## Three Equivalence Points (Carbonate System)

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

 (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 N-protic acid.

Due to the existence of three major carbonate species (CO2, HCO3-, CO3-2) we distinguish between three equivalence points (EP):23

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

As indicated here, each EP is characterized by a specific pH value. In addition, on the right-hand 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) CT  =  [CO2] + [HCO3-] + [CO3-2]  =  DIC

Aim.  Our aim is to calculate and plot equivalence points in pH-CT diagrams 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

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 self-ionization of water (defined by Kw = [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 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. Let’s abbreviate the carbonate ion 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 and the total amount CT of H2CO3 (n=0), NaHCO3 (n=1), and Na2CO3 (n=2):

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

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 non-zero ionic strengths either activation corrections should be done 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 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 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 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 too. At CT values 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 that aquatic complexes, such like NaHCO3-, are taken into account (which have been ignored so far). This aquatic complex becomes especially relevant at 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 get for 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 results remain the same when Na is replaced by K or NH4. 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+] = [HCO3-] ⇒ EP0 [CO2] = [CO3-2] ⇒ EP1 [HCO3-] = [OH-] ⇒ EP2

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 CT = 10-3 M and one for 10-4 M) upwards into the pH-CT diagram where they constitute the small circles on EP curves. Here we observe a different behavior for the external equivalence points (EP0 and EP2) and for the internal equivalence point EP1. The two external EPs are CT-dependent:

 EP0: pH = 5.16 (for CT = 10-4 M) and pH = 4.68 (for CT = 10-3 M) EP2: pH = 9.86 (for CT = 10-4 M) and pH = 10.52 (for CT = 10-3 M)

But this is not true for EP1. The intersection points in the two lower diagrams are both at the same fixed pH value of ½ (pK1+pK2) = 8.34, which belongs to the high-CT limit of the red curve in the upper diagram. Thus, the simple relationship [CO2] = [CO3-2] does not work for small CT (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:

 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)