Mathematical background of buffer systems and alkalimetric titration (with examples for the carbonate system).

1. Buffer Systems
Alkalimetric Titration
aqion.de
updated 2017-05-24

2. Weak Acid H2A + Strong Base
(Mathematical Background)
Part 1

3. Buffer = weak acid + strong base:
diprotic acid (e.g. H2CO3)
strong base
(B+ = Na+, K+, NH4
+, ...)
H2A + n BOH = BnH2-nA + nH2O
n = stoichiometric coefficient as
continuous variable

4. Example (n as integer variable)
H2CO3 + nNaOH = NanH2-nCO3 + nH2O
Equivalence Points
n = 0: pure H2CO3 solution
n = 1: pure NaHCO3 solution
n = 2: pure Na2CO3 solution
www.slideshare.net/aqion/diprotic_acids_and_equivalence_points

5. speciation
(in water)
6 aqueous species
in chem. equilibrium
H2A + n BOH = BnH2-nA + nH2O
acid base
n = =
CB amount of base
CT amount of acid
B+
H+
OH-
A-2
HA-
H2A

6. Kw = {H+} {OH-} = 10-14
K1 = {H+} {HA-} / {H2A} (1st diss. step)
K2 = {H+} {A-2} / {HA-} (2nd diss. step)
CT = [H2A] + [HA-] + [A-2] (mass balance)
0 = [H+] + n[H2A] + (n-1)[HA-] + (n-2)[A-2] – [OH-]
Set of 5 equations: (BnH2-nA solution)
(proton balance)
{..} = activities, [..] = molar concentrations

7. Replace: Activities {..}  Concentrations [..]
Kw = [H+] [OH-]
K1 = [H+] [HA-] / [H2A]
K2 = [H+] [A-2] / [HA-]
CT = [H2A] + [HA-] + [A-2]
0 = [H+] – [OH-] + n[H2A] + (n-1)[HA-] + (n-2)[A-2]
This is valid for small ionic strengths (I0)
or conditional equilibrium constants.
Basic Set of Equations
6 variables (species) – 5 equations = 1 degree of freedom

8. x = [H+] = 10-pH
dissolved species (H2A, HA-, A-2)
[i] = [H2-i A-i] for i = 0,1,2
“pure-water alkalinity”
w = [OH-] – [H+] = Kw/x – x
ionization fractions
ai = [i]/CT for i = 0,1,2
pH = – log x
Abbreviations

9. insert
abbreviations
Kw = x (w+x)
K1 = x (a1/a0) a0 = [1 + K1/x + K1K2/x2]-1
K2 = x (a2/a1) a1 = [x/K1 + 1 + K2/x]-1
1 = a0 + a1 + a2 a2 = [x2/(K1K2) + x/K2 + 1]-1
0 = n – a1 – 2a2 – w/CT Ionization Fractions
Basic Set of Equations

10. Ionization Fractions
a0 = [1 + K1/x + K1K2/x2]-1 [H2A] = CT a0
a1 = [x/K1 + 1 + K2/x]-1 [HA-] = CT a1
a2 = [x2/(K1K2) + x/K2 + 1]-1 [A-2] = CT a2
x = [H+] = 10-pH
K1 = 10-6.35
K2 = 10-10.33
carbonic acid
mass balance
a0 + a1 + a2 = 1
pK1 pK2

11. Relationships between pH, CT, and n
positive root of
4th order equation in x=10-pH:
0 = x4 + {K1+ nCT} x3 + {K1K2+ (n-1)CTK1– Kw} x2
+ K1 {(n-2)CTK2– Kw} x – K1K2Kw
pH(CT,n)
n(CT,pH) n = a1 + 2a2 + w/CT
CT (n,pH) CT = w/(n – a1 – 2a2)
see plots of these 3 nonlinear equations

12. Example:
CarbonateSystem

13. Relationships between pH, CT, and Alk = nCT
positive root of
4th order equation in x=10-pH:
0 = x4 + {K1+ Alk} x3 + {K1K2+ (Alk – CT)K1– Kw} x2
+ K1 {(Alk – 2CT)K2– Kw} x – K1K2Kw
pH(CT,Alk)
Alk (CT,pH) Alk = CT (a1 + 2a2) + w
CT (Alk,pH) CT = (Alk – w)/(a1 + 2a2)
see plots of these 3 nonlinear equations

14. CarbonateSystem
(nreplacedbyAlkalinity)
Alk = nCT

15. Alk = CT (a1 + 2a2) + w
Effect of Self-Ionization of Water
w influences the behavior at low and high pH.

16. Buffer Intensity
Part 2

17. normalized:  = dn/dpH [unitless]
non-normalized: C = dCB/dpH [mol/L]
Given:
Titration Curve n(CT,pH) = a1+2a2+w/CT
amount of base: CB = nCT
C = CT
Buffer Intensity
derivative d/dpH

18. Buffer Intensity
 
T
2
0202
C
x2w
)aa()aa(10ln
pHd
dn 

Derivative of Buffer Intensity










T
3
21
2
2
2
C
w
)x/K1K/x(
)x(f
)10(ln
)pH(d
nd
pHd
d
with f(x) = (x/K1-K2/x) (x/K1 + K2/x – 1 + 8K2/K1)
n = a1 + 2a2 + w/CT Titration Curve

19. buffered ( large)
pH
non-buffered
( small)
n=CB/CT
The steeper the slope the higher is the
buffer intensity  = dn/dpH,
i.e. the higher is the resistance to pH changes.

20. optimal
buffer range
(local) maximum of 
zero of d/dpH
titration curve
Example:
Carbonate System
CT = 10 mM

21. CT = 10 mM
Example: Carbonate System
titration curve is ever-increasing  is always positive
Le Châtelier: Every solution resists pH changes.

22. Carbonate
System
approximation
without self-ionization
of water (w= 0)
exact description
with self-ionization
of water (w 0)

23. at pH = pK1 6.35 maximum  optimal buffer range
at pH = ½(pK1+pK2) 8.34 minimum
at pH = pK2 10.33 maximum  optimal buffer range
Local Extrema of Buffer Intensity 
(= zeros of d/dpH)
Example: Carbonate System
Note: This is an approximation
(which ignores the self-ionization of water).

24. Example:
Carbonate System
(exact description)
Variation of CT

25. www.aqion.de/site/184 (EN)
Ref
www.aqion.de/site/61 (DE)
www.aqion.de/file/acid-base-systems.pdf

26. Appendix
(Kaleidoscope of Diagrams)

27. CT = (Alk – w)/(a1 + 2a2) Carbonate System

28. CT = w/(n – a1 – 2a2) Carbonate System

29. n = a1 + 2a2 + w/CT Carbonate System

30. Alk = CT (a1 + 2a2) + w Carbonate System

31. CT = w/(n – a1 – 2a2) Carbonate System

32. CT = (Alk – w)/(a1 + 2a2) Carbonate System