Acid-Base Systems -- a Mathematical Toolkit
Simple Closed-Form Equations
Given: | N-protic acid | HNA |
or zwitterionic acid | HNA+Z with Z ≥ 1 |
specified by N dissociation constants: K1, K2 to KN.
The pH-dependence of the acid is then characterized by the following easy-to-plot functions:1
(1.1) | buffer capacity: | |
(1.2) | buffer intensity: | |
(1.3) | 1st derivative of β: |
Notation and abbreviations:2
(2.1) | activity of H+: | x ≡ {H+} = 10-pH |
(2.2) | equivalent fraction: | n ≡ CB/CT |
(2.3) | total amount of acid: | CT |
(2.4) | total amount of strong base: | CB |
(2.5) | pure-water balance: | w(x) ≡ [OH-] – [H+] ≈ Kw/x – x |
(2.6) | self-ionization of H2O: | Kw = 10-14 (at 25 °C) |
(2.7) | charge of highest protonated acid species: | Z |
Common acids are characterized by Z = 0; only zwitterionic acids (e.g. amino acids) are identified by Z ≥ 1. The parameter Z itself enters only 1.1, where it acts as a constant offset. The buffer intensity β and its derivative in 1.2 and (1.3) are both independent of Z.
Moments YL. The building blocks from which the formulas are built are the so-called moments Y1, Y2 and Y3. They are weighted sums over ionization fractions aj from j=0 to N:
(3) | YL(x) ≡ |
In particular, we have:
(3.1) | Y0 = a0 + a1 + … + aN = 1 | ⇒ | mass balance | |
(3.2) | Y1 = a1 + 2a2 + … + N aN | ⇒ | enters buffer capacity | |
(3.3) | Y2 = a1 + 4a2 + … + N2aN | ⇒ | enters buffer intensity β | |
(3.4) | Y3 = a1 + 8a2 + … + N3aN | ⇒ | enters 1st derivative of β |
Ionization Fractions. To recapitulate: An N-protic acid is completely specified by the acid’s N acidity constants K1, K2 to KN. The ionization fractions aj rely on these acidity constants in the following way:
(4) | with |
where kj are cumulative equilibrium constants (as products of the acidity constants Kj):
(5) | k0 = 1, k1 = K1, k2 = K1K2, … kN = K1K2…KN |
The aj’s are the smallest building blocks of our math toolkit.
Speciation. The equilibrium distribution of the acid species [j] is closely related to the ionization fractions aj:
(6) | [j] = CT aj | where | [j] ≡ [HN-j AZ-j] | (for j = 0,1, … , N) |
Here, [0] represents the highest protonated species and [N] the fully de-protonated species. The symbol j is an integer that also indicates the charge of the species (common acids are characterized by Z=0):
(7) | charge of species [j]: | zj = Z – j |
Summary. The following scheme summarizes the modular structure of the equations that describe the acid-base system:
What we call “titration curve” is just the (normalized) buffer capacity. It represents the equivalent fraction as function of pH: n = n(pH).3 The 1st derivative of the buffer capacity is the buffer intensity:
(8) | buffer intensity β ≡ |
Equivalence Points (EP). Equation (1.1) can be used to plot curves of EPs and semi-EPs into pH-CT diagrams.
Examples for N = 1, 2, and 3
Let’s apply the analytical formulas to four common acids, which are characterized by the following acidity constants (where pKj = –lg Kj):
Acid | Formula | Type | pK1 | pK2 | pK3 |
---|---|---|---|---|---|
acetic acid | CH3COOH | HA | 4.76 | ||
carbonic acid4 | H2CO3 | H2A | 6.35 | 10.33 | |
phosphoric acid | H3PO4 | H3A | 2.15 | 7.21 | 12.35 |
citric acid | C6H8O7 | H3A | 3.13 | 4.76 | 6.4 |
Diagrams
Ionization Fractions aj based on 4:
Moments YL based on 3:
[Note: For the monoprotic acid HA — shown in the top-left diagram — all four curves are identical.]
Buffer Capacities (i.e. “titration curves”) based on 1.1 for different amounts CT of acid:5
The Y1-curve (in dark blue) describes the asymptotic case of infinitely large CT (i.e. highly-concentrated acids).
Buffer Intensity β (green curves) based on 1.2 and its first derivative (red curves) based on 1.3:
In addition: buffer capacities (i.e. titration curves) are shown as blue curves.
Polynomials for x = 10-pH
1.1 calculates n (or the acid amount CT) for a given pH, which is indeed a nice and simple formula. Unfortunately, the inverse task, calculating the pH for a given amount of CT (or n), is not so simple. It leads to a polynomial of high degree, namely degree N+2:
(9) |
The (high) degree of the polynomial is independent of whether we set n=0 or not (where n refers to the amount of strong base: n = CB/CT).
The problem is that there is no algebraic expression for solving polynomials of degree higher than 4, no matter how hard we try. Thus, numerical root-finding methods should be applied.
Example N=1. The monoprotic acid represents the simplest case, where the sum in 9 runs only over two terms, j = 0 and 1. With k0 = 1 and k1 = K1 we get a cubic equation:
(9a) | 0 = x3 + {K1 + nCT} x2 + {(n-1)CT – Kw} x – K1Kw |
Example N=2. For diprotic acids, the polynomial becomes a quartic equation. Now the sum in 9 runs over three terms, j = 0, 1 and 2. With k0 = 1, k1 = K1 and k2 = K1K2 it yields:
(9b) | 0 = x4 + {K1 + nCT} x3 + {K1Kw + (n-1)CTK1 – Kw} x2 | |
+ K1 {(n-1)CTK1 – Kw} x – K1K2Kw |
Final Note
The presented math toolkit provides a better understanding of the acid-base system. But it cannot replace numerical models like PhreeqC or aqion, which are able to handle real-world problems including activity corrections, aqueous complex formation, etc.
Remarks & Footnotes
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The mathematical derivation is provided as Review (2021), Lecture (2023) and/or PowerPoint (2017). ↩
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Square brackets [..] denote molar concentrations (in contrast to activities, which are expressed by curly braces {..}). ↩
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In textbooks, the equivalent fraction is often abbreviated by f. ↩
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In hydrochemistry, it is common practice to use the composite carbonic acid, H2CO3* = CO2(aq) + H2CO3 instead of the true carbonic acid. ↩
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Negative values of n mimic the withdrawal of the strong base from the solution. It is equivalent to the addition of a strong monoprotic acid (e.g. HCl). ↩