Acid-Base Systems – A Mathematical Toolkit
Simple Closed-Form Equations
||N-protic acid HNA
||(specified by N dissociation constants: K1, K2 to KN)
The pH dependence of this acid is then characterized by the following easy-to-plot functions:
Notation and abbreviations:
||x = [H+] = 10-pH
||w(x) ≡ [OH-] – [H+] = Kw/x – x
||Kw = 10-14 (at 25 °C)
||n ≡ CB/CT
||total amount of acid:
||total amount of strong base:
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
||Y2 = a1 + 4a2 + … + N2aN
||buffer intensity β
||Y3 = a1 + 8a2 + … + N3aN
||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.
Summary. The following scheme summarizes the modular structure of the general equations:
What we call ‘titration curve’ is the equivalent fraction as function of pH: n = n(pH). It describes nothing else than the (normalized) buffer capacity. 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 used to plot curves EPs and semi-EPs into pH-CT diagrams.
Examples for N = 1, 2 and 3
The formulas above 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 simplest acid HA – in the top-left diagram – all four curves are identical.]
Titration curves based on 1.1 for different amounts of acid, CT. The Y1-curve (in dark blue) describes the asymptotic case of infinitely large CT (i.e. pure, non-diluted acid).
Buffer intensity β (green curves) based on 1.2 and its first derivative (red curves) based on 1.3. In addition: titration curves (in blue).
Remarks & Footnotes
[last modified: 2017-09-10]