of each constituent in each phase and AI….BII are
their chemical potentials. Conserving component A, dnAII = - dnAI, and because component B is
not exchanged, dnBI = dnBII = 0. Applying these constraints to the above equation yields:
 AI =  AII
*
condensed phases are either liquids or solids
(2.45)
18
A similar equation applies to component B. Equation (2.45) is the multicomponent
generalization of the equilibrium condition for two coexisting phases of a pure substance, namely
gI = gII, where g is the molar Gibbs free energy (Eq (2.18)).
If phase I is a gas and phase II a liquid (or solid), the equilibrium criterion of Eq (2.45)
becomes Ag = AL. Using Eq (2.44) for Ag and the combination of Eqs (2.36) and (2.37) for
AL, the equilibrium condition becomes:
 g oAg  g A L 
pA

 exp  


γ AxA
RT


Equation (2.18a) shows that the right hand side of the above equation is the saturation pressure
of pure liquid A, so that:
p A  γ A x A p sat , A
(2.46)
This equation forms the basis for all analyses of phase equilibria in multicomponent
systems involving a gas phase. A formula similar to Eq (2.46) applies to component B, and to all
other components if the system contains more than two species. It equally valid for a solid
solution of A and B, provided that Eq (2.19b) is used for psat,A instead of Eq (2.19a). The
dependence of pA on composition is in general nonlinear because the activity coefficient of A in
the condensed phase, A, is a function of composition if the solution is nonideal and not infinitely
dilute in A.
If the solution is ideal, A = 1 for all xA, and Eq (2.46) reduces to Raoult’s law:
pA = xA psat,A
(2.47)
Component B also obeys Raoult’s law because Eq (2.38) shows that if A = 1, then B = 1 as
well.
In nonideal solutions, A  constant  1 as the solution becomes dilute in A. Equation
(2.46) reduces to Henry’s law:
pA = kHAxA
(2.48)
where the Henry’s law constant, kHA, is the product of the composition-independent activity
coefficient of A in solution and the saturation pressure of pure A. In the concentration range
where A follows Henry’s law, component B must obey Raoult’s law. This is a consequence of
the Gibbs-Duhem equation, Eq (2.38).
Figure 2.3 shows typical examples of nonideal solution behavior. The curves represent
Eq (2.47) for positive and negative deviations from ideality (i.e., A > 1 or A < 1). The limiting
cases of Raoult’s and Henry’s law are shown as dashed lines.
Equilibrium between condensed phases (solid and liquid) in two-component systems is
displayed as a binary phase diagram. These are plots of regions of a single phase or two phases
with temperature as the ordinate and composition as the abcissa. Because the gas phase is
19
ignored in this representation, the effect of pressure on the phase diagram is small and is
generally neglected. In two-phase regions, Eq (2.45), and the analogous equation for component
B, serve to determine the compositions of the two phases. The chemical potentials of A and B in
Fig. 2.3 Equilibrium pressures of component A over an A-B solution. The temperature
is fixed.
each phase are expressed in terms of composition by the equation obtained by eliminating the
activity between Eqs (2.36) and (2.37).
To illustrate this method of constructing phase diagrams, melting in an ideal system is
considered. Phase I is solid and phase II is liquid. All four activity coefficients are unity. With
these restrictions, the conditions of equilibrium become:
gAL + RTlnxAL = gAS + RTlnxAS
(2.49a)
gBL + RTlnxBL = gBS + RTlnxBS
(2.49b)
The differences gAL – gAS and gBL – gBS are the free energy changes on melting of the pure
species. These are related to the melting properties of A and B by Eq (2.23). Since xAL+ xBL = 1
and xAS+ xBS = 1, the above equations contain two unknowns. Solving yields:
xBL 
1  e
e   e
and
 1  e 
x BS  e    
e e 
(2.50)

T  Δ h MB

β  1 
 TMB  RT
(2.51)
with

T  Δ h MA

α  1 
 TMA  RT
20
Figure 2.4 shows the phase diagram for an ideal binary system calculated from Eqs (2.50) using
values of the melting properties of metals U and Zr. The upper line (representing T Vs xBL) is
called the liquidus. All points lying above this line are completely liquid. Similarly, all points
below the lower curve (the solidus, or T Vs xBS) are completely solid. In the region bounded by
the solidus and the liquidus, two phases coexist.
2200
2200
2000
2000
liq uid (L )
1800
1800
E
C
S+ L
B
1600
1600
sol id so lutio n (S )
P
A
D
1400
1400
1200
1200
0
0.2
0.4
0.6
0.8
Mole frac tion Zr (c om p onen t B
B ), x
1
0
0.2
0.4
0.6
0.8
1
Mole frac tion Zr (c om p onen t B
B ), x
Fig. 2.4 Phase diagram of the U-Zr binary system with ideal behavior in both liquid and solid
The left-hand panel of Fig. 2.4 represents the actual phase diagram. In the right-hand
panel, the horizontal and vertical lines are superimposed on the phase diagram in order to
illustrate important characteristics of the melting process. If the solid solution with a composition
xB = 0.4 is heated, the intersection of the vertical line with the solidus (at point A) shows that the
first liquid appears at 1630 K and has a composition xBL = 0.21 (at point B). As the temperature
is increased to 1700 K, the system lies at point P. Here a liquid phase with composition xBL =
0.31 (point C) and a solid phase with xBS = 0.49 (point D) coexist. The fraction of the mixture
present as liquid at point P is obtained from the mole balance known as the lever rule:
Fraction liquid at point P =
x  xB
PD
0.49  0.4
 BS

 0.50
C D x BS  x BL 0.49  0.31
(2.52)
Upon heating from point P, the last solid disappears at T = 1790 K (point E). Melting of this
binary system at this particular overall composition is spread over a 160 K temperature range.
At the opposite extreme from the ideal solution treated above is a system with strong
deviations from ideality in the solid phases. In a limiting version of such a system, the solid
phases are so nonideal, with positive deviations from ideality, that there is no solubility of A in B
or B in A. That is, molecules of B and A repel each other so strongly that solutions of one in the
other are not possible. On the other hand, A-rich and B-rich liquid solutions exist and may even
exhibit negative deviations from ideality (attraction between A and B on the molecular level).
21
For pure solid A in equilibrium with an A-rich liquid, Eq (2.45) becomes:
g AS  g AL  RT ln(γ AL x AL )
(2.53)
Solving for xAL, the A-rich portion of the phase diagram is expressed by:
x AL  e α / γ
AL
(2.54)
where  is the temperature-dependent function given by Eq (2.51). To complete this portion of
the phase diagram, AL must be known as a function of temperature and composition (e.g., by Eq
(2.39) if the liquid obeys regular solution theory).
An entirely analogous treatment applies to the portion of the phase diagram in which pure
solid B coexists with a B-rich liquid. The equilibrium condition is given by Eq (2.53) with A
replaced by B, and the composition-temperature equation is:
x BL  e β / γ
BL
(2.55)
with  given by Eq (2.51). Again, BL must be known as a function of temperature and
composition.
The gold-silicon binary system is representative of systems that exhibit this type of phase
behavior. This phase diagram is shown in Fig. 2.5. With A = Au, the Au-rich liquidus in the
22
Fig. 2.5 The gold-silicon phase diagram
figure is a plot of Eq (2.54) and the Si-rich liquidus represents Eq (2.55). The two liquidus curves
intersect at a point called the eutectic (Greek for “lowest melting”). At this point, three phases
coexist: the two pure solids and the liquid of the eutectic composition. At lower temperatures,
only the two pure solids are present
At this juncture, it is constructive to examine how binary phase diagrams relate to the
phase rule (introduced in Sect. 2.5). For a two component system, Eq (2.24) permits
F = 4 – P degrees of freedom. Since the diagrams deal only with condensed phases, they are
minimally affected by total pressure. Ignoring the total pressure reduces the number of degrees
of freedom by one, thereby allowing 3 – P properties to be independently varied. In a single
phase (P = 1) portion of the phase diagram, two degrees of freedom are permitted. These are the
temperature T and the composition, represented by the mole fraction of one of the constituents,
say xB. Single-phase regions appear as areas in the phase diagram.
In two-phase zones (P = 2), only one system property can be specified. Fixing the
temperature, for example, determines the compositions of the two coexisting condensed phases.
These temperature-composition relationships appear in the phase diagram as lines (or curves)
called phase boundaries. A three-phase system (P = 3) has no degrees of freedom and is
represented by a point on the phase diagram. An example of this is the eutectic point in Fig. 2.5.
The distinction between overall compositions and the compositions of individual phases is
essential to understanding phase diagrams. For single-phase zones, the two are identical. When
two phases coexist, the compositions of the individual phases are different from the overall
composition. The latter is the mole-weighted average of the compositions of the two phases (i.e.,
the lever rule, see Eq (2.52)).
When A and B are mutually soluble in each other, the phase diagram takes on the general
appearance of the one shown in Fig. 2.6. Compared to Fig. 2.5, two single-phase regions, labeled
c
temperature
L
o
+L
L +

e
b

n
m
a
 + 
mole fraction
Fig. 2.6 Generic eutectic phase diagram
23
 and , have been added to the diagram. The  phase retains the crystal structure of pure A but
some component B is dissolved in it, usually by substituting for A atoms in the lattice. The
curves ab and bc represent the terminal solubility of B in A. Compositions along these curves are
the maximum concentrations of B that the  phase can sustain. Additional B added to the system
causes precipitation of a new phase, which is the solid  phase below the eutectic temperature or
the liquid above this temperature.
An analogous portion of the diagram is present on the right hand side of Fig. 2.6, which is
a more general version of Fig. 2.5. If point b in Fig. 2.6 is moved to the left-hand ordinate and
point n displaced to the right-hand ordinate, the phase diagram reduces to the one shown in Fig.
2.5.
The iron-uranium phase diagram of Fig. 2.7 exhibits two eutectic points located between
high-melting entities called intermetallic compounds. The vertical lines at uranium fractions of
0.33 and 0.86 represent the stable species Fe2U and FeU6, respectively. These are true
compounds, with fixed and invariant Fe/U ratios, that are crystallographically distinct from the
pure metals and have definite melting points. They form the boundaries of eutectic features in the
phase diagram. Except for the presence of three allotropes of iron (the , , and  phases), the
portion of the diagram between Fe and Fe2U is the same as in Fig. 2.5. The region from Fe2U to
FeU6 is exactly like Fig. 2.5. The zone between FeU6 and pure U is complex, owing to the
presence of the three phases of uranium (, , and ) and the limited solubility of Fe in these
phases. The -U, -U, and -U phases indicated on the extreme right of the diagram are analogs
of the  region in Fig. 2.6. Except in these three single-phase solids and the liquid region, a point
in Fig. 2.7 indicates the coexistence of two phases. The coexisting phases lie at the intersections
of a horizontal line through the point and the boundaries of adjacent single phases. The lever rule
gives the relative proportions of the two phases. For example, an overall composition of 50% U
Fig. 2.7 The iron-uranium phase diagram
at 1000oC consists of 24 mole % Fe2U and 76 mole % liquid of 55 mole % U.
24
2.8
Chemical Equilibrium
Chemical reactions entail exchange of elements between molecules. Reactions are
designated as homogeneous if all species involved are in a single phase, or heterogeneous if the
participants are in two or more phases. Typically homogeneous reactions occur in a gas or a
liquid such as water. A common heterogeneous reaction is metal oxidation, which involves a gas
(containing oxygen) and two solid phases (the metal and its oxide).
Equilibrium analysis of chemical reactions provides an equation relating the concentrations
of all participants. When supplemented with specified ratios of the elements involved, the
composition of the equilibrium system is fixed. These generalities can be made more specific by
considering the generic reaction between reactant species A and B to form product species C
and D:
A + bB = cC + dD
(2.56)
The coefficients of the reactant and product species are the stoichiometric numbers or balancing
numbers that serve to conserve elements on the two sides of the reaction. They are usually
chosen so that one of them is unity, as has been done for the coefficient of A in the above
reaction. The equal sign indicates that the four species are present at equilibrium. At equilibrium,
there is no distinction between reactants and products; Eq (2.56) could just as well have been
written with C and D on the left and A and B on the right.
As in any system constrained to constant temperature and pressure, the equilibrium of a chemical
reaction is attained when the Gibbs free energy of the system containing all species involved is a
minimum, or dG = 0. Extending the second equality in Eq (2.30) to include species C and D, this
criterion is:
Adn A  Bdn B  Cdn C  Ddn D  0
The changes in the mole numbers, dnA, …dnD, are related to each other by the balancing
numbers in Eq (2.56); For example, for every mole of A consumed, b moles of B disappear and c
and d moles of C and D, respectively, are produced. These stoichiometric restraints are
equivalent to: dnB = bdnA, dnC = -cdnA, and dnD = -ddnA. With them, the above equilibrium
condition reduces to:
(2.57)
μ A  bμ B  cμ C  dμ D
The equation applies to an equilibrium in a single phase or involving multiple phases. For a
homogeneous gas-phase reaction, the chemical potentials are expressed in terms of the partial
pressures by Eq (2.44), and Eq (2.57) becomes:
Kp 
p cC p dD
p A p bB
 Δ Go
 exp  
 RT




(2.58)
where KP is the equilibrium constant in terms of partial pressures. It depends on both
temperature and total pressure. The effect of the latter variable can be made explicit by replacing
25
the partial pressures using Dalton’s rule, Eq (2.43). With this substitution, the equilibrium
constant becomes:
KP = Kpm,
where
m=c+d–1–b
(2.59)
K 
x cC x dD
(2.60)
x A x bB
K is the equilibrium constant in terms of mole fractions, and is usually the preferred method for
expressing concentrations in a mixture. Equations (2.58) and (2.60) relating compositions to KP
and K are sometimes called the law of mass action.
Go in Eq (2.58) is the standard free energy change of the reaction:
Δ G o  cg oC  dg oD  g oA  bg oB
(2.61)
The g io are the molar free energies of the pure species at the specified temperature. The
superscript o indicates that g io is to be evaluated at 1 atm pressure. The standard free energy
change can be broken up into its enthalpy and entropy components:
Go = Ho –TSo
(2.62)
The basic data for evaluating equilibrium constants from Go is either the molar free energies g io
tabulated as functions of temperature or the standard enthalpy and entropy changes, Ho and
So, respectively. The latter are often independent of temperature, and so provide the most
compact way of characterizing a reaction. A third method of compressing the database is to
break to overall reaction into formation reactions, whereby a molecular species is created from
its elemental components. Go then becomes:
Go = cΔ G fC  dΔ G fD  Δ G fA  bΔ G fB
(2.63)
Armed with the equilibrium constant K, Eq (2.60) is applied to determine the
composition of the equilibrium gas. The closed system initially contains specified moles of
species A, B, C and D. In achieving equilibrium, the initial numbers of moles change to new
values. Let  be the number of moles of A reacted in achieving equilibrium. The changes in the
number of moles of B, C, and D are related to  by the stoichiometric coefficients of reaction
(2.56). Table 2.1 gives the initial and final (equilibrium) mole numbers. In the last column,
a) Table 2.1 Initial and Equilibrium mole numbers in gas-phase
reaction a + bB = cC + dD
Moles
Initial
A
n 0A
B
n 0B
C
n 0C
D
n 0D
Total
n 0T
26
Equilibrium
n 0A - 
n oB  bξ n 0B  bξ
n oC  cξ
n oD  dξ
n oT  mξ
n oT is the sum of the initial moles of the four species and m is the combination of the
stoichiometric coefficients shown in Eq (2.59). The mole fractions in the equilibrium system are
obtained by dividing the moles of A, B, C, and D in the last row by the total moles in this row.
Substituting these mole fractions into the law of mass action given by Eq (2.60) yields:
n
 n  dξ 
K
n  ξ n  bξ  n  mξ 
o
A
o
C
c
 cξ
o
B
o
D
b
d
o
T
m
(2.64)
This equation is solved for  (in general numerical solution is required) and from this result, the
mole fractions at equilibrium are calculated.
Example: Combustion of 1 mole of methane by 2 moles of oxygen at 2000 K. The reaction is:
CH4(g) + 2O2(g) = CO2(g) + 2H2O(g). The letter g in parentheses following each species indicates that the
species is a gas. For this reaction, the exponent m of Eq (2.59) is zero, so there is no effect of pressure on
the equilibrium composition. At 2000 K, the standard free energy change of the reaction is Go = -191
kcal/mole, so that the equilibrium constant is, according to Eq (2.58): KP = K = 7.5x1020. .
Two approaches are available. First, a table similar to Table 2.1 can be constructed specifically for this
reaction. Second, Eq (2.64) can be specialized for this reaction. In the latter method, the input parameters
are: b = 2; c = 1; d = 2; m = 0 and n 0A  1; n oB  2; n oC  0; n oD  0 . Substituting these values into Eq
(2.64) yields:
 ξ 

1 ξ 
3
7.4x1020  
In this case, analytical solution is possible, yielding 1 -  = 1.1x10-7 . The corresponding mole fractions
at equilibrium are: xA = 4x10-8; xB = 8x10-8; xC = 0.333; xD = 0.667. The reaction goes essentially to
completion.
The class of heterogeneous reactions in which an element reacts with a diatomic gas to
form a compound is both of practical importance and amenable to simple thermodynamic
analysis. Reactions in this category include oxidation, nitriding, and hydriding of metals and
halogenation of the electronic material silicon. The simplicity of the thermodynamics stems from
the immiscibility of the reactants and products. Consider oxidation of a metal to form a dioxide
of the metal:
M(s) + O2(g) = MO2(s)
(2.65)
The letter s in parentheses indicates a solid phase, in this case two solid phases because
the metal and its oxide are essentially insoluble in each other. Since M and MO2 are practically
pure substances, their chemical potentials are equal to their molar Gibbs free energies. The
chemical potential of oxygen gas is dependent on its partial pressure, and is given by Eq (2.44).
With b = c =1 and d = 0, the general criterion, Eq (2.57), becomes:
27
goM  goO 2  RT ln pO 2  goMO2
(2.66)
Rearranging this equation into a more convenient form gives:
pO 2  e G
o
/ RT
 e  S
o
/ R H o / RT
e
(2.67a)
or
RT ln pO 2  G o  Ho  TSo
(2.67b)
G o  goMO2  goO 2  goM
(2.68)
where
Equations (2.67a) and (2.67b) are plotted in Fig. 2.8. These plots are called stability
diagrams because the lines separate regions in which only one of the two phases is present. The
line represents the p O 2 - T combinations where both the metal and its oxide coexist. The oxidemetal stability diagram is a solid-phase analog of the p-T phase diagram of a single substance
such as water, where lines separate existence regions of solid, liquid, and vapor phases (see
Fig.2.2 ).
MO2 stable
RTlnp O2
lnpO2
MO2 stable
slope = Ho/R
M stable
slope = - So
M stable
intercept = Ho
1/T
T
(a)
(b)
Fig. 2.8 Stability diagrams for the M + MO2 Couple.
Example: Powdered nickel metal is contacted with a flowing mixture of CO2 and CO at 1 atm total
pressure in a furnace at 2000 K. The quantity of metal is limited, but because of continual flow, the
quantity of the gas mixture is unlimited. Therefore, the oxygen pressure established in the gas phase is
imposed on the metal, and determines whether or not it oxidizes. At what CO2/CO ratio do both Ni and
NiO coexist?
The O2 pressure in the gas is fixed by the equilibrium: 2CO(g) + O2(g) = CO2(g). At 2000 K, the
equilibrium constant is KP = 4.4x105. The law of mass action for this equilibrium reaction is:
p O gas  (p CO
2
2
/ p CO ) 2 / K P
The nickel/nickel oxide equilibrium reaction is: 2Ni + O2 = 2NiO, for which
Go = -46 kJ/mole at 2000 K. According to Eq (2.67a), for coexisting Ni and NiO in the solid phase, the
oxygen pressure is:
p O solid  exp Δ G o / RT   6.3  10 2 atm
2
The mixed solid and the mixed gas are in equilibrium when ( p O 2 )gas = ( p O 2 )solid. From the above
equations, this condition yields the required ratio of CO2 to CO in the gas:
28
 solid  4.4  105 6.3  102   166
pCO 2 / pCO  K P pO 2
2.9
Aqueous Electrochemistry and Ionic Reactions
The chemical reactions among aqueous ions or between ions and solids or gases in contact
with the solution are called electrochemical reactions. Such reactions are the source of some of
the most important materials problems in the nuclear industry. Corrosion of the metal
components in the core and the primary circuit of light water reactors is a prime example of
practical electrochemistry. Another example is the attack of buried nuclear wastes by
groundwater in geologic repositories.
The elements that, as ions, take part in electrochemical reactions must have more than
one valence state, or oxidation state, that are stable in, or in contact with, water. Iron, for
example, commonly occurs in the elemental (Fe0), ferrous (Fe2+), and ferric (Fe3+) oxidation
states. Hydrogen occurs as the diatomic molecule H2 dissolved in water or as the hydrogen ion
H+. The forms of oxygen include OH- and the dissolved gas O2. Both hydrogen and oxygen
appear in H2O, which often takes part in electrochemical reactions. For an element to participate
in an electrochemical reaction, it must change valence state in the reaction. For example, the
overall reaction:
M + zH+ = Mz+ + ½zH2
(2.69)
involves oxidation of the elemental metal M to its z+ oxidation state. This ion is dissolved in
water. Accompanying oxidation of one element is reduction of another. In this case, hydrogen is
reduced from its 1+ state in solution to its elemental state as H2, which appears either in solution
or in the gas phase. Positive ions in solution are called cations. Negative ions, called anions,
which must accompany the positive ions to maintain electrical neutrality of the solution, are not
explicitly included in Eq (2.69). The reason for this is that most anions exhibit only one charge
state in aqueous solution. If an ionic species is not capable of changing oxidation state, it cannot
participate in the electrochemical reaction. Typical anions include Cl-, NO 3 , SO 24 and PO 34 .
Many electropositive elements are immune from electrochemical effects because they exhibit
only one stable oxidation state. Thus, sodium is always present in solution as Na+; reduction to
the element in water is not possible. The phases occupied by the species in the above overall
reaction are not indicated, but are usually evident: the metal M is a solid; H2 is a gas; both
cations are in solution.
In most electrochemical reactions, the exchange of the electrons is not physically
manifest because it occurs in an intimate mixture of reactants and products. However, in a device
known as an electrochemical cell, participants in a reaction are physically separated in a manner
that makes it possible for the electron transfer process to be observed, utilized for doing work, or
measured to provide information useful in understanding processes such as corrosion. This is
accomplished by separating the overall reaction into half cell reactions, in which the oxidation
and reduction portions are shown explicitly. For example, the half-cell reactions corresponding
to reaction (2.69) are:
29
M = Mz+ + ze
(2.70a)
H2 = 2H+ + 2e
(2.70b)
The overall reaction is (2.70a) less ½z of (2.70b). Figure 2.9 illustrates an electrochemical cell in
which the half-cell reactions take place in individual compartments, called electrodes. The inert
metal electrode permits reaction (2.70b) to proceed efficiently. The bridge separating the two
compartments permits anions to move in order to maintain electrical neutrality.
The two metal electrodes are connected by a wire connected to one of three devices.
Shown in the figure is a meter that measures the voltage, or EMF, between the two electrodes. In
this mode, no current flows and the system is at equilibrium. If instead a battery is placed in the
line and the applied voltage is different from the equilibrium EMF, a current can be caused to
flow. In this case, the metal is either electroplated or corroded, depending on the direction of
current flow. The half cell in which oxidation occurs is called the anode and the one supporting
reduction is the cathode.

H2
electrode,
metal M
Mz+
H+
bridge
inert metal
electrode
Fig. 2.9 An aqueous electrochemical cell with an active metal half cell and a hydrogen half
cell
Finally, the voltmeter shown in Fig. 2.9 can be replaced by an electric motor that is driven by the
electric current flowing from one electrode to the other. With this device, current flowing in the
external circuit can be utilized to perform work, and the cell functions as a battery.
From a thermodynamic point of view, the great utility of the equilibrium electrochemical
cell is that its EMF is proportional to the difference in the free energies of the mixtures in the two
electrodes – that is,  is a direct measure of G of the overall cell reaction. This connection is
established by considering a nonequilibrium cell and equating the electrical work (cell voltage
times the charge transferred) to the maximum possible work in a process at constant temperature
and pressure (equal to the change in the Gibbs free energy, Eq (2.16)). The result is:
30
G = -zF
(2.71)
G is the change in Gibbs free energy between the product and reactant sides of the overall
reaction when z moles of electrons are transferred. F is Faraday’s constant, which is the product
of Avogadro’s number and the electronic charge. Convenient units of F are 96500
Coulombs/mole, or 23.06 kcal/mole-Volt.
The cell in Fig. 2.9 is called an equilibrium cell because no current flows and no change
occurs with time. However, this does not mean that the ion concentrations in the two electrodes
are those that would be found if the two half-cell solutions were part of the same solution (which
is equivalent to short-circuiting the cell in Fig. 2.9). If the cell were short-circuited, the
concentrations of Mz+ and H+ would adjust until the equilibrium criterion of Eq (2.57) was
satisfied. In the open-circuit configuration of Fig. 2.9, on the other hand, the ion concentrations
can be arbitrarily fixed and the EMF (or, equivalently, G) reflects the imbalance of the
chemical potentials in the two half cells:
z
G  (M z )  (H 2 )  (M)  z(H  )
(2.72)
2
The chemical potential of H2 depends on the pressure according to Eq (2.44) (the chemical
potential of H2 dissolved in water is the same as that of the gas, see Eq (2.45)). The chemical
potential of the pure metal M is equal to its molar free energy. The chemical potential of the ions
in solution are related to the concentrations by:
(i) = o(i) + RTlnci
(2.73)
where ci is the concentration of species i in moles per liter of solution (molarity, denoted by M).
The standard state for species i is the ion in a 1 M solution. Nonideality due to interaction of the
ions with the surrounding water molecules is contained in the chemical potential in this standard
state, denoted by o(i). All that is required is that the ion-water interaction be independent of ci,
which is acceptable as long as the solution is not too concentrated in species i.
Eliminating G from Eq (2.72) by means of Eq (2.71) and expressing the chemical
potentials on the right hand side of Eq (2.72) according to the preceding discussion yields the
dependence of the cell EMF on the concentrations in solution:
z/2
RT  c M z  p H 2 
ln 

zF  c z  
H


(2.74)
o = -zFo
(2.75)
1
 o   o (M z )  zg oH 2  g oM  z o (H  )
2
(2.76)
ε ε
o
where
and
The quantity o in Eq (11.26) is the standard electrode potential of the M/Mz+ half cell. It
is termed the “standard” EMF because: i) the concentrations of M+ is 1 molar and ii) the M/Mz+
31
half cell is referenced to the standard hydrogen electrode (acronym SHE), which is an H2/H+
half cell in which c H+ =1 M and p H 2 = 1 atm. The basic database for aqueous electrochemistry
consists of the standard potentials, o. Since most aqueous systems are at room temperature (with
the notable exception of the coolant in light water reactors), a single table at 25oC suffices to
accommodate the entire database. An abridged table of standard electrode potentials is given in
Table 2.2.
Table 2.2 Standard Electrode Potentials at 25oC
Half-cell Reaction
o, Volts
Involving gases
1.
H2O = ½O2 + 2H+ + 2e
2.
2OH- = ½O2 + H2O + 2e
3.
H2 = 2H+ + 2e
Involving metals
4.
Au = Au3+ + 3e
5.
Cu = Cu2+ + 2e
6.
Ni = Ni2+ + 2e
7.
Fe = Fe2+ + 2e
8.
Na = Na+ + e
-1.229
-0.401
0
-1.498
-0.337
0.250
0.440
2.714
Involving only ions
9.
Fe2+ = Fe3+ + e
10. U4+ + 2H2O = UO 2 + 4H+ + 2e
2
11. Pu4+ + 2H2O = PuO 2 + 4H+ + 2e
2
12.
13.
Pu3+ = Pu4+ + e
Cu+ = Cu2+ + e
Involving solid oxides or hydroxides
14. UO2(s) = UO 2+ + 2e
2
15.
16.
17.
Fe(s) + H2O = FeO(s) + 2H+ + 2e
2Cu(s) + 2OH- = Cu2O(s) + H2O + 2e
Cu(s) + 2OH- = Cu(OH)2(s) + 2e
-0.771
-0.338
-1.043
-0.98
-0.16
0.43
0.03
0.36
0.22
Often a standard electrode potential not found in Table 2.2 can be derived from reactions
that appear there. The method involves converting o to o using Eq (2.75), combining the latter
algebraically to determine o of the desired half cell reaction, and finally converting back to o.
Example: Find o for the half-cell reaction Fe = Fe+3 + 3e
This half-cell reaction is the sum of reactions 7 and 9 in Table 2.2. The standard free energy
changes for these reactions and for the desired one are:
32
Δμ o (Fe / Fe 2 )  2Fε o7
Δμ o (Fe 2 / Fe3 )  Fε 9o
Δμ o (Fe / Fe3 )  3Fε o
Summing the first two free energy changes gives the third:
Δμ o (Fe / Fe 2+ ) + Δμ o (Fe 2+ / Fe 3+ ) = Δμ o (Fe / Fe 3+ )
Or, in terms of standard electrode potentials:
2
1
2
1
 o   o7   9o  (0.44)  (0.77)  0.036 V
3
3
3
3
An important extension of the information in Table 2.2 is to the electrode potential of half
cells wherein the ion concentrations are different from the standard value of 1 M. The reference
is still the standard hydrogen electrode. This effect can be deduced by generalizing Eq (2.74). To
convert this equation to the M/Mz+ potential relative to the SHE, c H is set equal to 1 M and
p H 2 to 1 atm. What remains is c Mz  in the argument of the logarithm. This represents the
oxidized part of the M/Mz+ couple; no concentration appears for the reduced component M
because this represents a pure metal. In general, the eff ect of nonstandard concentrations in a
half cell on the potential relative to the SHE is given by:
ε ε
o

RT  product of concentrations on the right hand side of the half cell reaction 
ln 

zF  product of concentrations on the left hand side of the half cell reaction 
(2.77)
As written in Table 2.2, the right hand sides of the half-cell reactions contain the oxidized
portion of the couple and the left hand sides contain the reduced part. Equation (2.77) is known
as the Nernst equation. At 25oC, RT/F = 0.059 V.
Example: Calculate the EMF (relative to the SHE ) of half-cell 10 in Table 2.2 for the following
concentrations: c H  =0.01 M, c UO2  = 1.0 M, c U4 = 0.1 M.
2
The specific form of Eq (2.77) for this half cell is:
 c 4  c 2
0.059
 H UO2
 
log 
2
 c U 4

o

 0.014  1.0 
0.059


  0.60 V


0
.
338

log



2
0
.
1




The Nernst equation, expressed in its general form by Eq (2.77), has at least three very
important applications: i) it gives the voltage (or EMF) of an an electrochemical cell consisting
of two arbitrary half cells; ii) it provides a means of converting the data in Table 2.2 to the
equilibrium constant of ionic reactions in a single solution; iii) it offers a systematic method for
assessing the tendency for metals to corrode in water.
The last of these applications is deferred until Chap. XX . The first two are closely related
and can be explained using the illustrative reaction wherein tetravalent plutonium is reduced to
the trivalent state by addition of ferrous ion:
33
Fe2+ + Pu4+ = Fe3+ + Pu3+
(2.78)
This reaction can be considered either as an overall reaction for an electrochemical cell
consisting of electrodes with the half-cell reactions:
Fe2+ = Fe3+ + e
Pu3+ = Pu4+ + e
(2.79)
or as an equilibrium reaction with all four ions in the same solution. These two interpretations
are shown schematically in Fig. 2.10.
Fe2+, Fe3+

Pu3+, Pu4+
Fe2+, Fe3+
Pu3+, Pu4+
Pu
Fe
SHE
ii) single solution
i) electrochemical cell
Fig. 2.10 Two ways of interpreting the plutonium-iron reaction in solution.
As components of an electrochemical cell, the EMFs of the Fe and Pu half-cells related to
the SHE are, using Eq (2.77):
ε
Pu
ε
o
Pu
 c 4
 0.059 log  P u
 c 3
 Pu




o
Fe
 c 3
 0.059 log  Fe
 c 2
 Fe




(2.80)
ε
Fe
ε
When the Fe and Pu electrodes are joined in an electrochemical cell, which is effectively a
battery, Fig. 2.10 indicates that the EMF of this cell is:
ε ε
Fe
ε
Pu
 (ε
o
Fe
ε
o
Pu )
c
3 c
3

 0.059 log  Pu Fe
 c Pu4  c Fe2 






(2.81)
Example: calculate the EMF of a plutonium-iron battery in which the Pu half cell contains equal
concentrations of trivalent and tetravalent plutonium and the iron half cell consists of a 5:1 concentration
ratio of Fe2+ to Fe3+.
The electrode reactions are numbers 9 and 12 in Table 2.2. From Eq (2.80), the Nernst potentials are:
Pu = -0.98 V and Fe = -0.73 V. The overall cell potential is  = -0.73 – (-0.98) = 0.25 V.
34
The electrochemistry of ions contained in a single solution (right hand diagram in Fig.
2.10) is equivalent to short-circuiting the connection between the Fe and Pu half cells in the left
hand panel. Instead of specified ion concentrations in each half cell that determines , setting
 = 0 fixes the ratio of ion concentrations in the two half cells, or equivalently, in the single
solution. With  = 0 in Eq (2.81), the law of mass action for reaction (2.78) is:
K
c Fe3 c Pu3
c Fe2  c Pu4 
 10 ε Fe ε Pu / 0.059
o
o
(2.82)
The important feature of the standard electrode potentials in Table 2.2 is their use in calculating
equilibrium constants for ionic reactions in aqueous solutions. Using the ε oFe and ε oP u values for
half-cell reactions 9 and 12 in this table in Eq (2.82) gives K =3600. Determination of the
concentrations of all four ions requires additional information. For example specifying that a 1 M
Fe2+ solution is mixed with a 0.5 M Pu4+ solution is sufficient, along with the first equality in Eq
(2.82) and a table similar to Table 2.1, to determine all four ion concentrations at equilibrium in
the combined solution.