M Alkalinity and P Alkalinity
Three Types of Alkalinity
Alkalinities are classified according to the endpoint of titration with strong acid:^{1}
(1a)  M alkalinity = Alk = total alkalinity  (endpoint: CO_{2} EP) 
(1b)  P alkalinity = carbonate alkalinity  (endpoint: HCO_{3}^{} EP) 
(1c)  caustic alkalinity  (endpoint: CO_{3}^{2} EP) 
Here, “M” refers to the pH indicator methylorange (endpoint 4.2 to 4.5); “P” refers to the pH indicator phenolphthalein (endpoint 8.2 to 8.3).^{2} “M alkalinity” is what we usually call “alkalinity” or “general alkalinity” or “total alkalinity”.
For a simple carbonate system (i.e. a pure CO_{2} solution with a background electrolyte, but without any other weak acids or bases) alkalinities are defined by:^{3}
(2a)  M alkalinity  =  ( [OH^{}] – [H^{+}] ) + [HCO_{3}^{}] + 2 [CO_{3}^{2}] 
(2b)  P alkalinity  =  ( [OH^{}] – [H^{+}] ) + [CO_{3}^{2}] – [H_{2}CO_{3}^{*}] 
(2c)  caustic alkalinity  =  ( [OH^{}] – [H^{+}] ) – [HCO_{3}^{}] – 2 [H_{2}CO_{3}^{*}] = 2P – M 
From these three alkalinities it is sufficient to measure M and P alkalinity only; the caustic alkalinity is then given by 2P – M.
Once M alkalinity and P alkalinity is known we are able to calculate the total amount of dissolved inorganic carbon:
(3)  M – P = [H_{2}CO_{3}^{*}] + [HCO_{3}^{}] + [CO_{3}^{2}] = DIC 
The equations above are restricted to carbonate systems without other weak acids or bases. The formula for the general case is given here, which is used in standard hydrochemistry programs (including aqion). The calculated M and P alkalinities are then displayed in output tables and simple overviews.
Alkalinity Relationships (Approximations)
Things become simpler at pH > 4.2, i.e. above the Malkalinity titration endpoint. In this pH range (which is typical for natural waters), the two species H_{2}CO_{3}^{*} and H^{+} almost vanish and Eqs. (2a) and (2b) reduce to:
(4a)  M alkalinity  ≈  [OH^{}] + [HCO_{3}^{}] + 2 [CO_{3}^{2}] 
(4b)  P alkalinity  ≈  [OH^{}] + [CO_{3}^{2}] 
Five special cases are obtained after inserting the following five conditions into Eqs.(4a) and (4b):
(5a)  [OH^{}] = 0 ,  [CO_{3}^{2}] = 0  ⇒  [HCO_{3}^{}] = M  and  P = 0  
(5b)  [OH^{}] = 0  ⇒  [HCO_{3}^{}] = M – 2P  and  P = [CO_{3}^{2}]  
(5c)  [OH^{}] = 0 ,  [HCO_{3}^{}] = 0  ⇒  [CO_{3}^{2}] = M/2  and  P = [CO_{3}^{2}]  
(5d)  [HCO_{3}^{}] = 0  ⇒  [CO_{3}^{2}] = M – P  and  [OH^{}] = 2P – M  
(5e)  [HCO_{3}^{}] = 0 ,  [CO_{3}^{2}] = 0  ⇒  [OH^{}] = M  and  P = M 
These results can be rearranged into the following table where 5a corresponds to column P = 0, 5b to column P < M/2, 5c to column P = M/2, 5d to column P > M/2, and 5e to column P = M:
P = 0  P < M/2  P = M/2  P > M/2  P = M  

[OH^{}]  0  0  0  2P–M  M 
2 · [CO_{3}^{2}]  0  2P  M  2(M–P)  0 
[HCO_{3}^{}]  M  M–2P  0  0  0 
[OH^{}] + 2 [CO_{3}^{2}] + [HCO_{3}^{}]  M  M  M  M  M 
The bottom row represents the sum in each column. According to Eq.(4a) the sum of the first column embodies the M alkalinity. This equality is fulfilled across the whole table.
The table above is often used by watertreatment professionals to estimate the content of HCO_{3}^{} and CO_{3}^{2} from measured M and P alkalinities. In contrast to these approximations, modern hydrochemistry programs provide the complete/exact carbonate speciation based on thermodynamic data – for the whole pH range and for any aqueous solution.
Two remarks:
1. In the equations above, [HCO_{3}^{}] represents the free hydrogen carbonate HCO_{3}^{} plus all hydrogencarbonate complexes such as CaHCO_{3}^{+}. Similarly, [CO_{3}^{2}] represents the free carbonate CO_{3}^{2} plus all carbonate complexes such as CaCO_{3}.
2. The quantities in square brackets are given in molar units such as mol/L or mmol/L. Thus, the quantity “2 · [CO_{3}^{2}]” in the first column of the table stands for carbonate in eq/L or meq/L (which is the same as val/L or mval/L).
Alkalinity and Proton Balance
The three alkalinities defined in 2 rely on the concept of proton balance. Proton balance states that – relative to a specific proton reference levels (PRL) – the amount of excess protons is equal to the amount of deficient protons:
(6)  proton balance:  0 = deficient protons – excess protons 
Let’s abbreviate the carbonate ion by A^{2} = CO_{3}^{2}, then we get for each reference level H_{2n }A^{n} (distinguished by n = 0, 1, and 2) the proton balance
(7)  0 = ([OH^{}] – [H^{+}]) – n [H_{2}A] – (n1) [HA^{}] – (n2) [A^{2}]  at PRL of H_{2n }A^{n} 
Thus the stage is set to define the alkalinity as follows: Alkalinity is the deviance from proton balance. There are three types of PRL and, hence, there are three types of alkalinity:
(8)  alkalinity of type n = deficient protons – excess protons  at PRL of H_{2n }A^{n} 
or, more explicitly,
(9)  alkalinity of type n = ([OH^{}] – [H^{+}]) – n [H_{2}A] – (n1) [HA^{}] – (n2) [A^{2}] 
where
n = 0:  PRL H_{2}CO_{3}  ⇔  M alkalinity 
n = 1:  PRL HCO_{3}^{}  ⇔  P alkalinity 
n = 2:  PRL CO_{3}^{2}  ⇔  caustic alkalinity 
Equation (9) is nothing else than a compact notation of 2a to (2c). Equation (9) also tells us that the alkalinity vanishes when the proton balance is exactly fulfilled. This, in fact, is the case for the following solutions:
pure H_{2}CO_{3} solution:  M alkalinity = 0 
pure NaHCO_{3} solution:  P alkalinity = 0 
pure Na_{2}CO_{3} solution:  caustic alkalinity = 0 
The statements remain valid when Na is replaced by K or NH_{4}.
Remarks & Footnotes

To be more precise, one should distinguish between titration endpoint (where the indicator changes its color) and equivalence point (or stoichiometric point, where the molar amount of acid and base is exactly equal). ↩

Note that equivalence points are not rigid quantities; they vary with the total content of carbonate (DIC) in the solution – as shown in this example. ↩

The asterisk on H_{2}CO_{3}^{*} symbolizes the composite carbonic acid. ↩