Acid-Base Systems

A
aqionaqion
Acid-Base Systems
Simple Analytical Formulas
aqion.de
updated 2017-08-30
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
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
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
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
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:
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
The Elegance of
Ionization
Fractions aj
(as the smallest
Building Blocks)
pK2 pK3
H3A (citric acid)
pH
pK1
pK2 pK3pK1
H3A (phosphoric acid)
Let’s start
with the derivation ..
Polyprotic Acid HNA
(The complete Set of Equations)
Part 1
the general case
(N = 1, 2, 3, ...)
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
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)
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+]
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
+, ...)
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
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).
Solving the Set of Equations
(Notation & Assumptions)
Part 2
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
Assumptions
Replace
Activities {..}  Concentrations [..]
This is legitimate for:
– small ionic strengths, I0 (dilute systems)
– or conditional equilibrium constants
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
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
pK1 pK2
pK2 pK3
pK1
HA (acetic acid)
H2A (carbonic acid)
H3A (citric acid)
pHpH
pK1
pK2 pK3pK1
H3A (phosphoric acid)
Ionization Fractions aj
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
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
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








Applications
(Titration & Buffer Intensity)
Part 3
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
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
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
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
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
pK1 pK2
pK2 pK3pK1
HA (acetic acid)
H2A (carbonic acid) H3A (phosphoric acid)
H3A (citric acid)
pHpH
CT = 1 mM
pK1 pK2 pK2
pK2 pK3pK1
HA (acetic acid)
H2A (carbonic acid) H3A (phosphoric acid)
H3A (citric acid)
pHpH
CT = 10 mM
pK1 pK2 pK2
pK2 pK3pK1
HA (acetic acid)
H2A (carbonic acid) H3A (phosphoric acid)
H3A (citric acid)
pHpH
CT = 100 mM pK1
pK1 pK2 pK1 pK2 pK3
pK1 pK2 pK3pK1
HA (acetic acid)
H2A (carbonic acid) H3A (phosphoric acid)
H3A (citric acid)
pHpH
CT  
Ref
www.aqion.de/file/acid-base-systems.pdf
www.aqion.de/site/38 (EN)
www.aqion.de/site/34 (DE)
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Acid-Base Systems

  • 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
  • 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
  • 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  