Acid-Base Systems -- A Mathematical Toolkit
Simple Closed-Form Equations
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:
Notation and abbreviations:
||x = [H+] = 10-pH
||n ≡ CB/CT
||total amount of acid:
||total amount of strong base:
||w(x) ≡ [OH-] – [H+] = Kw/x – x
||Kw = 10-14 (at 25 °C)
||charge of highest protonated acid species:
Common acids are characterized by Z = 0; only zwitterionic acids (e.g. amino acids) are identified by Z ≥ 1. The parameter Z enters only 1.1 as a constant shift of the buffer capacity.
Moments YL. The main building blocks of the formulas are the so-called moments Y1, Y2 and Y3, which are defined as weighted sums over ionization fractions aj from j=0 to N:
In particular, we have:
||Y0 = a0 + a1 + … + aN = 1
||Y1 = a1 + 2a2 + … + N aN
||enters buffer capacity
||Y2 = a1 + 4a2 + … + N2aN
||enters buffer intensity β
||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 dissociation or acidity constants K1, K2 to KN. The ionization fractions aj rely on these acidity constants in the following way:
where kj are the cumulative equilibrium constants (as products of acidity constants):
||k0 = 1, k1 = K1, k2 = K1K2, … kN = K1K2…KN
The aj’s are the smallest building blocks of our mathematical toolkit.
Speciation. The equilibrium distribution of the acid species [j] is closely related to the ionization fractions:
||[j] = CT aj
||[j] ≡ [HN-j AZ-j]
||(for j = 0,1, … , N)
Here,  represents the highest protonated species and [N] the fully deprotonated species. The symbol j is an integer that also indicates the charge of the species:
||charge of species [j]:
||zj = Z – j
Summary. The following scheme summarizes the modular structure of the general equations:
What we call ‘titration curve’ is just the (normalized) buffer capacity. It represents the equivalent fraction as function of pH: n = n(pH). The 1st derivative of the buffer capacity is the buffer intensity:
||buffer intensity β = buffer capacity = n(pH)
Equivalence Points (EP). Equation (1.1) can also be used to plot curves EPs and semi-EPs into pH-CT diagrams.
Examples for N = 1, 2 and 3
The analytical formulas will be applied to four common acids, which are characterized by the following acidity constants (where pKj = –lg Kj):
Ionization Fractions aj based on 4:
Moments YL based on 3:
[Note: For the monoprotic acid HA – 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:
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.
The presented mathematical toolkit provides a better understanding of the acid-base system. But it will and can never replace numerical models like PhreeqC or aqion, which are able to handle real-world problems (including activity corrections, aqueous complex formation, etc.).
Remarks & Footnotes
[last modified: 2018-03-24]