# Polyprotic Acids and Beyond—An Algebraic Approach

## Abstract

**:**

_{N}A

^{+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

_{N}A 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

_{N}A) 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.

_{N}A 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].

^{−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

_{N}A

^{+Z}(amino acids, NTA, EDTA, etc.), which embeds all “ordinary acids” as a special subclass characterized by Z = 0.

- 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?

^{−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.

_{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.

_{1}+ 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 EP

_{0}).

#### 1.3. Structure of this Report

_{1}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.

## 2. Mathematical Framework

#### 2.1. The 1-Component System: N-Protic Acid (H_{N}A)

#### 2.1.1. Notation

^{+}ions (more precisely: H

_{3}O

^{+}) when dissolved in water:

^{+}+ A

^{−}

_{2}O(l) = H

_{3}O

^{+}(aq) + A

^{−}(aq). An N-protic acid H

_{N}A dissolves into N + 1 species:

1 undissociated species: | H_{N}A^{0} | (uncharged) |

N dissociated species: | H_{N−1}A^{−1}, …, HA^{−(N−1)}, A^{−N} | (anions) |

_{N−j}A

^{−j}] for j = 0, 1, 2, … N

_{j}= 0 − j). Thus, the neutral, undissociated species H

_{N}A

^{0}is abbreviated with [0]. Henceforth we skip the superscript 0 in H

_{N}A

^{0}.

^{+}):

_{T}:

_{T}= [H

_{N}A]

_{T}should not be confused with the molar concentration of the undissociated species [H

_{N}A].

concentrations: | denoted by square brackets | [j] |

activities | denoted by curly braces | {j} |

_{j}:

_{j}[j]

_{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.

^{+}is abbreviated with x; it is linked to the pH value via:

^{+}} = 10

^{−pH}⇔ pH = −lg x

_{2}O = H

^{+}+ OH

^{−}with K

_{w}= {H

^{+}}{OH

^{−}}

_{w}= 1.0 × 10

^{−14}at 25 °C. This yields [OH

^{−}] ≈ {OH

^{−}} = K

_{w}/x.

_{H}≈ 1):

_{T}is the term which couples the polyprotic acid to H

_{2}O. This term vanishes in the so-called

_{T}limit: C

_{T}≫ w(x) (or C

_{T}→ ∞)

_{2}O” 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

_{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:

_{a}= − lg K

_{a}

_{a}, the stronger the acid—quite the opposite to a K

_{a}-based ranking.

_{1}(= K

_{a}), a diprotic acid H

_{2}A by two acidity constants (K

_{1}, K

_{2}), and a triprotic acid by three acidity constants (K

_{1}, K

_{2}, K

_{3}):

1st dissociation step: H_{3}A = H^{+} + H_{2}A^{−} K_{1} = {H^{+}}{H_{2}A^{−}}/{H_{3}A} |

2nd dissociation step: H_{2}A^{−} = H^{+} + HA^{−2} K_{2} = {H^{+}}{HA^{−2}}/{H_{2}A^{−}} |

3rd dissociation step: HA^{−2} = H^{+} + A^{−3} K_{3} = {H^{+}}{A^{−3}}/{HA^{−2}} |

H_{3}A = H^{+} + H_{2}A^{−} k_{1} = K_{1} |

H_{3}A = 2H^{+} + HA^{−2} k_{2} = K_{1}K_{2} |

H_{3}A = 3H^{+} + A^{−3} k_{3} = K_{1}K_{2}K_{3} |

^{+}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:

_{j}= 0. Equation (13) also includes the trivial case of the undissociated acid (j = 0), which is k

_{0}= 1. In our shortened notation, the last line is k

_{j}= x

^{j}{j}/{0}.

#### 2.1.3. Ionization Fractions (Degree of Dissociation)

_{N}A 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:

_{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:

_{1}, K

_{2}, … K

_{N}, which are encapsulated in the cumulative equilibrium constants (as products of K

_{j}-values):

_{0}= 1, k

_{1}= K

_{1}, k

_{2}= K

_{1}K

_{2}, … k

_{N}= K

_{1}K

_{2}…K

_{N}

_{j}< 1. The latter makes them excellent candidates for probabilities (as will be discussed later in Section 2.5.3).

semi-Eps | (blue circles): | a_{j} = a_{j−1} ≈ ½ | (inflection points of a_{j}) |

EPs | (yellow circles): | a_{j} = 1 − 2a_{j−1} (≈ 1) | (maximum of a_{j}) |

_{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 (H

_{2}A as titrant vs. H

_{2}A as analyte) and Section 3.3.4 (open vs. closed CO

_{2}system).

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

_{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, a

_{0}and a

_{1}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 ±∞.]

_{0}and a

_{N}attain the maximum value 1:

strongly acidic: | pH < 0 | (or x → ∞): | a_{0} = 1, | all other a_{j} = 0 | (20) |

strongly alkaline: | pH > 14 | (or x → 0): | a_{N} = 1, | all other a_{j} = 0 | (21) |

#### 2.1.5. Moments Y_{L}

_{L}is defined as the weighting sum over a

_{j}:

Y_{0} = a_{0} + a_{1} + … + a_{N} = 1 | ⇒ mass balance | (23) | |

Y_{1} = a_{1} + 2a_{2} + … + N a_{N} | ⇒ enters buffer capacity | in Equation (68) | (24) |

Y_{2} = a_{1} + 4a_{2} + … + N^{2} a_{N} | ⇒ enters buffer intensity β | in Equation (69) | (25) |

Y_{3} = a_{1} + 8a_{2} + … + N^{3} a_{N} | ⇒ enters 1st derivative of β | in Equation (70) | (26) |

_{1}is built up from the ionization fractions a

_{j}(of the triprotic phosphoric acid). Figure 6 displays the pH dependence of Y

_{1}to Y

_{4}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.

_{L}are positive, monotonic (increasing) functions, living in the range 0 < Y

_{L}≤ N

^{L}(where the equal sign applies only for Y

_{0}). Their asymptotic behavior results from Equations (20) and (21):

Y_{L}(pH → −∞) = 0 | or | Y_{L}(x → +∞) = 0 |

Y_{L}(pH → +∞) = N^{L} | or | Y_{L}(x → 0) = N^{L} |

#### 2.2. The 3-Component Acid–Base System

#### 2.2.1. Basic Set of Equations

- pure water H
_{2}O - N-protic acid H
_{N}A (with amount C_{T}= [H_{N}A]_{T}) - strong base BOH (with amount C
_{B}= [BOH]_{T})

^{+}+ 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

^{+}).

K_{w} = {H^{+}} {OH^{−}} | (self-ionization of H_{2}O) | (27) |

K_{1} = {H^{+}} {H_{N−1}A^{−}}/{H_{N}A} | (1st diss. step) | (28) |

K_{2} = {H^{+}} {H_{N−2}A^{−2}}/{H_{N-1}A^{−}} | (2nd diss. step) | (29) |

⋮ | ||

K_{N} = {H^{+}} {A^{−N}}/{HA^{−(N− 1)}} | (Nth diss. step) | (30) |

C_{T} = [H_{N}A] + [H_{N−1}A^{−}] + … + [A^{−N}] | (mass balance) | (31) |

C_{B} = [H_{N−1}A^{−}] + 2 [H_{N−2}A^{−2}] + … + N [A^{−N}] + [OH^{−}] − [H^{+}] | (charge balance) | (32) |

_{B}(amount of base) to change the pH (known as titration).

_{B}, we prefer the normalized or unitless quantity

_{N}A + H

_{2}O” (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.

_{A}= [HX]

_{T}(for example HCl, whereby X

^{−}= Cl

^{−}). In this way, we are able to generalize Equation (33):

_{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).

_{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 NaHCO

_{3}(or Na

_{2}CO

_{3}), the titration will start at n = 1 (or n = 2).

_{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)

- 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 Y
_{1}for summation over j a_{j}defined in Equation (24); - divide Equations (31) and (32) by C
_{T}.

w | = K_{w}/x − x | (self-ionization of H_{2}O) | (35) |

a_{1} | = (k_{1}/x) a_{0} | (1st diss. step) | (36) |

a_{2} | = (k_{2}/x^{2}) a_{0} | (2nd diss. step) | (37) |

⋮ | |||

a_{N} | = (k_{N}/x^{N}) a_{0} | (Nth diss. step) | (38) |

1 | = a_{0} + a_{1} + … + a_{N} | (39) | |

n | = Y_{1} + w/C_{T} | (40) |

_{T}—the effect of water, and Y

_{1}—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 = Y

_{1}(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].

_{T}values are plotted in Figure 10. The diagrams also include the high-C

_{T}limit, n = Y

_{1}(as dark-blue curves).

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

_{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

^{−pH}, the two forward tasks are easily obtained from Equation (41):

_{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}= 1 and k

_{1}= K

_{1}we get a cubic equation:

^{3}+ {K

_{1}+ nC

_{T}} x

^{2}− {(n − 1)(C

_{T}K

_{1}+ K

_{w})} x − K

_{1}K

_{w}

_{0}= 1, k

_{1}= K

_{1}, and k

_{2}= K

_{1}K

_{2}we obtain a quartic equation:

^{4}+ {K

_{1}+ nC

_{T}} x

^{3}+ {K

_{1}K

_{2}+ (n − 1) C

_{T}K

_{1}− K

_{w}} x

^{2}+ K

_{1}{(n − 2) C

_{T}K

_{2}− K

_{w}} x − K

_{1}K

_{2}K

_{w}

_{2}= 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:

^{4}+ {K

_{1}+ C

_{B}} x

^{3}+ {K

_{1}K

_{2}+ (C

_{B}− C

_{T}) K

_{1}− K

_{w}} x

^{2}+ K

_{1}{(C

_{B}− 2C

_{T}) K

_{2}− K

_{w}} x − K

_{1}K

_{2}K

_{w}

_{N}A+H

_{2}O). 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).

_{0}= 1 and h

_{N+2}= −K

_{w}K

_{1}K

_{2}...K

_{N}. [Note: k

_{j}is per definition zero for negative values of j and for j > N].

#### 2.3. Equivalence Points

_{N}A”, 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_{N}A)

^{+}) relates an acid species to its conjugate base—see Equation (3). In this context, Equation (49) relates conjugate acid–base pairs:

EP: | [j − 1] = [j + 1] | ⇒ | a_{j−1}(x) = a_{j+1}(x) | (51) |

semi-EP: | [j − 1] = [j] | ⇒ | a_{j−1}(x) = a_{j}(x) | (52) |

_{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:

EP_{n}: | [n − 1] = [n + 1] | (for integer n = 0, 1, … N) | (53) |

semi-EP_{n}: | [n − ½] = [n + ½] | (for half-integer n = ½, 3/2, … N − ½) | (54) |

_{1}is often introduced as the equilibrium state for which [CO

_{2}] = [CO

_{3}

^{−2}] applies. Another example is the triprotic acid (N = 3) in Figure 12, which dissolves into 3 + 1 species: H

_{3}A, H

_{2}A

^{−1}, HA

^{−2}, and A

^{−3}. There are 2 × 3 + 1 equivalence points EP

_{n}with n = 0, ½, 1, … 3.

external (or outer) equivalence points EP_{0} and EP_{N} (only two!) |

internal (or inner) equivalence points all other EP_{n} (for ½ ≤ n ≤ N − ½) |

^{+}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).

_{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:

_{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).

_{n}is the midpoint between two adjacent semi-EPs.

_{0}and pH

_{N}depend sensitively on C

_{T}. This relationship can only be expressed in implicit form:

_{0}and EP

_{3}at the extreme left and right positions. For C

_{T}→ ∞ the two external EPs drift apart: pH

_{0}→ 0 and pH

_{3}→ 14, while all internal EPs remain fixed at the position dictated by the pK values in Equation (55).

_{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

_{B}/C

_{T}becomes an integer or half-integer value:

EP_{n} | ⇔ | C_{B}/C_{T} = n | for n = 0, 1, … N | (58) |

semi-EP_{n} | ⇔ | C_{B}/C_{T} = n | for n = ½, 3/2, … N − ½ | (59) |

_{1}with n = 1. The particular case n = 0 (or EP

_{0}) refers to the base-free 2-component system with pH

_{0}as the equilibrium pH of the acid with amount C

_{T}dissolved in pure water.

_{n}and pH

_{n}is dictated by Equation (41), that is:

_{2}CO

_{3}with integer and half-integer values of n on the titration curve n = n(pH). More examples are given in Section 3.2.

_{T}≫ w, the last term in Equation (60) vanishes, leading to the 1-component approach:

_{1}(pH

_{n}) − n = 0

_{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].

_{N}A 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

_{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:

_{ref }or ΔC

_{B}= C

_{B}− n

_{ref}C

_{T}

_{n=j}(with integer j), that is n

_{ref}= j, yielding:

_{B}= C

_{B}(x) − j·C

_{T}(with j = 0, 1, …)

_{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 EP

_{0}.

_{j}(at pH

_{j}):

_{n=j}= C

_{B}(pH) − j·C

_{T}

_{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

_{0}= C

_{B}(pH)

_{n}= − [ANC]

_{n}

_{B}(pH) = n C

_{T}yields:

_{n=j}= { Y

_{1}(x) − j } C

_{T}+ w(x)

^{−pH}) to a particular EP

_{n}(titration endpoint) is:

_{0}= { Y

_{1}(x) − 0 } C

_{T}+ w(x) = { Y

_{1}(x) C

_{T}+ w(x)} − 0·C

_{T}

_{1}= { Y

_{1}(x) − 1 } C

_{T}+ w(x) = { Y

_{1}(x) C

_{T}+ w(x)} − 1· C

_{T}

_{2}= { Y

_{1}(x) − 2 } C

_{T}+ w(x) = { Y

_{1}(x) C

_{T}+ w(x)} − 2· C

_{T}

_{T}= 10 mM). The small circles at pH

_{0}= 4.2, pH

_{1}= 8.2, and pH

_{2}= 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 pH

_{0}(blue curve), pH

_{1}(green curve), and pH

_{2}(red curve). Of course, the highest amount (blue curve) is required to attain the lowest pH, namely pH

_{0}= 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

normalized buffer intensity: | β = dΔn/dpH = dn/dpH | (unitless) | (69) |

buffer intensity: | β_{C} = dC_{B}/dpH = β C_{T} | (in mM) | (70) |

_{T}= const (standard case), otherwise we have to use dC

_{B}/dpH = β C

_{T}+ n(dC

_{T}/dpH)].

_{C}over a definite pH interval (usually starting from an equivalence point EP

_{n}):

_{2}CO

_{3}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):

EP_{n} | (integer n) | ⇔ | minimum buffer intensity β |

semi-EP_{n} | (half-integer n) | ⇔ | maximum buffer intensity β |

_{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

_{j}and β

_{j}differ significantly:

_{j}(single-step deprotonation) as follows:

_{1}= 1/K

_{N}, β

_{2}= (K

_{N−1}K

_{N})

^{−1}, and β

_{N}= 1/k

_{N}. Note that β

_{j}is not the reciprocal of k

_{j}, because β

_{j}k

_{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).

_{j}):

_{j}express the probability that j protons are released from its fully protonated state H

_{N}A. 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: | a_{j} = (k_{j} /x^{j}) a_{0} | (80) |

j protons bound: | ã_{j} = a_{N-j} = (k_{N−j}/x^{N−j}) a_{0} = (β_{j} x^{j}) a_{N} = (β_{j} x^{j}) ã_{0} | (81) |

_{0}= ã

_{N}and ã

_{0}= 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

_{N}A 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)}= (α

_{1}, α

_{2}, … 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

_{N}A (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

α^{(0)} = (0, 0, … 0): A^{−N} | fully dissociated state (“empty”) | j = N protons released |

α^{(2^N−1)} = (1, 1, … 1): H_{N}A | undissociated state (fully occupied) | j = 0 protons released |

^{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

_{N}A = [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}.

_{1}= κ

_{1}. 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

_{0}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 a

_{0}and a

_{N}embody cumulative distribution functions which gives the area under the probability density functions f

_{0}and f

_{N}:

_{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 f

_{0}and f

_{N}correspond to the inflection points of a

_{0}and a

_{N}. The asymptotic behavior of a

_{0}and a

_{N}dictates that the probability densities are automatically normalized to 1:

_{0}and a

_{N}are opposite.

_{0}, which is the pH derivative of f

_{0}, is also known as the protonation probability of an isolated site, i.e., the probability that a monoprotic acid remains undissociated:

_{1}mimics the Fermi energy (except for a prefactor (k

_{B}T)

^{−1}consisting of the Boltzmann constant and temperature).

_{j}, as a number between 0 and 1, represents the probability that j protons are released (at a given pH). The average number $\u27e8\mathrm{j}\u27e9$ of protons released is then obtained by averaging over all ionization fractions a

_{j}:

_{1}embodies the average number of released protons. Note that $\u27e8\mathrm{j}\u27e9$ = [j]

_{avg}/C

_{T}is a real number between 0 and N, as displayed by the Y

_{1}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:

#### 2.5.4. Partition Functions and Moments in Statistics

_{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:

_{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}/a

_{0}= k

_{N}/x

^{N}.

^{+}yields (with µ

_{H+}= RT ln x):

_{1}, and the average number of bound protons is a

_{0}.

_{1}⟩ = Y

_{1}− Y

_{1}= 0. Most interesting are the moments for L = 2, 3, and 4:

#### 2.5.5. Decoupled Sites Representation and Simms Constants

_{i}as its equilibrium constant). Applying this to the partition functions in Equations (89) and (90) using Equations (78) and (79), we obtain:

_{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:

_{j}encompasses 2

^{N}− 1 single terms (g

_{1,}g

_{2}, g

_{3}, g

_{1}g

_{2}, g

_{2}g

_{3}, g

_{1}g

_{3}, g

_{1}g

_{2}g

_{3}for N = 3, for example), each of them represents one microscopic equilibrium constant. For N = 1 we have the trivial result:

_{1}= g

_{1}= k

_{1}= 1/β

_{1}

#### 2.5.6. Polyprotic Acids as Mixtures of Monoprotic Acids

_{1}+ w/C

_{T}in Equation (41), it is sufficient to focus on Y

_{1}. Applying DSR from Section 2.5.5 for the polyprotic acid, we get:

_{j}based on K

_{j}values.

_{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)).

_{N}A 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:

_{1}≫ K

_{2}≫ … ≫ K

_{N}⇔ K

_{1}≈ g

_{1}, K

_{2}≈ g

_{2}, … K

_{N}≈ g

_{N}

#### 2.5.7. The World of Acidity Constants

_{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 K

_{1}(making K

_{1}the largest value). For K

_{2}, 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: K

_{1}> K

_{2}> … > K

_{N}.

_{j}decreases from K

_{1}= Ng to K

_{N}= g/N. This can also be expressed by pK

_{N}− pK

_{1}= 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:

_{1}> K

_{2}> … > 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).

_{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

_{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:

_{T}= [HA] + [A

^{−}] or 1 = a

_{0}+ a

_{1}

_{1}> K

_{2}> K

_{3}> … or pK

_{1}< pK

_{2}< pK

_{3}< …

_{a}by the 1st dissociation constant K

_{1}. From Equation (17) we then get:

_{1}values (which are the inflection points of a

_{0}).

#### 3.1.2. Weak Acids vs. Diluted Acids

_{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 K_{a} ↔ large K_{a} |

dilute acid ↔ concentrated acid | ⇔ | small C_{T} ↔ large C_{T} |

#### 3.1.3. Strong Polyprotic Acids (Simplification)

^{+}is always released). Since the amount of the undissociated species is zero, [0] = 0 and a

_{0}= 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 K

_{1}(keeping in mind that K

_{1}is a large number). That is good news, because K

_{1}of strong acids is often not known precisely enough.

_{j}in Equation (18) and the ionization fractions a

_{j}in Equation (17) simplify as follows:

_{1}= 1, k

_{2}= K

_{2}, k

_{3}= K

_{2}K

_{3}, etc.

_{1}= 1, we get a quadratic equation:

^{2}+ (n − 1) C

_{T}x − K

_{w}

^{3}+ {(n − 1) C

_{T}+ K

_{2}} x

^{2}− {(n − 2) C

_{T}K

_{2}− K

_{w}} x − K

_{2}K

_{w}

_{1}= 0.

#### 3.1.4. Mixtures of Acids

_{a}, C

_{b}, C

_{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:

_{1}is replaced by the generalized moment Ỹ

_{1}as a superposition of the individual acid’s Y

_{1}:

_{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.

_{phos}= C

_{carb}= C

_{T}/2. The first moment Ỹ

_{1}of the two-acid system is displayed as the blue curve in the upper diagram of Figure 26. It is simply the sum of Y

_{1}

^{(phos)}and Y

_{1}

^{(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 (Y

_{1}) 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

_{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.

_{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:

_{T}→ ∞

_{2}O”: C

_{T}→ 0

_{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 “H

_{2}O” overtakes the rule when C

_{T}drops below 10

^{−7}M, which is just the amount of H

^{+}and OH

^{−}in pure water.

_{2}O” is at the position where the nominator in Equation (117) becomes zero (which is exactly at pH = 7): EP of pure H

_{2}O ⇔ 0 = w(x) ⇔ C

_{T}= 0.

#### 3.2.2. EPs as Inflection Points of Titration Curves

_{T}≫ w), the formulas for the buffer capacity and its pH derivative in Equations (72) and (73) simplify:

_{j}values, while the Y

_{L}’s are given by (see Appendix C.1):

Y_{L} = ½ {(j − 1)^{L} + j^{2}} | for semi-EP_{n} at pK_{j} | (n = j − ½) | (122) |

Y_{L} = j^{L} | for EP_{n} at pH_{j} ≡ ½ (pK_{j} + pK_{j+1}) | (n = j) | (123) |

_{L}’s into Equation (120) yields for

semi-EP_{n}: | β(pK_{j}) = ¼ (ln 10) = 0.576 | (maximum) | (124) |

EP_{n}: | (pH_{j}) = 0 | (minimum) | (125) |

_{L}values into Equation (121) results—after doing some school algebra—in exactly the required condition for inflection points:

_{1}, pK

_{2}and pK

_{3}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.

_{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

- Approach 1: “piecewise log-scale approximation” for lg a
_{j} - Approach 2: “midpoint approximation” for a
_{j}

_{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.

_{j}provides a “curve” consisting of several linear functions in pH (straight lines):

_{j}≈ (j − i) pH + (pk

_{i}− pk

_{j}) for the ith interval

_{i}= pK

_{1}+ pK

_{2}+ … + pK

_{i}and pk

_{0}= 0. This approach is shown (by the colored dashed lines) in the upper-left diagram in Figure 29.

_{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

_{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} |

mineral acidity: | [H-Acy] | = [BNC]_{n=0} |

CO_{2} acidity: | [CO_{2}-Acy] | = [BNC]_{n=1} |

acidity: | [Acy] | = [BNC]_{n=2} |

[ANC]_{0} = − [BNC]_{0} | ⇒ | [Alk] = − [H-Acy] |

[ANC]_{1} = − [BNC]_{1} | ⇒ | [P-Alk] = − [CO_{2}-Acy] |

[ANC]_{2} = − [BNC]_{2} | ⇒ | [OH-Alk] = − [Acy] |

_{0}= C

_{B}= nC

_{T}

_{T}as follows:

_{0}− [ANC]

_{1}= C

_{T}(= DIC)

#### 3.3.2. pH as Reference Point of ANC and BNC

_{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 EP

_{0}and EP

_{1}of the carbonate system:

EP_{0}: pH ≈ 4.3 |

EP_{1}: pH ≈ 8.2 |

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]) |

#### 3.3.3. Acid–Base Titration with H_{2}CO_{3} as Titrant

- var A
- 100 mM H
_{2}CO_{3}solution is titrated by a strong base/acid (NaOH and HCl) - var B
- 100 mM NaOH solution is titrated by H
_{2}CO_{3}

_{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 | C_{T} = const | with C_{T} = 100 mM |

var B | C_{T} = (C_{B} − w)/Y_{1} | with C_{B} = 100 mM |

_{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.

#### 3.3.4. Open vs. Closed CO_{2} System

_{2}system, the solution is in equilibrium with the CO

_{2}of the atmosphere. Let us compare it with the closed system:

- var A
- var C
- titration of 100 mM H
_{2}CO_{3}solution as “open CO_{2}system”

_{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.

_{2}between the aqueous and gas phase: CO

_{2}(aq) is proportional to CO

_{2}(g), whereas CO

_{2}(aq) is the undissociated acid H

_{2}CO

_{3}, i.e., the uncharged species [0]. Thus, we can write:

_{H}· P with K

_{H}= 10

^{−1.47}M/atm (at 25 °C)

_{2}pressure. Using [0] = C

_{T}a

_{0}, we obtain:

_{T}value:

var A | C_{T} = const | with C_{T} = 100 mM |

var C | C_{T} = K_{H} P/a_{0} | with P = 0.00039 atm (= 10^{−3.408} atm) |

_{2}is sucked out of the atmosphere (which increases the C

_{T}exponentially). [Note: In var C, pH < 5 is not available in practice.]

_{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

^{c}K.

^{c}K. In the literature, there are several compilations for

^{c}K; one example is given in Table 8.

^{c}K). 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

_{T}for real systems more accurately.

_{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 NaHCO

_{3}

^{−}and Na

_{2}CO

_{3}(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 Na

_{2}CO

_{3}(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

_{2}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):

_{3}, and so on).

#### 4.1.2. Zwitterions as Diprotic Acids

_{1}and K

_{2}) 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.

_{j}= Z − j. It confirms the statement that the offset Z equals the charge of the highest protonated species (j = 0): Z = z

_{0}.

#### 4.1.3. Model Extension for Zwitterionic Acids (H_{N}A^{+Z})

_{N}A

^{+Z}has N + 1 species:

_{N−}j A

^{Z−j}] with charge z

_{j}= Z − j (for j = 0 to N)

^{+}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.

_{2}O, H

_{N}A, 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:

_{j}is the charge of the acid species j:

_{1}= Y

_{1}(x), the average charge z

_{av}is pH dependent. Examples for z

_{av}are given in Table 9 (last line).

^{+}] + [H

^{+}] − [OH

^{−}] + total charge of acid

^{+}] 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:

_{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)

_{2}-CH

_{2}-COOH), which we abbreviate as HA with A = Gly. Its structural formula has the shortest side chain (R = H). The three species are:

[0] = [H_{2}A^{+}] = [H_{2}Gly^{+}]: | NH_{3}-CH^{+}_{2}-COOH | (glycinium cation) |

[1] = [HA] = [HGly]: | NH_{3}^{+}-CH_{2}-COO^{−} | (neutral zwitterion) |

[2] = [A^{−}] = [Gly^{−}]: | NH_{2}-CH_{2}-COO^{−} | (glycinate anion) |

_{2}A

^{+}, it is a diprotic acid specified by two acidity constants (which are compared to carbonic acid, also a diprotic acid):

glycine: | pK_{1} = 2.35 | pK_{2} = 9.78 |

carbonic acid: | pK_{1} = 6.35 | pK_{2} = 11.33 |

_{0}, a

_{1}, a

_{2}) 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)

_{N}A

^{+Z}with higher values of N and/or Z. Examples are given in Table 12 and Figure 36. To recap: N is the number of H

^{+}, and Z is the positive charge of the highest protonated species.

#### 4.1.6. EDTA

_{6}A

^{+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 a

_{0}to a

_{6}based on Equation (17).

#### 4.1.7. Equivalence Points

_{n}and pH

_{n}, originally defined in Equation (60), can easily be extended to zwitterionic acids by taking into account the offset Z:

_{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 H

_{2}CO

_{3}(shown in Figure 15) is the occurrence of EPs with negative integer and half-integer values of n.

_{T}diagrams are obtained after rearranging of Equation (139) to the form:

#### 4.1.8. Isoionic vs. Isoelectric Points

_{av}of the polyprotic acid is zero (the net charge of the solution is always zero):

isoionic point: pH_{0} | (= pH of 2-component system “acid + H_{2}O”) | |

isoelectric point: pI = pH at which z_{av} = 0 | (= pH of 1-component system “acid”) |

_{av}in Equation (136):

_{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:

_{1}= 0 ⇔ pI = ½ (pK

_{1}+ pK

_{2})

#### 4.2. Surface Complexation

#### 4.2.1. Definition

1st dissociation step: | ≡sOH_{2}^{+} = H^{+} + ≡sOH | K_{1} | (143) |

2nd dissociation step: | ≡sOH = H^{+} + ≡sO^{−} | K_{2} | (144) |

_{1}and K

_{2}are so-called intrinsic equilibrium constants. Examples are given in Table 14. Similar to the case of the zwitterionic acid H

_{2}A

^{+}, there are N + 1 = 3 (surface) species:

[0] = [≡sOH_{2}^{+}] | cationic |

[1] = [≡sOH] | neutral |

[2] = [≡sO^{−}] | anionic |

_{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

_{0}, a

_{1}, and a

_{2}, based on Equation (17), are shown in Figure 42; they illustrate the species distribution on kaolinite (in the absence of Coulomb interactions).

_{a}= 0) defines the PZC:

_{pzc}= ½ (pK

_{1}+ pK

_{2})

^{+}and OH

^{−}(pristine surface).

## 5. Summary

_{N}A

^{+Z}include ordinary acids as a special case (Z = 0). They are specified by N acidity constants: K

_{1}, K

_{2}, … K

_{N}. The acid–base system is then characterized by:

_{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 EP

_{0}). A classification of equivalence points is provided in Section 2.3 and Figure 16.

_{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.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Abbreviations and Symbols

a_{j} | ionization fractions, a_{j} = [j]/C_{T} | – |

α | vector of microstates: α^{(ν)} = (α_{1}, α_{2}, … α_{N}) | – |

Acy | acidity | mol/L |

Alk | total alkalinity (M-alkalinity) | mol/L |

ANC | acid-neutralizing capacity | mol/L |

BNC | base-neutralizing capacity | mol/L |

β | buffer intensity (normalized), β = dn/dpH | – |

β_{C} | buffer intensity, β_{C} = dC_{B}/dpH = C_{T} β | mol/L |

β_{j} | cumulative formation constants | (mol/L)^{−j} |

γ_{j} | activity correction for species j | – |

C_{A} | concentration of strong monoprotic acid C_{A} = [HX]_{T} | mol/L |

C_{B} | concentration of strong monoacidic base C_{B} = [BOH]_{T} | mol/L |

C_{T} | total concentration of N-protic acid: C_{T} = [H_{N}A]_{T} | mol/L |

δj | fluctuation of j around its mean: δj ≡ j − ⟨j⟩ | – |

EP_{j} | equivalence point (for integer j) | – |

EP_{n} | equivalence point (for integer and half-integer n) | – |

f_{0}, f_{N} | probability density functions (related to a_{0} and a_{N}) | – |

G | molar Gibbs energy | J/mol |

g_{i} | Simms constant | mol/L |

γ_{j} | activity correction for species j | – |

I | ionic strength | mol/L |

j | index denoting the aqueous species j, j = 0, 1, … N | – |

[j] | molar concentration of aqueous species j: [j] = [H_{N-j}A^{−j}] | mol/L |

{j} | activity of aqueous species j: {j} = {H_{N-j}A^{−j}} | mol/L |

⟨j⟩ | mean value of j (real number between 0 and N) | – |

K_{a} | acidity constant (general abbreviation) | mol/L |

^{c}K_{a} | conditional acidity constant (non-thermodynamic quantity) | mol/L |

K_{j} | acidity constant of dissociation step j | mol/L |

K_{H} | Henry constant | (mol/L)/atm |

k_{j} | cumulative acidity constant, e.g., k_{j} = K_{1}K_{2}...K_{j} | (mol/L)^{j} |

K_{w} | equilibrium constant of autoprotolysis (self-ionization of H_{2}O) | (mol/L)^{2} |

κ_{ν} | microscopic equilibrium constant | – |

µ_{H+} | chemical potential of H^{+} | mol/L |

n | equivalent fraction of titration, n = (C_{B} − C_{A})/C_{T} | – |

N | number of protons (H^{+}) of the N-protic acid H_{N}A | – |

ν | label for microstates | – |

P | pressure | atm |

pH | = −lg {H^{+}} = −lg x | – |

pH_{j} | = ½ (pK_{j} + pK_{j+1}) as pH of equivalence point EP_{j} | – |

pH_{n} | pH of EP_{n} | – |

pk_{j} | = −lg k_{j} | – |

pK_{j} | = −lg K_{j} | – |

pH_{pzc} | pH of PZC (point of zero charge) | – |

pI | isoelectric point | – |

$\mathcal{Q}$ | partition function (based on reference state H_{N}A) | – |

R | universal gas constant | J (mol K)^{−1} |

≡s | “empty” surface (in surface complexation) | – |

σ^{2} | variance | – |

x | activity of H^{+}, x = {H^{+}} | mol/L |

Y_{L} | L^{th} moment constructed from a_{j}: Y_{L} = ∑_{j} j^{L} a_{j} | – |

w(x) | “pure water balance”: w ≡ [OH^{−}] − [H^{+}] = K_{w}/x − x | mol/L |

Z | charge of highest protonated acid species | – |

z_{av} | average charge of acid (= Z_{T}/C_{T}) | – |

z_{j} | charge of acid species j: z_{j} = Z − j | – |

Z_{T} | total charge of acid | – |

$\mathcal{Z}$ | partition function (based on reference state A^{−N}) | – |

## Appendix B. Activity Models

#### Appendix B.1. Activity vs. Concentration

_{2}O molecules. In this way, ions behave chemically like they are less concentrated than they actually are (or measured). This effective concentration, which is available for reactions, is called activity:

_{j}:

_{j}[j]

_{j}= 1 ⇒ {j} = [j]

_{j}corrects for electrostatic shielding by other ions, γ

_{j}depends on the ionic strength:

_{j}, multivalent ions contribute particularly strongly to the ionic strength.

#### Appendix B.2. Activity Corrections

_{j}, α

_{j}

^{0}, and b

_{j}). The parameters A and B depend on temperature T and the dielectric constant ε. For water at 25 °C, we have (with ε = ε

_{r}ε

_{0}= 78.54·8.854·10

^{−12}J

^{−1}C

^{2}m

^{−1}):

^{6}(εT)

^{−3/2}= 0.5085 M

^{−1/2}and B = 3.29 nm

^{−1}M

^{−1/2}

## Appendix C. Mathematical Relationships

#### Appendix C.1. Index Transformations and Special Values for EPs

_{j}, and the ionization fractions a

_{j}. In contrast, the index n is a parameter of the 3-component system that confines the equivalent fraction n = C

_{B}/C

_{T}to integer and half-integer values. The transformation rule between both index types is simple:

_{j}, ionization fractions a

_{j}, and moments Y

_{L}.

semi-EP_{n}for Half-Integer n (j = n + ½) | EP_{n}for Integer n (j = n) |
---|---|

pH_{n} = pK_{j} | pH_{n} = pH_{j} ≡ ½ (pK_{j} + pK_{j+1}) |

a_{j} = a_{j−1} ≈ ½ | a_{j} = 1 − 2a_{j−1} (≈1) |

Y_{L} = ½ {(j−1)^{L} + j^{L}} | Y_{L} ≈ j^{L} |

Y_{L}(pK_{1}) = ½ | Y_{L}(pH_{1}) = 1 |

Y_{L}(pK_{2}) = ½ (1 + 2^{L}) | Y_{L}(pH_{2}) = 2^{L} |

Y_{L}(pK_{3}) = ½ (2^{L} + 3^{L}) | Y_{L}(pH_{3}) = 3^{L} |

Y_{1}(pK_{j}) = j − ½ | Y_{1}(pH_{j}) = j |

Y_{2}(pK_{j}) = j (j − 1) + ½ | Y_{2}(pH_{j}) = j^{2} |

#### Appendix C.2. pH Derivatives

## References

- Stumm, W.; Morgan, J.J. Aquatic Chemistry, 3rd ed.; John Wiley & Sons: New York, NY, USA, 1996; ISBN 978-0-471-51185-4. [Google Scholar]
- Morel, F.M.M.; Hering, J.G. Principles and Applications of Aquatic Chemistry; Wiley: New York, NY, USA, 1993; ISBN 978-0-471-54896-6. [Google Scholar]
- Appelo, C.; Postma, D. Geochemistry, Groundwater and Pollution, 2nd ed.; Routledge: Boca Raton, FL, USA, 2010; ISBN 978-0-415-36428-7. [Google Scholar]
- Butler, J.N. Ionic Equilibrium: Solubility and pH Calculations, Rev; Wiley: New York, NY, USA, 1998; ISBN 0471585262. [Google Scholar]
- Harris, D.C. Quantitative Chemical Analysis, 8th ed.; Freeman Custom Publishing: New York, NY, USA, 2010; ISBN 1429264845. [Google Scholar]
- Hunter, K.A. Acid-Base Chemistry of Aquatic Systems: An Introduction to the Chemistry of Acid-Base Equilibria with Emphasis on the Carbon Dioxide System in Natural Waters; Department of Chemistry, University of Otago: Dunedin, New Zealand, 1998. [Google Scholar]
- Kahlert, H.; Scholz, F. Acid-Base Diagrams; Springer: Berlin/Heidelberg, Germany, 2013; ISBN 978-3-642-37901-7. [Google Scholar]
- Pardue, H.L.; Odeh, I.N.; Tesfai, T.M. Unified Approximations: A New Approach for Monoprotic Weak Acid-Base Equilibria. J. Chem. Educ.
**2004**, 81, 1367. [Google Scholar] [CrossRef] - Casado-Riobó, A. A More Rational Treatment of the Acid-Base Equilibria Applying the Proton Condition in Equilibrium. How the Protons in Equilibria are Additive: Dissolutions of Acids. Crit. Rev. Anal. Chem.
**2005**, 35, 289–299. [Google Scholar] [CrossRef] - Hofmann, A.F.; Soetaert, K.; Middelburg, J.J.; Meysman, F.J.R. AquaEnv: An Aquatic Acid–Base Modelling Environment in R. Aquat Geochem.
**2010**, 16, 507–546. [Google Scholar] [CrossRef] [Green Version] - Parkhust, D.L.; Appelo, C. Description of Input and Examples for PHREEQC Version 3: A Computer Program for Speciation, Batch-Reaction, One-Dimensional Transport, and Inverse Geochemical Calculations; US Geological Survey: Denver, CO, USA, 2013.
- Kulik, D.A.; Wagner, T.; Dmytrieva, S.V.; Kosakowski, G.; Hingerl, F.F.; Chudnenko, K.V.; Berner, U.R. GEM-Selektor geochemical modeling package: Revised algorithm and GEMS3K numerical kernel for coupled simulation codes. Comput. Geosci.
**2012**. [Google Scholar] [CrossRef] [Green Version] - Koukkari, P.; Pajarre, R. A Gibbs energy minimization method for constrained and partial equilibria. Pure Appl. Chem.
**2011**, 83, 1243–1254. [Google Scholar] [CrossRef] - King, D.W.; Kester, D.R. A general approach for calculating polyprotic acid speciation and buffer capacity. J. Chem. Educ.
**1990**, 67, 932. [Google Scholar] [CrossRef] - Weltin, E. Calculating Equilibrium Concentrations for Stepwise Binding of Ligands and Polyprotic Acid-Base Systems: A General Numerical Method to Solve Multistep Equilibrium Problems. J. Chem. Educ.
**1993**, 70, 568. [Google Scholar] [CrossRef] - Rodriguez, J.L.; Loparo, K.A. Modeling and identification of pH processes. In Proceedings of the 2004 American Control Conference, Boston, MA, USA, 30 June–2 July 2004; IEEE: Piscataway, NJ, USA, 2004; Volume 6, pp. 5483–5488, ISBN 0-7803-8335-4. [Google Scholar]
- Hurek, J.; Nackiewicz, J. A Simple Method for the Consecutive Determination of Protonation Constants through Evaluation of Formation Curves. J. Chem. Educ.
**2013**, 90, 604–608. [Google Scholar] [CrossRef] - Mioni, R.; Mioni, G. A Mathematical Model of pH, Based on the Total Stoichiometric Concentration of Acids, Bases and Ampholytes Dissolved in Water. Scand. J. Clin. Lab. Investig.
**2015**, 75, 452–469. [Google Scholar] [CrossRef] - Gambi, A.; Toniolo, R. Acid-Base Logarithmic Diagrams with Computer Algebra Systems. ChemTexts
**2016**, 2, 9. [Google Scholar] [CrossRef] [Green Version] - Simms, H.S. Dissociation of Polyvalent Substances II. Relation of Constants to Chemical Structure. J. Am. Chem. Soc.
**1926**, 48, 1251–1261. [Google Scholar] [CrossRef] - Pehrsson, L. Acid-base titrations by stepwise addition of equal volumes of titrant with special reference to automatic titrations—II Theory of titration of mixtures of acids, polyprotic acids, acids in mixture with weak bases, and ampholytes. Talanta
**1976**, 23, 781–788. [Google Scholar] [CrossRef] - de Levie, R. The Formalism of Titration Theory. Chem. Educator
**2001**, 6, 272–276. [Google Scholar] [CrossRef] - Jano, I.; Hardcastle, J.; Jano, L.A.; Bates, K.R.; McCreary, H.E. General Equation for Determining the Dissociation Constants of Polyprotic Acids and Bases from Additive Properties: Part III. Calculation of Errors. Anal. Chim. Acta
**2001**, 428, 309–321. [Google Scholar] [CrossRef] - Khan, A. A simple method for analysis of titration curves of polyprotic acids. Nucleus
**2015**, 51, 448–454. [Google Scholar] - Asuero, A.G. Buffer Capacity of a Polyprotic Acid: First Derivative of the Buffer Capacity and pKa Values of Single and Overlapping Equilibria. Crit. Rev. Anal. Chem.
**2007**, 37, 269–301. [Google Scholar] [CrossRef] - Asuero, A.G.; Michałowski, T. Comprehensive Formulation of Titration Curves for Complex Acid-Base Systems and Its Analytical Implications. Crit. Rev. Anal. Chem.
**2011**, 41, 151–187. [Google Scholar] [CrossRef] - Michałowski, T.; Asuero, A.G. New Approaches in Modeling Carbonate Alkalinity and Total Alkalinity. Crit. Rev. Anal. Chem.
**2012**, 42, 220–244. [Google Scholar] [CrossRef] - Michałowska-Kaczmarczyk, A.M.; Michałowski, T. Dynamic Buffer Capacity in Acid-Base Systems. J. Solut. Chem.
**2015**, 44, 1256–1266. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kaczmarczyk, A.M.M.; Kucab, A.S.; Asuero, A.G.; Michalowski, T. The Simms Constants as Parameters in Hyperbolic Functions Related to Acid-Base Titration Curves. J. Chem. Appl. Chem. Eng.
**2018**, 2. [Google Scholar] [CrossRef] - Michalowska-Kaczmarczyk, A.M.; Michalowski, T. Dynamic buffer capacity versus alkalinity. Formulation in terms of Simms constants idea. Biomed. Sci. Eng.
**2020**, 6, 1–9. [Google Scholar] [CrossRef] - Schellman, J.A. Macromolecular binding. Biopolymers
**1975**, 14, 999–1018. [Google Scholar] [CrossRef] - Wyman, J.; Gill, S.J. Binding and Linkage: Functionnal Chemistry of Biological Molecules; University Science Books: Mill Valley, CA, USA, 1990; ISBN 9780935702569. [Google Scholar]
- Onufriev, A.; Case, D.A.; Ullmann, G.M. A novel view of pH titration in biomolecules. Biochemistry
**2001**, 40, 3413–3419. [Google Scholar] [CrossRef] [PubMed] - Ullmann, G.M. Relations between Protonation Constants and Titration Curves in Polyprotic Acids: A Critical View. J. Phys. Chem. B
**2003**, 107, 1263–1271. [Google Scholar] [CrossRef] - Allison, J.D.; Brown, D.S.; Novo-Gradac, K.J. MINTEQA2/PRODEFA2, A Geochemical Assessment Model for Environmental Systems: User Manual Supplement for Version 4.0; EPA: Washington, DC, USA, 1998. [Google Scholar]
- Ball, J.W.; Nordstrom, D.K. WATEQ4F—User’s Manual with Revised Thermodynamic Data Base and Test Cases for Calculating Speciation of Major, Trace and Redox Elements in Natural Waters; US Geological Survey: Denver, CO, USA, 1991. [Google Scholar]
- Reif, F. Fundamentals of Statistical and Thermal Physics; McGraw-Hill: Boston, MA, USA, 1965; ISBN 978-0-07-051800-1. [Google Scholar]
- Kittel, C. Introduction to Solid State Physics, 4th ed.; John Wiley & Sons Ltd.: Hoboken, NJ, USA, 1971; ISBN 978-0-471-14286-7. [Google Scholar]
- Millero, F.J. Thermodynamics of the Carbon Dioxide System in the Oceans. Geochim. Cosmochim. Acta
**1995**, 59, 661–677. [Google Scholar] [CrossRef] - RES³T- Rossendorf Expert System for Surface and Sorption Thermodynamics. Available online: https://www.hzdr.de/db/res3t.login (accessed on 15 August 2019).

**Figure 3.**pH dependence of ionization fractions for four acids (small circles mark equivalence points).

**Figure 4.**Each ionization fraction a

_{j}has its own domain in the pH interval between two adjacent pK

_{j}values. (HA—acetic acid, H

_{2}A—carbonic acid, H

_{3}A—phosphoric acid).

**Figure 5.**Construction of Y

_{1}(blue curve in bottom diagram) from summation over weighted ionization fraction a

_{j}. (Example: phosphoric acid as a triprotic acid).

**Figure 6.**pH dependence of the moments Y

_{1}to Y

_{4}for four acids. In the case of monoprotic acids (top-left diagram), all Y

_{L}are equal, i.e., the four curves cover each other.

**Figure 8.**Titration curves for 1, 10, and 100 mM H

_{2}CO

_{3}. (

**Left**): n = n(pH); (

**right**): pH = pH(n).

**Figure 9.**Set of equations describing the 3-component acid–base system (H

_{2}O, H

_{N}A, strong base). The three components are coupled via the charge-balance equation.

**Figure 10.**Titration curves (i.e., buffer capacities) for four acids with different amounts of C

_{T}(including the high-C

_{T}limit n(pH) = Y

_{1}). Plots are based on Equation (41).

**Figure 11.**Functional relationship between pH and C

_{T}for four acids (and n = 0). (

**Left**): calculation based on Equation (43); (

**right**): calculation based on the polynomial (44).

**Figure 15.**Relationship EP

_{n}⇔ pH

_{n}. EPs and semi-EPs are located at integer and half-integer values of n on the titration curve. Calculation for 100 mM H

_{2}CO

_{3}.

**Figure 19.**Normalized buffer capacity (blue), buffer intensity (green), and dβ/dpH (red) for four acids and C

_{T}→ ∞.

**Figure 21.**Micro- and macrostates of a triprotic acid and the corresponding (cumulative) equilibrium constants.

**Figure 22.**Ionization fractions a

_{j}(dashed curves) and probability density functions f

_{0}and f

_{N}.

**Figure 23.**Interrelation between different types of equilibrium constants of a polyprotic acid H

_{N}A.

**Figure 24.**Undissociated fraction a

_{0}for strong and weak acids. Strong acids are completely dissociated in the pH range above pH ≈ 0.

**Figure 27.**pH–C

_{T}diagrams with trajectories of EPs and semi-EPs for four acids. Calculations are based on Equation (117).

**Figure 28.**Trajectories of EPs and semi-EPs for phosphoric acid (H

_{3}PO

_{4}) for the uncoupled (

**top**) and coupled system (

**bottom**).

**Figure 29.**Ionization fractions for phosphoric acid in two approximations (dashed lines) vs. exact description based on Equation (17).

**Figure 30.**Titration curve n = n(pH) for carbonic acid (C

_{T}= 10 mM) with equivalence points and the corresponding types of alkalinity and acidity.

**Figure 31.**Species distribution of H

_{2}CO

_{3}as a function of pH for var A (

**left**) and var B (

**right**). Top diagrams: concentrations in linear scale; bottom diagrams: logarithmic scale.

**Figure 32.**Species distribution of H

_{2}CO

_{3}as a function of pH for var A (

**left**) and var C (

**right**). Top diagrams: concentrations in linear scale; bottom diagrams: logarithmic scale.

**Figure 33.**pH–C

_{T}diagrams of the carbonic-acid system. (

**Left)**: H

_{2}CO

_{3}in pure water (solid lines) vs. H

_{2}CO

_{3}in seawater (dashed lines). (

**Right)**: closed-form expression (117) vs. numerical model (dots).

**Figure 35.**Titration curves n = n(pH) for the amino acid glycine. (

**Left**): calculations based on Equation (138) for four amounts of C

_{T}. (

**Right**): comparison with numerical program PhreeqC (dots).

**Figure 37.**Titration curves n(pH), buffer capacity β and its derivative dβ/dpH for four zwitterionic acids (from Table 12). Calculations for C

_{T}→ ∞.

**Figure 39.**

**(Left)**: Ionization fractions of EDTA in linear scale (top) and logarithmic scale (bottom). (

**Right)**: Titration curves of EDTA for three values of C

_{T}based on Equation (138). The red dots represent numerical calculations with PhreeqC. [Note: Complexation with Na is ignored.]

**Figure 41.**(

**Left**) pH–C

_{T}diagrams for glycine. (

**Right**): enlarged section around EP

_{0}showing the isoelectric and isoionic points as a function of C

_{T}.

**Figure 42.**Ionization fractions representing the surface-species distribution on kaolinite (in the absence of Coulomb interactions).

**Figure 43.**pH dependence of surface charge for four surfaces (in the absence of Coulomb interactions).

**Figure 44.**Three mathematically equivalent descriptions of the N-protic acid–base system. The algebraic set of N + 3 equations is defined in Equations (27)–(32).

**Table 1.**pK values for four N-protic acids at 25 °C. (The composite carbonic acid is the sum of the unionized species CO

_{2}(aq) and the pure acid: H

_{2}CO

_{3}* = CO

_{2}(aq) + H

_{2}CO

_{3}; to simplify the notation, we omit the asterisk (*) on H

_{2}CO

_{3}* throughout the paper).

N | Acid | Formula | Type | pK_{1} | pK_{2} | pK_{3} | Ref. |
---|---|---|---|---|---|---|---|

1 | acetic acid | CH_{3}COOH | HA | 4.76 | [35] | ||

2 | (composite) carbonic acid | H_{2}CO_{3} | H_{2}A | 6.35 | 10.33 | [36] | |

3 | phosphoric acid | H_{3}PO_{4} | H_{3}A | 2.15 | 7.12 | 12.35 | [35] |

3 | citric acid | C_{6}H_{8}O_{7} | H_{3}A | 3.13 | 4.76 | 6.4 | [35] |

Type 1 (S Shaped) | Type 2 (Bell Shaped) | |
---|---|---|

ionization fraction | a_{0} and a_{N} | a_{1}, a_{2}, … a_{N−1}(does not exist for 1-protic acids) |

domain (pH interval) | pH < pK_{1} (for a_{0}) pH > pK _{N} (for a_{N}) | pK_{j} < pH < pK_{j+1} |

maximum at pH | −∞ (for a_{0}) +∞ (for a _{N}) | ½ (pK_{j} + pK_{j+1}) |

strongly acidic (pH → 0) strongly alkaline (pH → 14) | a_{0} = 1, a_{N} = 0 a _{0} = 0, a_{N} = 1 | a_{j} = 0 a _{j} = 0 |

integral (area below curve) | infinite | finite |

statistical meaning (see Section 2.5.3) | cumulative distribution function | – |

associated equivalence points | two external EPs: EP _{0} and EP_{N} | N − 1 internal EPs: EP _{1}, EP_{2}, … EP_{N-1} |

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

N | Acid | pH_{1/2} | pH_{1} | pH_{3/2} | pH_{2} | pH_{5/2} |
---|---|---|---|---|---|---|

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 |

N Acidity Constants | N + 1 Cumulative Constants for | N Simms Constants | 2^{N} − 1 Microscopic Constants | |
---|---|---|---|---|

Dissociation | Association | |||

K_{1,} K_{2}, …, K_{N} | k_{0} = 1 | β_{0} = 1 | g_{1,} g_{2}, …, g_{N} | g_{1,} g_{2}, g_{3}, …g _{1}g_{2}, g_{2}g_{3}, …g _{1}g_{2}g_{3}, …g _{1}g_{2}⋯g_{N} |

${\mathrm{k}}_{\mathrm{j}}={\displaystyle \prod _{\mathrm{i}=1}^{\mathrm{j}}{\mathrm{K}}_{\mathrm{i}}}$ | ${\mathsf{\beta}}_{\mathrm{j}}={\displaystyle \prod _{\mathrm{i}=\mathrm{N}+1-\mathrm{j}}^{\mathrm{N}}\frac{1}{{\mathrm{K}}_{\mathrm{i}}}}$ | (implicit functions of Equations (101) and (102)) |

Strong Acid | Weak Acid | |
---|---|---|

acidity constant | K_{a} ≫ 1 | K_{a} ≤ 1 |

pK_{a} = −lg K_{a} | pK_{a} < 0 | pK_{a} > 0 |

[H^{+}] ≈ {H^{+}} = 10^{−pH} | [H^{+}] ≈ C_{T} | [H^{+}] ≪ C_{T} |

undissociated acid | [HA] ≈ 0 or a_{0} ≈ 0 | [HA] ≈ C_{T} or a_{0} ≈ 1 |

dissociated acid | [A^{−}] ≈ C_{T} or a_{1} ≈ 1 | [A^{−}] ≪ C_{T} or a_{1} ≪ 1 |

Degree of Strength | Degree of Dilution | |||||
---|---|---|---|---|---|---|

determined by | acidity constant K_{a} | amount of acid C_{T} | ||||

relationships | weak acid small K _{a}(positive pK _{a} | ↔ ↔ ↔ | strong acid large K _{a}negative pK _{a}) | diluted acid small C _{T} | ↔ ↔ | concentr. Acid large C _{T} |

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) |

Function/Curve | semi-EP_{n}(half-integer n = j − ½) at pK _{j} | EP_{n}(integer n = j) at pH _{j} |
---|---|---|

titration curve n = n(pH) (normalized buffer capacity) | inflection points | inflection points |

buffer intensity β = dn/dpH | maxima | minima |

dβ/dpH = d^{2}n/dpH^{2} | zeros | zeros |

ionization fraction a_{j} | inflection point | maximum |

**Table 8.**Thermodynamic and conditional equilibrium constants for H

_{2}CO

_{3}in pure water and seawater (at 25 °C, 1 atm);

^{c}K values from [39].

Thermodynamic K (Pure Water, I = 0) | Conditional ^{c}K (Seawater, I = 0.7 M) | |
---|---|---|

pK_{1} | 5.18 | 6.0 |

pK_{2} | 10.33 | 9.1 |

pK_{w} | 14.0 | 13.9 |

Diprotic Acid | Zwitterion (Simplest Amino Acid) | |||||
---|---|---|---|---|---|---|

1st dissociation | H_{2}A = H^{+} + HA^{−} | K_{1} | H_{2}A^{+} = H^{+} + HA | K_{1} | ||

2nd dissociation | HA^{−} = H^{+} + A^{−2} | K_{2} | HA = H^{+} + A^{−} | K_{2} | ||

species | H_{2}A | HA^{−} | A^{−2} | H_{2}A^{+} | HA | A^{−} |

abbreviation | [0] | [1] | [2] | [0] | [1] | [2] |

ionic fraction a_{j} | a_{0} | a_{1} | a_{2} | a_{0} | a_{1} | a_{2} |

charge z_{j} | 0 | −1 | -2 | 1 | 0 | −1 |

average charge z_{av} | (0·[H_{2}A] − 1·[HA^{−}] − 2·[A^{−2}])/C_{T} | (1·[H_{2}A^{+}] − 0·[HA] − 1·[A^{−}])/C_{T} |

Diprotic Acid (Z = 0) | Zwitterion (Z = 1) | ||||
---|---|---|---|---|---|

[0]: | [H_{2}A] | (neutral) | [H_{2}A^{+}] | (cation) | highest protonation |

[1]: | [HA^{−}] | (anion) | [HA] | (neutral) | |

[2]: | [A^{−2}] | (anion) | [A^{−}] | (anion) | fully deprotonated |

Ordinary Acid Z = 0 | Zwitterion 1 ≤ Z < N | |||
---|---|---|---|---|

j = 0 | [0] = [H_{N}A^{+Z}] | highest protonation | neutral | cation |

j = Z | [Z] = [H_{N-Z}A] | undissociated acid | neutral, [Z] = [0] | neutral |

j = N | [N] = [A^{−(N−Z)}] | fully deprotonated | anion | anion |

Acid | N | Z | pK_{1} | pK_{2} | pK_{3} | pK_{4} | pK_{5} | pK_{6} | [j = 0] | [j = Z] | [j = N] |
---|---|---|---|---|---|---|---|---|---|---|---|

carbonic acid | 2 | 0 | 6.35 | 11.33 | H_{2}A | H_{2}A | A^{−2} | ||||

glycine | 2 | 1 | 2.35 | 9.778 | H_{2}A^{+} | HA | A^{−1} | ||||

glutamic acid | 3 | 1 | 2.16 | 4.30 | 9.96 | H_{3}A^{+} | H_{2}A | A^{−2} | |||

NTA | 4 | 1 | 1.0 | 2.0 | 2.942 | 10.28 | H_{4}A^{+} | H_{3}A | A^{−3} | ||

EDTA | 6 | 2 | 0 | 1.5 | 2.16 | 3.119 | 6.281 | 10.94 | H_{6}A^{+2} | H_{4}A | A^{−4} |

Isoionic Point | Isoelectric Point (pI) |
---|---|

pH of pure acid | pH at which the net charge of all acid species is zero |

depends on C_{T}(approaches pI-value for C _{T} → ∞and pH = 7 for C _{T} → 0) | independent of C_{T} |

Z − Y_{1}(pH) = w(pH)/C_{T} | Z − Y_{1}(pH) = 0 |

exists for ordinary acids and zwitterions | exists only for zwitterions |

Kaolinite | Mica | Goethite | Gibbsite | |
---|---|---|---|---|

pK_{1} | 2.52 | 6.01 | 7.13 | 7.53 |

pK_{2} | 5.88 | 7.65 | 9.60 | 9.87 |

pH_{pzc} = ½ (pK_{1}+pK_{2}) | 4.20 | 6.83 | 8.37 | 8.70 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kalka, H.
Polyprotic Acids and Beyond—An Algebraic Approach. *Chemistry* **2021**, *3*, 454-508.
https://doi.org/10.3390/chemistry3020034

**AMA Style**

Kalka H.
Polyprotic Acids and Beyond—An Algebraic Approach. *Chemistry*. 2021; 3(2):454-508.
https://doi.org/10.3390/chemistry3020034

**Chicago/Turabian Style**

Kalka, Harald.
2021. "Polyprotic Acids and Beyond—An Algebraic Approach" *Chemistry* 3, no. 2: 454-508.
https://doi.org/10.3390/chemistry3020034