AcidBase Systems  A Mathematical Toolkit
Simple ClosedForm Equations
Given:  Nprotic acid H_{N}A 
(specified by N dissociation constants: K_{1}, K_{2} to K_{N}) 
The pH dependence of this acid is then characterized by the following easytoplot functions:^{1}
(1.1)  titration curve:  
(1.2)  buffer intensity:  
(1.3)  1^{st} derivative of β: 
Notation and abbreviations:^{2}
(2.1)  H^{+} concentration:  x = [H^{+}] = 10^{pH} 
(2.2)  purewater balance:  w(x) ≡ [OH^{}] – [H^{+}] = K_{w}/x – x 
(2.3)  selfionization constant:  K_{w} = 10^{14} (at 25 °C) 
(2.4)  equivalent fraction:  n ≡ C_{B}/C_{T} 
(2.5)  total amount of acid:  C_{T} 
(2.6)  total amount of strong base:  C_{B} 
Moments Y_{L}. The main building blocks of the formulas are the socalled moments Y_{1}, Y_{2} and Y_{3}, which are defined as weighted sums over ionization fractions a_{j} from j=0 to N:
(3)  Y_{L}(x) ≡ j^{ L} a_{j}(x) 
In particular, we have:
(3.1)  Y_{0} = a_{0} + a_{1} + … + a_{N} = 1  ⇒  mass balance  
(3.2)  Y_{1} = a_{1} + 2a_{2} + … + N a_{N}  ⇒  titration curve  
(3.3)  Y_{2} = a_{1} + 4a_{2} + … + N^{2}a_{N}  ⇒  buffer intensity β  
(3.4)  Y_{3} = a_{1} + 8a_{2} + … + N^{3}a_{N}  ⇒  1^{st} derivative of β 
Ionization Fractions. To recapitulate: An Nprotic acid is completely specified by the acid’s N dissociation or acidity constants K_{1}, K_{2} to K_{N}. The ionization fractions a_{j} rely on these acidity constants in the following way:
(4)  with 
where k_{j} are the cumulative equilibrium constants (as products of acidity constants):
(5)  k_{0} = 1, k_{1} = K_{1}, k_{2} = K_{1}K_{2}, … k_{N} = K_{1}K_{2}…K_{N} 
The a_{j}’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).^{3} It describes nothing else than the (normalized) buffer capacity. The 1^{st} 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 semiEPs into pHC_{T} 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 pK_{j} = –lg K_{j}):
Acid  Formula  Type  pK_{1}  pK_{2}  pK_{3} 

acetic acid  CH_{3}COOH  HA  4.76  
carbonic acid^{4}  H_{2}CO_{3}  H_{2}A  6.35  10.33  
phosphoric acid  H_{3}PO_{4}  H_{3}A  2.15  7.21  12.35 
citric acid  C_{6}H_{8}O_{7}  H_{3}A  3.13  4.76  6.4 
Diagrams
Ionization Fractions a_{j} based on 4.
Moments Y_{L} based on 3. [Note: For the simplest acid HA – in the topleft diagram – all four curves are identical.]
Titration curves^{5} based on 1.1 for different amounts of acid, C_{T}. The Y_{1}curve (in dark blue) describes the asymptotic case of infinitely large C_{T} (i.e. pure, nondiluted 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

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

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

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

In hydrochemistry, it is common practice to use the composite carbonic acid, H_{2}CO_{3}^{*} = CO_{2}(aq) + H_{2}CO_{3} instead of the true carbonic acid. ↩

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). ↩