Equivalence Points -- Systematics & Classification

The concept of equivalence points (EP) runs like a golden thread through acid-base theory and applications.1 There are different types of equivalence points. We provide a classification of EPs and semi-EPs 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 HNA (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 titrated to neutralize the acid; it is also named (normalized) buffer capacity.

1 defines the equivalence point EP1, because n = CB/CT = 1. But this is only one single equivalence point, probably the best known one. An N-protic acid HNA has much more to offer. Firstly, there are N (integer) equivalence points: EP1, EP2 to EPN. Plus, EP0 that corresponds to the base-free system. In addition, there are semi-EPs matching half-integer values of the CB/CT-ratio. Taken all together, there is a whole family of equivalence points 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

These 2N+1 equivalence points characterize an N-protic acid system 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 ascending order:

    pH0, pH1/2, pH1, pH3/2, 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 an analytical formula 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 combines three components (subsystems):

• Y1 component HNA (based on N acidity constants K1 to KN)
• w component H2O (based on the self-ionization constant Kw)
• n component “strong base” (as amount CB included in n = CB/CT)

They are linked together via the law of charge balance. Plotting the equivalent fraction as a function of pH yields the (normalized) titration curve, n = n(pH), as shown in the diagram below (for 100 mM H2CO3).

EPn ⇔ pHn.  Inserting integer and half-integer values for n into 5 returns the equivalence points and their pHn values — marked by small circles on the titration curve. Since H2CO3 is a 2-protic acid there are altogether 2×2+1 = 5 equivalence points.

equivalence points: EPs and semi-EPs as located on the titration curve n=n(pH) for the carbonic acid

5 is suitable 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 math 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):

equivalence points for 100 mM H2CO3: EPs and semi-EPs located on the titration curve and buffer intensity beta

The blue titration curve, n = n(pH), represents the buffer capacity. The pH-derivative of the buffer capacity is the buffer intensity β = dn/dpH — here plotted as green curve. The maxima/minima of the green curve are located at points where the slope of the blue curve is largest/smallest.

EPn as Trajectories 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):

trajectories of equivalence points in pH-CT diagrams for the general case

These diagrams were generated in Excel using as input the acidity constants from the table below. [The dashed curves represent approximations based on simplified equations (without considering the self-ionization of H2O). We call it the “pure acid” limit.]

Elegance and Simplicity of the “Pure Acid” Limit (CT → ∞)

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

(7)   n = Y1(pH) or Y1(pH) – n = 0

The fascinating thing about this equation is that it establishes the direct link between the pH values of EPn and the acidity constants:4

(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

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:

EPs and semi-EPs of triprotic acid on pH scale

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.  The alternative definition of EPs in 9 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)

For more details about this special topic see here.

Coupling of two Subsystems

Let’s explain the general behavior of the EPn trajectories 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.

In the top diagram we have two isolated components (or subsystems) located at both ends of the CT scale:

• 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 the two 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.

trajectories of equivalence points in ph_CT diagrams: from pure-acid (isolated system) to general case (coupled system)


1.  Equivalence points are special equilibrium states in which the equivalent fraction n = CB/CT is 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 math relationship EPn ⇔ pHn is given by n = Y1(pH) + w(pH)/CT, where Y1 describes the component “acid” and w the component “water”.

4.  The equivalent fraction n = Y1(pH) + w(pH)/CT (titration curve) represents the buffer capacity. Its pH-derivative is the buffer intensity β = 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: Y1(pH) = n. It establishes 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 the equality of species concentrations — see 9a and (9b). (Example: In carbonate systems EP1 is often introduced as the equilibrium state for which [CO2] = [CO3-2] applies.)

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 the integer 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]

Using the abbreviation

(A3)   x  ≡  [H+]  =  10-pH

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.  The set of A1a to (A1c) provides 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 component “acid” defined in A1a to (A1c) with the component “H2O” (i.e. the self-ionization of water). Both subsystems are coupled via the law of charge balance.1 It yields the central closed-form equation:

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

with 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 components: (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. More details are presented in the review (2021) and/or lecture (2023).  2

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

  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. Mathematically, the equivalence between 7 and 8 is strictly valid for diprotic acids only, but remains a very good approximation for polyprotic acids with N ≥ 3. 

  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(aq) + H2CO3

[last modified: 2024-01-10]