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IntroductionCarbonate Chemistry
pH Calculation
Chemical Equilibria and pH Calculations
Guy Munhoven
Institut d’Astrophysique et de Geophysique (B5c Build.)Room 0/13
eMail: [email protected]: 04-3669771
24th February 2021
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Plan
Chemistry of the carbon dioxide system
Chemical equilibria
pH scales
Conservative state variables:dissolved inorganic carbon and alkalinity
Carbonate: calculation
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Processus and exchanges
2
2
2
3
−
500 − 1000
2−
3 3
=
VEGETATION
& SOIL
170 000−200 00010.6 − 12.5
WeatheringCarbonate Rock~ 3 000
6 000 − 7 000
deepsurface
in 1012 mol CReservoir sizes
Fluxes 12in 10 mol C/yr
COVolcanic
2 − 10COWeathering
CO
HCORiverine
32 − 38
organic C 70 − 120
CaCO3
3CaCO deposition
Deep−sea14 − 20
3CaCO depositionShallow water
14.5
CO + HCO + CO
(preindustrial 50 000)
OCEAN
LITHOSPHERE
ATMOSPHERE
~ 8 000
21.4 − 25.5
70 000
3 250 000
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Carbonate Chemistry
Dissolution of atmospheric CO2 in water
CO2(g) CO2(aq)
CO2(aq) + H2O H2CO3
H2CO3 HCO−3 + H+
HCO−3 CO2−3 + H+
Actually[H2CO3]
[H2CO3] + [CO2(aq)]�
For practical usage, we define
CO∗2(aq) = H2CO3 + CO2(aq).
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Carbonate Chemistry
Equilibrium system actually used:
CO2(g) CO∗2(aq)
CO∗2(aq) + H2O HCO−3 + H+
HCO−3 CO2−3 + H+
Equilibrium relationships
K ∗H =[CO∗2(aq)]
fCO2
(Henry’s Law)
K ∗1 =[H+] [HCO−3 ]
[CO∗2(aq)]
K ∗2 =[H+] [CO2−
3 ]
[HCO−3 ]
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
pK values of the equilibrium constants
pK :=− log10(K ), by analogy with pH :=− log10([H+])
Consider, e. g., the equilibrium between CO2(aq) and HCO−3 ina solution containing dissolved CO2:
K ∗1 =[H+] [HCO−3 ]
[CO∗2(aq)]
When [CO∗2(aq)] = [HCO−3 ] (→ equivalence point), we have
K ∗1 = [H+] ⇔ pK ∗1 = pH
⇒ equivalence points located at the pK values
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Stoichiometric vs. Thermodynamic Constants
K ∗H, K ∗1 and K ∗2 are stoichiometric constants as they linkconcentrations
The corresponding thermodynamic equilibrium constantsKH, K1 and K2
link activities instead of concentrationsonly depend on temperature and pressurehave been determined for a large number of reactions
The activity {A} and the concentration [A] of a chemicalspecies A are related by the activity coefficient γA
{A}= γA[A]
γA depends on the chemical composition of the solution
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Chemical Composition of Seawater
Composition ofone kilogram ofaverage seawater(S = 35)
Solute molNa+ 0.46900Mg2+ 0.05282Ca2+ 0.01028K+ 0.01021Sr2+ 0.00009Cl− 0.54588SO2−
4 0.02823HCO−3 0.00186Br− 0.00084CO2−
3 0.00019B(OH)−4 0.00008F− 0.00007B(OH)3 0.00033
After Millero (1982)
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Activity Coefficients
Influence of activity coefficients not negligible
Ion γ
Na+ 0.666Cl− 0.668H+ 0.590HCO−3 0.570CO2−
3 0.039
Conditions:
seawater at 25◦C and S = 35
After Zeebe and Wolf-Gladrow(2003, Tab. 1.1.3)
Two ways to address this complication
calculation of γ values from solute interaction models⇒ difficult and tediousempirical determination of stoichiometric coefficients includingeffets of γ, as a function of temperature, pressure and salinity⇒ adopted in practice
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
pH Scales
Classically pH =− log10[H+]
However, even in freshwater solutions,free H+ ions present only in negligible amounts:most are complexed by water molecules
In seawater, this complexing extends to other solutes as well
In seawater, it would be best to adopt pH =− log10{H+}⇒ useless as {H+} cannot be individually measured
Definition of operational pH scales that take into account thepresence of extra ions able to release H+ ions
Motivations essentially experimentally oriented
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
pH Scales: Free, Total, . . .
Free Scale – based upon [H+]F, the concentrationof free and hydrated H+ ions
Total Scale – takes into account the role of HSO−4 :
pHT := − log10 [H+]T
[H+]T := [H+]F(1 +ST/KS)
where
ST = [SO2−4 ] + [HSO−4 ] is the total sulphate concentration
KS =[H+]F[SO2−
4 ]
[HSO−4 ]is the dissociation constant of HSO−4
[H+]T ' [H+]F + [HSO−4 ]
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
pH Scales: . . . Seawater
Seawater Scale – takes into account the roles of HSO−4 andHF:
pHSWS := − log10 [H+]SWS
[H+]SWS := [H+]F(1 +ST/KS +FT/KF)
where
ST and KS as for the Total Scale
FT = [HF] + [F−] is the total concentration of fluorine
KF =[H+]F[F−]
[HF]is the dissociation constant of HF
[H+]SWS ' [H+]F + [HSO−4 ] + [HF]
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Carbonate Speciation
Why are these precisions important?
Stoichiometric dissociation acid dissociation constant (such asK ∗1 and K ∗2 , e. g.) have the same units as [H+]⇒ need to know on which pH scale these constants are given
Dialogue between modellers and experimentalists easier ifconcepts used in common are known and agreed upon
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Carbonate Chemistry
Let CT = [CO∗2(aq)] + [HCO−3 ] + [CO2−3 ]. Equilibrium relationships
lead to the following speciation relationships
[CO∗2(aq)]
CT=
[H+]2
[H+]2 +K ∗1 [H+] +K ∗1K∗2
[HCO−3 ]
CT=
K ∗1 [H+]
[H+]2 +K ∗1 [H+] +K ∗1K∗2
[CO2−3 ]
CT=
K ∗1K∗2
[H+]2 +K ∗1 [H+] +K ∗1K∗2
⇒ pH plays a central role for thespeciation of the CO2-HCO−3 -CO2−
3 system
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Speciation: Bjerrum Plot
2 4 6 8 10 12pH
−10
−8
−6
−4
−2
log 1
0(co
nc(m
ol/k
g))
T=25°CP=0 barS=35
CO*2(aq)
HCO−3
CO2−3
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Speciation: Bjerrum Plot
2 4 6 8 10 12pH
−10
−8
−6
−4
−2
log 1
0(co
nc(m
ol/k
g))
pK1* =
5.8
5
pK2* =
8.9
2T=25°CP=0 barS=35
CO*2(aq)
HCO−3
CO2−3
Points d’equivalence
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Speciation: Bjerrum Plot
2 4 6 8 10 12pH
−10
−8
−6
−4
−2
log 1
0(co
nc(m
ol/k
g))
T=25°CP=0 bar
S=35
S=0
CO*2(aq)
HCO−3
CO2−3
Seawater – freshwater
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Speciation: Temperature and Pressure Effects
2 4 6 8 10 12pH
−10
−8
−6
−4
−2
log 1
0(co
nc(m
ol/k
g))
S=35
T=25°CP=0 bar
T=1°CP=0 bar
T=1°CP=300 bar
CO*2(aq)
HCO−3
CO2−3
Temperate and cold surface waters, deep water (3000 m)
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Carbonate Chemistry
Special Roles of Different Species
CO2(aq): air-sea exchange
CO2−3 : carbonate dissolution
Measurables
CO2(aq): by IR absorption (under favourable conditions)
pH: after consideration of all the complications
CT: by degassing via acidification
Alkalinity: titration with a strong acid (e. g., HCl)
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
State Variables of the Carbonate System
H+ (or pH) and CO2(aq) (or pCO2) are the only speciesparticipating in the carbonate equilibria that can be directlymeasured
Neither H+ nor pCO2 are conservative:
variations are not only controlled by sources and sinks in thesystem, but also by other state variables of the system(temperature, pressure) or other solutes, . . .
⇒ pH and pCO2 are unsuitableas state variables in models
CT is conservative and measurable
4 unknowns and 2 equilibrium relationships would require asecond conservative and measurable parameter
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Alkalinity: a first tour
Alkalinity measures the capacity of a solution to neutralizeacid to the bicarbonate equivalence point (where[HCO−3 ] = [H+]), also called second equivalence point
Measured by titration of a sample with a strong acid(generally HCl) until the equivalence point is reached;the titration curve (evolution of pH as a function of theadded amount of acid) has an inflection point at this point,which must be determined with precision
The alkalinity of the sample is then defined as the moleequivalent of acid added to reach the equivalence point
⇒ at the equivalence point, alkalinity will have been reducedto zero
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Alkalinity: Exact Definition
“The total alkalinity of a natural water is thus defined asthe number of moles of hydrogen ion equivalent to theexcess of proton acceptors (bases formed from weak acidswith a dissociation constant K ≤ 10−4.5, at 25 ◦C and zeroionic strength) over proton donors (acids with K > 10−4.5)in one kilogram of sample.”
(Dickson, 1981)
AT := ∑i [proton acceptori ]−∑j [proton donorj ]
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Alkalinity Contributions: Carbonic Acid Example
Carbonic Acid H2CO3
H2CO3 HCO−3 + H+, pKC1 = 6.3
pKC1 > 4.5⇒ base is an acceptor, contributing +[HCO−3 ]
Bicarbonate ion HCO−3
HCO−3 CO2−3 + H+, pKC2 = 10.3
pKC2 > 4.5⇒ base is an acceptor, contributing +2× [CO2−3 ]:
by accepting a proton, the base CO2−3 is converted to HCO−3 ,
another acceptor, which must also be accounted for.
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Alkalinity Contributions: Phosphoric Acid Example
Orthophosphoric Acid H3PO4
H3PO4 H2PO−4 + H+, pKP1 = 2.1
pKP1 < 4.5⇒ acid is a donor and contributes −[H3PO4]
Dihydrogen phosphate H2PO−4
H2PO−4 HPO2−4 + H+, pKP2 = 7.2
pKP2 > 4.5⇒ base is an acceptor and contributes +[HPO2−4 ]
Hydrogen phosphate HPO2−4
HPO2−4 PO3−
4 + H+, pKP3 = 12.7
pKP3 > 4.5⇒ base is an acceptor, contributing +2× [PO3−4 ]
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Alkalinity
Acide pKA Type provided Species H+ eq/mol
H2O 14.0 acceptor OH− [OH−]H2CO3 6.3 acceptor HCO−3 [HCO−3 ]HCO−3 10.3 acceptor CO2−
3 2× [CO2−3 ]
B(OH)3 9.2 acceptor B(OH)−4 [B(OH)−4 ]HSO−4 2.0 donor HSO−4 −[HSO−4 ]HF 3.2 donor HF −[HF]H+ — donor H+ −[H+]
H3PO4 2.1 donor H3PO4 −[H3PO4]
H2PO−4 7.2 acceptor HPO2−
4 [HPO2−4 ]
HPO2−4 12.7 accepteur PO3−
4 2× [PO3−4 ]
H4SiO4 9.7 acceptor H3SiO−4 [H3SiO
−4 ]
H2S 7.0 acceptor HS− [HS−]HS− 12.0 acceptor S2− 2× [S2−]NH+
4 9.3 acceptor NH3 [NH3]
Compiled from data reported by Dickson (1981)
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Alkalinity in Detail
We thus obtain the following expression for alkalinity
AT = [HCO−3 ] + 2× [CO2−3 ] + [B(OH)−4 ] + [OH−]
+ [HPO2−4 ] + 2× [PO3−
4 ] + [H3SiO−4 ]
+ [NH3] + [HS−] + 2× [S2−] + . . .
− [H+]F− [HSO−4 ]− [HF]− [H3PO4]− . . .
where the . . . stand for the concentrations of additional negligibleproton donors and acceptors.
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Alkalinity in Practice
Alkalinity can generally be approximated to excellent precision by
AT ' [HCO−3 ] + 2× [CO2−3 ] + [B(OH)−4 ] + [OH−]− [H+
T ]
Often, it is even sufficient to adopt
AT ' [HCO−3 ] + 2× [CO2−3 ] + [B(OH)−4 ]
However, under certain particular conditions, it may be necessaryto take additional contributors into account, such as, e. g., theconjugate bases of phosphoric or silicic acids
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Alkalinity: a Few Comments
Alkalinity is a complex concept, with an opaque definition
In the literature, there are alternative definitions based uponelectroneutrality, that define alkalinity as being equal to thecharge difference between conservative cations and anions
Alkalinity defined this way
is also conservative (by construction);neglects contributions from non charged bases (e. g., NH3) thatmay be important under some conditions (e. g., anoxic waters)is equal to total alkalinity up to a sum of total concentrations(total phosphate, ammonium, sulphate), that are often, butnot always, negligiblemakes the concept even more confusing
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
Total Alkalinity: Properties
Total alkalinity is conservativeaffected by the precipitation and the dissolution of minerals
CaCO3 � Ca2+ + CO2−3
is not affected by the dissolution of gaseous CO2 in water
CO2(g) + H2O HCO−3 + H+
mixing two water samples, with masses M1 and M2, and totalalkalinities A1 et A2, resp., produce a mixture of massM = M1 +M2 and total alkalinity A, such thatMA = M1A1 +M2A2
The dominant alkalinity fraction in the most natural waters iscarbonate alkalinity
AC = [HCO−3 ] + 2× [CO2−3 ]
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
CT and AT in the Ocean
1900 2000 2100 2200 2300 2400CT (µmol kg-1)
2250
2300
2350
2400
2450
2500
AT (
µeq
kg-1
)
NPSW
NPDW
SPSW
SPDW
NASW
NADWSASW
SADW
NISW
NIDW
SISW
SIDWAASWAADW
DIC: Dissolved Inorganic Carbon
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Chemical EquilibriapH ScalesSpeciationAlkalinity
CT and AT in the Ocean: Origin of Gradients
BOTTOM
120°W 100°W20°W 160°W40°E 140°W60°E 100°E20°E 160°E80°E 60°W60°W 80°W180°40°W 120°E 140°E0°80°W
14
9
14
80°S
80°N
60°N
40°N
20°N
0°
60°S
20°S
40°S
DEEPCURRENT
LEVEL CURRENTUPPER
RENTLEVEL CUR
UPPER
UPPER LEVEL CURRENTDEEP CURRENT
UPPER LEVEL
BOTTOM
DEEP
RENTCURTOMBOT
4
5
14
10 4
14
4
4
14
18
4
4
4
20
20
10
24
14
10
10
10
Verticalgradients PacificIndianAtlantic
Sea Norvegian
Following Broecker and Peng (1982)
Inter-BasinGradients
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Formulation du problemeGeneral Acid-Base ReactionsProcedure
Calculating pH from CT and AT
Posing the problem
AT and CT are known
BT supposed to be conservative (proportional to S)
temperature, salinity and pressure given
determine
solution pH[CO∗2(aq)], [HCO−3 ], [CO2−
3 ] (speciation)CO2 partial pressure in the atmospherein equilibrium with the solution (pCO2)
Select an appropriate approximation, such as, e. g.,
AT ' [HCO−3 ] + 2× [CO2−3 ] + [B(OH)−4 ]
Express each concentration as a function of [H+] . . .
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Formulation du problemeGeneral Acid-Base ReactionsProcedure
Digression: General Acid-Base Reactions
Dissociation reaction of a general acid HnA
HnA H+ + Hn−1A− K ∗A1 =[H+][Hn−1A−]
[HnA]
Hn−1A− H+ + Hn−2A2− K ∗A2 =[H+][Hn−2A2−]
[Hn−1A−]
......
HA(n−1)− H+ + An− K ∗An =[H+][An−]
[HA(n−1)−]
K ∗A1, K ∗A2, . . .K ∗An (stoichiometric) equilibrium constants
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Formulation du problemeGeneral Acid-Base ReactionsProcedure
Digression: General Acid-Base Reactions
K ∗A1=[H+][Hn−1A−]
[HnA]⇒ K ∗A1 =
[H+][Hn−1A−]
[HnA]
K ∗A2=[H+][Hn−2A2−]
[Hn−1A−]⇒ K ∗A1K
∗A2 =
[H+]2[Hn−2A2−]
[HnA]
K ∗A3=[H+][Hn−3A3−]
[Hn−2A2−]⇒ K ∗A1K
∗A2K
∗A3 =
[H+]3[Hn−3A3−]
[HnA]
......
K ∗An=[H+][An−]
[HA(n−1)−]⇒ K ∗A1K
∗A2 . . .K
∗An =
[H+]n[An−]
[HnA]
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Formulation du problemeGeneral Acid-Base ReactionsProcedure
Digression: General Acid-Base Reactions
Let CA = [HnA] + . . .+ [An−] be the concentration of totaldissolved HnA (dissociated or not)
K ∗A1[H+]n−1 =[H+]n[Hn−1A−]
[HnA]
K ∗A1K∗A2[H+]n−2 =
[H+]n[Hn−2A2−]
[HnA]...
K ∗A1K∗A2 . . .K
∗An =
[H+]n[An−]
[HnA]
[H+]n+K ∗A1[H+]n−1 +K ∗A1K
∗A2[H
+]n−2 + . . .+K ∗A1K∗A2 · · ·K
∗An =
[H+]nCA
[HnA]
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Formulation du problemeGeneral Acid-Base ReactionsProcedure
Digression: General Acid-Base Reactions
The fractions of non dissociated acid and of the dissociated formsHn−1A−, Hn−2A2−, . . . , An− are, respectively,
[HnA]
CA=
[H+]n
[H+]n+K ∗A1[H+]n−1 +K ∗A1K
∗A2[H
+]n−2 + . . .+K ∗A1K∗A2 · · ·K
∗An
[Hn−1A−]
CA=
K ∗A1[H+]n−1
[H+]n+K ∗A1[H+]n−1 +K ∗A1K
∗A2[H
+]n−2 + . . .+K ∗A1K∗A2 · · ·K
∗An
...
[An−]
CA=
K ∗A1K∗A2 · · ·K
∗An
[H+]n+K ∗A1[H+]n−1 +K ∗A1K
∗A2[H
+]n−2 + . . .+K ∗A1K∗A2 · · ·K
∗An
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Formulation du problemeGeneral Acid-Base ReactionsProcedure
Calculating pH from CT and AT
Processing of the AT terms related to the carbonate system
Boric Acid
B(OH)3 + H2O B(OH)−4 + H+, K ∗B =[H+][B(OH)−4 ]
[B(OH)3]
hence
[B(OH)−4 ] =K ∗B
[H+] +K ∗BBT
So:
AT =K ∗1 [H+] + 2K ∗1K
∗2
[H+]2 +K ∗1 [H+] +K ∗1K∗2
CT +K ∗B
[H+] +K ∗BBT
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Formulation du problemeGeneral Acid-Base ReactionsProcedure
Calculating pH from CT and AT
K ∗1 [H+] + 2K ∗1K∗2
[H+]2 +K ∗1 [H+] +K ∗1K∗2
CT +K ∗B
[H+] +K ∗BBT−AT = 0
Equation of the form f ([H+]) = 0, where [H+] > 0 and
f strictly decreasing with [H+]f bounded: −AT < f ([H+]) < 2CT +BT−AT
⇒ no root for AT ≤ 0 or AT ≥ 2CT +BT;one and only one positive root for 0 < AT < 2CT +BT.
For 0 < AT < 2CT +BT, the root has intrinsic boundsand the equation can be reliably solved for [H+] witha hybrid Newton-Raphson–bisection method(convergence guaranteed)
Reference: Munhoven (2013)
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Formulation du problemeGeneral Acid-Base ReactionsProcedure
Calculating pH from CT and AT in General
K ∗1 [H+] + 2K ∗1K∗2
[H+]2 +K ∗1 [H+] +K ∗1K∗2
CT +K ∗B
[H+] +K ∗BBT +
K ∗W[H+]
− [H+]−AT = 0
Equation of the form f ([H+]) = 0, where [H+] > 0 and
f strictly decreasing with [H+]f unbounded: sup = +∞, inf =−∞
⇒ one and only one positive root for any AT.
Root has intrinsic bounds
Equation can (again) be reliably solved for [H+] by the hybridNewton-Raphson–bisection method (convergence guaranteed)
Reference: Munhoven (2013)
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Formulation du problemeGeneral Acid-Base ReactionsProcedure
Calculating pH from CT et de AT
Concentrations [CO∗2(aq)], [HCO−3 ], [CO2−3 ] can now be
calculated from the speciation relationships and [H+]
pCO2 is calculated from Henry’s Law
pCO2 ' fCO2 =[CO∗2(aq)]
K ∗H
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Formulation du problemeGeneral Acid-Base ReactionsProcedure
Calculating pH and Chemical Speciation in General
Determine the total concentrations of all the acid-basesystems present
Chose an adequate approximation for Total Alkalinity
Use the speciation relationships to convert the expression forAT to an equation in [H+]
Solve that equation (please refer to Munhoven (2013) forrequirements and methods)
Calculate the speciation of all the systems present from thespeciation relationships
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Formulation du problemeGeneral Acid-Base ReactionsProcedure
pCO2 et [CO2−3 ] as a Function of CT and AT
1.90 2.00 2.10 2.20 2.30 2.40 2.50Dissolved Inorganic Carbon (mmol/kg-SW)
2.20
2.30
2.40
2.50
2.60
2.70
Alk
alin
ity (
meq
/kg-
SW)
400
400
500
500
600
600
800
800
1000
1000
2000 30
00
4000 50
00 6000
9010
020
0
200
300
300
t = 20◦C, P = 0 bar, S = 35
pCO2 (µatm)
1.90 2.00 2.10 2.20 2.30 2.40 2.50Dissolved Inorganic Carbon (mmol/kg-SW)
2.20
2.30
2.40
2.50
2.60
2.70
Alk
alin
ity (
meq
/kg-
SW)
50
50
100
100
200
200
300
300
400
t = 1◦C, P = 300 bar, S = 35
[CO2−3 ] (µmol/kg-SW)
Guy Munhoven Chemical Equilibria and pH Calculations
IntroductionCarbonate Chemistry
pH Calculation
Formulation du problemeGeneral Acid-Base ReactionsProcedure
References cited and recommended
Broecker W. S. and Peng T.-H. (1982) Tracers in the Sea, Eldigio Press,Palisades, NY. 690 pp.
Dickson A. G. and Goyet C. (1994) Handbook of Methods for the Analysis ofthe Various Parameters of the Carbon Dioxide System in Sea Water (Version 2),ORNL/CDIAC-74.
Dickson A. G. (1981) An exact definition of total alkalinity and a procedure forthe estimation of alkalinity and total inorganic carbon from titration data.Deep-Sea Res., 28A(6):609–623.
Dickson A. G. (1984) pH scales and proton-transfer reactions in saline mediasuch as sea water. Geochim. Cosmochim. Acta 48:2299–2308.
Gruber N. and J. Sarmiento (2006) Ocean Biogeochemical Dynamics. PrincetonUniversity Press, Princeton, NJ. 503 pp.
Munhoven G. (1997) Modelling Glacial-Interglacial Atmospheric CO2 Variations:The Role of Continental Weathering. Universite de Liege. 273 pp.(http://www.astro.ulg.ac.be/~munhoven/en/PhDThesis.pdf).
Munhoven G. (2013) Mathematics of the total alkalinity-pH equation – pathwayto robust and universal solution algorithms: the SolveSAPHE package v1.0.1.Geoscientif. Model Dev. 6, 1367–1388.
Zeebe R. et D. Wolf-Gladrow (2003) CO2 in Seawater: Equilibrium, Kinetics,Isotopes. Elsevier, Amsterdam. 346 pp.
Guy Munhoven Chemical Equilibria and pH Calculations