Proton Balance Equations

The method for carrying out any pH calculation centers around the proton balance equation (PBE). This equation permits you to emphasize the chemical aspects of the problem before the introduction of complex mathematical terms. It, together with all of the equilibrium and balance (EjOj =1) equations, satisfies the general requirement that, in order to solve a problem with n unknowns, at least n independent equations are needed. 

The PBE matches the concentrations of species which have released protons with those which have consumed protons. 

Let us first consider the PBE for H2O, not only because it is the simplest, but also since it will be involved in all other PBEs as well. 

The reason the hydroxide ion concentration measures the concentration of protons released is that when water acts as an acid, i.e., a proton releasing species, it produces a hydroxide ion for every proton released. When water acts as a base, it forms one H30 for every proton consumed. The hydronium ion concentration, abbreviated as [H ], is a measure of the proton consumption of water. 

To obtain the PBE for any aqueous solution, the PBE for water is used as a starting point since water is always present. In a solution of a monoprotic acid, HX (strong or weak), the dissociation of HX releases one proton which is measured by the X formed in the same process. Thus, the PBE for HX is  

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Another constraint equation often used in equilibrium problems is the proton balance equation (PBE) (Pankow, 1991). It provides a means of keeping account of protons in the system. A PBE can be formulated by writing an MBE in which the concentration of each species in the EPM table is multiplied by the stoichiometric coefficient of H in the EPM table. For example, the PBE of a diprotic acid H2L where the components that define the species are H2L and H+ would be... 

Let us next consider the case of strong electrolytes MX (where M is a cation other than H and X is an anion other than OH, since these cases have already been considered above). Considering the neutral solution of MX as a starting point, the dissociation into M and X neither consumes nor releases protons. One or both of these ions, however, may subsequently consume or release protons. Thus, a solution of a salt such as NaCl will have the same proton balance equation as that... 

Graphical-Algebraic Solution of the Proton Balance Equation... 

In acid-base titration, the appropriate concentration variable is [H" ] or, most commonly, pH. We have already developed all the necessary equations earlier (Chapter 4). The only difference is that now, instead of writing a proton balance equation (PBE) for a single set of conditions, these apply to a whole family of points, i.e., those involved in the entire titration. 

Two principal features characterize the systematic approach to equilibrium calculations used in this book a) Expressing concentrations of every species by the product of a (a fraction of that species of all others in the same system. These fractions are a function of only the critical variable (e.g., pH), the relevant equilibrium constants, and C, the total concentration of the component, and b) Describing the equilibrium condition by a single balance equation, e.g., the proton balance equation (PBE), the ligand balance equation, etc. This results ultimately in a description of the equilibrium condition of the solution by one equation with a single concentration variable, i.e., in an implicit solution. 

The various species with acid-base properties can be related to each other by the equations expressing their equilibria and by equations expressing conservation of matter material, charge, and proton balancing equations. When initial (analytical) concentrations of solutes are given, these equations serve to determine as many unknowns as we have equations. Molarities, not activities, must be used in these balances, so we require the conditional (molarity) equilibrium constants appropriate to the solution under consideration. 

Then the material balance equations can be simplified to allow the reaction quotients Qa and Qa2 to be approximated from the experimentally observed protonic surface charge <7q ... 

Figure 5. Titration of a suspension of Ti02 in KNOj at three different concentrations. CTq is the protonic surface charge, calculated from the acid-base mole balance equations, specific surface area of 20 nr g" and the solids...

Although convenient to use in equations, the symbol H+ (aq) does not really represent the structure of the ion present in aqueous solution. As a bare hydrogen nucleus (proton) with no electron nearby, H + is much too reactive to exist by itself. Rather, the H+ attaches to a water molecule, giving the more stable hydronium ion, H30 +. We ll sometimes write H+(aq) for convenience, particularly when balancing equations, but will more often write H30+(fl(/) to represent an aqueous acid solution. Hydrogen chloride, for instance, gives C (aq) and H30+(d /) when it dissolves in water. 

To predict the direction of reaction, use the balanced equation to identify the proton donors (acids) and proton acceptors (bases), and then use Table 15.1 to identify the stronger acid and the stronger base. When equal concentrations of reactants and products are present, proton transfer always occurs from the stronger acid to the stronger base. 

Step 2. Write balanced equations for all possible proton-transfer reactions. 

If additional salt is added to yield a concentratiCtii, the free acid precipitates, and the pH of the solution increases owing to protons being removed from the solution. By utilizing the mass balance and the charge balance equations for the solutions, they derived an expression for the proton activity ... 

Proton balance and electrical neutrality. For bulk solutions in their natural condition the overall charge of all the soluble chemical species is zero, therefore, this constraint can be imposed if it is not possible to use an MBE. The example in the section on carbonate equilibria (Section 5.2.6.4) provides an example of the use of an electrical neutrality equation (ENE) to calculate pEL... 

A suitable oxidant is cation 9.14a,b, derived from a,(3-unsaturated ketone 9.13 by protonation under strongly acidic conditions in the absence of water. Quenching of this cation with a hydride ion (from the C4 position of 9.11) produces the saturated ketone 9.15. The balanced equation is shown below. 

The fact that protons are used as a motive force can contribute to an increase in the pH of the soil solution, due to the uptake by bacteria. Moreover, the pH can also be increased due to the fact that reactions involve the transformation of oxalic acid into carbonic acid, i.e. of a strong (pKi = 1.25, pK2 = 4.27) to a weak (pKi = 6.35, pK2= 10.33) acid (Braissant et al., 2002). This alkalinization facilitates precipitation of calcium carbonate (CaCOs). By modifying a general equation for oxalate metabolism by oxalatrophic bacteria (Harder et al., 1974), the following balanced equation can be proposed ... 

When a hydrogen ion is added to a species, the process is called protonation. Write a balanced equation for the protonation of a general alcohol and a general ether. 

Next, substitute into the charge-balance equation to place it in the proton condition, its proper form for plotting. The equation thus becomes... 

Next, consider the buffer capacity of carbonic acid species at constant COj pressure. In the previous calculation /3 was obtained by taking the derivative of an equation for in the proton condition that had been obtained from the solution charge-balance equation. We will instead base the calculation of ji directly on the equation for Cg, differentiating in parts. Thus, we write... 

The equation of a titration curve of an acid or base may be derived, taking into account the charge-balance equation for the solution, the total acidity (C ) or alkalinity (Q) equation, and deriving an equation in terms of stability constants, known concentrations and volumes, and the pH. This resultant equation, which can be solved, is said to be in the proton condition. Explain this statement as it applies to a simple titration curve. 

In the equilibrium constant, the superscript asterisk indicates that the dissolution reaction is written in terms of protons. The subscript following K (zero in this case) is the number of hydroxyl ions associated with AF+ in the reaction. According to Nordstrom et al. (1990), Kq = 10. Reactions similar to (7.35) can be written in which kaolinite is dissolved to form each of the aluminum hydroxy complexes listed in mass-balance equation (7.22) for total aluminum. These reactions can be generated by successively adding the cumulative reactions for the Al-hydroxy complexes—Eqs. (7.23) to (7.26)—to Eq. (7.35). For example, adding Eqs. (7.35) and (7.23) and their log values for 25°C, we obtain ... 

E23.1 (a) Balanced equation. All lanthanoids are very electropositive and reduce water (or protons) to H2 while being... 

It is characteristic of nuclear reactions of the type occurring in nuclear reactors that the sum of the number of neutrons and protons in the reactants equals the sum in the products. The same is true of the charge of the reactants and products. Consequently, in balancing nuclear reactions, the sum of the s of the reactants must equal the sum of the. 4 s of the products and the sum of the Z s of the reactants must equal the sum of the Z s of the products. As an example of a balanced equation for a nuclear reaction, we may consider one of the fission reactions that occurs when absorbs a neutron ... 

Balancing Redox Reactions in Acidic Solution When a redox reaction occurs in acidic solution, H2O molecules and ions are available for balancing. Even though we ve usually used H30 to indicate the proton in water, we use H" in this chapter because it makes the balanced equations less complex. 

In the calculation of protolytic equilibria, the ionic product of water, equations for dissociation constants of acids and bases, equations of analytical concentrations and equations of electroneutrality or proton balance are taken for the starting point. Due to the difficulty of the numerical calculation of pH these systems are generally solved by graphical methods. 

Mathematical equations for the titration curves shown in Figure 5a are given in Appendix I. These equations can be derived from charge-balance (or proton balance) conditions. Equations 3 and 5 of Appendix II illustrate that the shift in the titration curve at any pH is related quantitatively to the extent of specific adsorpion (Equations 17 and 18). The latter can be calculated independently of adsorption measurements of the shifts (17). 

The alkalimetric titration curve (a) permits the calculation of the surface charge as a function of pH (c) caused by the disturbance of the proton balance and the microscopic acidity constants as a function of the charge (b) intrinsic constants are obtained by extrapolation to zero surface charge (b). With the help of Equation 26, the surface potential either as a function of charge (d) or pH may be calculated. Experimental data are for y-AhOs in NaClOi solutions (17). 

The proton condition is a special type of mass balance equation on protons. It is an essential component of equilibrium problem solving if either or OH are involved in the equilibria. It is used in this section only to solve problems related to solutions made up in the laboratory. In later sections and chapters it is used to solve natural water problems. 

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