The Proton Condition

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. 

The proton mass balance is established with reference to a zero level or a reference level for protons (the proton reference level, PRL). The species having protons in excess of the PRL are equated with the species having less protons than the PRL. The PRL is established as the species with which the solution was prepared. 

Determine the proton condition when HA is added to water. 

Note that HA and HgO, the species at the PRL do not appear in the proton condition. 

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To facilitate understanding, Eq. (v) was derived on the basis of charge balance it can be derived directly on the basis of the proton condition (using H20, and =AIOH as a reference). 

Instead of the proton condition, the electroneutrality equation can be used. Independent of chaise type, a combination of electroneutrality and concentration condition gives the proton condition. Na in NaHX or Na2X is used as a symbol of a nonprotolyzable cation (Li, K, . . . ). 

There are alternative algebraic functions. Instead of writing the electro-neutrality equation, we can derive a relation called the proton condition. If we made our solution from pure H2O and HB, after equilibrium has been reached the number of excess protons must be equal to the numtier of proton deficiencies. Excess of deficiency of protons is counted with respect to a zero level reference condition representing the species that were added, that is, H2O and HB. The number of excess protons is equal to [H ] the number of proton deficiencies must equal [B ] -I-[OH ]. This proton condition gives, as in equation iva, [H" ] = [B ] -f lOH-]. 

We note that equation 40b is identical to the proton condition (iva) of Example 3.3a that is, the choice of HB as a component is equivalent to our earli( r notion of HB and H2O as reference level. In examining equations 39 and 40, we see that there are effectively four unknown species activities (B, HB, OH, and H ), the activity of H2O being constant for the dilute solution. As n Example 3.3a, there are four independent relationships 39a, 39d, 40a, and 40b, the remaining expressions being irrelevant for our conditions. Although water is always a component in our tableaux, we will omit it from subsequent tableaux for simplicity, the concentration of H2O being effectively constant for dilute solutions. 

Approximations in the proton condition similar to those used in the previous examples are not permissible in this case. However, an approximation might be possible in the concentration condition. Because Ac is a weak base, [Ac ] > [HAc] and [Ac ] = C. If we combine this approximation with the proton condition and the two equilibrium constants, we obtain... 

The same graph can be used to compute the equilibrium concentrations of a 10" M solution of NaA, In this case the proton condition is... 

The proton conditions of equations 48 and 49 correspond to the two equivalence points in acid-base titration systems. The half-titration point is usually (not always) given by pH = pAT. Thus the qualitative shape of the titration curve can be sketched readily along these three points (Figure 3,3a). 

It is obvious from the graph that no approximation in the proton condition is possible. In order to find the point where the proton condition is fulfilled, we move slightly to the right of the intersection of log[HAc] with log[OH"] and find by trial and error where the proton condition is fiilfilled, that is, at... 

The reference level is defined by the composition of a pure solution of HA in H2O (/ = 0 [ANC] = 0), which is defined by the proton condition, [H l = [A-] -i- [OH ]. (In this and subsequent equations, the charge type of the eicid is unimportant the equation defining the net proton excess or deficiency can always be derived from a combination of the concentration condition and the condition of electroneutrality.) Thus in a solution containing a mixture of HA and NaA, [ANC] is a conservative capacity parameter. It must be expressed in concentrations (and not activities). Addition of HA (a species defining the reference level) does not change the proton deficiency and thus does not affect [ANC]. 

The base-neutralizing capacity of the phosphoric acid system with reference to the equivalence point, / = 2 (solution of Na2HP04 with the proton condition 2[H3P04] -h [H2P04 ] -h [H ] = [POj ] -h [OH" ]), is given by... 

Figure 4.5. Aqueous carbonate equilibrium constant Pcoz- Water is equilibrated with the atmosphere pco2 10 atm), and the pH is adjusted with strong base or strong acid. Equations 7-9 with the constants (25°C) pATn = 1.5, pAT, = 6.3, pAT2 = 10.25, and pAT (hydration of CO2) = -2.8 have been used. The pure CO2 solution is characterized by the proton condition [H ] = [HCO ] + 2[C03 ] + [OH ] (see point P) and the equilibrium concentrations -log[H J = -log[HCO ] = 5.65 log[C02aq] = -log[H2C03 ] = 5.0 -log[H2C03] = 7.8 and -log[C03 ] = 8.5.

Example 4.2. Pristine Rctinwater Pure water in equilibrium with the atmosphere is characterized by Tableau 4.2. In Figure 4.5 the proton condition (or the charge balance) is given by equation viii of Tableau 4.2. This condition is fulfilled essentially at point P, where [H" ] = [HCO ], where pH = 5.65 (25°C). 

One has to distinguish between the ion concentration (or activity) as an intensity factor and the availability of the ion reservoir as given by the H-acidity or the deficiency of ions, or the alkalinity. Alkalinity and acidity are very important concepts although there are different ways to define these capacity factors, all definitions essentially relate to the proton condition at a given reference level. For the carbonate system, alkalinity [Aik] refers conceptually to the proton condition with reference to H2CO, H2O ... 

Figure 5.5. NH3 species in an open system with Pnh, = constant = 5 x 10 atm (25°C). The proton condition ([NH4 ] 4- [H ] = [OH ], or approximately [NHa ] = [OH ]) corresponds to the condition of a pure ammonia solution (no other acid or base added to the system).

A pure S02 Water system at this p cn characterized by pH == 4.8, where the proton condition is [H ] =s [HS03 ]. At higher pH values, that is, upon addition of a base, the solubility becomes very large. At low pH values the solubility is relatively small. 

As shown in Table 7.6 (Type 1), the composition is defined by the proton condition and two independent variables such as temperature and partial pressure of CO2, Pcoi- When TOTH = 0, this system is constant and independent of concentration (isothermal dilution or evaporation) as long as all three phases remain in equilibrium with each other. 

Our first goal is to replace the terms in Eq. (5.33) with terms that are only functions of [H ]. This is described as putting Eq, (5.33) in the proton condition. The resultant equation can then be solved for the solution pH. We will use equilibrium expres.sions to develop an equation in the proton condition. 

Substituting into Eq. (5.33) now provides us with our equation in the proton condition... 

This last equation is now in the proton condition. This means that we have eliminated all terms other than Cg, H as pH, and known concentrations and stability constants, allowing us to plot the equation as Cg versus pH. Assuming Cl = 0.01 eq/L, we now introduce values of H and solve for Cg. The results are plotted in Fig. 5.5. The figure shows the inflection point and endpoint at pH 7.0. 

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... 

This equation shows the contribution of each species, but is not solved readily unless substitutions are made to put it in the proton condition. 

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. 

Our goal is to put this equation in the proton condition. We will do so dealing with each successive right-hand term in Eq. (6.16). 

Consider the standard form of (4.2-1), i.e., with all non-zero terms moved to the left-hand side of the expression [H+] - [A-] - [OH ] = 0. Use cell J155 to calculate that left-hand side of the standard form, which in this case would read = B155-1155-D155. The proton condition corresponds to the pH that will make 1155 zero. 

To see that for yourself, somewhere else on the spreadsheet use three cells, which we will here call cells a, b and c. In cell a place a reasonable guess value for the [H+] this is the value that will be adjusted to make the proton condition zero. The proton condition in standardform should go in cell , i.e., it should code for[H+] - [A-] - [OH-] = [H+] -CKal ([H+] + Ka) - 10-14 / [H+], The third cell, c, is merely for your convenience, to calculate the pH from [H+]. Note that we here do not need to use any approximations. 

PRL = HgO and MA (or, equivalently. A", since completely dissociates from A and does not affect the proton condition). 

State the proton condition for a solution of phosphoric acid, H3PO4, in pure water. 

State the proton condition for sodium bicarbonate, NaHCOa, added to water. Solution... 

A profoji or hydroxyl mass balance also may be used to arrive at the proton condition. This approach can be applied to the more complicated situation that arises when both MA and HA are added to the same solution. However, these more-complicated situations are more readily solved by obtaining the proton condition from a combination of the charge balance and the mass balance, as is shown in Section 4-4-4. 

The charge balance and mass balance equations can be combined to give the proton condition. For complex solutions this is generally the best way to determine proton conditions. For example, when C moles of MA are added per liter of solution, assuming complete dissociation of MA, we obtain the following results. 

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