## Equivalence Points – Systematics & Classification

The concept of equivalence points (EP) runs like a golden thread through acid-base theory and applications. There are different types of equivalence points.1 We provide a classification of EPs and semi-EPs. This is done for the general case of N-protic acids, HNA.

Definition of EPn

An equivalence point is a special equilibrium state at which chemically equivalent quantities of acid and bases have been mixed:

 (1) equivalence point: [acid]T  =  [base]T

The square brackets (with the small subscript “T”) denote the total molar amount of acid or base. In what follows, ‘base’ stands for a strong monoacidic base BOH,2 whereas ‘acid’ can be any N-protic acid (either strong or weak). For convenience we introduce CT and CB:

 (2a) total amount of N-protic acid: CT  =  [HNA]T (2b) total amount strong base: CB  =  [BOH]T

The ratio of both quantities is called

 (3) equivalent fraction: n = CB / CT

It represents the fraction of strong base that is added to an acid in titration. Thus, 1 defines the equivalence point EP1, because n = CB/CT = 1. But this is only one single equivalence point, probably the best known one. This concept can easily be adopted for other types of EPs.

In the case of an N-protic acid, HNA, we have (at first glance) N equivalence points EP1, EP2 to EPN based on integer values of n. In addition there are other important members: Firstly, EP0 which corresponds to the trivial case of a base-free system. Secondly, semi-EPs which correspond to half-integer values of the CB/CT-ratio. Taken all together we obtain a whole family of equivalence points EPn for both integer and half-integer values of n:

 (4a) EPn: CB / CT = n for n = 0, 1, … N (4b) semi-EPn: CB / CT = n for n = ½, 3/2, … N-½

In total there are 2N+1 equivalence points to characterize an N-protic acid sufficiently. It makes no sense to extend this set by more types of EPs.

Each EPn, as a special equilibrium state, is characterized by exactly one pH value: EPn ⇔ pHn. On the pH scale they are arranged in the following sequence:

 pH0, pH1/2, pH1, pH3/2, … pHN

But what is the concrete mathematical relationship between EPn and pHn?

Correspondence between EPn and pHn

The central quantity to start our consideration is the equivalent fraction n defined in 3. Remarkably enough, there is indeed a mathematical closed-form expression for the equivalent fraction as function of pH:

 (5) $n \ = \ Y_1(pH) \,+\, \dfrac{w(pH)}{C_T}$ (for details see Appendix B)

This equation consists of two parts or components (subsystems):

 • Y1 describes the component HNA (based on N acidity constants K1 to KN) • w describes the component H2O (based on the self-ionization constant Kw)

Both components (subsystems) are linked together via the proton balance in 5. Plotting the equivalent fraction as a function of pH yields the (normalized) titration curve, n = n(pH).

EPn ⇔ pHn.  Inserting integer and half-integer values for n into 5 yields just the equivalence points and the corresponding pHn values – marked by small circles in the diagram below. (The diagram displays the titration curve of 100 mM H2CO3 as the prototype of a diprotic acid).

In fact, 5 is amenable to simple hand calculations. It works for any N-protic acid and requires as input: (i) the acidity constants K1 to KN and (ii) the amount of acid CT.

[Remark. Though 5 provides an exact mathematical relationship between EPn and pHn, it does not allow you to isolate the pH variable on one side of the equation.3 Thus, an explicit formula, into which you enter an integer or half-integer value of n and just get pHn, does not exist.]

EPn as Extrema of the Buffer Intensity

EPs and semi-EPs emerge in acid-base titrations as extrema points of the buffer intensity:

 EPn (integer n) ⇔ minimum buffer intensity β semi-EPn (half-integer n) ⇔ maximum buffer intensity β

This behavior is illustrated for the same alkalimetric titration as in the previous diagram (CT = 100 mM H2CO3):

The blue titration curve, n = n(pH), is also known as buffer capacity. The pH-derivative of the buffer capacity is just the buffer intensity β = dn/dpH – here depicted as green curve. In fact, the maxima/minima of the green curve are located at points where the slope of the blue curve is largest/smallest.

EPn as Curves in pH-CT Diagrams

Equation (5) can be rearranged into the form

 (6) $C_T = \dfrac{w(pH)}{n-Y_1(pH)}$

Now, it’s easy to plot all EPn as curves into a pH-CT diagram (one curve for one integer or half-inter value of n):

These plots were generated in Excel using as input the acidity constants from the table below. [The dashed curves represent approximations based on simplified equations (where the self-ionization of H2O is neglected).]

The Elegant Simplicity of the ‘Pure Acid’ Approach (CT → ∞)

The general approach simplifies drastically if we neglect the second term in 5. This is legitimate for high enough values of CT because then the term w/CT tends to zero and can be neglected. What remains is the simple formula for the single component ‘acid’:

 (7) n = Y1(pH) or n – Y1(pH) = 0

The fascinating thing about this equation is that it predicts the direct relationship between the pH values of EPn and the acidity constants:

 (8a) EPn ⇔ pHn = ½ (pKn + pKn+1) for integer n = 1, 2, … N-1 (8b) semi-EPn ⇔ pHn = pKn+1/2 for half-integer n = ½, 3/2, … N-½

This is a mathematical fact that relies on the striking features of the ionization fractions.4

Equations (8a) and (8b) are valid for so-called internal EPs only, thereby excluding the two external equivalence points EP0 and EPN. Since the internal EPs do not depend on CT they appear as vertical dashed lines in the pH-CT diagrams shown above.

Equality of Species.  In the pure-acid case there is an alternative definition of equivalence points based on the equality of ‘neighbor’ and ‘next-to-neighbor’ acid species. For example, a triprotic acid encompasses the following set of EPs:

 EP0 [H+] = [H2A-] EP1/2 [H3A] = [H2A-] ⇔ pH1/2 = pK1 EP1 [H3A] = [HA-2] ⇔ pH1 = ½ (pK1+pK2) EP3/2 [H2A-] = [HA-2] ⇔ pH3/2 = pK2 EP2 [H2A-] = [A-3] ⇔ pH2 = ½ (pK2+pK3) EP5/2 [HA-2] = [A-3] ⇔ pH5/2 = pK3 EP3 [HA-2] = [OH-]

On the pH scale they are arranged as follows:

In short-hand notation,5 this can be generalized for any N-protic acid (valid for all internal EPs):

 (9a) EPn ⇔ [n-1] = [n+1] for integer n = 1, 2, … N-1 (9b) semi-EPn ⇔ [n-½] = [n+½] for half-integer n = ½, 3/2, … N-½

Application.  Just the alternative definition of EPs is often applied in textbooks for the carbonate system:

 (10a) EP0 ⇔ [H+] = [HCO3-] (also known as CO2 EP) (10b) EP1 ⇔ [CO2] = [CO3-2] (also known as HCO3- EP) (10c) EP2 ⇔ [HCO3-] = [OH-] (also known as CO3-2 EP)

Coupling of two Subsystems

Let’s explain the general behavior of the EPn curves in pH-CT diagrams by the example of phosphoric acid (as a triprotic acid). It is illustrated in the figure below that consists of two diagrams.

At first we have two separate components or subsystems located at both ends of the CT scale (shown in the top diagram):

 • pure H2O at CT = 0 with one EP at pH 7 • pure acid at CT → ∞ with EPs defined in 8a and (8b)

The bottom diagram displays what happens when both subsystems are linked together. Starting at pH 7 the curves fan out when CT increases until they fit the ‘pure-acid’ values at the top of the chart. The whole choreography is determined by 5.

Summary

1.  Equivalence points are special equilibrium states where the equivalent fraction n = CB/CT becomes an integer or half-integer value.

2.  An N-protic acid has in total 2N+1 equivalence points EPn defined in 4a and (4b). The trivial case EP0 refers to the base-free system.

3.  The mathematical relationship EPn ⇔ pHn is given by n = Y1(pH) + w(pH)/CT, where Y1 describes the acid and w the water.

4.  The equivalent fraction n = Y1(pH) + w(pH)/CT (titration curve) describes the buffer capacity. Its pH-derivative is the buffer intensity &beta = dn/dpH. EPs are extrema of β:

 EPn (integer n) ⇔ minimum buffer intensity β semi-EPn (half-integer n) ⇔ maximum buffer intensity β

5.  In the limit of undiluted acids (CT → ∞) the general relationship simplifies to: n = Y1(pH). This equation yields the direct link between pHn and the acidity constants:

 pHn = ½ (pKn + pKn+1) for integer n (EPn) pHn = pKn+1/2 for half-integer n (semi-EPn)

6.  In the limit of undiluted acids (CT → ∞) there is an alternative definition of EPs based on equal species concentrations – see 9a and (9b). (Example: In carbonate systems EP1 is often introduced as the equilibrium state where [CO2] = [CO3-2].)

Appendix A – Polyprotic Acids (HNA)

An N-protic acid is characterized by N acidity constants K1 to KN. Each acidity constant acts as an equilibrium constant in a series of dissociation reactions:

 (A1a) HNA = H+ + HN-1A- K1 = [H+] [HN-1A-] / [HNA] (A1b) HN-1A- = H+ + HN-2A-2 K2 = [H+] [HN-2A-2] / [HNA-] ⋯ (A1c) HA-(N-1) = H+ + A-N KN = [H+] [A-N] / [HA-(N-1)]

The subsequent release of H+ generates N+1 acid species. To shorten the notation we abbreviate (the molar concentration of) these species by [j],5 where j runs from 0 to N. By the way, the index j labels the negative electrical charge of the acid species. Thus, [0] abbreviates the dissolved but undissociated, neutral species [HNA].6 All species together add up to the total amount of acid:

 (A2) CT  =  [0] + [1] + … + [N]

In addition we introduce the abbreviation

 (A3) x  ≡  [H+]  =  10-pH

In this notation, the law-of-mass-action formulas in A1 get a particular simple form:

 (A4) $K_j \ = \ \dfrac{x \cdot [j]}{[j-1]}$ (Generalized Henderson-Hasselbach Equation)

Ionization Fractions.  What the set of mathematical equations in A1a to (A1c) actually provide is the pH dependence of each acid species. The best way to make it vivid relies on normalized acid-species concentrations known as

 (A5) ionization fractions: aj ≡ [j] \ CT for j = 0 to N

They are simple analytical expressions based on the acidity constants K1 to KN. Starting out from

 (A6a) $a_0 = \left( 1+\dfrac{K_1}{x} +\dfrac{K_1K_2}{x^2}+\cdots +\dfrac{K_1K_2\cdots K_N}{x^N} \right)^{-1}$

all other aj can be iteratively obtained:

 (A6b) $a_j = \left( \dfrac{K_j}{x} \right ) a_{j-1}$

Recall that the set of equations in (A1a) to (A1c) and the set of equations in (A6a) and (A6b) are mathematically equivalent.

Moments YL. It is useful to introduce so-called moments YL, which are weighted sums over the entire set of ionization fractions aj:

 (A7) YL(x)  ≡  $\sum\limits_{j=0}^{N}\$ j L aj(x)

The most important representatives that are used as building blocks of other relevant quantities are:

 (A8a) Y0  =  a0 + a1 + … + aN  =  1 ⟹ mass balance (cf. A2) (A8b) Y1  =  a1 + 2a2 + … + N aN ⟹ titration curve: n = n(pH) (A8c) Y2  =  a1 + 4a2 + … + N2 aN ⟹ buffer intensity β

In particular, A8b embodies the heart of the general equation in Appendix B. Again: each YL comprises the information contained in the set of equations (A1a) to (A1c).

pKj Values.  Acids are specified by their acidity constants Kj. Because they vary by orders of magnitude it is often practical to switch to pKj = –lg Kj. Typical values of four common acids are:

Acid Formula Type pK1 pK2 pK3
acetic acid CH3COOH HA 4.76
carbonic acid7 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

Appendix B – General Equation for HNA + H2O

The general case combines the subsystem ‘pure acid’ defined in A1a to (A1c) with the subsystem ‘H2O’ (i.e. the self-ionization of water). Both subsystems are coupled via the proton balance (as described in pdf or ppt). It yields the central closed-form equation:

 (B1) $n \ = \ Y_1(x) \,+\, \dfrac{w(x)}{C_T}$ with   x = [H+] = 10-pH

and the following abbreviations:

 (B2) w(x)  =  Kw/x – x (with Kw = 10-14 at 25 °C) (B3) Y1(x)  =  a1 + 2a2 + … + N aN

Y1 characterizes the N-protic acid; it was introduced in A8b.

[Remark. To be more accurate, the fundamental equation (B1) interlinks even three subsystems: (i) the pure acid, (ii) the pure water, and (iii) the strong base encapsulated in n = CB/CT.]

Examples.  Applying B1 for a mono-, di- and triprotic acid yield the following simple equations:

 (B4a) HA: n  =  a1 + (Kw/x – x)/CT (B4b) H2A: n  =  a1 + 2a2 + (Kw/x – x)/CT (B4c) H3A: n  =  a1 + 2a2 + 3a3 + (Kw/x – x)/CT

Note that the ionization fractions aj for the mono-, di- and triprotic acid differ (due to the different number of acidity constants each acid-type has). In particular, A6a yields:

 (B5a) HA: a0  =  (1 + K1/x )-1 (B5b) H2A: a0  =  (1 + K1/x + K1K2/x2 )-1 (B5c) H3A: a0  =  (1 + K1/x + K1K2/x2 + K1K2K3/x3 )-1

from which via A6b all other aj follow.

Remarks & Footnotes

1. A short PowerPoint lecture is here

2. BOH abbreviates strong bases such like NaOH, KOH or NH4OH.

3. 5 is a polynomial in pH of high order. In particular: For an N-protic acid we have a polynomial of order N+2 in x = 10-pH. To solve such polynomials numerical root-finding methods should be applied.

4. Aside from a strict mathematical proof you can check it easily: Enter the corresponding pK values (acidity constants) into Y1 in 7 then you will obtain integer or half-integer values for n.

5. [j] ≡ [HN-jAj 2

6. Do not confuse the dissolved acid-species [HNA] with the total amount of acid. The latter is abbreviated by [HNA]T

7. In hydrochemistry, it is common practice to use the composite carbonic acid, H2CO3* = *CO2