Overview Activity Chem Equilibrium Acids Carbonate System Mineral Phases

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 K_N.

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 Y_L. 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 a_j 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 K_N. The ionization fractions a_j 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 k_j are cumulative equilibrium constants (as products of the acidity constants K_j):

(5)

k0 = 1,    k1 = K1,    k2 = K1K2,    …    kN = K1K2…KN

The a_j’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 a_j:

(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:

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 1^st 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-C_T 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 pK_j = –lg K_j):

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

Diagrams

Ionization Fractions a_j based on 4:

Moments Y_L based on 3:

[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 C_T of acid:⁵

The Y₁-curve (in dark blue) describes the asymptotic case of infinitely large C_T (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.

Polynomials for x = 10^-pH

1.1 calculates n (or the acid amount C_T) for a given pH, which is indeed a nice and simple formula. Unfortunately, the inverse task, calculating the pH for a given amount of C_T (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 = C_B/C_T).

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