Three Equivalence Points (Carbonate System)
Equivalence points (EP) represent special equilibrium states at which the amount of acid is equal to the amount of base^{1}
(1)  amount of acid = amount of base 
In the following we focus on the carbonate system which is just the vehicle that controls the pH in most natural waters. The present approach fits into a broader view that classifies sets of equivalence points of an Nprotic acid.
Due to the existence of three major carbonate species (CO_{2}, HCO_{3}^{}, CO_{3}^{2}) we distinguish between three equivalence points (EP):^{2}^{3}
(2a)  CO_{2}  EP: pH of H_{2}CO_{3} solution (acid)  ⇔  [H^{+}] = [HCO_{3}^{}]  
(2b)  HCO_{3}^{}  EP: pH of NaHCO_{3} solution (ampholyte)  ⇔  [CO_{2}] = [CO_{3}^{2}]  
(2c)  CO_{3}^{2}  EP: pH of Na_{2}CO_{3} solution (base)  ⇔  [HCO_{3}^{}] = [OH^{}] 
As indicated here, each EP is characterized by a specific pH value. In addition, on the righthand side there is an alternative definition of those EPs based on equal species concentrations. The latter, however, is an approximation, though a very good one (as long as we do not apply it to very dilute systems).
The pH value of an equivalence point is generally not a fixed quantity; it depends on the total amount of dissolved carbonates (also known as DIC):
(3a)  C_{T} = [CO_{2}] + [HCO_{3}^{}] + [CO_{3}^{2}] = DIC 
Aim. Our aim is to calculate and plot equivalence points in pHC_{T} diagrams using three different approaches (from simplest to more sophisticated and accurate):
 Case 1 Approximate formula based on ionization fractions (a_{0}, a_{1}, a_{2})
 Case 2 Exact relationship between pH and total concentration C_{T} of reactant
 Case 3 Numerical model with activity corrections (aqion or PhreeqC)
The three approaches, together with their underlying assumptions, can be summarized as follows:
selfionization of water  activity corrections  formation of complexes  

Case 1  no  no  no  
Case 2  yes  no  no  
Case 3  yes  yes  yes 
Case 3 represents the best and most realistic description; the other two cases are more or less good approximations. The table above unveils the vulnerability of the first two approaches. The selfionization of water (defined by K_{w} = [H^{+}] [OH^{}] = 10^{14}) becomes relevant for highly diluted acids, thus ignoring it in Case 1, the description will fail for very small values of C_{T}. On the other hand, Case 2 will fail for high Na_{2}CO_{3} concentrations (i.e. high ionic strengths), because activity corrections are ignored. Let’s start.
Notation. Let’s abbreviate the carbonate ion by A^{2} = CO_{3}^{2}. C_{T} is then given by
(3b)  C_{T} = [H_{2}A] + [HA^{}] + [A^{2}] 
Additional information is presented as PowerPoint.
Case 1 – Approximate Formula based on Ionization Fractions
In textbooks, the molar concentration of the three carbonate species is commonly described by
(4)  [H_{2}A] = C_{T} a_{0}  [HA^{}] = C_{T} a_{1}  [A^{2}] = C_{T} a_{2} 
based on three ionization fractions (and x = [H^{+}] = 10^{pH}):
(5a)  a_{0} = ( 1 + K_{1}/x + K_{1}K_{2}/x^{2} )^{1}  
(5b)  a_{1} = ( x/K_{1} + 1 + K_{2}/x )^{1}  =  (K_{1}/x) a_{0}  
(5c)  a_{2} = ( x^{2}/(K_{1}K_{2}) + x/K_{2} + 1 )^{1}  =  (K_{2}/x) a_{1} 
with the equilibrium constants K_{1} = 10^{6.35} and K_{2} = 10^{10.33}. The conditions defined in 2 yield simple formulas for C_{T} as a function of pH = –log x:
(6a)  [H^{+}] = [HA^{}]  ⇒  x = C_{T} a_{1}  ⇒  C_{T} = x^{2}/K_{1} + x + K_{2}  
(6b)  [H_{2}A] = [A^{2}]  ⇒  a_{0} = a_{2}  ⇒  x^{2} = K_{1}K_{2} for all C_{T}  
(6c)  [HA^{}] = [OH^{}]  ⇒  C_{T} a_{1} = K_{w}/x  ⇒  C_{T} = (K_{w}/x^{2}) (x^{2}/K_{1} + x + K_{2}) 
The equations (written in brown color) are shown as dashed curves in the diagram below. Please note that this approach is an approximation that works for almost all practical cases, but fails at extremely low concentrations, i.e. for C_{T} ≤ 10^{7} M (because the selfionization of water is ignored). The discrepancies emerge when compared with exact calculations in the next diagram.
Case 2 – Exact Relationship between C_{T} and pH
There is an exact relationship between pH and the total amount C_{T} of H_{2}CO_{3} (n=0), NaHCO_{3} (n=1), and Na_{2}CO_{3} (n=2):
(7) 
where x = [H^{+}] = 10^{pH}. In the diagram below, the exact results are displayed as three solid curves, which are compared with the approximations in 6. The latter are the dashed curves taken from the diagram above.
Seawater. The above results are valid for zero ionic strength, I = 0. For nonzero ionic strengths either activation corrections should be done or all thermodynamic equilibrium constants should be replaced by conditional constants, K → ^{c}K. Seawater has I ≈ 0.7 M, which is just on the upper bound of the validity range of common activity models. In oceanography, chemists prefer conditional equilibrium constants ^{c}K (at 25, 1 atm):^{4}
thermodynamic K (pure water, I=0)  conditional ^{c}K (seawater)  

pK_{1}  6.35  6.00  
pK_{2}  10.33  9.10  
pK_{W}  14.0  13.9 
The next diagram compares the results calculated by 7 for both the standard case (solid lines based on thermodynamic equilibrium constants K) and seawater (dashed lines based on conditional constants ^{c}K). The solid curves are identical with the solid curves displayed in the diagram above.
Case 3 – Numerical Model with Activity Corrections
Equation (7) is based on the assumption that activities (that enter the lawofmass action) are replaced by concentrations. Modern hydrochemistry programs do not adhere to those restrictions; they always perform activity corrections. In this way, they are more accurate in predicting the relationship between pH and a given C_{T}.
The diagram below compares the results of the closedform equation (7) (solid lines) with numericalmodel predictions (dots). An example calculation with aqion for C_{T} = 10^{3} M is presented here.
[Note: With increasing C_{T} the ionic strength increases too. At C_{T} values between 1 and 10 M Na_{2}CO_{3} (i.e. the most upper part of the green curve) we are just outside the applicability range of aqion.]
Yet another advantage of the numerical model is that aquatic complexes, such like NaHCO_{3}^{}, are taken into account (which have been ignored so far). This aquatic complex becomes especially relevant at high concentrations of Na_{2}CO_{3}.
Example. Natural waters are usually in the DIC range between 1 to 10 mM, i.e. 10^{3} to 10^{2} M. In particular, at 1 mM we get for the equivalence points (taken from here):
1 mM  H_{2}CO_{3} solution:  pH = 4.68 
1 mM  NaHCO_{3} solution:  pH = 8.27 
1 mM  Na_{2}CO_{3} solution:  pH = 10.52 
Note: The results remain the same when Na is replaced by K or NH_{4}. This is because all strong bases, such like NaOH or KOH, behave the same in water, i.e. they dissociate completely.
Equivalence Points and Carbonate Speciation
As already stated in 2, EPs of the carbonate system are usually introduced as equilibrium states at which the amount of two species coincide:
[H^{+}] = [HCO_{3}^{}]  ⇒  EP_{0} 
[CO_{2}] = [CO_{3}^{2}]  ⇒  EP_{1} 
[HCO_{3}^{}] = [OH^{}]  ⇒  EP_{2} 
In Bjerrum plots these are just points of intersection of two concentration curves. The figure below maps those intersections from the two lower diagrams (one for C_{T} = 10^{3} M and one for 10^{4} M) upwards into the pHC_{T} diagram where they constitute the small circles on EP curves.
Here we observe a different behavior for the external equivalence points (EP_{0} and EP_{2}) and for the internal equivalence point EP_{1}. The two external EPs are C_{T}dependent:
EP_{0}:  pH = 5.16  (for C_{T} = 10^{4} M)  and  pH = 4.68  (for C_{T} = 10^{3} M)  
EP_{2}:  pH = 9.86  (for C_{T} = 10^{4} M)  and  pH = 10.52  (for C_{T} = 10^{3} M) 
But this is not true for EP_{1}. The intersection points in the two lower diagrams are both at the same fixed pH value of ½ (pK_{1}+pK_{2}) = 8.34, which belongs to the highC_{T} limit of the red curve in the upper diagram. Thus, the simple relationship [CO_{2}] = [CO_{3}^{2}] does not work for small C_{T} (and fails for very dilute acids).
Alkalinity and Acid Neutralizing Capacity (ANC)
The three equivalence points of the carbonate system are interrelated with the concept of ANC and/or alkalinities:
CO_{2}  EP  ⇔  ANC to 4.3 … 4.5  ⇔ M alkalinity 
HCO_{3}^{}  EP  ⇔  ANC to 8.2 … 8.4  ⇔ P alkalinity 
CO_{3}^{2}  EP  ⇔  ANC to 10.5 … 10.8  ⇔ caustic alkalinity 
Remarks & References

The equivalence point (stoichiometric point) should be distinguished from the titration endpoint (where the indicator changes its color). Both are not exactly the same. ↩

CO_{2} is an abbreviation for the composite carbonic acid H_{2}CO_{3}^{*}, which is the sum of dissolved CO_{2}(aq) and a tiny amount of true carbonic acid H_{2}CO_{3}. ↩

Equivalence points are tightly related to the concept of proton reference level (PRL). ↩

Millero, F.J.: The thermodynamics of the carbonic acid system in the oceans. Geochimica et Cosmochimica Acta 59, 661–667 (1995) ↩