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: \(n(x) = \left( Y_1 - Z \right) + \dfrac{w}{C_T}\)
(1.2) buffer intensity: \(\beta(x) \equiv \dfrac{\textrm{d} n}{\textrm{d}\,\textrm{pH}} = (\ln 10) \left( Y_2 - Y_1^2 + \dfrac{w+2x}{C_T} \right)\)
(1.3) 1st derivative of β: \(\dfrac{\textrm{d}\beta}{\textrm{d}\,\textrm{pH}} = (\ln 10)^2 \left( Y_3 - 3Y_1Y_2 + 2Y_1^3 + \dfrac{w}{C_T} \right)\)

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)  ≡  \(\sum\limits_{j=0}^{N}\,\) j L aj(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)   \(a_j = \left( \dfrac{k_j}{x^j} \right ) \,a_0\) with \(a_0 = \left( 1+\dfrac{k_1}{x} +\dfrac{k_2}{x^2}+\cdots +\dfrac{k_N}{x^N} \right)^{\!-1}\)

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:

set of mathematical equations for polyprotic acids

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  ≡  \(\mathrm{\dfrac{d}{d\,pH}}\) buffer capacity  =  \(\mathrm{\dfrac{d}{d\,pH}}\) n(pH)

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


Ionization Fractions aj based on 4:

ionization fractions of four common acids as a function of pH

Moments YL based on 3:

moments Y1 to Y4 of four common acids as a function of pH

[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

titration curves of four common acids

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:

buffer intensity of four common acids as a function of pH

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)   \(0 \ =\ \sum\limits_{j=0}^{N}\left \{\, x^2 + (n-j) \,C_T\,x -K_w\, \right\}\, k_j \,x^{N-j}\)

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

  1. The mathematical derivation is provided as Review (2021), Lecture (2023) and/or PowerPoint (2017). 

  2. Square brackets [..] denote molar concentrations (in contrast to activities, which are expressed by curly braces {..}). 

  3. In textbooks, the equivalent fraction is often abbreviated by f

  4. In hydrochemistry, it is common practice to use the composite carbonic acid, H2CO3* = CO2(aq) + H2CO3 instead of the true carbonic acid. 

  5. 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). 

[last modified: 2024-01-16]