Equivalence Points – Systematics & Classification
The concept of equivalence points (EP) runs like a golden thread through acidbase theory and applications. There are different types of equivalence points.^{1} We provide a classification of EPs and semiEPs. This is done for the general case of Nprotic acids, H_{N}A.
Definition of EP_{n}
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 Nprotic acid (either strong or weak). For convenience we introduce C_{T} and C_{B}:
(2a)  total amount of Nprotic acid:  C_{T} = [H_{N}A]_{T}  
(2b)  total amount strong base:  C_{B} = [BOH]_{T} 
The ratio of both quantities is called
(3)  equivalent fraction:  n = C_{B} / C_{T} 
It represents the fraction of strong base that is added to an acid in titration. Thus, 1 defines the equivalence point EP_{1}, because n = C_{B}/C_{T} = 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 Nprotic acid, H_{N}A, we have (at first glance) N equivalence points EP_{1}, EP_{2} to EP_{N} based on integer values of n. In addition there are other important members: Firstly, EP_{0} which corresponds to the trivial case of a basefree system. Secondly, semiEPs which correspond to halfinteger values of the C_{B}/C_{T}ratio. Taken all together we obtain a whole family of equivalence points EP_{n} for both integer and halfinteger values of n:
(4a)  EP_{n}:  C_{B} / C_{T} = n  for n = 0, 1, … N  
(4b)  semiEP_{n}:  C_{B} / C_{T} = n  for n = ½, 3/2, … N½ 
In total there are 2N+1 equivalence points to characterize an Nprotic acid sufficiently. It makes no sense to extend this set by more types of EPs.
Each EP_{n}, as a special equilibrium state, is characterized by exactly one pH value: EP_{n} ⇔ pH_{n}. On the pH scale they are arranged in the following sequence:
pH_{0},  pH_{1/2},  pH_{1},  pH_{3/2},  …  pH_{N} 
But what is the concrete mathematical relationship between EP_{n} and pH_{n}?
Correspondence between EP_{n} and pH_{n}
The central quantity to start our consideration is the equivalent fraction n defined in 3. Remarkably enough, there is indeed a mathematical closedform expression for the equivalent fraction as function of pH:
(5)  (for details see Appendix B) 
This equation consists of two parts or components (subsystems):
• Y_{1}  describes the component H_{N}A  (based on N acidity constants K_{1} to K_{N}) 
• w  describes the component H_{2}O  (based on the selfionization constant K_{w}) 
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).
EP_{n} ⇔ pH_{n}. Inserting integer and halfinteger values for n into 5 yields just the equivalence points and the corresponding pH_{n} values – marked by small circles in the diagram below. (The diagram displays the titration curve of 100 mM H_{2}CO_{3} as the prototype of a diprotic acid).
In fact, 5 is amenable to simple hand calculations. It works for any Nprotic acid and requires as input: (i) the acidity constants K_{1} to K_{N} and (ii) the amount of acid C_{T}.
[Remark. Though 5 provides an exact mathematical relationship between EP_{n} and pH_{n}, 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 halfinteger value of n and just get pH_{n}, does not exist.]
EP_{n} as Extrema of the Buffer Intensity
EPs and semiEPs emerge in acidbase titrations as extrema points of the buffer intensity:
EP_{n}  (integer n)  ⇔  minimum buffer intensity β  
semiEP_{n}  (halfinteger n)  ⇔  maximum buffer intensity β 
This behavior is illustrated for the same alkalimetric titration as in the previous diagram (C_{T} = 100 mM H_{2}CO_{3}):
The blue titration curve, n = n(pH), is also known as buffer capacity. The pHderivative 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.
EP_{n} as Curves in pHC_{T} Diagrams
Equation (5) can be rearranged into the form
(6) 
Now, it’s easy to plot all EP_{n} as curves into a pHC_{T} diagram (one curve for one integer or halfinter 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 selfionization of H_{2}O is neglected).]
The Elegant Simplicity of the ‘Pure Acid’ Approach (C_{T} → ∞)
The general approach simplifies drastically if we neglect the second term in 5. This is legitimate for high enough values of C_{T} because then the term w/C_{T} tends to zero and can be neglected. What remains is the simple formula for the single component ‘acid’:
(7)  n = Y_{1}(pH)  or  n – Y_{1}(pH) = 0 
The fascinating thing about this equation is that it predicts the direct relationship between the pH values of EP_{n} and the acidity constants:
(8a)  EP_{n}  ⇔  pH_{n} = ½ (pK_{n} + pK_{n+1})  for integer n = 1, 2, … N1 
(8b)  semiEP_{n}  ⇔  pH_{n} = pK_{n+1/2}  for halfinteger n = ½, 3/2, … N½ 
This is a mathematical fact that relies on the striking features of the ionization fractions.^{4}
Equations (8a) and (8b) are valid for socalled internal EPs only, thereby excluding the two external equivalence points EP_{0} and EP_{N}. Since the internal EPs do not depend on C_{T} they appear as vertical dashed lines in the pHC_{T} diagrams shown above.
Equality of Species. In the pureacid case there is an alternative definition of equivalence points based on the equality of ‘neighbor’ and ‘nexttoneighbor’ acid species. For example, a triprotic acid encompasses the following set of EPs:
EP_{0}  [H^{+}] = [H_{2}A^{}]  
EP_{1/2}  [H_{3}A] = [H_{2}A^{}]  ⇔  pH_{1/2} = pK_{1}  
EP_{1}  [H_{3}A] = [HA^{2}]  ⇔  pH_{1} = ½ (pK_{1}+pK_{2})  
EP_{3/2}  [H_{2}A^{}] = [HA^{2}]  ⇔  pH_{3/2} = pK_{2}  
EP_{2}  [H_{2}A^{}] = [A^{3}]  ⇔  pH_{2} = ½ (pK_{2}+pK_{3})  
EP_{5/2}  [HA^{2}] = [A^{3}]  ⇔  pH_{5/2} = pK_{3}  
EP_{3}  [HA^{2}] = [OH^{}] 
On the pH scale they are arranged as follows:
In shorthand notation,^{5} this can be generalized for any Nprotic acid (valid for all internal EPs):
(9a)  EP_{n}  ⇔  [n1] = [n+1]  for integer n = 1, 2, … N1 
(9b)  semiEP_{n}  ⇔  [n½] = [n+½]  for halfinteger n = ½, 3/2, … N½ 
Application. Just the alternative definition of EPs is often applied in textbooks for the carbonate system:
(10a)  EP_{0}  ⇔  [H^{+}] = [HCO_{3}^{}]  (also known as CO_{2} EP)  
(10b)  EP_{1}  ⇔  [CO_{2}] = [CO_{3}^{2}]  (also known as HCO_{3}^{} EP)  
(10c)  EP_{2}  ⇔  [HCO_{3}^{}] = [OH^{}]  (also known as CO_{3}^{2} EP) 
For more details about this special topic see here.
Coupling of two Subsystems
Let’s explain the general behavior of the EP_{n} curves in pHC_{T} 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 C_{T} scale (shown in the top diagram):
• pure H_{2}O  at C_{T} = 0  with one EP at pH 7 
• pure acid  at C_{T} → ∞  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 C_{T} increases until they fit the ‘pureacid’ 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 = C_{B}/C_{T} becomes an integer or halfinteger value.
2. An Nprotic acid has in total 2N+1 equivalence points EP_{n} defined in 4a and (4b). The trivial case EP_{0} refers to the basefree system.
3. The mathematical relationship EP_{n} ⇔ pH_{n} is given by n = Y_{1}(pH) + w(pH)/C_{T}, where Y_{1} describes the acid and w the water.
4. The equivalent fraction n = Y_{1}(pH) + w(pH)/C_{T} (titration curve) describes the buffer capacity. Its pHderivative is the buffer intensity &beta = dn/dpH. EPs are extrema of β:
EP_{n}  (integer n)  ⇔  minimum buffer intensity β  
semiEP_{n}  (halfinteger n)  ⇔  maximum buffer intensity β 
5. In the limit of undiluted acids (C_{T} → ∞) the general relationship simplifies to: n = Y_{1}(pH). This equation yields the direct link between pH_{n} and the acidity constants:
pH_{n} = ½ (pK_{n} + pK_{n+1})  for integer n  (EP_{n})  
pH_{n} = pK_{n+1/2}  for halfinteger n  (semiEP_{n}) 
6. In the limit of undiluted acids (C_{T} → ∞) there is an alternative definition of EPs based on equal species concentrations – see 9a and (9b). (Example: In carbonate systems EP_{1} is often introduced as the equilibrium state where [CO_{2}] = [CO_{3}^{2}].)
Appendix A – Polyprotic Acids (H_{N}A)
An Nprotic acid is characterized by N acidity constants K_{1} to K_{N}. Each acidity constant acts as an equilibrium constant in a series of dissociation reactions:
(A1a)  H_{N}A  =  H^{+} + H_{N1}A^{}  K_{1} = [H^{+}] [H_{N1}A^{}] / [H_{N}A]  
(A1b)  H_{N1}A^{}  =  H^{+} + H_{N2}A^{2}  K_{2} = [H^{+}] [H_{N2}A^{2}] / [H_{N}A^{}]  
⋯  
(A1c)  HA^{(N1)}  =  H^{+} + A^{N}  K_{N} = [H^{+}] [A^{N}] / [HA^{(N1)}] 
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 [H_{N}A].^{6} All species together add up to the total amount of acid:
(A2)  C_{T} = [0] + [1] + … + [N] 
In addition we introduce the abbreviation
(A3)  x ≡ [H^{+}] = 10^{pH} 
In this notation, the lawofmassaction formulas in A1 get a particular simple form:
(A4)  (Generalized HendersonHasselbach 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 acidspecies concentrations known as
(A5)  ionization fractions:  a_{j} ≡ [j] \ C_{T}  for j = 0 to N 
They are simple analytical expressions based on the acidity constants K_{1} to K_{N}. Starting out from
(A6a) 
all other a_{j} can be iteratively obtained:
(A6b) 
Recall that the set of equations in (A1a) to (A1c) and the set of equations in (A6a) and (A6b) are mathematically equivalent.
Moments Y_{L}. It is useful to introduce socalled moments Y_{L}, which are weighted sums over the entire set of ionization fractions a_{j}:
(A7)  Y_{L}(x) ≡ j^{ L} a_{j}(x) 
The most important representatives that are used as building blocks of other relevant quantities are:
(A8a)  Y_{0} = a_{0} + a_{1} + … + a_{N} = 1  ⟹  mass balance (cf. A2)  
(A8b)  Y_{1} = a_{1} + 2a_{2} + … + N a_{N}  ⟹  titration curve: n = n(pH)  
(A8c)  Y_{2} = a_{1} + 4a_{2} + … + N^{2} a_{N}  ⟹  buffer intensity β 
In particular, A8b embodies the heart of the general equation in Appendix B. Again: each Y_{L} comprises the information contained in the set of equations (A1a) to (A1c).
pK_{j} Values. Acids are specified by their acidity constants K_{j}. Because they vary by orders of magnitude it is often practical to switch to pK_{j} = –lg K_{j}. Typical values of four common acids are:
Acid  Formula  Type  pK_{1}  pK_{2}  pK_{3} 

acetic acid  CH_{3}COOH  HA  4.76  
carbonic acid^{7}  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 
Appendix B – General Equation for H_{N}A + H_{2}O
The general case combines the subsystem ‘pure acid’ defined in A1a to (A1c) with the subsystem ‘H_{2}O’ (i.e. the selfionization of water). Both subsystems are coupled via the proton balance (as described in pdf or ppt). It yields the central closedform equation:
(B1)  with x = [H^{+}] = 10^{pH} 
and the following abbreviations:
(B2)  w(x) = K_{w}/x – x  (with K_{w} = 10^{14} at 25 °C)  
(B3)  Y_{1}(x) = a_{1} + 2a_{2} + … + N a_{N} 
Y_{1} characterizes the Nprotic 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 = C_{B}/C_{T}.]
Examples. Applying B1 for a mono, di and triprotic acid yield the following simple equations:
(B4a)  HA:  n = a_{1}  +  (K_{w}/x – x)/C_{T}  
(B4b)  H_{2}A:  n = a_{1} + 2a_{2}  +  (K_{w}/x – x)/C_{T}  
(B4c)  H_{3}A:  n = a_{1} + 2a_{2} + 3a_{3}  +  (K_{w}/x – x)/C_{T} 
Note that the ionization fractions a_{j} for the mono, di and triprotic acid differ (due to the different number of acidity constants each acidtype has). In particular, A6a yields:
(B5a)  HA:  a_{0} = (1 + K_{1}/x )^{1}  
(B5b)  H_{2}A:  a_{0} = (1 + K_{1}/x + K_{1}K_{2}/x^{2} )^{1}  
(B5c)  H_{3}A:  a_{0} = (1 + K_{1}/x + K_{1}K_{2}/x^{2} + K_{1}K_{2}K_{3}/x^{3} )^{1} 
from which via A6b all other a_{j} follow.
Remarks & Footnotes

BOH abbreviates strong bases such like NaOH, KOH or NH_{4}OH. ↩

5 is a polynomial in pH of high order. In particular: For an Nprotic acid we have a polynomial of order N+2 in x = 10^{pH}. To solve such polynomials numerical rootfinding methods should be applied. ↩

Aside from a strict mathematical proof you can check it easily: Enter the corresponding pK values (acidity constants) into Y_{1} in 7 then you will obtain integer or halfinteger values for n. ↩

Do not confuse the dissolved acidspecies [H_{N}A] with the total amount of acid. The latter is abbreviated by [H_{N}A]_{T}. ↩

In hydrochemistry, it is common practice to use the composite carbonic acid, H_{2}CO_{3}^{*} = *CO_{2} ↩