Electroneutrality equation

Electroneutrality At all times the crystal must remain electrically neutral. Equations (7.9) and (7.10) define the formation of charged species, so that the appropriate electroneutrality equation is... 

As the pressure diminishes far below that of the stoichiometric point, the number of metal vacancies and holes will continue to fall and the electroneutrality equation chosen, Eq. (7.16), will no longer be representative. A more appropriate form of the electroneutrality equation for the low-pressure region ignores the minority defects, which are now metal vacancies and holes, to give... 

This central, medium pressure, region of the diagram displays the electroneutrality equation approximation [V J = [Vj J at the top. In this region the compound... 

The low-pressure region displays the electroneutrality equation approximation [e ] = 2[Vx ]. Electrons predominate so that the material is an n-type semiconductor in this regime. In addition, the conductivity will increase as the partial pressure of the gaseous X2 component decreases. The number of nonmetal vacancies will increase as the partial pressure of the gaseous X2 component decreases, and the phase will display a metal-rich nonstoichiometry opposite to that in the high-pressure domain. Because there is a high concentration of anion vacancies, easy diffusion of anions is to be expected. 

As in the case of ionic defects, the form of this diagram can easily be modified to smooth out the abmpt changes between the three regions by including intermediate electroneutrality equations. 

In general, there is little problem in extending the concepts just outlined to more complex materials. The procedure is to write down the equations specifying the various equilibria point defect formation, electronic defect formation, the oxidation reaction, and the reduction reaction. These four equations, only three of which are independent, are augmented by the electroneutrality equation. Two examples will be sketched for the oxides Cr2C>3 and Ba2In2Os. 

The approximations to use depend upon the pressure regime and the values of the equilibrium constants. This oxide is an insulator under normal conditions, and so, in the middle region of the diagram, Schottky equilibrium is dominant, that is, > Ke and the electroneutrality equation is approximated by... 

In the oxidized region, holes and cation vacancies are preferred, so the electroneutrality equation is approximated by... 

The approximations inherent in Brouwer diagrams can be bypassed by writing the appropriate electroneutrality equation as a polynomial equation and then solving this numerically using a computer. (This is not always a computationally trivial task.) To illustrate this method, the examples given in Sections 7.5 and 7.6, the MX system, will be rewritten in this form. 

It is necessary to write the electroneutrality equation in terms of just two variables, a defect type and the partial pressure, to obtain a polynomial capable of solution. For example, the equation for the concentration of holes, [h ], is obtained thus Electroneutrality ... 

The addition of either donors or acceptors will, however, upset the charge balance, and these must be included in the electroneutrality equation. Consider donor doping by a trivalent ion D3+ due to reaction with D2X3 to introduce D defects, once again assuming that Frenkel defects are not important. The original electroneutrality Eq. (7.12) ... 

The four Eqs. [(8.3)—(8.6)] are simplified using chemical and physical intuition and appropriate approximations to the electroneutrality Eqs. (8.7) and (8.10). Brouwer diagrams similar to those given in the previous chapter can then be constructed. However, by far the simplest way to describe these equilibria is by way of polynomials. This is because the polynomial appropriate for the doped system is simply the polynomial equation for the undoped system, together with one extra term, to account for the donors or acceptors present. For example, following the procedure described in Section 7.9, and using the electroneutrality equation for donors, Eq. (8.9), the polynomial appropriate to donor doping is ... 

Figure 8.19 Schematic representation of the variation of the defect concentrations in BaPrj- YbjOs-s as a function of dopant concentration, assuming fixed temperature, water, and oxygen pressure. The electroneutrality equation used is [h ] = [YbPr]. [Adapted from data in S. Mimuro, S. Shibako, Y. Oyama, K. Kobayashi, T. Higuchi, S. Shin, and S. Yamaguchi, Solid State Ionics, 178, 641-647 (2007).]...

The five equations to be solved read (the electroneutrality equation is written first)... 

The initial calcium guess is calculated from the electroneutrality equation... 

Excluding activity coefficients, three relationships are required in addition to the nine thermodynamic equilibria in order to calculate concentrations of the 12 unknown species. These relationships are the mass balances for magnesium and chloride, and the electroneutrality equation. 

Substituting from the electroneutrality Equations (87) and (88) transforms Equation (93) into... 

Case 4. The raindrop continues on its journey, now flowing into a limestone (calcite) aquifer. Here it again comes to equilibrium with calcite, but under closed system conditions. The only knowns are the initial composition of the solution and that it must react to equilibrium with calcite. This is a particularly difficult case for an exact solution. Both mass balance and electroneutrality equations must be used. Let us write these out again. 

In many ceramics, intrinsic and extrinsic disorder, as well as the disorder due to nonstoichiometry, have to be considered. Independent of the dopant level, the mass action laws of intrinsic disorder, of the e-h equilibrium and of the reaction with the surrounding phase are valid in thermodynamic equilibrium. Together with the electroneutrality equation... 

For broad partial pressure regimes, however, only two mobile defects, or one mobile defect and the dopant, play a role in the electroneutrality equation (Eq. (11)). In theses cases simple partial pressure dependences, namely power laws with concentrations being proportional top 02)", can be calculated (see e.g. Refs. [57, 80, 89, 90]). One example of such a simplified situation has been examined above (Eq. (10)). 

Considering different pairs of majority defects, all relationships between defect concentrations and partial pressure can be constructed from simplified situations, and this leads to so-called Brouwer diagrams. Figs. 2a and 3a show such Brouwer diagrams for a pure oxide MO with Schottky disorder, and for a Schottky-disordered oxide with a negative dopant. (Please notice that the exact curves calculated from the complete electroneutrality equation (Eq. (11)) exhibit smooth transitions rather than sharp bends.)... 

This same basic approach (equilibrium, material balance, and electroneutrality equations) applies to the calculation of the concentration of OH in a solution of a weak base at equilibrium. The resulting equation is ... 

As illustrated in Section 2.1.2, the titration curve of a weak acid with a strong base can be constructed in terms of pH versus volume of the strong base. Consider a weak acid (e.g., HA) of volume Va its concentration Ca is titrated with a strong base (e.g., NaOH) of volume Vb and concentration Cb. The mass balances and electroneutrality equation can be given by ... 

Substituting [Na+] and [Ba ] from the mass balance equation into the electroneutrality equation yields ... 

It is interesting to find the pH of a solution containing the salt of a weak acid (HA) and a weak base (B) where they have the same concentration. The equilibrium, mass balance, and electroneutrality equations can be given as ... 

The same equilibrium and electroneutrality equations shown above are used with the pKal = 2.23, pKa2 = 7.21, and pKa3 = 12.32. However, mass balance equations are different for each salt ... 

The electroneutrality equation is the same as Equation (2.127c). Rearrangement of Equation (2.127b) for [HPO 2] and Equation (2.130) for [Na+] and substitution into Equation (2.127c) gives ... 

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