## Three Equivalence Points (Carbonate System)

Equivalence points (EP) represent points at which the amount of acid is equal to the amount of base1

 (1) amount of acid  =  amount of base

The carbonate system controls the pH in most natural waters. Due to the existence of three major carbonate species (CO2, HCO3-, CO3-2) we distinguish between three equivalence points (EP):23

 (2a) [H+] = [HCO3-] ⇒ CO2 EP  =  pH of pure H2CO3 solution (acid) (2b) [CO2] = [CO3-2] ⇒ HCO3- EP  =  pH of pure NaHCO3 solution (ampholyte) (2c) [HCO3-] = [OH-] ⇒ CO3-2 EP  =  pH of pure Na2CO3 solution (base)

The pH value (or equivalence point) is not a fixed quantity; it depends on the total amount of dissolved carbonates (also known as DIC):

 (3a) CT  =  [CO2] + [HCO3-] + [CO3-2]  =  DIC

In the following we calculate equivalence points (as a function of CT) using three different approaches (from simplest to more sophisticated and accurate):

• Case 1  Approximate formula based on ionization fractions (a0, a1, a2)
• Case 2  Exact relationship between pH and total concentration CT 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:

self-ionization of water activity corrections formation of complexes
Case 1 no no no
Case 2 yes no no
Case 3 yes yes yes

In fact, case 3 represents the best and most realistic description; the other two cases are more or less good approximations, and the table above just unveils the vulnerability of the first two approaches. The self-ionization of water (defined by Kw = {H+} {OH-} = 10-14) becomes relevant for low H+ concentrations, thus ignoring it, case 1 will fail for the description of very low values of CT. On the other hand, case 2 will fail for high Na2CO3 concentrations (i.e. high ionic strengths), because activity corrections are ignored. Let’s start.

Notation. To shorten the notation the carbonate ion will be abbreviated by A-2 = CO3-2. CT is then given by

 (3b) CT  =  [H2A] + [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) [H2A] = CT a0 [HA-] = CT a1 [A-2] = CT a2

based on three ionization fractions (and x = [H+] = 10-pH):

 (5a) a0   =   [ 1 + K1/x + K1K2/x2 ]-1 (5b) a1   =   [ x/K1 + 1 + K2/x ]-1 = (K1/x) a0 (5c) a2   =   [ x2/(K1K2) + x/K2 + 1 ]-1 = (K2/x) a1

with the equilibrium constants K1 = 10-6.35 and K2 = 10-10.33. The conditions defined in 2 yield simple formulas for CT as a function of pH = –log x:

 (6a) [H+] = [HA-] ⇒ x = CT a1 ⇒ CT = x2/K1 + x + K2 (6b) [H2A] = [A-2] ⇒ a0 = a2 ⇒ x2 = K1K2   for all CT (6c) [HA-] = [OH-] ⇒ CT a1 = Kw/x ⇒ CT = (Kw/x2) (x2/K1 + x + K2)

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 CT ≤ 10-7 M (because the self-ionization of water is ignored). The discrepancies emerge when compared with exact calculations in the next diagram.

Case 2 – Exact Relationship between CT and pH

There is an exact relationship between pH = –log [H+] and the total amount CT of H2CO3 (n=0), NaHCO3 (n=1) and Na2CO3 (n=2):

 (7) $C_T = \left(x-\dfrac{K_w}{x}\right) \ \left(\dfrac{1+2K_2/x} {x/K_1 + 1 + K_2/x} - n\right)^{-1}$

where x = [H+]. 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 non-zero ionic strengths either activation corrections should be included or all thermodynamic equilibrium constants should be replaced by conditional constants, K → cK. 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 cK (at 25, 1 atm):4

thermodynamic K (pure water, I=0)   conditional cK (seawater)
pK1 6.35   6.00
pK2 10.33   9.10
pKW 14.0   13.9

The diagram below 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 cK). 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 law-of-mass 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 CT.

The diagram below compares the results of the closed-form equation (7) (solid lines) with the numerical-model predictions (dots). An example calculation with aqion for CT = 10-3 M is presented here.

[Note: With increasing CT the ionic strength increases as well. At values of CT between 1 and 10 M Na2CO3 (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 the inclusion of additional aquatic complexes such like NaHCO3- (which were ignored so far). Just this aquatic complex becomes especially relevant for high concentrations of Na2CO3.

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 have the equivalence points (taken from here):

 1 mM H2CO3 solution: pH =   4.68 1 mM NaHCO3 solution: pH =   8.27 1 mM Na2CO3 solution: pH = 10.52

Note: The same results are obtained when Na is replaced by K or NH4. This is because all strong bases, such like NaOH and KOH, obey the same behavior in water: they dissociate completely.

Equivalence Points and Carbonate Speciation

The idea behind the concept of equivalence points becomes particularly evident when plotting the CT = CT(pH) diagrams together with the species distributions. As stated above, equivalence points are pH values at which the amount of two molar concentrations coincide. Let’s consider two equivalence points that are defined by:

 EP   H2CO3: [H+] = [HCO3-] EP   Na2CO3: [HCO3-] = [OH-]

The corresponding pH values depend on CT as the total concentration of H2CO3 or Na2CO3, as shown below in the first diagram. Thus, what we call an EP is one curve in the pH-CT-diagram. In the first diagram, two points on the left and two points on the right EP-curve are selected:

 EP   H2CO3: pH = 5.16 (for CT = 10-4 M) and pH = 4.68 (for CT = 10-3 M) EP   Na2CO3: pH = 9.86 (for CT = 10-4 M) and pH = 10.52 (for CT = 10-3 M)

The selected points refer to the species distribution in the two lower diagrams (one for CT = 10-3 M and one for CT = 10-3 M). They are just points of intersection, that is, points where two concentrations coincide.

The curves in the diagrams are calculated with the “numerical model” aqion.

Alkalinity and Acid Neutralizing Capacity (ANC)

The three equivalence points of the carbonate system are interrelated with the concept of ANC and/or alkalinities:

 CO2 EP ⇔ ANC to 4.3 … 4.5 ⇔   M alkalinity HCO3- EP ⇔ ANC to 8.2 … 8.4 ⇔   P alkalinity CO3-2 EP ⇔ ANC to 10.5 … 10.8 ⇔   caustic alkalinity

Remarks & References

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

2. CO2 is an abbreviation for the composite carbonic acid H2CO3*, which is the sum of dissolved CO2(aq) and a tiny amount of true carbonic acid H2CO3

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

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