Polyprotic Acids and Beyond—An Algebraic Approach

Abstract

For an N-protic acid–base system, the set of nonlinear equations (i.e., mass action and balance laws) provides a simple analytical solution/formula for any integer N ≥ 1. The approach is applicable for the general case of zwitterionic acids H_NA^+Z (e.g., amino acids, NTA, and EDTA), which includes (i) the “ordinary acids” as a special case (Z = 0) and (ii) surface complexation. Examples are presented for N = 1 to 6. The high-N perspective allows the classification of equivalence points (including isoionic and isoelectric points). Principally, there are two main approaches to N-protic acids: one from hydrochemistry and one “outside inorganic hydrochemistry”. They differ in many ways: the choice of the reference state (either H_NA or A^−N), the reaction type (dissociation or association), the type/nature of the acidity constants, and the structure of the formulas. Once the (nonlinear) conversion between the two approaches is established, we obtain a systematics of acidity constants (macroscopic, microscopic, cumulative, and Simms). Finally, from the viewpoint of statistical mechanics (canonical isothermal–isobaric ensemble), buffer capacities, buffer intensities, and higher pH derivatives are actually fluctuations in the form of variance, skewness, and kurtosis.

1. Introduction

1.1. State of the Art

The subject of acid–base equilibria is covered in an extensive bibliography, usually focusing on mono- and diprotic acids, which is the entry point to understanding chemical equilibrium reactions per se. The list of classical textbooks and monographies is long and stems from various subfields, such as hydrochemistry [1,2,3] as well as general and analytical chemistry [4,5,6,7]. This is accompanied with several modeling approaches, e.g., [8,9,10].

For the general case of aquatic systems (as mixtures of any number of acids and bases plus solid and gaseous phases), there are two prototypes of numerical approaches: (i) models that are based on the law of mass action (e.g., PhreeqC [11] and many others) and (ii) models that are based on Gibbs energy minimization (GEM) [12,13].

The mathematical description of polyprotic acids (with N ≥ 3) is a special topic that extends the traditional view on acid–base reactions. Algebraic equations are presented in [14,15,16,17,18,19,20]; their application to titration and buffer capacities can be found in [21,22,23,24] and in [25,26,27,28,29,30]. Many of the developments come from areas outside conventional hydrochemistry (e.g., org/bio/med chemistry), which is the playground of proton-binding macromolecules such as nucleic acids and fulvic/humic acids. This also encourages a statistical description [31,32,33,34].

There are two principal ways of mathematical description: (i) the hydrochemical approach (based on dissociation reactions with reference state H_NA) and (ii) the approach employed in organic and biochemistry (based on association reactions with reference state A^−N). The present review follows the first approach; the interrelation with the second approach is established in Section 2.5. The latter provides an overview of different types of equilibrium constants (macroscopic, cumulative, microscopic) used today in acid–base theory—see Section 2.5.7.

The question whether a polyprotic acid H_NA can be represented as a sum of N monoprotic acids is answered in Section 2.5.5 and Section 2.5.6 by factorizing the partition function of the canonical isothermal–isobaric ensemble. This is equivalent to the “quasiparticle-like” concept of N noninteracting proton-binding sites, known as decoupled sites representation (DSR) [33,34].

Titration and buffer capacities are summarized in an excellent review article by Asuero and Michałowski [26]. This and other papers [24,25,26,27,28,29,30] rely on association reactions (with reference state A^−N), so one has to be careful when comparing the formulas with the present approach. Additionally, most of these papers take the dilution during titration into account by explicitly using the volume of the titrant. This effect is ignored in this review to keep the formulas simple.

1.2. Motivation

Today, computers solve nonlinear systems numerically in the shortest time with high quality, which is a great help in dealing with complex real-world tasks (and we are grateful for that). However, by delegating everything to computers, we sometimes lose the overview of the underlying principles and functional relationships (digital data are too incomplete/imprecise to understand the deeper aspects of reality).

Starting from the laws of mass action and mass/charge balance, a mathematical solution is provided in the form of simple and smooth analytical formulas for acid–base reactions. This is performed for the general case of N-protic acids, where N can be any integer (N ≥ 1). The approach applies to the broad class of zwitterionic acids H_NA^+Z (amino acids, NTA, EDTA, etc.), which embeds all “ordinary acids” as a special subclass characterized by Z = 0.

There are at least five questions:

• Why do we need equations/formulas for N > 3?
• Is the approach mathematically strict/rigorous?
• What is the difference to standard approaches celebrated in textbooks?
• What does it mean to be a simple and smooth analytical formula?
• This report contains more than 100 formulas. What is the central formula?

Answer 1. Usually (and this is the first that comes to mind) N is a small number: 1, 2 or 3 for monoprotic, diprotic and triprotic acids. However, in reality, there are compounds with more protons (e.g., EDTA with N = 6 in Section 4.1.6 or other macromolecules in biochemistry and/or mixtures thereof). Moreover, when treating N as a variable integer, the equations tell us things that might otherwise be non-obvious (e.g., classification of equivalence points in Section 2.3).

Answer 2. The mathematical derivation is strict/rigorous. For this purpose, it is assumed that the activities (that enter the mass action laws) are replaced by molar concentrations, which is justified either for dilute systems or for non-dilute systems using conditional equilibrium constants (c.f. seawater example in Section 3.3.5). Deviations in the analytical model from numerical activity-based calculations are discussed in Section 3.3.6.

Answer 3. In standard textbooks [1,2,3,4,5,6], an algebraic solution of the acid–base problem is usually provided for diprotic acids (N = 2) in implicit form, namely as a polynomial of degree 4 in x = 10^−pH (quartic equation!)—see, e.g., [1] (p. 107) or Equation (46) below. That is the common way to handle the acid–base problem. In the general case of N-protic acids, this procedure leads to polynomials of order N + 2, where, for N > 4, there is principally no algebraic solution (according to the Abel–Ruffini theorem). This dilemma will be avoided in the present approach. However, before we start, let us explain this in another way.

In titration, a titrant (strong base of amount C_B) is added to the analyte (N-protic acid with amount C_T), resulting in a certain pH value. So, one is tempted to write the pH as a function of C_T and C_B (or n = C_B/C_T), that is: pH = pH (analyte, titrant) = pH (C_T,C_B) or pH = pH (C_T, n). However, as mentioned above, these relations cannot be expressed in the form of an explicit function; they only exist implicitly in the form of polynomials of degree N + 2 (see Equation (44)). So we put the whole thing “from head to toe” by providing a strict algebraic solution in the form of the explicit function n = n(C_T, pH), or shorter n = n(x). The polynomials are then considered as the inverse task in Section 2.2.3.

Answer 4. “Simple” means that the analytical formula(s) is (are) easy to handle in a Lego-like manner from plain constructs, as summarized in Section 5 at the end of this report. There is no need for programming or root-solving methods. All results/diagrams in this report were created with Excel, and anyone can easily reproduce them.

“Smooth” means that the analytical formula n = n(x)—as well as its building blocks—are infinitely derivable functions that offer calculus in the form of pH derivatives and integrals (a feature that is not possible for numerical solutions/data). The pH derivatives convert buffer capacities to buffer intensities; pH derivatives are also used to identify equivalence points (EPs) as local minima/maxima and/or inflection points.

Answer 5. The central formula is the equivalent fraction n = Y₁ + w/C_T (symbols are explained in the text), which contains all information about the acid–base system in condensed form. As sketched in Figure 1, depending on whether n is a specific discrete number or a real function, different aspects appear: the equivalence points (as special/exceptional equilibrium states) or buffer capacities as “distances” between two equilibrium states. In this report, the function n(x) appears under several names: equivalence fraction, titration function/curve, and normalized buffer capacity (because it measures the distance to EP₀).

1.3. Structure of this Report

The report consists of three main parts/sections: (i) the mathematical framework (see also Figure 2), (ii) its application (including the discussion of alternative concepts and common approximations), and (iii) the extension of the approach to zwitterions and surface complexation.

The goal of Section 2 is to bundle a set of N + 3 nonlinear equations into an analytical formula. We start with the 1-component system in Section 2.1, i.e., the N-protic acid itself, which is fully determined by N acidity constants K₁ to K_N. In Section 2.2, the 1-component system is extended to a 3-component acid–base system, which opens the door to the description of acid–base titrations. After discussion/classification of equivalence points in Section 2.3, buffer capacities and buffer intensities are introduced in Section 2.4. Regarding the connection between the three subsystems in Figure 2, the numbers 1/C_T and C_B/C_T act as “coupling constants” of the acid to the water (autoprotolysis) and to the base. Section 2.5 embeds the present mathematical approach into alternative concepts “outside” traditional hydrochemistry.

In Section 2 and Section 3, example calculations are performed for four common acids listed in

Table 1. pK values for four N-protic acids at 25 °C. (The composite carbonic acid is the sum of the unionized species CO₂(aq) and the pure acid: H₂CO₃* = CO₂(aq) + H₂CO₃; to simplify the notation, we omit the asterisk (*) on H₂CO₃* throughout the paper).

N	Acid	Formula	Type	pK1	pK2	pK3	Ref.
1	acetic acid	CH3COOH	HA	4.76			[35]
2	(composite) carbonic acid	H2CO3	H2A	6.35	10.33		[36]
3	phosphoric acid	H3PO4	H3A	2.15	7.12	12.35	[35]
3	citric acid	C6H8O7	H3A	3.13	4.76	6.4	[35]

. In fact, the beauty and elegance of the acid–base behavior/equations is best illustrated in diagrams, and this is used extensively in this report. In Section 4, the mathematical description is applied beyond the realm of ordinary acids to zwitterions and surface complexation.

2. Mathematical Framework

2.1. The 1-Component System: N-Protic Acid (H_NA)

2.1.1. Notation

An acid is a proton donor; it releases H⁺ ions (more precisely: H₃O⁺) when dissolved in water:

HA = H⁺ + A⁻

(1)

which is a shorthand for HA(aq) + H₂O(l) = H₃O⁺(aq) + A⁻(aq). An N-protic acid H_NA dissolves into N + 1 species:

1 undissociated species:	HNA0	(uncharged)
N dissociated species:	HN−1A−1, …, HA−(N−1), A−N	(anions)

To simplify the notation, we abbreviate the molar concentrations of the N + 1 acid species with:

[j] ≡ [H_N−jA^−j]   for   j = 0, 1, 2, … N

(2)

The integer j also labels the electric charge of the species (z_j = 0 − j). Thus, the neutral, undissociated species H_NA⁰ is abbreviated with [0]. Henceforth we skip the superscript 0 in H_NA⁰.

In each successive dissociation step, j is enhanced by 1 (due to the release of one H⁺):

jth dissociation step:   [j − 1] → [j]

The corresponding conjugate acid–base pair is composed of

$$\begin{array}{l}\mathrm{acid}:\hfill \\ \mathrm{conjugate} \mathrm{base}:\hfill \end{array}\begin{array}{l}[\mathrm{j}-1]\hfill \\ [\mathrm{j}]\hfill \end{array}\}\mathrm{of} \mathrm{j}\mathrm{th} \mathrm{dissociation} \mathrm{step}$$

(3)

The sum of all species is the acid’s total amount C_T:

$${\mathrm{C}}_{\mathrm{T}}\equiv {[{\mathrm{H}}_{\mathrm{N}}\mathrm{A}]}_{\mathrm{T}}={\displaystyle \sum _{\mathrm{j}=0}^{\mathrm{N}}[\mathrm{j}]}=[0]+[1]+\dots +[\mathrm{N}](\mathrm{mass}\hspace{0.17em}\mathrm{balance})$$

(4)

Note: The total concentration C_T = [H_NA]_T should not be confused with the molar concentration of the undissociated species [H_NA].

In chemical thermodynamics, one has to distinguish between molar concentrations and activities:

concentrations:	denoted by square brackets	[j]
activities	denoted by curly braces	{j}

As discussed in Appendix B, activities are “effective concentrations” calculated from the molar concentrations using semi-empirical activity corrections γ_j:

{j} = γ_j [j]

(5)

The activity corrections γ_j increase with the ionic strength I of the solution. In ideal or nearly-ideal solutions (i.e., diluted systems), we have I ≈ 0 and γ_j ≈ 1, so the activities and concentrations become equal.

The activity of H⁺ is abbreviated with x; it is linked to the pH value via:

x ≡ {H⁺} = 10^−pH  ⇔  pH = −lg x

(6)

Using x instead of pH simplifies the formulas considerably. For the sake of simplicity, we speak about pH dependence when a function or quantity depends on x, f = f(x).

The self-ionization of water (autoprotolysis) is defined by

H₂O = H⁺ + OH⁻   with   K_w = {H⁺}{OH⁻}

(7)

and K_w = 1.0 × 10⁻¹⁴ at 25 °C. This yields [OH⁻] ≈ {OH⁻} = K_w/x.

In this context, we introduce the quantity w(x) (in combination with the approximation γ_H ≈ 1):

$$\mathrm{w}(\mathrm{x})\equiv [{\mathrm{OH}}^{-}]-[{\mathrm{H}}^{+}]=\frac{{\mathrm{K}}_{\mathrm{w}}}{\mathrm{x}}-\frac{\mathrm{x}}{{\mathsf{\gamma }}_{\mathrm{H}}}\approx \frac{{\mathrm{K}}_{\mathrm{w}}}{\mathrm{x}}-\mathrm{x}$$

(8)

For pure water, we have w = 0. As we will see later, the quotient w/C_T is the term which couples the polyprotic acid to H₂O. This term vanishes in the so-called

high-C_T limit:   C_T ≫ w(x)   (or C_T → ∞)

(9)

and removes the component “H₂O” from the 3-component acid–base system, as shown in Figure 2. A list of all abbreviations and symbols is given in Appendix A.

2.1.2. Acidity Constants and Dissociation Reactions

The equilibrium constant of reaction (1) is called acidity constant; it exists in (at least) two modifications:

$$\begin{array}{}\mathrm{(10)}& (\mathrm{thermodynamic}) \mathrm{acidity} \mathrm{constant}:\hfill & {\mathrm{K}}_{\mathrm{a}}=\frac{\left\{{\mathrm{H}}^{+}\right\}\left\{{\mathrm{A}}^{-}\right\}}{\left\{\mathrm{H}\mathrm{A}\right\}}& (\mathrm{based}\hspace{0.17em}\mathrm{on}\hspace{0.17em}\mathit{activities})\mathrm{(11)}& \mathit{conditional} \mathrm{acidity} \mathrm{constant} (\mathrm{of} \mathrm{mixed} \mathrm{type}):& {}^{\mathrm{c}}\mathrm{K}{}_{\mathrm{a}}=\frac{\left\{{\mathrm{H}}^{+}\right\}[{\mathrm{A}}^{-}]}{[\mathrm{H}\mathrm{A}]}& (\mathrm{based}\hspace{0.17em}\mathrm{on}\hspace{0.17em}\mathit{concentrations})\end{array}$$

Both equations represent the law of mass action. The value of K_a signifies the strength of the acid (strong acids: K_a large; weak acids: K_a small). The negative decadic logarithm of K_a is abbreviated by:

pK_a = − lg K_a

(12)

The smaller the pK_a, the stronger the acid—quite the opposite to a K_a-based ranking.

A monoprotic acid HA is characterized by one acidity constant K₁ (= K_a), a diprotic acid H₂A by two acidity constants (K₁, K₂), and a triprotic acid by three acidity constants (K₁, K₂, K₃):

1st dissociation step:    H3A = H+ + H2A−     K1 = {H+}{H2A−}/{H3A}

2nd dissociation step:   H2A− = H+ + HA−2   K2 = {H+}{HA−2}/{H2A−}

3rd dissociation step:    HA−2 = H+ + A−3      K3 = {H+}{A−3}/{HA−2}

The dissociation reactions can also be written as:

H3A = H+ + H2A−     k1 = K1

H3A = 2H+ + HA−2   k2 = K1K2

H3A = 3H+ + A−3      k3 = K1K2K3

The first representation describes the step-by-step release of one H⁺ in each dissociation step (it is the way nature works); the second representation, by contrast, relates each dissociated species to the undissociated acid by a cumulative H⁺ release. Extending it from 3- to N-protic acids, we obtain a general formula for the cumulative acidity constants:

$${\mathrm{k}}_{\mathrm{j}}=\{\begin{array}{l}1\hfill \\ {\mathrm{K}}_1{\mathrm{K}}_2\cdots {\mathrm{K}}_{\mathrm{j}}={\left\{{\mathrm{H}}^{+}\right\}}^{\mathrm{j}}\{{\mathrm{H}}_{\mathrm{N}-\mathrm{j}}{\mathrm{A}}^{-\mathrm{j}}\}/\{{\mathrm{H}}_{\mathrm{N}}\mathrm{A}\}\hfill \end{array}\begin{array}{l} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j}=0\hfill \\ \mathrm{f}\mathrm{o}\mathrm{r} 1\le \mathrm{j}\le \mathrm{N}\hfill \end{array}$$

(13)

For values of j outside this range (i.e., either for negative j or for j > N), we set k_j = 0. Equation (13) also includes the trivial case of the undissociated acid (j = 0), which is k₀ = 1. In our shortened notation, the last line is k_j = x^j {j}/{0}.

Equation (13) together with the mass balance (4) provides the set of N + 1 equations for the description of N-protic acids. In shorthand notation (and ignoring the trivial case j = 0), this set becomes:

$$\begin{array}{}\mathrm{(14)}& \mathrm{N}\hspace{0.17em}\mathrm{dissociation}\hspace{0.17em}\mathrm{reactions}\hspace{0.17em}(\mathrm{j}=1 \mathrm{to} \mathrm{N}):\hfill & \{\mathrm{j}\}=\left(\frac{{\mathrm{k}}_{\mathrm{j}}}{{\mathrm{x}}^{\mathrm{j}}}\right)\{0\}\hfill \mathrm{(15)}& \mathrm{mass}\hspace{0.17em}\mathrm{balance}:\hfill & {\mathrm{C}}_{\mathrm{T}}={\displaystyle \sum _{\mathrm{j}=0}^{\mathrm{N}}[\mathrm{j}]}=[0] {\displaystyle \sum _{\mathrm{j}=0}^{\mathrm{N}} \frac{{\mathrm{k}}_{\mathrm{j}}}{{\mathrm{x}}^{\mathrm{j}}}}\hfill \end{array}$$

In the next Section, using the approximation {j} ≈ [j], these equations are converted into analytical formulas for ionization fractions.

2.1.3. Ionization Fractions (Degree of Dissociation)

The N-protic acid H_NA dissolves into N + 1 acid species denoted by [j], where j runs from 0 to N. Instead of the molar concentrations [j], it is convenient to use unitless or intensive quantities, known as ionization fractions:

$${\mathrm{a}}_{\mathrm{j}} \equiv \frac{[\mathrm{j}]}{{\mathrm{C}}_{\mathrm{T}}}\mathrm{for} \mathrm{j}=0, 1, 2,\dots \mathrm{N}$$

(16)

The pH dependence of a_j as a function of x (= 10^−pH) follows directly from Equations (14) and (15). Dividing them by C_T and using the approximation {j} ≈ [j] yields:

$${\mathrm{a}}_{\mathrm{j}}=\left(\frac{{\mathrm{k}}_{\mathrm{j}}}{{\mathrm{x}}^{\mathrm{j}}}\right) {\mathrm{a}}_0 \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} {\mathrm{a}}_0={\left(1+\frac{{\mathrm{k}}_1}{\mathrm{x}}+\frac{{\mathrm{k}}_2}{{\mathrm{x}}^2}+\dots +\frac{{\mathrm{k}}_{\mathrm{N}}}{{\mathrm{x}}^{\mathrm{N}}}\right)}^{-1}={\left({\displaystyle \sum _{\mathrm{j}=0}^{\mathrm{N}} \frac{{\mathrm{k}}_{\mathrm{j}}}{{\mathrm{x}}^{\mathrm{j}}}}\right)}^{-1}$$

(17)

The ionization fractions are completely specified by the N acidity constants K₁, K₂, … K_N, which are encapsulated in the cumulative equilibrium constants (as products of K_j-values):

k₀ = 1,   k₁ = K₁,   k₂ = K₁K₂,   …   k_N = K₁K₂…K_N

(18)

Figure 3 displays the ionization fractions as a function of pH for four acids. For any chosen value of x (or pH), the sum of all ionization fractions equals 1:

$$1={\mathrm{a}}_0+{\mathrm{a}}_1+\dots +{\mathrm{a}}_{\mathrm{N}}={\displaystyle \sum _{\mathrm{j}=0}^{\mathrm{N}}{\mathrm{a}}_{\mathrm{j}}(\mathrm{x})}(\mathrm{mass}\hspace{0.17em}\mathrm{balance})$$

(19)

Ionization fractions are smooth, infinitely differentiable functions of x or pH with values between 0 and 1: 0 < a_j < 1. The latter makes them excellent candidates for probabilities (as will be discussed later in Section 2.5.3).

The points where two curves intersect (marked with small circles in Figure 3) are special equilibrium states called equivalence points (EP)—cf. Section 2.3. The actual values at these points are:

semi-Eps	(blue circles):	aj = aj−1 ≈ ½	(inflection points of aj)
EPs	(yellow circles):	aj = 1 − 2aj−1 (≈ 1)	(maximum of aj)

Universality. Ionization fractions have the nice feature that they are independent of the acid’s total amount C_T; they are intensive (non-extensive) variables. Regardless of the assumed C_T (either constant or pH dependent, such as in open systems), the curves/shapes of the ionization fractions remain the same (as in Figure 3). Examples for this “universality” are given in Section 3.3.3 (H₂A as titrant vs. H₂A as analyte) and Section 3.3.4 (open vs. closed CO₂ system).

2.1.4. Two Types of Ionization Fractions: S Shaped vs. Bell Shaped

The acidity constants in the form of pK_j values subdivide the entire pH domain into N + 1 distinct intervals, as shown in Figure 4. The jth interval is the subdomain in which the ionization fraction a_j exercises its full dominance—see right diagrams in Figure 4. As indicated by colors, there are two types of curves: (i) S-shaped curves (sigmoid curves) in the 0th and the Nth interval at the opposite ends of the pH scale (red color) and (ii) bell-shaped curves in all other intervals (blue color), with their maxima in the middle of the interval. [Note 1: For N = 1, a₀ and a₁ represent prominent functions: logistic functions or the Fermi–Dirac distribution—see Section 2.5.3. Note 2: The S-shaped curves appear as the two halves of a bell-shaped curve when the opposite ends are glued together at ±∞.]

Table 2. Two types of ionization fractions.

	Type 1 (S Shaped)	Type 2 (Bell Shaped)
ionization fraction	a0 and aN	a1, a2, … aN−1 (does not exist for 1-protic acids)
domain (pH interval)	pH < pK1 (for a0) pH > pKN (for aN)	pKj < pH < pKj+1
maximum at pH	−∞ (for a0) +∞ (for aN)	½ (pKj + pKj+1)
strongly acidic (pH → 0) strongly alkaline (pH → 14)	a0 = 1, aN = 0 a0 = 0, aN = 1	aj = 0 aj = 0
integral (area below curve)	infinite	finite
statistical meaning (see Section 2.5.3)	cumulative distribution function	–
associated equivalence points	two external EPs: EP0 and EPN	N − 1 internal EPs: EP1, EP2, … EPN-1

contrasts the two types of ionization fractions. Their otherness implies the distinction between external (outer) and internal (inner) equivalence points in Section 2.3.

At the opposite ends of the pH scale, the two ionization fractions a₀ and a_N attain the maximum value 1:

strongly acidic:	pH < 0	(or x → ∞):	a0 = 1,	all other aj = 0	(20)
strongly alkaline:	pH > 14	(or x → 0):	aN = 1,	all other aj = 0	(21)

This behavior is an important fact to identify them as cumulative distribution functions in Section 2.5.3. [Note: The pH scale does not end at 0 or 14, but also extends to pH < 0 and pH > 14 (in theory, even up to −∞ and to +∞).]

2.1.5. Moments Y_L

The simplest constructs we can build from ionization fractions are so-called moments. The Lth moment Y_L is defined as the weighting sum over a_j:

$${\mathrm{Y}}_{\mathrm{L}}\equiv {\displaystyle \sum _{\mathrm{j}=0}^{\mathrm{N}} {\mathrm{j}}^{ \mathrm{L}}} {\mathrm{a}}_{\mathrm{j}}$$

(22)

In the next Sections, the moments become the building blocks of several relevant quantities (while their statistical significance is discussed in Section 2.5.4):

Y0 = a0 + a1 + … + aN = 1	⇒ mass balance		(23)
Y1 = a1 + 2a2 + … + N aN	⇒ enters buffer capacity	in Equation (68)	(24)
Y2 = a1 + 4a2 + … + N2 aN	⇒ enters buffer intensity β	in Equation (69)	(25)
Y3 = a1 + 8a2 + … + N3 aN	⇒ enters 1st derivative of β	in Equation (70)	(26)

The example in Figure 5 demonstrates how the “titration curve”-like Y₁ is built up from the ionization fractions a_j (of the triprotic phosphoric acid). Figure 6 displays the pH dependence of Y₁ to Y₄ for four acids. Note that in the trivial case of monoprotic acids (top-left diagram), all moments are equal, i.e., the four Y_L curves cover each other.

The moments Y_L are positive, monotonic (increasing) functions, living in the range 0 < Y_L ≤ N ^L (where the equal sign applies only for Y₀). Their asymptotic behavior results from Equations (20) and (21):

YL(pH → −∞) = 0	or	YL(x → +∞) = 0
YL(pH → +∞) = NL	or	YL(x → 0) = NL

Note: The introduction of moments is not a new idea. Similar constructs can be found in ligand theory [32]. The formulas, however, differ due to differently chosen reference states (dissociation vs. association reactions).

2.2. The 3-Component Acid–Base System

2.2.1. Basic Set of Equations

The acid–base system is made up of three components:

• pure water H₂O
• N-protic acid H_NA (with amount C_T = [H_NA]_T)
• strong base BOH (with amount C_B = [BOH]_T)

The term strong means that the base dissociates completely: BOH = B⁺ + OH⁻. This implies two things: first, [B⁺] = C_B; second, no extra equilibrium constant is needed for the dissociation of the base. Here, BOH, for example, stands for NaOH or KOH (i.e., B⁺ = Na⁺ or K⁺).

The 3-component system is characterized by

$$\mathrm{N}+4 \mathrm{species} \left(\mathrm{variables}\right): {\mathrm{H}}^{+}, \mathrm{O}{\mathrm{H}}^{-}, \underset{\mathrm{N}+1 \mathrm{acid} \mathrm{species}}{\underbrace{{\mathrm{H}}_{\mathrm{N}}\mathrm{A}, {\mathrm{H}}_{\mathrm{N}-1}{\mathrm{A}}^{-},\dots , {\mathrm{A}}^{-\mathrm{N}}}}, {\mathrm{B}}^{+}$$

For the mathematical description, we use a set of N + 3 equations:

Kw = {H+} {OH−}	(self-ionization of H2O)	(27)
K1 = {H+} {HN−1A−}/{HNA}	(1st diss. step)	(28)
K2 = {H+} {HN−2A−2}/{HN-1A−}	(2nd diss. step)	(29)
⋮
KN = {H+} {A−N}/{HA−(N− 1)}	(Nth diss. step)	(30)
CT = [HNA] + [HN−1A−] + … + [A−N]	(mass balance)	(31)
CB = [HN−1A−] + 2 [HN−2A−2] + … + N [A−N] + [OH−] − [H+]	(charge balance)	(32)

It is worth noting that this set of equations comprises two types of laws: mass-action laws (first N + 1 equations) and balance laws for mass and charge (last two equations). This embodies a kind of asymmetry: while the mass-action laws are based on activities, the mass balance and charge balance rely on molar concentrations.

Since we have N + 4 variables, but only N + 3 equations, the description is given one degree of freedom. Thus, we can vary the parameter C_B (amount of base) to change the pH (known as titration).

Instead of C_B, we prefer the normalized or unitless quantity

$$\mathrm{equivalent} \mathrm{fraction}:\hspace{1em}\hspace{1em}\mathrm{n}=\frac{{\mathrm{C}}_{\mathrm{B}}}{{\mathrm{C}}_{\mathrm{T}}} (\mathrm{n}\ge 0)$$

(33)

If n = 0, the 3-component system collapses to the 2-component system “H_NA + H₂O” (see Figure 2). Plotting n = n(pH) as a function of pH yields the titration curve. The corresponding formula, n = n(x), will be derived in the next Section.

The concept even works when the strong base is replaced by a strong monoprotic acid HX with amount C_A = [HX]_T (for example HCl, whereby X⁻ = Cl⁻). In this way, we are able to generalize Equation (33):

$$\mathrm{equivalent} \mathrm{fraction}:\hspace{1em}\hspace{1em}\mathrm{n}=\frac{{\mathrm{C}}_{\mathrm{B}}-{\mathrm{C}}_{\mathrm{A}}}{{\mathrm{C}}_{\mathrm{T}}} (\mathrm{for}\hspace{0.17em}\mathrm{any}\hspace{0.17em}\mathrm{real}\hspace{0.17em}\mathrm{n})$$

(34)

where either C_A or C_B is zero. For n > 0, it describes the alkalimetric titration (with a strong base); for n < 0, it describes the acidimetric titration (with a strong monoprotic acid). The integer and half-integer values of n are especially interesting because they constitute equivalence points (cf. Section 2.3).

The principle is explained in Figure 7, where a carbonate system with C_T = 100 mM is titrated. The special state n = 0 represents the pH of the base-free 2-component system. Starting from this point, the pH increases by adding NaOH (n > 0) and decreases by adding HCl (n < 0). The small circles at n = 0, 1, and 2 mark three equivalence points. Alternatively, taking 100 mM NaHCO₃ (or Na₂CO₃), the titration will start at n = 1 (or n = 2).

The titration curve from Figure 7 is shown in Figure 8 (blue color) together with curves for C_T = 1 mM and 10 mM. The only difference between the left and right diagram is that the x and y axis are interchanged; left: n = n(pH), right: pH = pH(n). All curves are calculated using Equation (41) below.

2.2.2. Analytical Formula (Titration Curves)

The starting point is the set of nonlinear Equations (27)–(32). Its algebraic solution leads to a simple analytical formula when the following procedure is applied:

• replace activities by concentrations: {j} ⇒ [j] in Equations (28)–(30);
• replace acid species by ionization fractions: [j] ⇒ a_j in Equations (28)–(32);
• replace {H⁺} by x and use w(x) defined in Equation (8);
• use Y₁ for summation over j a_j defined in Equation (24);
• divide Equations (31) and (32) by C_T.

In this way, the set of N + 3 Equations (27)–(32) becomes:

w	= Kw/x − x	(self-ionization of H2O)	(35)
a1	= (k1/x) a0	(1st diss. step)	(36)
a2	= (k2/x2) a0	(2nd diss. step)	(37)
⋮
aN	= (kN/xN) a0	(Nth diss. step)	(38)
1	= a0 + a1 + … + aN		(39)
n	= Y1 + w/CT		(40)

The essence of the entire set is now bundled in a single analytical formula (taken from the last line):

$$\mathrm{n}(\mathrm{x})={\mathrm{Y}}_1(\mathrm{x})+\frac{\mathrm{w}(\mathrm{x})}{{\mathrm{C}}_{\mathrm{T}}}$$

(41)

This formula encapsulates the information contained in all other equations, i.e., Equations (27)–(31). Each term in Equation (41) represents one of the three components/subsystems: n—the strong base, w/C_T—the effect of water, and Y₁—the pure acid. Figure 9 illustrates how the three components are cemented together via one equation: the charge-balance equation. In the high-C_T limit, the last term in Equation (41) vanishes and the formula simplifies to n = Y₁(x). [Note: Instead of the charge balance, one can also use an alternative concept known as proton balance, which is the balance between the species that have excess protons versus those that are deficient in protons. Both concepts lead to the same results].

Using Equation (41), titration curves for four acids with different C_T values are plotted in Figure 10. The diagrams also include the high-C_T limit, n = Y₁ (as dark-blue curves).

2.2.3. Forward and Inverse Task (Polynomials for x = 10^−pH)

The 3-component system is controlled by three “master variables”: pH, C_T, and n (or C_B), but only two of them can be chosen freely. From the viewpoint of Equation (41), the following tasks emerge:

• forward task 1: given pH and C_T ⇒ calculate n (or C_B)
• forward task 2: given pH and n (or C_B) ⇒ calculate C_T
• inverse task: given C_T and n (or C_B) ⇒ calculate pH

Using the shorthand x = 10^−pH, the two forward tasks are easily obtained from Equation (41):

$$\begin{array}{}\mathrm{(42)}& \mathrm{forward}\hspace{0.17em}\mathrm{task}\hspace{0.17em}1:\hfill & \mathrm{n}={\mathrm{Y}}_1(\mathrm{x})+\frac{\mathrm{w}(\mathrm{x})}{{\mathrm{C}}_{\mathrm{T}}}\hfill & \mathrm{or}\hfill & {\mathrm{C}}_{\mathrm{B}}={\mathrm{Y}}_1(\mathrm{x}) {\mathrm{C}}_{\mathrm{T}}+\mathrm{w}(\mathrm{x})\hfill \mathrm{(43)}& \mathrm{forward}\hspace{0.17em}\mathrm{task}\hspace{0.17em}2:\hfill & {\mathrm{C}}_{\mathrm{T}}=\frac{\mathrm{w}(\mathrm{x})}{\mathrm{n}-{\mathrm{Y}}_1(\mathrm{x})}\hfill & \mathrm{or}\hfill & {\mathrm{C}}_{\mathrm{T}}=\frac{{\mathrm{C}}_{\mathrm{B}}-\mathrm{w}(\mathrm{x})}{{\mathrm{Y}}_1(\mathrm{x})}\hfill \end{array}$$

These are explicit functions.

The inverse task to calculate the pH for a given C_T and C_B is intricate because an explicit function, such as pH = f(C_T,C_B), does not exist for N > 1. The only thing we can offer is an implicit function in the form of polynomials of degree N + 2 in x:

$$0={\displaystyle {\sum }_{\mathrm{j}=0}^{\mathrm{N}}}\left\{ {\mathrm{x}}^2+(\mathrm{n}-\mathrm{j} ) {\mathrm{C}}_{\mathrm{T}}\mathrm{x}-{\mathrm{K}}_{\mathrm{w}}\right\} {\mathrm{k}}_{\mathrm{j}}{\mathrm{x}}^{\mathrm{N}-\mathrm{j}}$$

(44)

Example N = 1. The monoprotic acid represents the simplest case, where the sum runs over two terms only: j = 0 and 1. With k₀ = 1 and k₁ = K₁ we get a cubic equation:

0 = x³ + {K₁ + nC_T} x² − {(n − 1)(C_TK₁ + K_w)} x − K₁K_w

(45)

Example N = 2. For a diprotic acid with k₀ = 1, k₁ = K₁, and k₂ = K₁K₂ we obtain a quartic equation:

0 = x⁴ + {K₁ + nC_T} x³ + {K₁K₂ + (n − 1) C_TK₁ − K_w} x² + K₁ {(n − 2) C_TK₂ − K_w} x − K₁K₂K_w

(46)

Setting K₂ = 0, this equation falls back to Equation (45). Setting n = 0, one obtains the formula for diprotic acids given in textbooks (e.g., [1] (p. 107)). Setting n = C_B/C_T yields an alternative form of the quartic equation:

0 = x⁴ + {K₁ + C_B} x³ + {K₁K₂ + (C_B − C_T) K₁ − K_w} x² + K₁ {(C_B − 2C_T) K₂ − K_w} x − K₁K₂K_w

(47)

Theoretically, the polynomial (44) can be used to calculate x or pH for any N. However, in practice, it is not quite easy. For polynomials of order N > 4, there is principally no algebraic solution, no matter how hard we try (due to the Abel–Ruffini theorem). Thus, numerical root-finding methods should be applied. However, this effort is not necessary if you want to plot titration curves with pH on the abscissa, because you can take the forward approach and then swap the axes.

An example is given in Figure 11 for the case n = 0 (i.e., the 2-component system H_NA+H₂O). The left diagram represents the forward task pH ⇒ C_T based on Equation (43); the right diagram represents the inverse task C_T ⇒ pH based on the polynomial (44) using root-finding methods. Alternatively, the right diagram can be created from the left diagram by interchanging the axes (as a cost-free by-product of the first calculation).

Remark. To show that the main polynomial is indeed of degree N + 2, you can transform Equation (44) into the following form:

$$0={\displaystyle {\sum }_{\mathrm{j}=0}^{\mathrm{N}+2} {\mathrm{h}}_{\mathrm{j}} {\mathrm{x}}^{\mathrm{N}+2-\mathrm{j}}}\mathrm{with}{\mathrm{h}}_{\mathrm{j}}={\mathrm{k}}_{\mathrm{j}}+{\mathrm{k}}_{\mathrm{j}-1} (\mathrm{n}+1-\mathrm{j}) {\mathrm{C}}_{\mathrm{T}}-{\mathrm{k}}_{\mathrm{j}-2} {\mathrm{K}}_{\mathrm{w}}$$

(48)

The first and the last coefficients of this polynomial are: h₀ = 1 and h_N+2 = −K_wK₁K₂...K_N. [Note: k_j is per definition zero for negative values of j and for j > N].

2.3. Equivalence Points

An equivalence point (EP) is a special equilibrium state at which chemically equivalent quantities of an acid and a base have been mixed:

equivalence point:  [acid] = [base]

(49)

In titration, it is the point where all of the acid/base has been neutralized. In the next Sections, we will extend this perspective/definition. Equivalence points can be introduced in two ways: (i) within the 1-component system “H_NA”, which leads to a widely used approach with simple formulas, and (ii) within the 3-component acid–base system, which provides a strict and more general definition of EPs (including the first definition as an approximation).

2.3.1. EPs of the 1-Component System (H_NA)

Even though the 1-component system does not involve a strong base, the concept in Equation (49) can still be applied, but in a modified form. For this purpose, recall that a polyprotic acid dissociates in successive steps, whereby each dissociation step (by releasing one H⁺) relates an acid species to its conjugate base—see Equation (3). In this context, Equation (49) relates conjugate acid–base pairs:

equivalence point:  [acid] = [conjugate base]

(50)

This gives rise to a whole series of equivalence points, as defined by

EP:	[j − 1] = [j + 1]	⇒	aj−1(x) = aj+1(x)	(51)
semi-EP:	[j − 1] = [j]	⇒	aj−1(x) = aj(x)	(52)

They are easily recognizable as intersection points of two curves in the ionization-fraction diagrams of Figure 3 (where semi-EPs are marked as small blue circles and EPs as small yellow circles).

An N-protic acid has N + 1 integer-valued EPs (the same number as the number of acid species) plus N semi-EPs. In total, there are 2N + 1 equivalence points, henceforth symbolized by EP_n, where n runs over all integer and half-integer values: n = 0, ½, 1, …, N − ½, N. The idea behind the index conversion from j to n becomes clear in the next Section where all equivalence points are related to the equivalent fraction n in Equation (33). To recall: In contrast to n, j is an integer (never a half-integer); j indicates the acid species [j], the acidity constants K_j, and the ionization fraction a_j. In the new notation, Equations (51) and (52) become:

EPn:	[n − 1] = [n + 1]	(for integer n = 0, 1, … N)	(53)
semi-EPn:	[n − ½] = [n + ½]	(for half-integer n = ½, 3/2, … N − ½)	(54)

where the simple transformation rule is applied: n = j for EPs and n = j − ½ for semi-EPs—see Appendix C.1).

Example for Equation (53): In carbonate systems, EP₁ is often introduced as the equilibrium state for which [CO₂] = [CO₃⁻²] applies. Another example is the triprotic acid (N = 3) in Figure 12, which dissolves into 3 + 1 species: H₃A, H₂A⁻¹, HA⁻², and A⁻³. There are 2 × 3 + 1 equivalence points EP_n with n = 0, ½, 1, … 3.

As already indicated in Section 2.1.4 and Table 2 (last row), there is another important subdivision that differentiates between so-called external and internal EPs:

external (or outer) equivalence points EP0 and EPN (only two!)

internal (or inner) equivalence points all other EPn (for ½ ≤ n ≤ N − ½)

The two external EPs are special, because they are linked to non-acid species H⁺ and OH⁻, as highlighted in red in Figure 12. In pH diagrams, they are positioned at the opposite ends of the entire EP_n sequence (see Figure 13, Figure 14 and Figure 15).

Each EP_n is characterized by a specific pH value called pH_n. The 2N − 1 internal EPs provide particularly simple formulas that establish a direct link to the acidity constants:

$$\mathrm{p}{\mathrm{H}}_{\mathrm{n}}=\{\begin{array}{l}\frac{1}{2} (\mathrm{p}{\mathrm{K}}_{\mathrm{n}}+\mathrm{p}{\mathrm{K}}_{\mathrm{n}+1})\hfill \\ \mathrm{p}{\mathrm{K}}_{\mathrm{n}+1/2}\hfill \end{array} \begin{array}{c}\iff \\ \iff \end{array} \begin{array}{l}\mathrm{E}{\mathrm{P}}_{\mathrm{n}}\hfill \\ {\text{semi-EP}}_{\mathrm{n}}\hfill \end{array} \begin{array}{l}(\mathrm{for}\hspace{0.17em}\mathrm{integer}\hspace{0.17em}\mathrm{n}, \mathrm{except}\hspace{0.17em}0\hspace{0.17em}\mathrm{and}\hspace{0.17em}\mathrm{N})\hfill \\ (\mathrm{for}\hspace{0.17em}\text{half-integer}\hspace{0.17em}\mathrm{n})\hfill \end{array}$$

(55)

This is obtained from Equations (51) and (52) by inserting Equation (17) into a_j−1 = a_j+1 and a_j−1 = a_j (then use the transformation rule: n = j for EPs, n = j − ½ for semi-EPs).

Table 3. Internal equivalence points (based on pK values in Table 1).

N	Acid	pH1/2	pH1	pH3/2	pH2	pH5/2
1	acetic acid	4.76
2	(composite) carbonic acid	6.35	8.34	10.33
3	phosphoric acid	2.15	4.68	7.21	9.78	12.35
3	citric acid	3.13	3.94	4.76	5.58	6.4

lists the internal equivalence points of four acids calculated with Equation (55). As shown on the pH scale in Figure 13, each EP_n is the midpoint between two adjacent semi-EPs.

For the two external EPs, such simple formulas do not exist. In contrast, pH₀ and pH_N depend sensitively on C_T. This relationship can only be expressed in implicit form:

$$\begin{array}{}\mathrm{(56)}& {\mathrm{EP}}_0:\hfill & [{\mathrm{H}}^{+}]=[1]\hfill & \Rightarrow \hfill & \mathrm{x}={\mathrm{C}}_{\mathrm{T}} {\mathrm{a}}_1\hfill & \Rightarrow \hfill & {\mathrm{C}}_{\mathrm{T}}=\frac{{\mathrm{x}}^2}{{\mathrm{K}}_1}\cdot \frac{1}{{\mathrm{a}}_0(\mathrm{x})}\hfill \mathrm{(57)}& {\mathrm{EP}}_{\mathrm{N}}:\hfill & [\mathrm{N}-1]=[{\mathrm{OH}}^{-}]\hfill & \Rightarrow \hfill & {\mathrm{C}}_{\mathrm{T}}{\mathrm{a}}_{\mathrm{N}-1}=\frac{{\mathrm{K}}_{\mathrm{w}}}{\mathrm{x}}\hfill & \Rightarrow \hfill & {\mathrm{C}}_{\mathrm{T}}=\frac{{\mathrm{K}}_{\mathrm{w}}{\mathrm{K}}_{\mathrm{N}}}{{\mathrm{x}}^2}\cdot \frac{1}{{\mathrm{a}}_{\mathrm{N}}(\mathrm{x})}\hfill \end{array}$$

The scheme in Figure 13 illustrates how the 2 × 3 + 1 equivalence points of a triprotic acid are marshalled in ascending order on the pH scale, with EP₀ and EP₃ at the extreme left and right positions. For C_T → ∞ the two external EPs drift apart: pH₀ → 0 and pH₃ → 14, while all internal EPs remain fixed at the position dictated by the pK values in Equation (55).

The same behavior is displayed in Figure 14, where the equivalence points of the carbonic acid (left) and the phosphoric acid (right) are plotted in pH–C_T diagrams. Again, the internal equivalence points (red color) are independent of C_T, and therefore straight vertical lines, whereas the two external EPs (blue and green) are not. The representation of the curves as dashed instead of solid lines is to remind us that these are approximations, valid for the 1-component system only. The general case will be discussed later and is shown in Figure 27.

2.3.2. EPs of the 3-Component Acid–Base System

In the most general sense, equivalence points are equilibrium states in which the equivalent fraction n = C_B/C_T becomes an integer or half-integer value:

EPn	⇔	CB/CT = n	for n = 0, 1, … N	(58)
semi-EPn	⇔	CB/CT = n	for n = ½, 3/2, … N − ½	(59)

An N-protic acid has a total of 2N + 1 equivalence points (i.e., the same number as in the 1-component system). In this context, the original idea in Equation (49) represents only one special case, namely EP₁ with n = 1. The particular case n = 0 (or EP₀) refers to the base-free 2-component system with pH₀ as the equilibrium pH of the acid with amount C_T dissolved in pure water.

The algebraic relation between EP_n and pH_n is dictated by Equation (41), that is:

$${\mathrm{EP}}_{\mathrm{n}} \iff {\mathrm{pH}}_{\mathrm{n}}:\mathrm{n}={\mathrm{Y}}_1(\mathrm{p}{\mathrm{H}}_{\mathrm{n}})+\frac{\mathrm{w}(\mathrm{p}{\mathrm{H}}_{\mathrm{n}})}{{\mathrm{C}}_{\mathrm{T}}} \mathrm{for} \mathrm{n}=0, ½, 1,\dots \mathrm{N}$$

(60)

Figure 15 shows all five equivalence points of the diprotic acid H₂CO₃ with integer and half-integer values of n on the titration curve n = n(pH). More examples are given in Section 3.2.

For C_T ≫ w, the last term in Equation (60) vanishes, leading to the 1-component approach:

Y₁(pH_n) − n = 0

(61)

This approximation works for C_T > 10⁻³ M, but fails miserably for very dilute acids when the autoprotolysis of water becomes dominant—cf. Figure 28. [Note: Equations (61) and (55) are equivalent for N ≤ 2, but deviate for higher N, though the deviation is very small].

The scheme in Figure 16 contrasts the 1-component approach based on Equation (61) with the general and strict approach based on Equation (60). While the first approach relies on the equality of species concentrations, the second relies on the equality of two real chemical components (the acid H_NA and the strong base). As C_T increases, the two approaches become more and more similar.

2.4. Buffer Capacities

2.4.1. ANC and BNC

An equilibrium state of the acid–base system with a given amount of C_T is fully specified by the parameter C_B or n = C_B/C_T. Buffer capacities are “distances” between two equilibrium states, expressed as the difference/deviation from a reference point:

Δn = n − n_ref  or ΔC_B = C_B − n_ref C_T

(62)

The reference point is usually an equivalence point EP_n=j (with integer j), that is n_ref = j, yielding:

Δn = n(x) − j  or  ΔC_B = C_B(x) − j·C_T  (with j = 0, 1, …)

(63)

If n_ref = 0, these equations collapse to Δn = n and ΔC_B = C_B. This legitimizes calling n and C_B buffer capacities; they measure the distance to EP₀.

Two types of buffer capacities are in common use: the acid-neutralizing capacity (ANC) and the base-neutralizing capacity (BNC). The ANC is the amount of basicity of the system that can be titrated with a strong acid to a chosen equivalence point EP_j (at pH_j):

[ANC]_n=j = C_B(pH) − j·C_T

(64)

The small subscript n in the symbol [ANC]_n refers to the chosen reference point, usually EP_n (with an integer n). In the special case of n = 0, which corresponds to the base-free 2-component system, the last term vanishes, and the ANC becomes

[ANC]₀ = C_B(pH)

(65)

The BNC is the exact opposite of ANC:

[BNC]_n = − [ANC]_n

(66)

The definition in Equation (64) leads to a simple formula for ANC. Using Equation (41) for C_B(pH) = n C_T yields:

[ANC]_n=j = { Y₁(x) − j } C_T + w(x)

(67)

For example, the amount of strong acid (say HCl) required to neutralize the system from startpoint x (= 10^−pH) to a particular EP_n (titration endpoint) is:

[ANC]₀ = { Y₁(x) − 0 } C_T + w(x) = { Y₁(x) C_T + w(x)} − 0·C_T

[ANC]₁ = { Y₁(x) − 1 } C_T + w(x) = { Y₁(x) C_T + w(x)} − 1· C_T

[ANC]₂ = { Y₁(x) − 2 } C_T + w(x) = { Y₁(x) C_T + w(x)} − 2· C_T

The three ANC curves are shown in the top diagram of Figure 17 for carbonic acid (C_T = 10 mM). The small circles at pH₀ = 4.2, pH₁ = 8.2, and pH₂ = 11.1 mark the corresponding EP_n. The curves display the amount of strong acid (normalized by C_T) required to remove the inherent basicity and to attain pH₀ (blue curve), pH₁ (green curve), and pH₂ (red curve). Of course, the highest amount (blue curve) is required to attain the lowest pH, namely pH₀ = 4.2. Negative ANC values indicate that the system’s acidity should be removed to attain the EP_n (which is the same as the addition of a strong base—see Equation (66) for BNC). Curves of BNC are shown in the bottom diagram, which is the mirror image of the top diagram.

2.4.2. Buffer Intensity

Let us start with the normalized buffer capacity in Equation (63). Using Equation (41) yields:

$$\mathrm{normalized} \mathrm{buffer} \mathrm{capacity}:\Delta \mathrm{n}(\mathrm{p}\mathrm{H})=\left\{{\mathrm{Y}}_1(\mathrm{p}\mathrm{H})-\mathrm{j}\right\}+\frac{\mathrm{w}(\mathrm{p}\mathrm{H})}{{\mathrm{C}}_{\mathrm{T}}}$$

(68)

Via pH derivation, we obtain:

normalized buffer intensity:	β = dΔn/dpH = dn/dpH	(unitless)	(69)
buffer intensity:	βC = dCB/dpH = β CT	(in mM)	(70)

[Note: The last equation is valid if C_T = const (standard case), otherwise we have to use dC_B/dpH = β C_T + n(dC_T/dpH)].

The acid-neutralizing capacity is re-established by integrating β_C over a definite pH interval (usually starting from an equivalence point EP_n):

$${[\mathrm{A}\mathrm{N}\mathrm{C}]}_{\mathrm{n}}={\displaystyle \underset{\mathrm{p}{\mathrm{H}}_{\mathrm{n}}}{\overset{\mathrm{p}\mathrm{H}}{\int }}{\mathsf{\beta }}_{\mathrm{C}}(\mathrm{p}{\mathrm{H}}^{\prime }) \mathrm{d} \mathrm{p}{\mathrm{H}}^{\prime }}$$

(71)

Using the pH derivatives in Appendix C.2, the following analytical formulas for the first and second pH derivative of Equation (68) are obtained:

$$\begin{array}{}\mathrm{(72)}& \mathrm{normalized} \mathrm{buffer} \mathrm{intensity}:\hfill & \mathsf{\beta }(\mathrm{x}) \equiv \frac{\mathrm{d}\mathrm{n}}{\mathrm{d} \mathrm{p}\mathrm{H}}=(\ln 10) \left({\mathrm{Y}}_2-{\mathrm{Y}}_1^2+\frac{\mathrm{w}+2\mathrm{x}}{{\mathrm{C}}_{\mathrm{T}}}\right)\hfill \mathrm{(73)}& 1\mathrm{st} \mathrm{derivative} \mathrm{of} \mathsf{\beta }:\hfill & \frac{\mathrm{d}\mathsf{\beta }}{\mathrm{d} \mathrm{p}\mathrm{H}}={(\ln 10)}^2 \left({\mathrm{Y}}_3-3{\mathrm{Y}}_1{\mathrm{Y}}_2+2{\mathrm{Y}}_1^3+\frac{\mathrm{w}}{{\mathrm{C}}_{\mathrm{T}}}\right)\hfill \end{array}$$

Figure 18 displays the titration curve (blue) together with the buffer intensity β (green) and its first derivative (red) for the H₂CO₃ system with C_T = 100 mM. The calculations are performed using Equations (68), (72), and (73). The small circles indicate the EPs and semi-EPs. The EPs are the extreme points of β (see also Table 7):

EPn	(integer n)	⇔	minimum buffer intensity β
semi-EPn	(half-integer n)	⇔	maximum buffer intensity β

A good pH buffer should mitigate pH changes when the system is attacked by a strong base or strong acid. This means that the pH change ΔpH should be small when n = C_B/C_T changes by Δn. In other words, the slope of the titration curve in Figure 18, Δn/ΔpH, should be large for maximum buffering capability. The buffer intensity, β = dn/dpH, is just the measure of this slope. Thus, the pH at the point where β reaches its maximum signals the optimal buffer range (bounded by pH_max ± 1). More examples are given in Figure 19 and Figure 20. Since each titration curve (blue) is an ever-increasing function, its pH derivative, i.e., the buffer intensity β, is always positive (green curves). [One interesting fact is that the number of zeros of the red curve differs in Figure 19 and Figure 20—this phenomenon will be discussed in Section 3.2.2].

2.5. Alternative and Statistical Approaches

2.5.1. Dissociation vs. Association Reactions

Acids can be described either by dissociation reactions (deprotonation), as shown in Section 2.1.2 and Equation (13), or as association reactions (protonation or complex formation). The corresponding cumulative (overall) equilibrium constants k_j and β_j differ significantly:

$$\begin{array}{}\mathrm{(74)}& \mathrm{dissociation}:\hfill & {\mathrm{H}}_{\mathrm{N}}\mathrm{A}=\mathrm{j} {\mathrm{H}}^{+}+{\mathrm{H}}_{\mathrm{N}-\mathrm{j}}{\mathrm{A}}^{-\mathrm{j}}\hfill & {\mathrm{k}}_{\mathrm{j}}={\displaystyle \frac{{\mathrm{x}}^{\mathrm{j}} [\mathrm{j}]}{[0]}}={\mathrm{K}}_1{\mathrm{K}}_2\dots {\mathrm{K}}_{\mathrm{j}}\hfill \mathrm{(75)}& \mathrm{association}:\hfill & \mathrm{j} {\mathrm{H}}^{+}+{\mathrm{A}}^{-\mathrm{N}}={\mathrm{H}}_{\mathrm{j}}{\mathrm{A}}^{-(\mathrm{N}-\mathrm{j})}\hfill & {\mathsf{\beta }}_{\mathrm{j}}={\displaystyle \frac{[\mathrm{N}-\mathrm{j}]}{{\mathrm{x}}^{\mathrm{j}} [\mathrm{N}]}}={\displaystyle \frac{{\mathrm{k}}_{\mathrm{N}-\mathrm{j}}}{{\mathrm{k}}_{\mathrm{N}}}}\hfill \end{array}$$

They are related to the acidity constants K_j (single-step deprotonation) as follows:

$$\begin{array}{}\mathrm{(76)}& {\mathrm{k}}_{\mathrm{j}}={\displaystyle \prod _{\mathrm{i}=1}^{\mathrm{j}}{\mathrm{K}}_{\mathrm{i}}}={\displaystyle \frac{{\mathsf{\beta }}_{\mathrm{N}-\mathrm{j}}}{{\mathsf{\beta }}_{\mathrm{N}}}}\hfill & \mathrm{and} {\mathrm{k}}_0=1\hfill \mathrm{(77)}& {\mathsf{\beta }}_{\mathrm{j}}={\displaystyle \prod _{\mathrm{i}=\mathrm{N}+1-\mathrm{j}}^{\mathrm{N}} \frac{1}{{\mathrm{K}}_{\mathrm{i}}}}={\displaystyle \frac{{\mathrm{k}}_{\mathrm{N}-\mathrm{j}}}{{\mathrm{k}}_{\mathrm{N}}}}\hfill & \mathrm{and} {\mathsf{\beta }}_0=1\hfill \end{array}$$

This relationship is sketched in Figure 21. In particular, we have β₁ = 1/K_N, β₂ = (K_N−1K_N)⁻¹, and β_N = 1/k_N. Note that β_j is not the reciprocal of k_j, because β_jk_j ≠ 1 (except for j = N). While dissociation reactions are preferred in hydrochemistry, association reactions are used in other fields (e.g., organic and biochemistry, ligand theory).

The formula in Equation (17) now acquires a kind of partner or mirror equation (based on β_j):

$$\begin{array}{}\mathrm{(78)}& {\displaystyle \frac{1}{{\mathrm{a}}_0}=1+\frac{{\mathrm{k}}_1}{\mathrm{x}}+\frac{{\mathrm{k}}_2}{{\mathrm{x}}^2}+\dots +\frac{{\mathrm{k}}_{\mathrm{N}}}{{\mathrm{x}}^{\mathrm{N}}}}={\displaystyle \sum _{\mathrm{j}=0}^{\mathrm{N}} \frac{{\mathrm{k}}_{\mathrm{j}}}{{\mathrm{x}}^{\mathrm{j}}}}\hfill \mathrm{(79)}& {\displaystyle \frac{1}{{\mathrm{a}}_{\mathrm{N}}}=1+{\mathsf{\beta }}_1\mathrm{x}+{\mathsf{\beta }}_2{\mathrm{x}}^2+\dots +{\mathsf{\beta }}_{\mathrm{N}}{\mathrm{x}}^{\mathrm{N}}}={\displaystyle \sum _{\mathrm{j}=0}^{\mathrm{N}} {\mathsf{\beta }}_{\mathrm{j}}{\mathrm{x}}^{\mathrm{j}}}\hfill \end{array}$$

The ionization fractions a_j express the probability that j protons are released from its fully protonated state H_NA. Conversely, we denote by ã_j = a_N−j the probability that j protons are bound to the fully deprotonated species A^−N. Thus, we have:

j protons released:	aj = (kj /xj) a0	(80)
j protons bound:	ãj = aN-j = (kN−j/xN−j) a0 = (βj xj) aN = (βj xj) ã0	(81)

Special cases are a₀ = ã_N and ã₀ = a_N. The last expression on the right-hand side represents the definition for the ionization fractions in [26] (where ã_j is abbreviated as f_j). The collection of ã_j provides an alternative set of ionization fractions from which acid–base theory can be developed.

2.5.2. Microstates vs. Macrostates

A polyprotic acid H_NA is a molecule with N proton-binding sites. Each site is capable of binding 1 proton; the corresponding site-variable has two states: α_i = 0 (empty) and 1 (occupied). In total, there are 2^N microstates α^(ν) = (α₁, α₂, … α_N), which form a statistical ensemble. The microstates can be grouped into N + 1 macrostates [j], characterized by the number j of protons released from the fully protonated state H_NA (undissociated acid). The number of microstates that form the macrostate [j] is equal to the number of microstates that form the macrostate [N − j] and is given by

$$\left(\begin{array}{c}\mathrm{N}\\ \mathrm{j}\end{array}\right)=\frac{\mathrm{N}!}{(\mathrm{N}-\mathrm{j})! \mathrm{j}!}$$

(82)

Among them are two microstates that are themselves macrostates, namely:

α(0) = (0, 0, … 0): A−N	fully dissociated state (“empty”)	j = N protons released
α(2^N−1) = (1, 1, … 1): HNA	undissociated state (fully occupied)	j = 0 protons released

One of these two states can be chosen as the reference state (with Gibbs energy G = 0). The other 2^N − 1 microstates are then coupled to this reference state by 2^N − 1 microscopic equilibrium constants κ_ν. In this report, the reference state is H_NA = [0] defined by j = 0; the corresponding reaction type is the dissociation in Equation (74). In contrast, for association reactions in Equation (75), the reference state is A^−N.

A monoprotic acid (N = 1) has two macrostates (being the two microstates defined above) and is completely determined by a single acidity constant K₁ = κ₁. The case N = 2 (diprotic acid) has four microstates (0, 0), (0, 1), (1, 0), and (1, 1), from which three macrostates are formed j = 0, 1, and 2, whereby the macrostate j = 1 includes the two microstates (0, 1) and (1, 0). For N = 3, the six microstates and three macrostates are shown in Figure 21 (which constitutes the third row of Pascal’s triangle).

2.5.3. Probability Distributions and Averages

The two ionization fractions associated with the two reference states, a₀ and a_N, are distinguished from all others by having a sigmoidal (S-shaped) curve, as discussed in Section 2.1.4. Their asymptotical values for pH = ±∞ are 0 and 1, so that a₀ and a_N embody cumulative distribution functions which gives the area under the probability density functions f₀ and f_N:

$${\mathrm{f}}_{\mathrm{j}} \equiv \frac{\mathrm{d}{\mathrm{a}}_{\mathrm{j}}}{\mathrm{d} \mathrm{p}\mathrm{H}}=\ln 10 \left(\mathrm{j}-{\mathrm{Y}}_1\right) {\mathrm{a}}_{\mathrm{j}} \iff {\mathrm{a}}_{\mathrm{j}}=\underset{-\infty }{\overset{\mathrm{p}\mathrm{H}}{\int }}{\mathrm{f}}_{\mathrm{j}}(\mathrm{p}{\mathrm{H}}^{\prime }) \mathrm{d} \mathrm{p}{\mathrm{H}}^{\prime } (\mathrm{for} \mathrm{j}=0 \mathrm{and} \mathrm{N})$$

(83)

Two examples are shown in Figure 22, where a_j and its antiderivative f_j are drawn with the same color (red for j = 0 and blue for j = N). The minima and maxima of f₀ and f_N correspond to the inflection points of a₀ and a_N. The asymptotic behavior of a₀ and a_N dictates that the probability densities are automatically normalized to 1:

$${\displaystyle {\int }_{-\infty }^{+\infty }{\mathrm{f}}_{ \mathrm{N}} \mathrm{d} \mathrm{p}\mathrm{H}}=-{\displaystyle {\int }_{-\infty }^{+\infty }{\mathrm{f}}_{ 0} \mathrm{d} \mathrm{p}\mathrm{H}}=1$$

(84)

Here, the minus sign reflects the fact that the slopes of a₀ and a_N are opposite.

Monoprotic acids are special. The two probability densities, shown as blue and red solid curves in the left diagram, are the same except for a minus sign:

$${\mathrm{f}}_{ \mathrm{N}=1}=\ln 10 \frac{{\mathrm{K}}_1/\mathrm{x}}{{\left(1+{\mathrm{K}}_1/\mathrm{x}\right)}^2}=-{\mathrm{f}}_{ 0}$$

(85)

a₀, which is the pH derivative of f₀, is also known as the protonation probability of an isolated site, i.e., the probability that a monoprotic acid remains undissociated:

$${\mathrm{a}}_0=\frac{1}{1+{\mathrm{K}}_1/\mathrm{x}}=\frac{1}{1+{10}^{\mathrm{p}\mathrm{H}-\mathrm{p}{\mathrm{K}}_1}}$$

(86)

This function, which is shown as the dashed red curve in the left diagram in Figure 22, bears a striking similarity to the Fermi–Dirac distribution in statistical physics [37,38], where pK₁ mimics the Fermi energy (except for a prefactor (k_BT)⁻¹ consisting of the Boltzmann constant and temperature).

The ionization fraction a_j, as a number between 0 and 1, represents the probability that j protons are released (at a given pH). The average number $⟨\mathrm{j}⟩$ of protons released is then obtained by averaging over all ionization fractions a_j:

$$⟨\mathrm{j}⟩=\frac{{\displaystyle {\sum }_{j=0}^{\mathrm{N}} \mathrm{j}\cdot {\mathrm{a}}_{\mathrm{j}}}}{{\displaystyle {\sum }_{\mathrm{j}=0}^{\mathrm{N}} {\mathrm{a}}_{\mathrm{j}}}}={\displaystyle \sum _{\mathrm{j}=0}^{\mathrm{N}} \mathrm{j}\cdot {\mathrm{a}}_{\mathrm{j}}}={\mathrm{Y}}_1$$

(87)

In other words, the first moment Y₁ embodies the average number of released protons. Note that $⟨\mathrm{j}⟩$ = [j]_avg/C_T is a real number between 0 and N, as displayed by the Y₁ curves in Figure 6. The result in Equation (87) can be generalized to any power L of a_j, interpreting the moments Y_L (originally introduced in Equation (22)) as expectation values:

$$⟨{\mathrm{j}}^{\mathrm{L}}⟩={\mathrm{Y}}_{\mathrm{L}}$$

(88)

2.5.4. Partition Functions and Moments in Statistics

The ensemble of N + 1 macrostates [j] is specified by N acidity constants K_j (or ionization fractions a_j). As in statistical mechanics, we can introduce a partition function, as a sum over all states [j], from which relevant “thermodynamic” quantities are then obtained by differentiation. Depending on the reference state (either [0] or [N]), there are two ways to define the partition function:

$$\begin{array}{}\mathrm{(89)}& {\displaystyle \mathrm{ref}. \mathrm{state} [0]={\mathrm{H}}_{\mathrm{N}}\mathrm{A}:\hfill & \mathcal{Q}\equiv \frac{[0]}{[0]}+\frac{[1]}{[0]}+\frac{[2]}{[0]}+\dots +\frac{[\mathrm{N}]}{[0]}={\displaystyle \sum _{\mathrm{j}=0}^{\mathrm{N}} \left(\frac{{\mathrm{a}}_{\mathrm{j}}}{{\mathrm{a}}_0}\right)}=\frac{1}{{\mathrm{a}}_0}\hfill }\mathrm{(90)}& {\displaystyle \mathrm{ref}. \mathrm{state} [\mathrm{N}]=\mathrm{A}{-}^{\mathrm{N}}:\hfill & \mathcal{Z}\equiv \frac{[\mathrm{N}]}{[\mathrm{N}]}+\frac{[\mathrm{N}-1]}{[\mathrm{N}]}+\dots +\frac{[0]}{[\mathrm{N}]}={\displaystyle \sum _{\mathrm{j}=0}^{\mathrm{N}} \left(\frac{{\mathrm{a}}_{\mathrm{j}}}{{\mathrm{a}}_{\mathrm{N}}}\right)}=\frac{1}{{\mathrm{a}}_{\mathrm{N}}}\hfill }\end{array}$$

The first approach is based on dissociation reactions (defined by cumulative acidity constants k_j), the second on association reactions (defined by β_j). In particular, the so-called binding polynomial $\mathcal{Z}$ was introduced in [32] for the description of ligands that bind to macromolecules. Both partition functions are interrelated by $\mathcal{Q}/\mathcal{Z}$= a_N/a₀ = k_N/x^N.

The two partition functions belong to the isothermal–isobaric ensemble (also called the canonical NPT ensemble, where N, P, T refer to particle number, pressure, and temperature) which leads straight to the Gibbs energy (J/mol):

$$\mathrm{G}=-\mathrm{R}\mathrm{T} \ln \mathcal{Q}$$

(91)

where R and T are the gas constant and temperature, respectively. In thermodynamics, the partial derivative of G with respect to the chemical potential μ equals the (average) particle number. Applying it to H⁺ yields (with µ_H+ = RT ln x):

$$⟨\mathrm{j}⟩={\left(\frac{\partial \mathrm{G}}{\partial {\mathsf{\mu }}_{{\mathrm{H}}^{+}}}\right)}_{\mathrm{P},\mathrm{T}}=-\frac{\mathrm{d}\ln \mathcal{Q}}{\mathrm{d}\ln \mathrm{x}}$$

(92)

The same applies to $\mathcal{Z}$, except for a minus sign. From Equations (89) and (90) and using Equation (A16), we then obtain:

$$\begin{array}{}\mathrm{(93)}& \mathrm{average} \mathrm{number} \mathrm{of} \mathit{released} \mathrm{protons}:\hfill & ⟨\mathrm{j}⟩=-\frac{\mathrm{d}\ln \mathcal{Q}}{\mathrm{d}\ln \mathrm{x}}=\frac{d\ln {\mathrm{a}}_0}{\mathrm{d}\ln \mathrm{x}}={\mathrm{Y}}_1\hfill \mathrm{(94)}& \mathrm{average} \mathrm{number} \mathrm{of} \mathit{bound} \mathrm{protons}:\hfill & \mathrm{N}-⟨\mathrm{j}⟩=\frac{\mathrm{d}\ln \mathcal{Z}}{\mathrm{d}\ln \mathrm{x}}=-\frac{\mathrm{d}\ln {\mathrm{a}}_{\mathrm{N}}}{\mathrm{d}\ln \mathrm{x}}=\mathrm{N}-{\mathrm{Y}}_1\hfill \end{array}$$

Comparing it to the outcome of Equation (87), everything falls into the right place nicely. Note: In the particularly simple case of monoprotic acids (N = 1), the average number of protons released is just a₁, and the average number of bound protons is a₀.

Higher-order derivatives of G (or ln Q) with respect to µ deliver fluctuation measures (in the form of variance, skewness and kurtosis). In particular, for the fluctuation of j around its mean, δj ≡ j − ⟨j⟩, we get:

$$⟨(\mathsf{\delta }\mathrm{j}{)}^{\mathrm{L}}⟩={(-)}^{\mathrm{L}} {\left(\frac{\mathrm{d}}{\mathrm{d}\ln \mathrm{x}}\right)}^{\mathrm{L}} \ln \mathcal{Q}=\frac{1}{{\left(\ln 10\right)}^{\mathrm{L}-1}} {\left(\frac{\mathrm{d}}{\mathrm{d} \mathrm{p}\mathrm{H}}\right)}^{\mathrm{L}-1}{\mathrm{Y}}_1 \mathrm{for} \mathrm{L} \ge 2$$

(95)

For L = 1, this expression is zero because ⟨δj⟩ = ⟨j − Y₁⟩ = Y₁ − Y₁ = 0. Most interesting are the moments for L = 2, 3, and 4:

$${\displaystyle \begin{array}{}\mathrm{(96)}& \mathrm{variance} {\mathsf{\sigma }}^2:\hfill & ⟨(\mathrm{j}-{\mathrm{Y}}_1{)}^2⟩={\mathrm{Y}}_2-{{\mathrm{Y}}_1}^2\hfill & ={\displaystyle \frac{1}{\ln 10}} \left({\displaystyle \frac{\mathrm{d} {\mathrm{Y}}_1}{\mathrm{d} \mathrm{p}\mathrm{H}}}\right)\hfill \mathrm{(97)}& \mathrm{skewness}:\hfill & ⟨(\mathrm{j}-{\mathrm{Y}}_1{)}^3⟩={\mathrm{Y}}_3-3{\mathrm{Y}}_2{\mathrm{Y}}_1+2{{\mathrm{Y}}_1}^3\hfill & ={\displaystyle \frac{1}{{\left(\ln 10\right)}^2}} \left({\displaystyle \frac{{\mathrm{d}}^2 {\mathrm{Y}}_1}{\mathrm{d} \mathrm{p}{\mathrm{H}}^2}}\right)\hfill \mathrm{(98)}& \mathrm{excess} \mathrm{kurtosis}:\hfill & ⟨(\mathrm{j}-{\mathrm{Y}}_1{)}^4⟩-3{\mathsf{\sigma }}^4=({\mathrm{Y}}_4-4{\mathrm{Y}}_3{\mathrm{Y}}_1+6{\mathrm{Y}}_2{{\mathrm{Y}}_1}^2-3{{\mathrm{Y}}_1}^4)-3{\mathsf{\sigma }}^4\hfill & ={\displaystyle \frac{1}{{\left(\ln 10\right)}^3}} \left({\displaystyle \frac{{\mathrm{d}}^3 {\mathrm{Y}}_1}{\mathrm{d} \mathrm{p}{\mathrm{H}}^3}}\right)\hfill \end{array}}$$

(These formulas can be verified using Equations (A18) through (A20) in Appendix C).

Skewness measures the lack of symmetry. Kurtosis is a measure of whether the distribution is heavy tailed or light tailed; “excess kurtosis” is kurtosis relative to the normal distribution (with a kurtosis of 3).

2.5.5. Decoupled Sites Representation and Simms Constants

In physics, microscopically complicated systems are often described by the concept of noninteracting quasiparticles (e.g., phonons, magnetons, excitons). This idea was adopted in the decoupled sites representation (DSR) [33,34] by replacing the polyprotic acid with N interacting proton-binding sites by a sum of N noninteracting quasisites. As we know from statistical mechanics, partition functions of noninteracting entities factorize into products of individual partition functions (each of them characterizing a virtual monoprotic acid with g_i as its equilibrium constant). Applying this to the partition functions in Equations (89) and (90) using Equations (78) and (79), we obtain:

$$\begin{array}{}\mathrm{(99)}& {\displaystyle \mathcal{Q}=1+\frac{{\mathrm{k}}_1}{\mathrm{x}}+\frac{{\mathrm{k}}_2}{{\mathrm{x}}^2}+\dots +\frac{{\mathrm{k}}_{\mathrm{N}}}{{\mathrm{x}}^{\mathrm{N}}}\hfill }={\displaystyle \prod _{\mathrm{i}=1}^{\mathrm{N}}}{\displaystyle \left(1+\frac{{\mathrm{g}}_{\mathrm{i}}}{\mathrm{x}}\right)}\hfill \mathrm{(100)}& {\displaystyle \mathcal{Z}=1+{\mathsf{\beta }}_1\mathrm{x}+{\mathsf{\beta }}_2{\mathrm{x}}^2+\dots +{\mathsf{\beta }}_{\mathrm{N}}{\mathrm{x}}^{\mathrm{N}}\hfill }={\displaystyle \prod _{\mathrm{i}=1}^{\mathrm{N}}}\left({\displaystyle 1+\frac{\mathrm{x}}{{\mathrm{g}}_{\mathrm{i}}}}\right)\hfill \end{array}$$

The N equilibrium constants g_i, also known as Simms constants [20,29,30], differ from the N acidity constants K_j of the real polyprotic acid (which are deprotonation constants of the jth proton). The relationship can be established by multiplying out the N products in (99) and (100). After a coefficient comparison, the conversion between the cumulative equilibrium constants and Simms constants is found to be:

$$\begin{array}{}\mathrm{(101)}& {\mathrm{k}}_1={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}} {\mathrm{g}}_{\mathrm{i}}},\hfill & {\mathrm{k}}_2={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}} {\displaystyle \sum _{\mathrm{j}<\mathrm{i}}^{\mathrm{N}}}{\mathrm{g}}_{\mathrm{i}}{\mathrm{g}}_{\mathrm{j}}},\hfill & \dots \hfill & {\mathrm{k}}_{\mathrm{N}}={\displaystyle \prod _{\mathrm{i}=1}^{\mathrm{N}} {\mathrm{g}}_{\mathrm{i}}}\hfill \mathrm{(102)}& {\mathsf{\beta }}_1={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}} \frac{1}{{\mathrm{g}}_{\mathrm{i}}}},\hfill & {\mathsf{\beta }}_2={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}} {\displaystyle \sum _{\mathrm{j}<\mathrm{i}}^{\mathrm{N}}}\frac{1}{{\mathrm{g}}_{\mathrm{i}}{\mathrm{g}}_{\mathrm{j}}}},\hfill & \dots \hfill & {\mathsf{\beta }}_{\mathrm{N}}={\displaystyle \prod _{\mathrm{i}=1}^{\mathrm{N}} \frac{1}{{\mathrm{g}}_{\mathrm{i}}}}\hfill \end{array}$$

The full set of equations for the k_j encompasses 2^N − 1 single terms (g_1, g₂, g₃, g₁g₂, g₂g₃, g₁g₃, g₁g₂g₃ for N = 3, for example), each of them represents one microscopic equilibrium constant. For N = 1 we have the trivial result:

monoprotic acid:   K₁ = g₁ = k₁ = 1/β₁

(103)

Note that a quasisite is a collective phenomenon formed by the interaction of all (real) proton-binding sites.

2.5.6. Polyprotic Acids as Mixtures of Monoprotic Acids

Since the description of acid–base equilibria is centered on n = Y₁ + w/C_T in Equation (41), it is sufficient to focus on Y₁. Applying DSR from Section 2.5.5 for the polyprotic acid, we get:

$${\mathrm{Y}}_1^{}={\mathrm{Y}}_1^{ (\mathrm{mono} 1)}+{\mathrm{Y}}_1^{ (\mathrm{mono} 2)}+\cdots +{\mathrm{Y}}_1^{ (\mathrm{mono} \mathrm{N})}$$

(104)

whereby

$${\mathrm{Y}}_1^{ (\mathrm{mono} \mathrm{j})}={\mathrm{a}}_1^{(\mathrm{mono} \mathrm{j})}=\frac{1}{1+\mathrm{x}/{\mathrm{g}}_{\mathrm{j}}}\left(={\displaystyle \sum _{\mathrm{i}=\mathrm{j}}^{\mathrm{N}} {\mathrm{a}}_{\mathrm{i}}^{}}\right)$$

(105)

The last equation in brackets indicates the relation to the ionization fractions a_j based on K_j values.

In fact, the mathematical treatment of a sum of monoprotic acids is much easier to handle than the polyprotic acid as a whole. However, the use of virtual acidity constants g_i instead of the K_j shifts the problem to a complicated relationship between the g_i and K_j values (cf. Equations (101) and (102)).

The criterion for representing the H_NA by a mixture of N monoprotic acids (each of amount C_T and one acidity constant K_j) is that all K_j values should be well “separated” from each other:

K₁ ≫ K₂ ≫ … ≫ K_N  ⇔  K₁ ≈ g₁, K₂ ≈ g₂, … K_N ≈ g_N

(106)

This assertion is not immediately obvious, but you can clarify it easily for N = 2 or 3, for example.

2.5.7. The World of Acidity Constants

Polyprotic acids are specified by the N acidity constants K_j (or pK_j values), which describe the step-by-step dissociation without any indication from which specific site the H⁺ is released. So, any of all the N binding sites can contribute to K₁ (making K₁ the largest value). For K₂, as the second dissociation step, the proton comes from any one of the remaining N − 1 sites, and so on. This implies the order by size: K₁ > K₂ > … > K_N.

Consider a molecule with identical quasisites of strength g. We then have:

$${\mathrm{k}}_{\mathrm{j}}=\left(\begin{array}{c}\mathrm{N}\\ \mathrm{j}\end{array}\right)\mathrm{g} \mathrm{and} {\mathrm{K}}_{\mathrm{j}}=\frac{\mathrm{N}-\mathrm{j}+1}{\mathrm{j}} \mathrm{g}$$

(107)

where K_j decreases from K₁ = Ng to K_N = g/N. This can also be expressed by pK_N − pK₁ = 2 lg N, which is the minimum separation between the first and the last dissociation constant. For a diprotic and triprotic acid it is 0.60 and 0.95, respectively. The minimum requirement for two adjacent acidity constants is:

$$\frac{{\mathrm{K}}_{\mathrm{j}}}{{\mathrm{K}}_{\mathrm{j}+1}}=\frac{\mathrm{j}+1}{\mathrm{j}} \frac{\mathrm{N}-\mathrm{j}+1}{\mathrm{N}-\mathrm{j}}$$

(108)

This implies once again the sequence: K₁ > K₂ > … > K_N. Conversely, the more the g_i deviate from each other, the more the K_j (and pK_j) values drift apart until the condition in Equation (106) is reached (typical for inorganic acids).

The nonlinear relationships between the different types of acidity constants is summarized in

Table 4. Macroscopic and microscopic acidity constants.

N Acidity Constants	N + 1 Cumulative Constants for		N Simms Constants	2N − 1 Microscopic Constants
	Dissociation	Association
K1, K2, …, KN	k0 = 1	β0 = 1	g1, g2, …, gN	g1, g2, g3, … g1g2, g2g3, … g1g2g3, … g1g2⋯gN
	${\mathrm{k}}_{\mathrm{j}}={\displaystyle \prod _{\mathrm{i}=1}^{\mathrm{j}}{\mathrm{K}}_{\mathrm{i}}}$	${\mathsf{\beta }}_{\mathrm{j}}={\displaystyle \prod _{\mathrm{i}=\mathrm{N}+1-\mathrm{j}}^{\mathrm{N}} \frac{1}{{\mathrm{K}}_{\mathrm{i}}}}$	(implicit functions of Equations (101) and (102))

and Figure 23. The link between the acidity constants K_j and Simms constant g_i is particularly tricky and obtainable via the cumulative constants (which are sums over all combinations of products of g_i). To extract the g_i values from a set of given acidity constants requires solving polynomials of order N.

3. Applications

3.1. More About Acids

3.1.1. Strong Acids vs. Weak Acids

Monoprotic Acids. Unlike weak acids, strong acids dissociate completely in water. Let us consider a monoprotic acid specified by K_a and the amount C_T ≡ [HA]_T (which is de facto the acid’s initial concentration before it dissolves). In the equilibrium state, the total concentration splits into an undissociated and a dissociated part:

C_T = [HA] + [A⁻]  or  1 = a₀ + a₁

(109)

The difference between strong and weak acids is summarized in

Table 5. Strong vs. weak acids (greatly simplified).

	Strong Acid	Weak Acid
acidity constant	Ka ≫ 1	Ka ≤ 1
pKa = −lg Ka	pKa < 0	pKa > 0
[H+] ≈ {H+} = 10−pH	[H+] ≈ CT	[H+] ≪ CT
undissociated acid	[HA] ≈ 0 or a0 ≈ 0	[HA] ≈ CT or a0 ≈ 1
dissociated acid	[A−] ≈ CT or a1 ≈ 1	[A−] ≪ CT or a1 ≪ 1

.

Polyprotic Acids. Protons are released sequentially one after the other, with the first proton being the fastest and most easily lost, followed by the second, then the third, etc. This yields the following ranking of acidity constants of a polyprotic acid (as discussed in Section 2.5.7):

K₁ > K₂ > K₃ > …  or  pK₁ < pK₂ < pK₃ < …

Examples are given in Table 1, Table 12, and Table 14.

The concept/distinction in Table 5 can also be applied for N-protic acids if we rename the acidity constant K_a by the 1st dissociation constant K₁. From Equation (17) we then get:

$$\mathrm{undissociated} \mathrm{fraction}:\hspace{1em}{\mathrm{a}}_0=\frac{[{\mathrm{H}}_{\mathrm{N}}\mathrm{A}]}{{\mathrm{C}}_{\mathrm{T}}}\approx \frac{1}{1+{\mathrm{K}}_1/\mathrm{x}}$$

(110)

The undissociated fraction as a function of pH is displayed in Figure 24 for several acids. As expected, strong acids are completely dissociated in real-world applications (pH ≥ 0). The small circles mark the position of the corresponding pK₁ values (which are the inflection points of a₀).

3.1.2. Weak Acids vs. Diluted Acids

A weak acid and a diluted acid are two different things. The first relies on the acidity constants K_a (which is a thermodynamic property of the acid that nobody can change), while the second relies on the amount C_T of a given acid:

weak acid ↔ strong acid	⇔	small Ka ↔ large Ka
dilute acid ↔ concentrated acid	⇔	small CT ↔ large CT

One cannot make a weak acid strong, but one can change the degree of dilution (or concentration).

Table 6. Degree of strength vs. degree of dilution.

	Degree of Strength			Degree of Dilution
determined by	acidity constant Ka			amount of acid CT
relationships	weak acid small Ka (positive pKa	↔ ↔ ↔	strong acid large Ka negative pKa)	diluted acid small CT	↔ ↔	concentr. Acid large CT
compares	two different acids			dilution of the same acid
describes	release of H+			dilution of H+
type	fundamental property (cannot be changed)			control parameter (can be changed)

summarizes the principal differences between the degree of strength and the degree of dilution. The basic idea is also depicted in Figure 25. For polyprotic acids, replace K_a by K₁.

3.1.3. Strong Polyprotic Acids (Simplification)

For the same N, the mathematical description of strong N-protic acids is simpler than that of weak acids. This is because strong acids never occur in the undissociated state at pH ≥ 0 (at least one H⁺ is always released). Since the amount of the undissociated species is zero, [0] = 0 and a₀ = 0, we can skip the calculation of the 1st dissociation step. Thus, we remove Equation (28) or Equation (36) from the set of N + 3 equations and ignore the acidity constant K₁ (keeping in mind that K₁ is a large number). That is good news, because K₁ of strong acids is often not known precisely enough.

The cumulative acidity constants k_j in Equation (18) and the ionization fractions a_j in Equation (17) simplify as follows:

k₁ = 1, k₂ = K₂, k₃ = K₂K₃, etc.

(111)

$${\mathrm{a}}_{\mathrm{j}}=\left(\frac{{\mathrm{k}}_{\mathrm{j}}}{{\mathrm{x}}^{\mathrm{j}-1}}\right) {\mathrm{a}}_1 \mathrm{with} {\mathrm{a}}_1={\left(1+\frac{{\mathrm{K}}_2}{\mathrm{x}}+\frac{{\mathrm{K}}_2{\mathrm{K}}_3}{{\mathrm{x}}^2}+\dots \right)}^{-1} \mathrm{for} \mathrm{j} \text{>} 1$$

(112)

Polynomials. For strong acids, the polynomial in Equation (44) becomes one degree less in x (i.e., the summation now starts with j = 1):

$$0={\displaystyle {\sum }_{\mathrm{j}=1}^{\mathrm{N}}}\left\{ {\mathrm{x}}^2+(\mathrm{n}-\mathrm{j} ) {\mathrm{C}}_{\mathrm{T}}\mathrm{x}-{\mathrm{K}}_{\mathrm{w}}\right\} {\mathrm{k}}_{\mathrm{j}}{\mathrm{x}}^{\mathrm{N}-\mathrm{j}}$$

(113)

Example N = 1. For a strong monoprotic acid, the sum in Equation (113) runs only over one term, j = 1. Using k₁ = 1, we get a quadratic equation:

0 = x² + (n − 1) C_T x − K_w

Note that this equation contains no acidity constant. Example N = 2. For a strong diprotic acid, the polynomial in Equation (113) becomes a cubic equation:

0 = x³ + {(n − 1) C_T + K₂} x² − {(n − 2) C_T K₂ − K_w} x − K₂K_w

This equation can also be obtained directly from Equation (46) by applying the condition x/K₁ = 0.

3.1.4. Mixtures of Acids

Until now we considered acid–base systems with one polyprotic acid. It is not difficult to extend the approach to mixtures of several polyprotic acids:

acid a + acid b + acid c + …   with amounts C_a, C_b, C_c, …

The total sum of all acids will be abbreviated by C_T = C_a + C_b + C_c + …. The equivalent fraction n = n(x), i.e., the titration curve, of the multi-acid system (plus a strong base of amount C_B = nC_T) is then described by:

$$\mathrm{n}={\tilde{\mathrm{Y}}}_1(\mathrm{x})+\frac{\mathrm{w}(\mathrm{x})}{{\mathrm{C}}_{\mathrm{T}}}$$

(114)

It has the same structure as Equation (41), except Y₁ is replaced by the generalized moment Ỹ₁ as a superposition of the individual acid’s Y₁:

$${\tilde{\mathrm{Y}}}_1={\mathrm{n}}_{\mathrm{a}}{\mathrm{Y}}_1^{(\mathrm{a})}+{\mathrm{n}}_{\mathrm{b}}{\mathrm{Y}}_1^{(\mathrm{b})}+{\mathrm{n}}_{\mathrm{c}}{\mathrm{Y}}_1^{(\mathrm{c})}+\dots \mathrm{with} \mathrm{coefficients} {\mathrm{n}}_{\mathsf{\alpha }}=\frac{{\mathrm{C}}_{\mathsf{\alpha }}}{{\mathrm{C}}_{\mathrm{T}}}$$

(115)

The generalized moments are built from the ionization fractions in the usual way (i.e., according to Equation (22)):

$${\tilde{\mathrm{Y}}}_{\mathrm{L}}^{(\mathsf{\alpha })}={\displaystyle \sum _{\mathrm{j}=1}^{{\mathrm{N}}_{\mathsf{\alpha }}}}{\mathrm{j}}^{ \mathrm{L}}{\mathrm{a}}_{\mathrm{j}}^{(\mathsf{\alpha })}$$

(116)

Here the ionization fractions a_j^(α) are determined by the (cumulative) acidity constants of the individual acid’s k_j^(α), according to Equation (17). The sum runs from j = 1 to N^(α), which is the number of protons of acid α = a, b, c.

Example. Given is a mixture of two acids: phosphoric acid plus carbonic acid with equal amounts: C_phos = C_carb = C_T/2. The first moment Ỹ₁ of the two-acid system is displayed as the blue curve in the upper diagram of Figure 26. It is simply the sum of Y₁^(phos) and Y₁^(carb). This curve approaches the value 5 when pH → 14, which is the degree of the two-acid system (N = 3 + 2 = 5). The bottom diagram in Figure 26 displays the individual ionization fractions of the two acids. To recall: the blue curve (Y₁) in the top diagram represents the “titration curve” in the high-C_T limit.

3.2. Equivalence Points and Ionization Fractions

3.2.1. Trajectories of EPs in pH–C_T Diagrams

Equation (60) can be rearranged into the form

$${\mathrm{C}}_{\mathrm{T}}=\frac{\mathrm{w}(\mathrm{p}\mathrm{H})}{\mathrm{n}-{\mathrm{Y}}_1(\mathrm{p}\mathrm{H})}$$

(117)

Now it is easy to plot all EP_n values as distinct curves/trajectories into a pH–C_T diagram (one curve for an integer or half-integer value of n). This is performed in Figure 27 for four acids. The dashed curves and lines are the approximations overtaken from Figure 14.

The genesis of the EP_n trajectories is explained in Figure 28, which consists of two diagrams. In the top diagram, there are the two uncoupled subsystems located at the opposite ends of the C_T scale:

1-component system “acid”: C_T → ∞

1-component system “H₂O”: C_T → 0

The bottom diagram shows the situation when the two isolated subsystems are coupled together. Starting at pH = 7, the curves fan out when C_T increases until they fit the “pure-acid” values at the top of the chart. The whole behavior is choreographed by Equation (117). The subsystem “H₂O” overtakes the rule when C_T drops below 10⁻⁷ M, which is just the amount of H⁺ and OH⁻ in pure water.

The behavior of the two isolated subsystems can be deduced from Equation (117) by setting either the nominator or the denominator to zero:

$${\mathrm{C}}_{\mathrm{T}}=\frac{\mathrm{w}(\mathrm{p}\mathrm{H})}{\mathrm{n}-{\mathrm{Y}}_1(\mathrm{p}\mathrm{H})}\iff \{\begin{array}{l}\mathrm{w}=0\hfill \\ \mathrm{n}-{\mathrm{Y}}_1=0\hfill \end{array}\begin{array}{c}\Rightarrow \\ \Rightarrow \end{array}\begin{array}{l}{\mathrm{C}}_{\mathrm{T}}=0\hfill \\ {\mathrm{C}}_{\mathrm{T}} \to \infty \hfill \end{array}$$

(118)

In math jargon, the corresponding pH values of the “pure acid” subsystem are the poles of Equation (117). On the other hand, the single EP of “pure H₂O” is at the position where the nominator in Equation (117) becomes zero (which is exactly at pH = 7): EP of pure H₂O ⇔ 0 = w(x) ⇔ C_T = 0.

3.2.2. EPs as Inflection Points of Titration Curves

The (internal) equivalence points are usually acknowledged as inflection points of titration curves. This statement, however, is not rigorously applicable. Mathematically, an inflection point is a point on a curve at which the curvature or concavity changes sign (plus to minus or vice versa). This happens at points where the 2nd derivative of the titration curve n(pH) is zero:

$$\mathrm{criterion} \mathrm{for} \mathrm{inflection} \mathrm{points}: \frac{{\mathrm{d}}^2\mathrm{n}}{\mathrm{d} \mathrm{p}{\mathrm{H}}^2}=\frac{\mathrm{d}\mathsf{\beta }}{\mathrm{d} \mathrm{p}\mathrm{H}}=0$$

(119)

1-Component System. For the 1-component system (i.e., C_T ≫ w), the formulas for the buffer capacity and its pH derivative in Equations (72) and (73) simplify:

$$\begin{array}{}\mathrm{(120)}& \mathsf{\beta }\left(\mathrm{x}\right)=(\ln 10) ({\mathrm{Y}}_2-{{\mathrm{Y}}_1}^2)\hfill \mathrm{(121)}& \frac{\mathrm{d}\mathsf{\beta }}{\mathrm{d} \mathrm{p}\mathrm{H}}=(\ln 10{)}^2 ({\mathrm{Y}}_3-3{\mathrm{Y}}_1{\mathrm{Y}}_2+2{{\mathrm{Y}}_1}^3)\hfill \end{array}$$

The 2N − 1 internal equivalence points are then fully determined by the pK_j values, while the Y_L’s are given by (see Appendix C.1):

YL = ½ {(j − 1)L + j2}	for semi-EPn at pKj	(n = j − ½)	(122)
YL = jL	for EPn at pHj ≡ ½ (pKj + pKj+1)	(n = j)	(123)

Inserting the Y_L’s into Equation (120) yields for

semi-EPn:	β(pKj) = ¼ (ln 10) = 0.576	(maximum)	(124)
EPn:	(pHj) = 0	(minimum)	(125)

Plugging the Y_L values into Equation (121) results—after doing some school algebra—in exactly the required condition for inflection points:

$$\frac{\mathrm{d}\mathsf{\beta }}{\mathrm{d} \mathrm{p}\mathrm{H}}({\mathrm{pK}}_{\mathrm{j}})=0\hspace{1em}\hspace{1em}\mathrm{and}\hspace{1em}\hspace{1em}\frac{\mathrm{d}\mathsf{\beta }}{\mathrm{d} \mathrm{p}\mathrm{H}}({\mathrm{pH}}_{\mathrm{j}})=0$$

(126)

The zeros of dβ/dpH for mono-, di-, and triprotic acids are shown as small circles in Figure 19. Of special interest are the semi-EPs at pK₁, pK₂ and pK₃ where the buffer intensity (green curves) reaches its maximum value 0.576, as predicted by Equation (124). At exactly these pK values the titration curve (blue) has its inflection points. These findings are summarized in

Table 7. Equivalence points as local extrema and inflection points.

Function/Curve	semi-EPn (half-integer n = j − ½) at pKj	EPn (integer n = j) at pHj
titration curve n = n(pH) (normalized buffer capacity)	inflection points	inflection points
buffer intensity β = dn/dpH	maxima	minima
dβ/dpH = d2n/dpH2	zeros	zeros
ionization fraction aj	inflection point	maximum

.

3-Component System. The additional terms (w + x)/C_T and w/C_T that enter the 3-component system in Equations (72) and (73) disturb the nice and simple picture discussed above. This can be seen if you compare Figure 19 (for the 1-component system) with Figure 20 (for the 3-component system): The function dβ/dpH (red curve) changes its shape and, thus, the position and number of its zeros. The deviation from the “ideal case” grows the more the acid is diluted (i.e., the smaller C_T). A strict and simple assignment between zeros, inflection points, and EPs is no longer possible.

3.2.3. Ionization Fractions—Two Approaches

The formula for the ionization fractions in Equation (17) can be approximated in two different ways:

• Approach 1: “piecewise log-scale approximation” for lg a_j
• Approach 2: “midpoint approximation” for a_j

Approach 1. This is an approximation for the logarithm of a_j, i.e., for lg a_j. It is exactly the approach used in textbooks as a graphical method for solving algebraic equations of equilibrium systems in double-log diagrams.

The approximate formula for lg a_j provides a “curve” consisting of several linear functions in pH (straight lines):

lg a_j ≈ (j − i) pH + (pk_i − pk_j)  for the ith interval

(127)

where pk_i = pK₁ + pK₂ + … + pK_i and pk₀ = 0. This approach is shown (by the colored dashed lines) in the upper-left diagram in Figure 29.

Approach 2. This approach is motivated by the fact that the curves in Figure 3 look pretty elementary, so one wonders if they can be described by a much simpler formula. This is indeed so; you can replace the exact formulas in Equation (17) by:

$${\mathrm{a}}_0 \approx {\left(1+\frac{{\mathrm{K}}_1}{\mathrm{x}}\right)}^{-1}, {\mathrm{a}}_{\mathrm{j}} \approx {\left(\frac{\mathrm{x}}{{\mathrm{K}}_{\mathrm{j}}}+1+\frac{{\mathrm{K}}_{\mathrm{j}+1}}{\mathrm{x}}\right)}^{-1} (\mathrm{for} \mathrm{j}=1 \mathrm{to} \mathrm{N}-1), \mathrm{and} {\mathrm{a}}_{\mathrm{N}} \approx {\left(\frac{\mathrm{x}}{{\mathrm{K}}_{\mathrm{N}}}+1\right)}^{-1}$$

The approach relies on no more than two (adjacent) pK values; all other pK values are ignored. Thus, the approach is exact for mono- and diprotic acids; for acids with N > 2, it deviates from the exact description, albeit insignificantly. The small deviations can only be recognized in logarithmic plots—as shown for phosphoric acid (N = 3) in the top right diagram in Figure 29.

Summary. The two approaches are complementary, as shown in Figure 29. Approach 1 offers a very clever approximation in log-plots but fails to reproduce the S-shaped and bell-shaped curves in pH-a_j diagrams (dashed curves in the bottom left diagram). Conversely, Approach 2 reproduces the a_j curves perfectly, but if we look more closely, we see deviations in the log-plots for values below 10⁻⁵ (dashed curves in the top-right diagram).

3.3. Alkalinity and Carbonate System

3.3.1. Alkalinity and Acidity

In carbonate systems, ANC is known as alkalinity and BNC as acidity. Again, we have to distinguish between different types of alkalinity and acidity depending on the reference point EP_n chosen. The carbonic acid has three integer-valued EPs; hence there are three types of alkalinity (cf. Figure 30):

total alkalinity (M alkalinity):	[Alk]	= [ANC]n=0
P alkalinity:	[P-Alk]	= [ANC]n=1
caustic alkalinity:	[OH-Alk]	= [ANC]n=2

Correspondingly, there are three types of acidity:

mineral acidity:	[H-Acy]	= [BNC]n=0
CO2 acidity:	[CO2-Acy]	= [BNC]n=1
acidity:	[Acy]	= [BNC]n=2

Alkalinity and acidity are complementary. From Equation (66) we get:

[ANC]0 = − [BNC]0	⇒	[Alk] = − [H-Acy]
[ANC]1 = − [BNC]1	⇒	[P-Alk] = − [CO2-Acy]
[ANC]2 = − [BNC]2	⇒	[OH-Alk] = − [Acy]

Of all three types of alkalinity, total alkalinity is the most important; it is given by

[Alk] ≡ [ANC]₀ = C_B = nC_T

(128)

Using Equation (67), the difference between total alkalinity (M-alkalinity) and P-alkalinity yields the amount of C_T as follows:

[Alk] − [P-Alk] = [ANC]₀ − [ANC]₁ = C_T (= DIC)

(129)

In carbonate systems, this is just the molar concentration of dissolved inorganic carbon (DIC).

3.3.2. pH as Reference Point of ANC and BNC

In Section 2.4.1, ANC and BNC have been defined with respect to an equivalence point EP_n. ANC and BNC can also be defined with respect to a particular pH value (which can be any chosen value). In practice it is common to use the pH of the equivalence points EP₀ and EP₁ of the carbonate system:

EP0: pH ≈ 4.3

EP1: pH ≈ 8.2

The two EPs are shown as yellow dots in Figure 30. The usefulness of this choice is that these are the pH values of common indicators: indicator methylorange (titration endpoint 4.2 to 4.5) and indicator phenolphthalein (titration endpoint 8.2 to 8.3). The measured amount of strong acid or base to reach these endpoints are called:

ANC to pH 4.3:	[ANC]pH 4.3	(≈ [Alk])
ANC to pH 8.2:	[ANC]pH 8.2	(≈ [P-Alk])
BNC to pH 4.3:	[BNC]pH 4.3	(≈ − [Alk])
BNC to pH 8.2:	[BNC]pH 8.2	(≈ − [P-Alk])

The measured “ANC to pH 4.3” corresponds to the total alkalinity (or M-alkalinity) of the system; the measured “ANC to pH 8.2” to the P-alkalinity. Here, the abbreviation “M” refers to the indicator methylorange and “P” to phenolphthalein.

3.3.3. Acid–Base Titration with H₂CO₃ as Titrant

During titration, a titrant is added to the analyte to reach the target pH or equivalence point. Two cases (which are opposite of each other) will be considered:

var A 

100 mM H₂CO₃ solution is titrated by a strong base/acid (NaOH and HCl)

var B 

100 mM NaOH solution is titrated by H₂CO₃

In var A C_T is kept fixed (and C_B is varied), while in var B C_B is kept fixed (and C_T is varied). The aim is to calculate the carbonate speciation as a function of pH. In both cases, we start with the ionization fractions a_j (based on Equation (17) and shown in the bottom left diagram in Figure 3), which are the same for var A and var B. From each a_j, we then get the species concentration by multiplication with C_T: [j] = C_T a_j. The main point is that var A and var B differ in the C_T value:

var A	CT = const	with CT = 100 mM
var B	CT = (CB − w)/Y1	with CB = 100 mM

where the last line is taken from Equation (43).

The obtained species distribution is displayed in Figure 31: var A (left) and var B (right diagrams). The gray curve represents the total concentration C_T as the sum of all carbonate species. The top and bottom diagrams only differ by the concentration scale: the y axis is linear or logarithmic, respectively.

Although both variants rely on exactly the same ionization fractions, the pH dependence of the species in var A and var B differ significantly. While the species distribution in var A (top left) replicates the shapes of the ionization fractions, var B is completely out of line. [Note: In var B, pH < 5 is not available in practice.]

3.3.4. Open vs. Closed CO₂ System

In an open CO₂ system, the solution is in equilibrium with the CO₂ of the atmosphere. Let us compare it with the closed system:

var A 

titration of 100 mM H₂CO₃ solution as “closed CO₂ system” (as in Section 3.3.3)

var C 

titration of 100 mM H₂CO₃ solution as “open CO₂ system”

As in Section 3.3.3, we start with the same ionization fractions a_j for both var A and var C, as shown in in the left bottom diagram in Figure 3. As in Section 3.3.3, the two variants differ only in the functional dependence of C_T, which will be derived now.

The “open system” is described by Henry’s law that partitions the CO₂ between the aqueous and gas phase: CO₂(aq) is proportional to CO₂(g), whereas CO₂(aq) is the undissociated acid H₂CO₃, i.e., the uncharged species [0]. Thus, we can write:

[0] = K_H · P    with   K_H = 10^−1.47 M/atm (at 25 °C)

(130)

and with P as the partial CO₂ pressure. Using [0] = C_T a₀, we obtain:

$$\mathrm{open} {\mathrm{CO}}_2 \mathrm{system}:{\mathrm{C}}_{\mathrm{T}}=\frac{{\mathrm{K}}_{\mathrm{H}}\mathrm{P}}{{\mathrm{a}}_0}$$

(131)

Thus, the two variants differ in the C_T value:

var A	CT = const	with CT = 100 mM
var C	CT = KH P/a0	with P = 0.00039 atm (= 10−3.408 atm)

The carbonate speciation is displayed in Figure 32: var A (left) and var C (right diagrams).

The conclusions are similar to Section 3.3.3. Although both variants rely on exactly the same ionization fractions, the pH dependence of the species in var A and var C is completely different. While the species distribution in var A (top left) replicates the shapes of the ionization fractions, var C does not. The more alkaline the solution becomes, the more CO₂ is sucked out of the atmosphere (which increases the C_T exponentially). [Note: In var C, pH < 5 is not available in practice.]

Resume. The three variants (var A, var B, var C) discussed in the last two sections exhibit the universality of the ionization fractions a_j (shown in the bottom left diagram in Figure 3). They are independent of the chosen model, i.e., the functional dependence of C_T.

3.3.5. Seawater

The analytical formulas in Equation (41) and Equation (117) are based on the assumption that activities could be replaced by concentrations, {j} → [j]. This is valid either for dilute systems with near-zero ionic strength (I ≈ 0), or for non-dilute systems when the thermodynamic equilibrium constants are replaced by conditional constants, K → ^cK.

Seawater has I ≈ 0.7 M, which is at the upper limit of the validity range of common activity models (as discussed in Appendix B). Hence, in oceanography, chemists prefer conditional equilibrium constants ^cK. In the literature, there are several compilations for ^cK; one example is given in

Table 8. Thermodynamic and conditional equilibrium constants for H₂CO₃ in pure water and seawater (at 25 °C, 1 atm); ^cK values from [39].

	Thermodynamic K (Pure Water, I = 0)	Conditional cK (Seawater, I = 0.7 M)
pK1	5.18	6.0
pK2	10.33	9.1
pKw	14.0	13.9

.

Figure 33 (left diagram) compares the results calculated by Equation (117) for both the standard case (solid lines based on thermodynamic equilibrium constants K) and seawater (dashed lines based on conditional constants ^cK). The solid curves in Figure 33 are identical to the solid curves for n = 0, 1, and 2 shown in the bottom left diagram in Figure 27.

3.3.6. From Ideal to Real Solutions

All calculations so far (except in Section 3.3.5) were performed for the ideal case (i.e., no activity corrections, no aqueous complexation). Modern hydrochemistry software does not adhere to those restrictions; they perform activity corrections per se. In this respect, they are able to predict the relationship between pH and a given C_T for real systems more accurately.

Given is a carbonic acid system titrated with NaOH. Figure 33 (right diagram) compares the results of the analytical formula in Equation (117) (solid lines) with the numerical-model predictions (dots) using PhreeqC [11]. (Similar results are obtained when NaOH is replaced by KOH).

As expected, deviations between the ideal and real case only occur at high C_T values. There are two reasons: (i) with rising C_T, the ionic strength increases; consequently, the activity corrections are large and cannot be ignored; and (ii) numerical models consider the formation of aquatic complexes such as NaHCO₃⁻ and Na₂CO₃(aq), which are absent in the analytical approach. The aquatic complexes become particularly relevant at high concentrations for n = 1 and 2. [Note: At very high values of C_T between 1 and 10 M Na₂CO₃ (i.e., the most upper part of the green curve), we exit the applicability range of common activation models.]

4. Beyond Ordinary Acids

4.1. Zwitterionic Acids

4.1.1. Zwitterions and Amino Acids

A zwitterion is a molecule with functional groups, of which at least one has a positive and one has a negative electrical charge. The net charge of the entire molecule is zero.

Amino acids are the best-known examples of zwitterions. They contain an amine group (basic) and a carboxylic group (acidic). The NH₂ group is the stronger base, and so it picks up H⁺ from the COOH group to form a zwitterion (i.e., the amine group deprotonates the carboxylic acid):

Here, R denotes the side chain (glycine: R = H, alanine: R = CH₃, and so on).

The (neutral) zwitterion is the usual form amino acids exist in water. Depending on the pH, there are two other forms—anionic and cationic:

This parallels the behavior of a diprotic acid:

When an amino acid dissolves in water, it acts as both an acid and a base. Unlike amphoteric compounds, which can only form either a cationic or an anionic species, a zwitterion has both ionic states simultaneously.

4.1.2. Zwitterions as Diprotic Acids

The acid–base behavior of the simplest zwitterions (with 1 amine group and 1 carboxylic group) resembles that of a diprotic acid, as summarized in

Table 9. Diprotic acids (N = 2): ordinary acid vs. zwitterion (simplest amino acid).

	Diprotic Acid			Zwitterion (Simplest Amino Acid)
1st dissociation	H2A = H+ + HA−		K1	H2A+ = H+ + HA		K1
2nd dissociation	HA− = H+ + A−2		K2	HA = H+ + A−		K2
species	H2A	HA−	A−2	H2A+	HA	A−
abbreviation	[0]	[1]	[2]	[0]	[1]	[2]
ionic fraction aj	a0	a1	a2	a0	a1	a2
charge zj	0	−1	-2	1	0	−1
average charge zav	(0·[H2A] − 1·[HA−] − 2·[A−2])/CT			(1·[H2A+] − 0·[HA] − 1·[A−])/CT

. In both cases, there are N = 2 dissociation steps (controlled by two acidity constants K₁ and K₂) and three species: [0], [1], and [2], where [0] refers to the highest protonated species. The number of H⁺ in the highest protonated species is N = 2 for both acid types.

The only difference is the offset Z = 1, where Z represents the positive charge of the highest protonated species (which equals the number of amine groups in the molecule). Ordinary acids have Z = 0. The offset determines the individual charge of species j: z_j = Z − j. It confirms the statement that the offset Z equals the charge of the highest protonated species (j = 0): Z = z₀.

Table 10. The three species of an ordinary diprotic acid vs. the simplest zwitterion.

	Diprotic Acid (Z = 0)		Zwitterion (Z = 1)
[0]:	[H2A]	(neutral)	[H2A+]	(cation)	highest protonation
[1]:	[HA−]	(anion)	[HA]	(neutral)
[2]:	[A−2]	(anion)	[A−]	(anion)	fully deprotonated

shows the different ionic characters (cationic, neutral, anionic) of the three species of an ordinary diprotic acid versus the simplest zwitterion.

4.1.3. Model Extension for Zwitterionic Acids (H_NA^+Z)

Zwitterionic acids extend the concept of ordinary acids to non-zero integers Z > 0. An N-protic zwitterionic acid H_NA^+Z has N + 1 species:

[ j ] ≡ [H_N−j A^Z−j]  with charge z_j = Z − j  (for j = 0 to N)

(132)

N is the number of H⁺ in the highest protonated species and Z is the positive charge of the highest protonated species. Of all N + 1 species, three species are particularly interesting, as listed in

Table 11. Three exceptional cases out of N + 1 acid species.

			Ordinary Acid Z = 0	Zwitterion 1 ≤ Z < N
j = 0	[0] = [HNA+Z]	highest protonation	neutral	cation
j = Z	[Z] = [HN-ZA]	undissociated acid	neutral, [Z] = [0]	neutral
j = N	[N] = [A−(N−Z)]	fully deprotonated	anion	anion

.

The central formula in Equation (41) interconnects three components (H₂O, H_NA, strong base) via the charge-balance equation. Thus, an extension of the acid–base model to zwitterions requires a redefinition of the charge-balance equation. For this purpose, we introduce two quantities:

$$\begin{array}{}\mathrm{(133)}& \mathrm{total} \mathrm{charge} \mathrm{of} \mathrm{acid}:\hfill & {\mathrm{Z}}_{\mathrm{T}}\equiv {\displaystyle {\sum }_{\mathrm{j}=0}^{\mathrm{N}} {\mathrm{z}}_{\mathrm{j}} [\mathrm{j}]}\hfill \mathrm{(134)}& \mathrm{average} \mathrm{charge} \mathrm{of} \mathrm{acid}:\hfill & {\mathrm{z}}_{\mathrm{a}\mathrm{v}}\equiv \frac{{\mathrm{Z}}_{\mathrm{T}}}{{\mathrm{C}}_{\mathrm{T}}}={\displaystyle {\sum }_{\mathrm{j}=0}^{\mathrm{N}} {\mathrm{z}}_{\mathrm{j} }{\mathrm{a}}_{\mathrm{j}}}\hfill \end{array}$$

where z_j is the charge of the acid species j:

$${\mathrm{z}}_{\mathrm{j}}=\mathrm{Z}-\mathrm{j}\mathrm{with} \{ \begin{array}{c}\mathrm{Z}=0\\ \mathrm{Z}\ge 1\end{array}\begin{array}{l}\mathrm{for} \mathrm{ordinary} \mathrm{acids}\hfill \\ \mathrm{for} \mathrm{zwitterionic} \mathrm{acids}\hfill \end{array}$$

(135)

Inserting Equation (135) into Equation (134) yields for the average charge:

$${\mathrm{z}}_{\mathrm{av}}={\displaystyle \sum _{j=0}^{\mathrm{N}} (\mathrm{Z}-\mathrm{j}) {\mathrm{a}}_{\mathrm{j}}}=\mathrm{Z} {\displaystyle \sum _{\mathrm{j}=0}^{\mathrm{N}} {\mathrm{a}}_{\mathrm{j}}}-{\displaystyle \sum _{\mathrm{j}=0}^{\mathrm{N}} \mathrm{j}{\mathrm{a}}_{\mathrm{j}}}=\mathrm{Z}-{\mathrm{Y}}_1$$

(136)

where Equations (23) and (24) are applied. Due to Y₁ = Y₁(x), the average charge z_av is pH dependent. Examples for z_av are given in Table 9 (last line).

Now let us turn to the charge balance:

0 = [B⁺] + [H⁺] − [OH⁻] + total charge of acid

(137)

As we already know, [B⁺] is equal to C_B = n C_T, and [H⁺] − [OH⁻] can be expressed by w(x). Then, after dividing by C_T, we obtain:

$$0=\mathrm{n}-\frac{\mathrm{w}}{{\mathrm{C}}_{\mathrm{T}}}+{\mathrm{z}}_{\mathrm{a}\mathrm{v}}=\mathrm{n}-\frac{\mathrm{w}}{{\mathrm{C}}_{\mathrm{T}}}+(\mathrm{Z}-{\mathrm{Y}}_1)$$

This provides the extension of Equation (41) to the general case of zwitterionic acids (Z ≥ 1):

$$\mathrm{n}=\left(\underset{}{\overset{}{{\mathrm{Y}}_1}}(\mathrm{x})-\mathrm{Z}\right)+\frac{\mathrm{w}(\mathrm{x})}{{\mathrm{C}}_{\mathrm{T}}}$$

(138)

The only difference to the original formula is the constant offset Z, which disappears when performing the pH derivative. Thus, the formulas for the buffer intensity β and its pH derivation in Equations (69) and (70) are the same for zwitterionic acids and ordinary acids.

To sum up, the description of ordinary acids and zwitterionic acids is based on the same formulas (except the offset Z). That is, everything relies on exactly the same building blocks (ionization fractions a_j and moments Y_L), as introduced in Section 2.1. The offset Z, as the new ingredient, only enters the titration formula (i.e., normalized buffer capacity) in Equation (138); the buffer intensity β and its pH derivative are independent of Z.

4.1.4. Glycine (Z = 1) vs. Carbonic Acid (Z = 0)

The simplest amino acid is glycine (NH₂-CH₂-COOH), which we abbreviate as HA with A = Gly. Its structural formula has the shortest side chain (R = H). The three species are:

[0] = [H2A+] = [H2Gly+]:	NH3+-CH2-COOH	(glycinium cation)
[1] = [HA] = [HGly]:	NH3+-CH2-COO−	(neutral zwitterion)
[2] = [A−] = [Gly−]:	NH2-CH2-COO−	(glycinate anion)

Due to the presence of H₂A⁺, it is a diprotic acid specified by two acidity constants (which are compared to carbonic acid, also a diprotic acid):

glycine:	pK1 = 2.35	pK2 = 9.78
carbonic acid:	pK1 = 6.35	pK2 = 11.33

Figure 34 shows the pH dependence of the three ionization fractions (a₀, a₁, a₂) of glycine and carbonic acid. Both diagrams are based on the same formulas given in Equation (17). Figure 35 displays titration curves n = n(pH) based on Equation (138). The right diagram compares the analytical formula for C_T = 100 mM with numerical results from PhreeqC calculations. The latter are more realistic due to activity corrections for HCl and NaOH (especially at high ionic strengths, i.e., for high values of |n|).

4.1.5. Polyprotic Zwitterions (N = 2 to 6)

In addition to glycine as the simplest zwitterion (N = 2, Z = 1), we now consider compounds H_NA^+Z with higher values of N and/or Z. Examples are given in

Table 12. Acidity constants for carbonic acid and zwitterionic acids (the pK values for EDTA are taken from [5]; they differ slightly from those in [17]).

Acid	N	Z	pK1	pK2	pK3	pK4	pK5	pK6	[j = 0]	[j = Z]	[j = N]
carbonic acid	2	0	6.35	11.33					H2A	H2A	A−2
glycine	2	1	2.35	9.778					H2A+	HA	A−1
glutamic acid	3	1	2.16	4.30	9.96				H3A+	H2A	A−2
NTA	4	1	1.0	2.0	2.942	10.28			H4A+	H3A	A−3
EDTA	6	2	0	1.5	2.16	3.119	6.281	10.94	H6A+2	H4A	A−4

and Figure 36. To recap: N is the number of H⁺, and Z is the positive charge of the highest protonated species.

Figure 37 and Figure 38 present calculations for four zwitterionic acids (from Table 9). The titration curves (blue) are based on Equation (138), while the corresponding buffer capacity β (green) and its pH derivative (red) are based on Equations (69) and (70). The small circles mark the zeros of dβ/dpH; they indicate (i) the extrema of β and (ii) the inflection points of the titration curves.

4.1.6. EDTA

Ethylenediaminetetraacetic acid (EDTA) is particularly interesting because it represents the acid with the highest number of dissociation steps in this report: N = 6 (specified by six acidity constants in Table 11). There are N + 1 = 7 acid species; the highest protonated species is H₆A⁺², and the fully deprotonated species is A⁻⁴. The two left diagrams in Figure 39 display the pH dependence of EDTA’s seven ionization fractions a₀ to a₆ based on Equation (17).

The right diagram in Figure 39 depicts titration curves for three C_T values based on Equation (138). The red dots represent numerical calculations with PhreeqC. [Note: In contrast to Figure 37 and Figure 38, which also contain EDTA titration curves, the x and y axes are interchanged here.]

4.1.7. Equivalence Points

The algebraic relationship between EP_n and pH_n, originally defined in Equation (60), can easily be extended to zwitterionic acids by taking into account the offset Z:

$${\mathrm{EP}}_{\mathrm{n}} \iff {\mathrm{pH}}_{\mathrm{n}}:\mathrm{n}=\left(\underset{}{\overset{}{{\mathrm{Y}}_1}}(\mathrm{p}{\mathrm{H}}_{\mathrm{n}})-\mathrm{Z}\right)+\frac{\mathrm{w}(\mathrm{p}{\mathrm{H}}_{\mathrm{n}})}{{\mathrm{C}}_{\mathrm{T}}}\mathrm{for}\mathrm{n}=-\mathrm{Z},\dots -½, 0, ½,\dots \mathrm{N}-\mathrm{Z}$$

(139)

The number of equivalence points remains the same: 2N + 1 (independent of Z). For Z = 0, this relation simplifies to Equation (60).

Figure 40 shows the titration curve of glycine for C_T = 500 mM (same as in the upper left diagram of Figure 38). The small circles at integer and half-integer values of n mark the assignment EP_n ⇔ pH_n. Since HGly acts as a 2-protic acid, there are 2×2 + 1 = 5 equivalence points in total. The main difference to the diprotic acid H₂CO₃ (shown in Figure 15) is the occurrence of EPs with negative integer and half-integer values of n.

pH–C_T diagrams are obtained after rearranging of Equation (139) to the form:

$${\mathrm{C}}_{\mathrm{T}}=\frac{\mathrm{w}(\mathrm{p}\mathrm{H})}{\mathrm{n}+\mathrm{Z}-{\mathrm{Y}}_1(\mathrm{p}\mathrm{H})}$$

(140)

This relationship is plotted in Figure 41—one curve for each integer and half-integer value of n. The dashed curves (straight lines) correspond to the 1-component system “acid” (high-C_T limit). This diagram should be compared with the plots for ordinary acids in Figure 27.

4.1.8. Isoionic vs. Isoelectric Points

The isoionic point is the pH of the pure, neutral polyprotic acid (i.e., when the neutral zwitterion is dissolved in water). The isoelectric point pI is the pH at which the average charge z_av of the polyprotic acid is zero (the net charge of the solution is always zero):

isoionic point: pH0	(= pH of 2-component system “acid + H2O”)
isoelectric point: pI = pH at which zav = 0	(= pH of 1-component system “acid”)

The formula for the isoionic point follows from Equation (139) by setting n = 0; the formula for the isoelectric point follows from the definition of z_av in Equation (136):

$$\begin{array}{}\mathrm{(141)}& \mathrm{isoionic} \mathrm{point}:\hfill & \mathrm{Z}-{\mathrm{Y}}_1(\mathrm{pH})=\frac{\mathrm{w}(\mathrm{p}\mathrm{H})}{{\mathrm{C}}_{\mathrm{T}}}\hfill \mathrm{(142)}& \mathrm{isoelectric}\mathrm{point}:\hfill & \mathrm{Z}-{\mathrm{Y}}_1\left(\mathrm{pH}\right)=0\hfill \end{array}$$

Note that both are implicit functions. With increasing C_T, the isoionic and isoelectric points approach each other until they become identical in the high-C_T limit. This behavior is illustrated in the right diagram of Figure 41. For a diprotic acid (e.g., glycine) we have Z = 1, which—after insertion into Equation (142)—leads to:

isoelectric point:   1 − Y₁ = 0 ⇔ pI = ½ (pK₁ + pK₂)

The difference between the isoionic and isoelectric points (even if it is numerically very small) is summarized in

Table 13. Isoionic vs. isoelectric point.

Isoionic Point	Isoelectric Point (pI)
pH of pure acid	pH at which the net charge of all acid species is zero
depends on CT (approaches pI-value for CT → ∞ and pH = 7 for CT → 0)	independent of CT
Z − Y1(pH) = w(pH)/CT	Z − Y1(pH) = 0
exists for ordinary acids and zwitterions	exists only for zwitterions

.

4.2. Surface Complexation

4.2.1. Definition

Surface complexation [1] becomes especially simple when Coulomb interactions are ignored. This approximation is valid and useful, e.g., in the vicinity of the point of zero charge (PZC).

Let us consider a surface hydroxyl group ≡sOH, where ≡s symbolizes the surface. The uptake and release of protons involves two dissociation steps (N = 2):

1st dissociation step:	≡sOH2+ = H+ + ≡sOH	K1	(143)
2nd dissociation step:	≡sOH = H+ + ≡sO−	K2	(144)

In the absence of Coulomb interactions, K₁ and K₂ are so-called intrinsic equilibrium constants. Examples are given in

Table 14. Intrinsic pK values and pH_pzc for surface complexation on four (clay) minerals [40].

	Kaolinite	Mica	Goethite	Gibbsite
pK1	2.52	6.01	7.13	7.53
pK2	5.88	7.65	9.60	9.87
pHpzc = ½ (pK1+pK2)	4.20	6.83	8.37	8.70

. Similar to the case of the zwitterionic acid H₂A⁺, there are N + 1 = 3 (surface) species:

[0] = [≡sOH2+]	cationic
[1] = [≡sOH]	neutral
[2] = [≡sO−]	anionic

The sum of all species is the total amount of surface sites C_T. The formulas for the buffer capacity and the buffer intensity of surface complexation are the same as for zwitterionic acids, namely Equations (69), (70) and (138) for N = 2 and Z = 1.

4.2.2. Example Calculations

The ionization fractions a₀, a₁, and a₂, based on Equation (17), are shown in Figure 42; they illustrate the species distribution on kaolinite (in the absence of Coulomb interactions).

The “average charge” defined in Equation (136) represents the

$$\mathrm{surface} \mathrm{charge}:\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{z}}_{\mathrm{av}}=1-{\mathrm{Y}}_1(\mathrm{pH})={\mathrm{a}}_0-{\mathrm{a}}_2=\frac{1}{{\mathrm{C}}_{\mathrm{T}}} \left( [\equiv \mathrm{s}{\mathrm{H}}_2{\mathrm{O}}^{+}]-[\equiv \mathrm{s}{\mathrm{O}}^{-}] \right)$$

(145)

This quantity is plotted in Figure 43 for the four surface types defined in Table 14. The pH at which the surface charge is zero (z_a = 0) defines the PZC:

pH_pzc = ½ (pK₁ + pK₂)

(146)

The PZC is equivalent to the isoelectric point when there is no adsorption of ions other than H⁺ and OH⁻ (pristine surface).

5. Summary

Polyprotic acids in the general form H_NA^+Z include ordinary acids as a special case (Z = 0). They are specified by N acidity constants: K₁, K₂, … K_N. The acid–base system is then characterized by:

$$\begin{array}{}\mathrm{(147)}& (\mathrm{normalized}) \mathit{buffer}\mathit{capacity}:\hfill & \mathrm{n}(\mathrm{x})=\left({\mathrm{Y}}_1-\mathrm{Z}\right)+\frac{\mathrm{w}}{{\mathrm{C}}_{\mathrm{T}}}\hfill \mathrm{(148)}& (\mathrm{normalized}) \mathit{buffer}\mathit{intensity}:\hfill & \mathsf{\beta }(\mathrm{x})\equiv \frac{\mathrm{d}\mathrm{n}}{\mathrm{d} \mathrm{p}\mathrm{H}}=(\ln 10) \left({\mathrm{Y}}_2-{\mathrm{Y}}_1^2+\frac{\mathrm{w}+2\mathrm{x}}{{\mathrm{C}}_{\mathrm{T}}}\right)\hfill \mathrm{(149)}& 1\mathrm{st} \mathrm{derivative} \mathrm{of} \mathsf{\beta }:\hfill & \frac{\mathrm{d}\mathsf{\beta }}{\mathrm{d} \mathrm{p}\mathrm{H}}={(\ln 10)}^2 \left({\mathrm{Y}}_3-3{\mathrm{Y}}_1{\mathrm{Y}}_2+2{\mathrm{Y}}_1^3+\frac{\mathrm{w}}{{\mathrm{C}}_{\mathrm{T}}}\right)\hfill \end{array}$$

Here, n(x) in Equation (147) represents the central formula from which the other two are obtained via pH derivation. The parameter Z enters only Equation (147), where it acts as a constant offset; the buffer intensity and its derivative are both independent of Z.

The function n(x) has different meanings/names: (i) equivalent fraction (n = C_B/C_T), (ii) titration function/curve and (iii) normalized buffer capacity. Discrete values of n, i.e., the integer and half-integer values of n, define the equivalence points EP_n and semi-EP_n. They mark the inflection points of the titration curve, the maxima/minima of β and the zeros of dβ/dpH. (Note: n(x) as buffer capacity measures the “distance” from x = 10^−pH to EP₀). A classification of equivalence points is provided in Section 2.3 and Figure 16.

The analytical formula for n(x) is the strict mathematical solution of the set of N + 3 nonlinear equations (N + 1 mass action and 2 balance equations) defined in Equations (27)–(32). As sketched in Figure 44, there are three mathematical equivalent representations of the N-protic acid–base system (all carrying the same information content): (i) the algebraic set of N + 3 equations, (ii) the equivalent-fraction formula as defined in Equation (41) or Equation (147), and (iii) the polynomial of degree N + 2 in Equation (44).

The main feature that makes the analytical formula n(x) so attractive is its easy calculability (no programming or root solver are required). As shown in Figure 45, its construction relies on a 3-layer hierarchy of building blocks: (i) the acidity constants themselves (the only input data), (ii) the ionization fractions that rely on the cumulative acidity constants k_j, and (iii) the moments Y_L that are weighted sums of a_j. These building blocks characterize the 1-component system “N-protic acid”. The step from the 1-component to the 3-component system requires two extra input parameters, C_T and C_B, to obtain n = n(pH). The pH derivatives of n(pH) are again analytical formulas which can easily be constructed from the same building blocks.

Another advantage: Once you have an analytical formula (explicit function), it can be “modified” according to all rules of mathematics: (i) you can differentiate/integrate the function as in Section 2.4.2, (ii) you can form superpositions (mixtures) as in Section 3.1.4, and (iii) you can invert the function as in Section 2.2.3.

Final Remark. The mathematical framework is based on three assumptions: (i) activities are replaced by concentrations, (ii) no aqueous complex formation, and (iii) no density effects (molality = molarity). These are the limits of the analytical model. Although the current approach deepens our understanding of the acid–base system, it can never replace numerical models such as PhreeqC, which are capable of handling real-world problems more accurately (including activity corrections, an arbitrary number of species and phases, and complex formation).