chemical equilibria and ph calculations

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


pH Calculation

Plan

Chemistry of the carbon dioxide system

Chemical equilibria

pH scales

Conservative state variables:dissolved inorganic carbon and alkalinity

Carbonate: calculation



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



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).



pH Calculation


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 ]



pH Calculation


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



pH Calculation


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



pH Calculation


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)



pH Calculation


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



pH Calculation


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



pH Calculation


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 ]



pH Calculation


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]



pH Calculation


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



pH Calculation


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



pH Calculation


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



pH Calculation



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



pH Calculation



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



pH Calculation


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)



pH Calculation


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)



pH Calculation


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



pH Calculation


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



pH Calculation


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 ]



pH Calculation


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.



pH Calculation


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 ]



pH Calculation


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)



pH Calculation


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.



pH Calculation


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



pH Calculation


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



pH Calculation


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 ]



pH Calculation


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



pH Calculation


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



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+] . . .



pH Calculation


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



pH Calculation



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]



pH Calculation



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]



pH Calculation



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



pH Calculation



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



pH Calculation



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)



pH Calculation


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)



pH Calculation


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



pH Calculation


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



pH Calculation


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)



pH Calculation


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.


http://www.astro.ulg.ac.be/~munhoven/en/PhDThesis.pdf

chemical equilibria and ph calculations

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