1

Chapter 2. Thermodynamics 2.1 Applications of Thermodynamics to Light-Water Reactor Materials ................................ 2 2.2 Basic Thermodynamic Properties and Laws....................................................................... 2 2.3 Criterion of Chemical and Phase Equilibrium.................................................................... 7 2.4 Single-component phase equilibria..................................................................................... 7 2.5 The Phase Rule ................................................................................................................. 11 2.6 Solution Thermodynamics ................................................................................................ 12 2.7 Binary Phase Equilibria .................................................................................................... 17 2.8 Chemical Equilibrium....................................................................................................... 24 2.9 Aqueous Electrochemistry and Ionic Reactions ............................................................... 28

2

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:

3

H = U + pV = enthalpy F = U - TS = Helmholz free energy (2.1) G = H - TS = Gibbs free energy The particular combination of the properties that these three function represent are natural

variables describing 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)

CuT

ChTV

VP

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 α and the coefficient of compressibility β involve the fractional changes in volume as temperature or pressure is increased:

α∂∂

β∂∂

=⎛⎝⎜

⎞⎠⎟ = −

⎛⎝⎜

⎞⎠⎟

1 1v

vT

vpp T

v (2.3)

Because the specific volume (or density) of a substance depends on both temperature and pressure, α and β 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 α and β are positive numbers, which accounts for the negative sign in the definition of β.

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).

4

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:

)TT(CTd)T(Cu)T(u oV

T

T Voo

−≅′′=− ∫

(2.4)

)TT(CTd)T(Ch)T(h oP

T

T Poo

−≅′′=− ∫

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:

ΔhM = 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:

Δhvap = 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, ΔhM and Δhvap 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 ΔX, which represents displacement of the system boundary. This action involves a quantity of work given by W = FΔX. If we multiply ΔX by A, the area over which the force acts, and divide F by A, the work equation becomes F = (F/A)(AΔX). Since F/A defines pressure p, and since the product AΔX is the volume change ΔV, the work involved can also be written as W = pΔV. This form of mechanical work done on (or by) the system is called “pV” work.

5

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:

ΔU = Q - W (2.7) where ΔU = 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:

ΔU + ΔUsurr = 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:

∫ ⎟⎠⎞

⎜⎝⎛≥

TQδSΔ (2.9)

6

where ΔS 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:

ΔsM =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:

Δsvap = sg - sL (2.11)

Both ΔsM and Δsvap 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:

ΔS+ΔSsurr ≥ 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 = δq - δw. For reversible changes, the second law (Eq (2.9)) is δq = Tds and if only expansion/contraction work is permitted, δw = 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.

7

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 = -δWext (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 (non-expansion) 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

8

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 = to the actual vapor at p = psat, the saturation pressure at temperature T, yields o

gg

gg = + RTlnpsat. The superscript o denotes the reference or standard state at 1 atm pressure. Equating gL and gg yields:

ogg

( ) ( )o

vapovap

ovapL

ogsat sΔThΔgΔggplnRT −−=−=−−= (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:

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛=

RThΔ

expRsΔ

expp vapovap

sat (2.19a)

A formula of the same mathematical form but with different coefficients ( instead of

and Δhsub in place of Δhvap); this is the sublimation pressure curve of the substance.

osubsΔ

ovapsΔ

⎟⎠⎞

⎜⎝⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

RThΔexp

RsΔexpp sub

osub

sat (2.19b)

The heats and entropies of sublimation and vaporization are related to the corresponding quantities for melting by:

Δhsub = Δhvap + ΔhM (2.20) Movap

osub sΔsΔsΔ +=

The superscript o is omitted from the entropy of melting because neither the entropies of the solid or liquid are pressure-sensitive.

9

103/T

0.4 0.8 1.2 1.6 2.0 2.4

log

p sat

-12

-10

-8

-6

-4

-2

0

CdZn

Ca

Pb

Mn

Sn

CrTi

PtW

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 -Δhsub/R for the solid-gas transition and -Δhvap/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 Δhsub by Δhvap. 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

10

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 = ΔgM = ΔhM - TMΔsM = 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(ΔgM), must be zero. Obtaining the difference in the liquid and solid free energy changes from the fundamental differential of Eq (2.15) yields:

d(ΔgM) = - ΔsMdTM + ΔvMdp = 0 (2.21b)

Eliminating ΔsM between Eqs (2.21a) and (2.21b) yields

M

MMM

hΔvΔT

dpdT

= (2.22)

here ΔhM is the enthalpy of fusion and ΔvM = vL - vs is the volume change on melting. For most materials, ΔvM 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.

triple point

critical pointSOLIDLIQUID

VAPOR/GASsublimation

vaporization

mel

ting

pres

sure

temperature

11

Fig. 2.2 Generic phase diagram of a pure substance A related aspect of the melting process is the variation of ΔgM 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 (two-component) phase diagrams requires ΔgM 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:

M

MM

p

M

ThΔsΔ

TgΔ

−=−=⎟⎠⎞

⎜⎝⎛

∂∂

where Eq (2.21a) has been used to eliminate ΔsM. Assuming ΔhM to be independent of temperature, this equation can be integrated to give:

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

MMM T

T1hΔ)T(gΔ (2.23)

If T > TM, ΔgM 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, two-component (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:

∑== iii nnwheren/nx (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:

BBAABBAA qxqxn/QqorqnqnQ +==+= (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:

Δsmix = -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:

BBAABBAA gxgxgorgngnG +=+= (2.28) where Ag and Bg 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. Ag 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), Ag depends on the solution composition. The same characteristics apply to Bg . The partial molar Gibbs free energy of a component in solution is called the chemical potential of the component, or μi = ig . With this replacement, Eq (2.28) becomes:

BBAABBAA μxμxgorμnμnG +=+= (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:

BBAABn,p,TB

An,p,TA

dnμdnμdGordnnGdn

nGdG

AB

+=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

= (2.30)

The partial derivatives in the first form of dG define the chemical potentials in the second form:

AB n,p,TBB

n,p,TAA n

GμnGμ ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

= (2.31)

This equation shows that μA 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 μA and μB 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 = μAdnA + nAdμA + μBdnB + nBdμB. Eliminating dG using Eq (2.30) and dividing by n to convert mole numbers to mole fractions yields:

0μdxμdx BBAA =+ (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 – TΔsmix = xAgA + xBgB + hex – Tsex – TΔsmix (2.33) Because gA and gB refer to the pure components, these terms and TΔsmix 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 room-temperature 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 = ΩxAxB (2.35)

where Ω 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 Ω 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:

μi = gi + Rtlnai (2.36) The activity tends to unity for pure i, where μi = 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:

γi = 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:

xAdlnγA + xBdlnγB = 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: 2AB

2BA xΩγlnRTandxΩγlnRT == (2.39)

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

BBB

AAA x

nn

pp

xn

np

p==== (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 μ 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 , 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:

gio

dgdp

RTp

i =

Integrating this equation from pi to 1 atm and identifying gi at pi with the chemical potential yields:

μi = + RTlnpi. (2.44) gio

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:

μ AIdnAI + μ BIdnBI + μ AIIdnAII + μ BIIdnBII = 0

where nAI …..nBII are the numbers of moles of each constituent in each phase and μAI….μBII 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:

μ AI = μ AII (2.45)

* condensed phases are either liquids or solids

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 μAg = μAL. Using Eq (2.44) for μAg and the combination of Eqs (2.36) and (2.37) for μAL, the equilibrium condition becomes:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −−=

RTgg

expxγ

p LAoAg

AA

A

Equation (2.18a) shows that the right hand side of the above equation is the saturation pressure of pure liquid A, so that:

A,satAAA pxγp = (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, γA, is a function of composition if the solution is nonideal and not infinitely dilute in A.

If the solution is ideal, γA = 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 γA = 1, then γB = 1 as well. In nonideal solutions, γA → 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., γA > 1 or γA < 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:

αβ

α

eeexBL −

−=

1 and ⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

= αβ

αβ

eee1exBS (2.50)

with

RThΔ

TT1β

RThΔ

TT1α MB

MB

MA

MA⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−= (2.51)

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

2000

1800

1600

1400

12000 0.2 0.4 0.8 1

Mole fraction Zr (component B ), xB

P

0.6

2200

2000

1800

1600

1400

12000 0.2 0.4 0.8 1

Mole fraction Zr (component B ), xB

0.6

liquid (L)

solid solution (S)

S + L

AB

CD

E

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 = 50.031.049.04.049.0

xxxx

DCDP

BLBS

BBS =−−

=−−

= (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:

)xγln(RTgg ALALALAS += (2.53)

Solving for xAL, the A-rich portion of the phase diagram is expressed by:

ALα

AL γ/ex −= (2.54)

where α is the temperature-dependent function given by Eq (2.51). To complete this portion of the phase diagram, γAL 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:

BLβ

BL γ/ex −= (2.55)

with β given by Eq (2.51). Again, γBL 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

e

a

c

m

o

tem

pera

ture

mole fraction

αβ

β

β

α

α +

+ LL +

L

a

b n

Fig. 2.6 Generic eutectic phase diagram

23

α and β, have been added to the diagram. The α 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 α phase can sustain. Additional B added to the system causes precipitation of a new phase, which is the solid β 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 α, γ, and δ 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 (α, β, and γ) and the limited solubility of Fe in these phases. The α-U, β-U, and γ-U phases indicated on the extreme right of the diagram are analogs of the β 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:

0dndndndn DDCCBBAA =μ+μ+μ+μ

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:

DCBA μdμcμbμ +=+ (2.57) 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:

⎟⎟⎠

⎞⎜⎜⎝

⎛−==

RTGΔexp

pppp

Ko

bBA

dD

cC

p (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)

bBA

dD

cC

xxxx

K = (2.60)

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. ΔGo in Eq (2.58) is the standard free energy change of the reaction:

oB

oA

oD

oC

o bggdgcgGΔ −−+= (2.61)

The are the molar free energies of the pure species at the specified temperature. The superscript o indicates that is to be evaluated at 1 atm pressure. The standard free energy change can be broken up into its enthalpy and entropy components:

oig

oig

ΔGo = ΔHo –TΔSo (2.62) The basic data for evaluating equilibrium constants from ΔGo is either the molar free energies tabulated as functions of temperature or the standard enthalpy and entropy changes, ΔHo and ΔSo, 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. ΔGo then becomes:

oig

ΔGo = (2.63) f

BfA

fD

fC GΔbGΔGΔdGΔc −−+

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 ξ 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 ξ 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 A B C D Total Initial 0

An 0Bn 0

Cn 0Dn 0

Tn

26

Equilibrium 0An - ξ ξbno

B − ξbn0B − ξcn o

C + ξdnoD + ξmno

T + oTn is the sum of the initial moles of the four species and m is the combination of the

sto ystem are ichiometric coefficients shown in Eq (2.59). The mole fractions in the equilibrium sobtained 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:

( ) ( )( )( ) ( )mo

Tbo

BoA

doD

coC

ξmnξbnξn

ξdnξcnK

+−−

++= (2.64)

This equation is solved for ξ (in general numerical solution is required) and from this result, the

xample:

mole fractions at equilibrium are calculated. E Combustion of 1 mole of methane by 2 moles of oxygen at 2000 K. The reaction is:

s hat the

are available. First, a table similar to Table 2.1 can be constructed specifically for this

7.4x1020

CH4(g) + 2O2(g) = CO2(g) + 2H2O(g). The letter g in parentheses following each species indicate tspecies 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 ΔGo = -191 kcal/mole, so that the equilibrium constant is, according to Eq (2.58): KP = K = 7.5x1020. . Two approaches 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 0n;0n;2n;1n o

DoC

oB

0A ==== . Substituting these values into Eq

(2.64) yields:

3

ξ1ξ

⎟⎠⎞

⎜⎝⎛−

=

this case, analytical solution is possible, yielding 1 - ξ = 1.1x10-7 . The corresponding mole fractions

The class of heterogeneous reactions in which an element reacts with a diatomic gas to form

d om

M(s) + O (g) = MO2(s) (2.65)

he letter s in parentheses indicates a solid phase, in this case two solid phases because tically

4).

Inat equilibrium are: xA = 4x10-8; xB = 8x10-8; xC = 0.333; xD = 0.667. The reaction goes essentially to completion.

a compound is both of practical importance and amenable to simple thermodynamic analysis. Reactions in this category include oxidation, nitriding, and hydriding of metals anhalogenation of the electronic material silicon. The simplicity of the thermodynamics stems frthe immiscibility of the reactants and products. Consider oxidation of a metal to form a dioxide of the metal:

2

Tthe metal and its oxide are essentially insoluble in each other. Since M and MO2 are pracpure 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 (2.4With b = c =1 and d = 0, the general criterion, Eq (2.57), becomes:

27

oMOO

oO

oM 222

gplnRTgg =++ (2.66) Rearranging this equation into a more convenient form gives:

RT/HR/SRT/GO

ooo

2eeep ΔΔ−Δ == (2.67a)

or ooo

O STHGplnRT2

Δ−Δ=Δ= (2.67b) where Δ (2.68) o

MoO

oMO

o gggG22−−=

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

2- T combinations where both the metal and its oxide coexist. The oxide-

metal 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 ).

Op

M stable

lnp O

2

RTl

npO

2

1/T

MO2 stable

slope = ΔHo/R

T

MO2 stable

M stable

slope = -ΔSo

intercept = ΔHo

(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:

( ) P2

COCOgasO K/)p/p(p22

=

The nickel/nickel oxide equilibrium reaction is: 2Ni + O2 = 2NiO, for which ΔGo

= -46 kJ/mole at 2000 K. According to Eq (2.67a), for coexisting Ni and NiO in the solid phase, the oxygen pressure is:

( ) ( ) atm103.6RT/GΔexpp 2osolidO2

−×==

The mixed solid and the mixed gas are in equilibrium when (2)gas = ( )solid. From the above

equations, this condition yields the required ratio of CO2 to CO in the gas: Op

2Op

28

( ) ( )( ) 166103.6104.4pKp/p 25solidOPCOCO 22

=××== −

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 , SO and PO . 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.

−3

−24

−34

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.

electrode,metal M

inert metalelectrode

H2

H+Mz+

bridge

ε

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, ε is a direct measure of ΔG 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

ΔG = -zFε (2.71)

ΔG 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, ΔG) reflects the imbalance of the chemical potentials in the two half cells:

)H(z)M()H(2z)M(G 2

z ++ μ−μ−μ+μ=Δ (2.72)

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:

μ(i) = μo(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 μo(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 ΔG 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:

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

+

+

zH

2/zHMo

c

pcln

zFRT-εε 2z

(2.74)

where Δμo = -zFεo (2.75)

and

)H(zgzg21)M( oo

MoH

zoo2

++ μ−−+μ=μΔ (2.76)

The quantity εo 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 =1 M and = 1 atm. The basic database for aqueous electrochemistry consists of the standard potentials, εo. 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.

+Hc2Hp

Table 2.2 Standard Electrode Potentials at 25oC

Half-cell Reaction εo, Volts Involving gases

1. H2O = ½O2 + 2H+ + 2e -1.229 2. 2OH- = ½O2 + H2O + 2e -0.401 3. H2 = 2H+ + 2e 0

Involving metals

4. Au = Au3+ + 3e -1.498 5. Cu = Cu2+ + 2e -0.337 6. Ni = Ni2+ + 2e 0.250 7. Fe = Fe2+ + 2e 0.440 8. Na = Na+ + e 2.714

Involving only ions 9. Fe2+ = Fe3+ + e -0.771 10. U4+ + 2H2O = + 4H+ + 2e +2

2UO -0.338

11. Pu4+ + 2H2O = + 4H+ + 2e +22PuO -1.043

12. Pu3+ = Pu4+ + e -0.98 13. Cu+ = Cu2+ + e -0.16

Involving solid oxides or hydroxides 14. UO2(s) = + 2e +2

2UO 0.43

15. Fe(s) + H2O = FeO(s) + 2H+ + 2e 0.03 16. 2Cu(s) + 2OH- = Cu2O(s) + H2O + 2e 0.36 17. Cu(s) + 2OH- = Cu(OH)2(s) + 2e 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 εo to Δμo using Eq (2.75), combining the latter algebraically to determine Δμo of the desired half cell reaction, and finally converting back to εo. Example: Find εo 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

o3oo9

32oo7

2o εF3)Fe/Fe(ΔμεF)Fe/Fe(ΔμεF2)Fe/Fe(Δμ −=−=−= ++++

Summing the first two free energy changes gives the third:

)Fe/Fe(Δμ=)Fe/Fe(Δμ+)Fe/Fe(Δμ +3o+3+2o+2o

Or, in terms of standard electrode potentials:

V036.0)77.0(31)44.0(

32

31

32 o

9o7

o =−+=ε+ε=ε

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, is set equal to 1 M and

to 1 atm. What remains is 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:

+Hc

2Hp +zMc

⎥⎦

⎤⎢⎣

⎡−=

reactioncellhalftheofsidehandlefttheonionsconcentratofproductreactioncellhalftheofsidehandrighttheonionsconcentratofproductln

zFRTεε o (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: =0.01 M, = 1.0 M, = 0.1 M. +Hc +2

2UOc +4Uc

The specific form of Eq (2.77) for this half cell is:

V60.01.0

0.101.0log2059.0338.0

c

cclog

2059.0 4

U

UO4Ho

4

22 −=⎟

⎟⎠

⎞⎜⎜⎝

⎛ ×−−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−ε=ε

+

++

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+

SHE

ε

εPuεFe

i) electrochemical cell

Fe2+, Fe3+

Pu3+, Pu4+

ii) single solution

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):

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

+

+

3

4

Pu

PuoPuPu c

clog059.0εε

(2.80)

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

+

+

2

3

Fe

FeoFeFe c

clog059.0εε

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:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−−=−=

++

++

24

33

FePu

FePuoPu

oFePuFe

cc

cclog059.0)εε(εεε (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: εPu = -0.98 V and εFe = -0.73 V. The overall cell potential is ε = -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 ε, setting ε = 0 fixes the ratio of ion concentrations in the two half cells, or equivalently, in the single solution. With ε = 0 in Eq (2.81), the law of mass action for reaction (2.78) is:

( ) 059.0/εε

PuFe

PuFe oPu

oFe

42

3310

cc

ccK −==

++

++ (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 and 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.

oFeε o

Puε