Derivation of mathematical closed-from equations for multiprotic acids, which allow the calculation of titration curves and buffer intensities (with Excel). Examples are given for acetic acid, carbonic acid, phosphoric acid, and citric acid.
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
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, I0 (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
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
