1. Acid-Base Systems
Simple Analytical Formulas
aqion.de
updated 2017-08-30

2. Acids can be investigated
• in the lab (titrations)
• with modern hydrochemistry software
(one click  pH)
• by chemical thermodynamics
(derive equations, plot pH curves)
• ...
Motivation
that’s our aim

3. Aim:
Simple closed-form equations for
• titration curves
• buffer intensity β
• 1st derivative of β
Motivation
examples on next slides
for any N-protic
acid HNA

4. pK1 pK2
pH
titration curve
buffer intensity β
1st derivative of β
Input • thermodynamic data: K1, K2
• amount of acid: CT = 100 mM
Outputequivalentfraction
Example
H2CO3

5. pK1 pK2 pK2
pK2 pK3pK1
HA (acetic acid)
H2A (carbonic acid) H3A (phosphoric acid)
H3A (citric acid)
pHpH
CT = 100 mM pK1
titration curve
buffer intensity β
1st derivative of β

6. T
1
C
w
Y 





 

T
2
12
C
x2w
YY303.2







T
3
1213
2
C
w
Y2YY3Y303.2
titration
curve
buffer
intensity β
1st derivative
of 
amount of acidthe building blocks
Y1 , Y2 , Y3
... and these are the formulas:

7. Building-Block Hierarchy
K1 , K2 , ... KN
k0 =1, k1=K1 , k2=K1K2 , ...
aj(x) =
YL(x) =moments (sums of aj)
x = 10-pH
)x(a
x
k
0j
j






)x(aj j
LN
0j
AcidHNA
ionization fractions
titration curves buffer intensity β 1st derivative of β
H2O: w = Kw/x – x
amount of acid CT
1
N
N
2
21
0
x
k
...
x
k
x
k
1a








acidity constants
cumulative constants

8. The Elegance of
Ionization
Fractions aj
(as the smallest
Building Blocks)
pK2 pK3
H3A (citric acid)
pH
pK1
pK2 pK3pK1
H3A (phosphoric acid)

9. Let’s start
with the derivation ..

10. Polyprotic Acid HNA
(The complete Set of Equations)
Part 1
the general case
(N = 1, 2, 3, ...)

11. Warm-Up Example: Triprotic Acid (N=3)
1st dissociation step: H3A = H+ + H2A- K1
2nd dissociation step: H2A- = H+ + HA-2 K2
3rd dissociation step: HA-2 = H+ + A-3 K3
stepwise equilibrium constants
cumulative equilibrium constants
total amount: CT  [H3A]T = [H3A] + [H2A-] + [HA-2] + [A-3]
H3A = H+ + H2A- k1 = K1
H3A = 2H+ + HA-2 k2 = K1K2
H3A = 3H+ + A-3 k3 = K1K2K3
number of variables (unknowns): N+3
H+ OH- H3A H2A- HA-2 A-3
requires N+3
equations

12. 0 = [H+] – [OH-] – [H2A-] – 2[A-2] – 3[A-3]
Kw = {H+} {OH-} = 10-14
k1 = {H+}1 {H2A-} / {H3A}
k2 = {H+}2 {HA-2} / {H3A}
k3 = {H+}3 {A-3} / {H3A}
mass balance
law of mass action
charge balance
CT = [H3A] + [H2A-] + [HA-2] + [A-3]
N+1 equations rely on
activities {..}
2 equations rely on
concentrations [..]
Set of N+3 Equations (for Triprotic Acid H3A)

13. Kw = {H+} {OH-} (self-ionization H2O)
k1 = {H+}1 {HN-1 A-} / {HNA}
k2 = {H+}2 {HN-2 A-2} / {HNA}
kN = {H+}N {A-N} / {HNA}
CT = [HNA] + [HN-1A-] + ... + [A-N] (mass balance)
0 = [H+] – [OH-] – [HN-1A-] – 2[HN-2A-2] – ... – N[A-N] (charge balance)
N+3 species (variables/unknowns): H+, OH-, HNA, HN-1A-, ... A-N
acid
N-protic acid HN A
N+1 acid species
N equations
Set of N+3 equations
given CT  the pH is determined, and vice versa
(0 degrees of freedom)
pH =  lg [H+]

14. To study pH dependences we add one degree of freedom
to the system: CB
For this purpose, only the last line in the set of N+3 equations
should be changed:
0 = [H+] – [OH-] – [HN-1A-] – 2[HN-2A-2] – ... – N[A-N]
CB = [H+] – [OH-] – [HN-1A-] – 2[HN-2A-2] – ... – N[A-N]
(chargebalance)
(protonbalance)
CB is the amount of strong base BOH = B+ + OH-
(where B+ = Na+, K+, NH4
+, ...)

15. N+4 species (variables): H+, OH-, HNA, HN-1A-, ... A-N, CB
N-protic acid HNA + Strong Base BOH
N+1 acid species
Set of N+3 equations
1 degree of freedom
amount
of base
Kw = {H+} {OH-} (self-ionization H2O)
k1 = {H+}1 {HN-1 A-} / {HNA}
k2 = {H+}2 {HN-2 A-2} / {HNA}
kN = {H+}N {A-N} / {HNA}
CT = [HNA] + [HN-1A-] + ... + [A-N] (mass balance)
CB = [H+] – [OH-] – [HN-1A-] – 2[HN-2A-2] – ... – N[A-N] (protonbalance)
acid
N equations

16. Acid Formula Type pK1 pK2 pK3
acetic acid CH3COOH HA 4.76
(composite) carbonic
acid
H2CO3 H2A 6.35 10.33
phosphoric acid H3PO4 H3A 2.15 7.21 12.35
citric acid C6H8O7 H3A 3.13 4.76 6.4
Example: Common Acids for N = 1, 2 and 3
pKj =  lg Kj
An acid is completely defined by these
thermodynamic data (equilibrium constants).

17. Solving the Set of Equations
(Notation & Assumptions)
Part 2

18. x = [H+] = 10-pH
dissolved species
[j] = [HN-jA-j] for j = 0,1, ... N
“pure-water balance”
w = [OH-] – [H+] = Kw/x – x
ionization fractions
aj = [j]/CT for j = 0,1, ... N
pH = – log x
Terms&Abbreviations
equivalence fraction
n = CB/CT

19. Assumptions
Replace
Activities {..}  Concentrations [..]
This is legitimate for:
– small ionic strengths, I0 (dilute systems)
– or conditional equilibrium constants

20. Kw = x(x+w) (self-ionization H2O)
k1 = x(a1/a0)
k2 = x(a2/a0)
kN = x(aN/a0)
1 = a0 + a1 + a2 + ... + aN (mass balance)
n = – w/CT – (a1 + 2a2 + ... + NaN) (protonbalance)
pure
acid
N equations
Set of N+3 equations
Abbreviations
Assumptions
HN A + Strong Base
0j
j
j a
x
k
a 






1
N
N
2
21
0
x
k
...
x
k
x
k
1a








Ionization Fractions (j = 0, 1, ... N)
with

21. n = w/CT + Y1
pure H2O N-protic acid
proton
balance
N+1equations
strong base
n = CB/CT
Kw = x (x+w) a1 = (k1/x) a0
a2 = (k2/x2) a0
. . .
aN = (kN/xN) a0
1 = a0 + a1 + a2 + ... + aN
Y1 = a1 + 2a2 + 3a3 + ... + NaN
couples three subsystems:
H2O, HNA, BOH

22. pK1 pK2
pK2 pK3
pK1
HA (acetic acid)
H2A (carbonic acid)
H3A (citric acid)
pHpH
pK1
pK2 pK3pK1
H3A (phosphoric acid)
Ionization Fractions aj

23. Moments YL
(Sums over aj)
)x(ajY j
L
N
0j
L 

Y1 = a1 + 2a2 + ... + NaN
Y0 = a0 + a1 + ... + aN = 1
ionization fractions
Y2 = a1 + 4a2 + ... + N2aN
Y3 = a1 + 8a2 + ... + N3aN
 mass balance
 titration function
 buffer intensity β
 1st derivative of β

24. pK1 pK2 pK1 pK2 pK3
pK1 pK2 pK3
pK1
HA (acetic acid)
H2A (carbonic acid) H3A (phosphoric acid)
H3A (citric acid)Moments YL (for L = 1 to 4)
pHpH

25. n = Y1(x) +
w(x)
CT
Acid HNA
analytical solution of the Set of N+3 equations
K1 , K2 , ... KN
k0 =1, k1=K1 , k2=K1K2 , ...
aj(x) =
YL(x) =
acidity constants
cumulative constants
ionization fractions
moments (sums)
x = 10-pH
)x(a
x
k
0j
j






j
j
L
)x(aj
LEGOSet
1
N
N
2
21
0
x
k
...
x
k
x
k
1a









26. Applications
(Titration & Buffer Intensity)
Part 3

27. Titration Curves
‘Pure-Acid Limit’
T
1
C
w
Y)pH(n 
Y1 = a1 + 2a2
H2A (carbonic acid)
pK1 pK2pH1
1Y)pH(n 
CT  
ionization fractions aj
Y1 = a1 + 2a2

28. n = Y1
pK1 pK2pH1
Y1 = a1 + 2 a2
n = Y1 + w/CT
Titration Curves
T
1
C
w
Y)pH(n 
H2A (carbonic acid)
CT =  (pure acid)
Variation of CT

29. pK1 pK2 pK1 pK2 pK3
pK1 pK2 pK3pK1
HA (acetic acid)
H2A (carbonic acid) H3A (phosphoric acid)
H3A (citric acid)
n(pH)
pH
n(pH)
pH

30. T
1
C
w
Y)x(n 





 

T
2
12
C
x2w
YY10ln
pHd
dn
)x(








T
3
1213
2
2
2
C
w
Y2YY3Y)10(ln
pHd
nd
pHd
d
titration
curve
buffer
intensity β
1st derivative
of 
x = 10-pH

31. pH
CT = 100 mM
CT = 10 mM
CT = 1 mM
n (pH)
 = dn/dpH
d/dpH Buffer Intensity β
& Co.
for the
Carbonate System H2CO3
CTincreasing

32. pK1 pK2
pK2 pK3pK1
HA (acetic acid)
H2A (carbonic acid) H3A (phosphoric acid)
H3A (citric acid)
pHpH
CT = 1 mM

33. pK1 pK2 pK2
pK2 pK3pK1
HA (acetic acid)
H2A (carbonic acid) H3A (phosphoric acid)
H3A (citric acid)
pHpH
CT = 10 mM

34. pK1 pK2 pK2
pK2 pK3pK1
HA (acetic acid)
H2A (carbonic acid) H3A (phosphoric acid)
H3A (citric acid)
pHpH
CT = 100 mM pK1

35. pK1 pK2 pK1 pK2 pK3
pK1 pK2 pK3pK1
HA (acetic acid)
H2A (carbonic acid) H3A (phosphoric acid)
H3A (citric acid)
pHpH
CT  

36. Ref
www.aqion.de/file/acid-base-systems.pdf
www.aqion.de/site/38 (EN)
www.aqion.de/site/34 (DE)