Equilibrium constant defect reaction

The assumption that the defects are noninteracting allows the law of mass action in its simplest form, with concentrations instead of activities, to be used for this purpose. In this case, the equilibrium constant K for this reaction is... 

To construct such a diagram, a set of defect reaction equations is formulated and expressions for the equilibrium constants of each are obtained. The assumption that the defects are noninteracting allows the law of mass action in its simplest form, with concentrations instead of activities, to be used for this purpose. To simplify matters, only one defect reaction is considered to be dominant in any particular composition region, this being chosen from knowledge of the chemical attributes of the system under consideration. The simplified equilibrium expressions are then used to construct plots of the logarithm of defect concentration against an experimental variable such as the log (partial pressure) of the components. The procedure is best illustrated by an example. 

The Gibbs energy of this defect reaction defines the degree of disorder through the equilibrium constant. 

The equilibrium concentration of intrinsic defects in a structure depends on temperature. For the Schottky defect, the equilibrium constant K for the defect-generation reaction is... 

Defect Reaction Equilibrium Constants. Recall that a Frenkel disorder is a self interstitial-vacancy pair. In terms of defect concentrations, there should be equal concentrations of vacancies and interstitials. Frenkel defects can occur with metal... 

As with all reactions, defect reactions are subject to the law of mass action [see Eq. (3.4) for more details), so an equilibrium constant, Kp, can be written ... 

The importance of interactions amongst point defects, at even fairly low defect concentrations, was recognized several years ago. Although one has to take into account the actual defect structure and modifications of short-range order to be able to describe the properties of solids fully, it has been found useful to represent all the processes involved in the intrinsic defect equilibria in a crystal (with a low concentration of defects), as well as its equilibrium with its external environment, by a set of coupled quasichemical reactions. These equilibrium reactions are then handled by the law of mass action. The free energy and equilibrium constants for each process can be obtained if we know the enthalpies and entropies of the reactions from theory or... 

The original assumption was that Schottky defect formation was preferred to the formation of electronic defects, and this is explicitly stated at the top of the figure. As Frenkel defect formation has been ignored altogether it is also possible to write Ks > IQ > Kp, where Kp would represent the equilibrium constant for a reaction that formed Frenkel defects in MX. Figure 3 contains all the assumptions made, as well as the way in which defect concentrations vary. Moreover, it is easy to see what changes will take place if the relative values of Ks and IQ are altered, and the graphical representation is able to present trends in a lucid fashion. 

We begin by noting that the bulk stoichiometry of an oxide is a function of the temperature and composition of the surrounding atmosphere [46]. For a hypothetical metal oxide (MO), we can write the appropriate defect chemical reactions in Kroger-Vink notation and equilibrium constants for congruent evaporation (k o). the production of vacancies by evaporation (k and ko), and Shottkey defect formation (kg) ... 

It is worth noting here that the exact solution of a set of nonlinear equations for more complicated equilibria is often unachievable. In such cases, the approximation method implying a simplification of the overall electroneutrality condition using the only pair of predominant defects can be useful. This approach can be illustrated on the basis of the above example of a Si crystal. As the equilibrium constants (Equations (3.15-3.17)) are functions of temperature, the concentrations of different defects can alter in different ways, depending on the value of the pre-exponential factor K° and the enthalpy of the defects reaction, AH . As a result, it is possible to choose a temperature range where the overall electroneutrality condition (Equation (3.18)) can be approximated by pairing the predominant defects. In this case, two possible approximations can be suggested ... 

Consider this factor carefully by again examining Table 2-2. Also given is the reaction producing the defect, with its corresponding equilibrium constant. In most cases, the deviation, 6, is presented in terms of the equilibrium constant and the partial pressure of the external gaseous reactant. 

KNs. The d.c. electrical conduction of KN3 in aqueous-solution-grown crystals and pressed pellets was studied by Maycock and Pai Verneker [127]. The room-temperature conductivity was found to be approximately 10" (ohm cm) in the pure material. Numerical values for the enthalpies of migration and defect formation were calculated from ionic measurements to be 0.79 0.05 and 1.43 0.05 eV (76 and 138 kJ/mole), respectively. In a subsequent paper [128], the results were revised slightly and the fractional number of defects, the cation vacancy mobility, and the equilibrium constant for the association reaction were calculated. The incorporation of divalent barium ions in the lattice was found to enhance the conductivity in the low-temperature region. Assuming the effect of the divalent cation was to increase the number of cation vacancies, the authors concluded that the charge-carrying species is the cation, and the diffusion occurs by means of a vacancy mechanism. 

It is interesting to compare an order of magnitude estimate of the experimental value of the equilibrium constant for this reaction (11) to those obtained from the work of Bengaard et al.(26) at nickel step defect sites and at nickel planes. Experimentally, the onset of carbon formation was found (11) to be at partial pressures of H2, H2S, and... 

The concentration of defects can be derived from statistical thermodynamics point of view, but it is more convenient treat the formation of defects as a chemical reaction, so that equilibrium constant of mass action can be applied. For a general reaction, in which the reactants A and B lead to products C and D, the equation is given by ... 

Fig. 2.7 Diagram summarizing the key elements of the Kroger-Vink notation for point defects in ionic solids. The formation of defects can be described with defect-chemical reactions and corresponding equilibrium constants...

Anti-Frenkel disorder similar to Frenkel disorder except that the interstitials are anions and vacancies are therefore in the anion sublattice. In Zr02 the reaction is 0 kS + 0[ and the anti-Frenkel equilibrium constant is K p = [ko ][On- This type of thermal defect is found in lattices that have a fluorite structure (CaF2, Zr02), which means that there are many large interstitial sites where the anions can be accommodated, but not the cations because their charge is larger, and they are less well screened from each other. 

On doping with chromium, nickel oxide that is initially p-type becomes less p-type conducting. Whether this permits making NiO n-type in this way depends on the equilibrium constants. The electronic conductivity also depends on the capture reactions of the charge carriers by the localized stationary defects. 

Whilst these are indistinguishable thermodynamically, they are distinguishable defect-chemically. Allowing Kp and K to denote the equilibrium constant for these reactions, respectively, they are ... 

K is the equilibrium constant for Frenkel disorder. This can be seen if we set the time derivative in eq. (6-2) equal to zero, since this is the condition for thermodynamic equilibrium. On the basis of the simple theory of homogeneous reactions, two limiting cases arise. 1. The electrostatic interaction between electrically charged defects at a separation 2 is negligible compared... 

At this point let us briefly consider the formation of associates. The formation of associates between cation vacancies and divalent impurities in alkali halides has already been given as an example. Such reactions are homogeneous solid state reactions, and so the relaxation time for the formation of associates can be calculated in a completely analogous manner to the calculation of the relaxation time for the equilibration of Frenkel defects. The result of such calculations is precisely the same as the result given in eq. (6-5). It is only necessary, in the case of association, to replace the concentration c (eq) = in the denominator by the nearly constant concentration of the corresponding majority defect. In general, in the case of the formation of defect associates, we can conclude that the equilibrium concentration is attained rapidly compared to the time required by defect reactions which occur at sites of repeatable growth. 

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