Acid-Base Systems -- A Mathematical Toolkit

Simple Closed-Form Equations

Given: 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:1

(1.1) titration curve:
(1.2) buffer intensity:
(1.3) 1st derivative of β:

Notation and abbreviations:2

(2.1) H+ concentration: x = [H+] = 10-pH
(2.2) pure-water balance: w(x) ≡ [OH-] – [H+] = Kw/x – x
(2.3) self-ionization constant: Kw = 10-14    (at 25 °C)
(2.4) equivalent fraction: n ≡ CB/CT
(2.5) total amount of acid: CT
(2.6) total amount of strong base: CB

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:

(3)   YL(x)  ≡  j L aj(x)

In particular, we have:

(3.1)   Y0  =  a0 + a1 + … + aN  =  1 mass balance
(3.2)   Y1  =  a1 + 2a2 + … + N aN titration curve
(3.3)   Y2  =  a1 + 4a2 + … + N2aN buffer intensity β
(3.4)   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:

(4)   with

where kj are the cumulative equilibrium constants (as products of acidity constants):

(5)   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:

set of mathematical equations for polyprotic acids

What we call ‘titration curve’ is the equivalent fraction as function of pH: n = n(pH).3 It describes nothing else than the (normalized) buffer capacity. The 1st derivative of the buffer capacity is the buffer intensity:

(6)   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):

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

Moments YL based on 3. [Note: For the simplest acid HA – in the top-left diagram – all four curves are identical.]

moments Y1 to Y4 of four common acids

Titration curves5 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).

titration curves of four common acids

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).

buffer intensity of four common acids

Remarks & Footnotes

  1. The mathematical derivation is provided as pdf and/or PowerPoint

  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 or the addition of a strong, monoprotic acid (e.g. HCl). 

[last modified: 2017-09-10]