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# Acid-Base Systems

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# Acid-Base Systems

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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.

#### Transcript

1. 1. Acid-Base Systems Simple Analytical Formulas aqion.de updated 2017-08-30
2. 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. 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. 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. 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. 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. 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. 8. The Elegance of Ionization Fractions aj (as the smallest Building Blocks) pK2 pK3 H3A (citric acid) pH pK1 pK2 pK3pK1 H3A (phosphoric acid)
10. 10. Polyprotic Acid HNA (The complete Set of Equations) Part 1 the general case (N = 1, 2, 3, ...)
11. 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. 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. 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. 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. 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. 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. 17. Solving the Set of Equations (Notation & Assumptions) Part 2
18. 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. 19. Assumptions Replace Activities {..}  Concentrations [..] This is legitimate for: – small ionic strengths, I0 (dilute systems) – or conditional equilibrium constants
20. 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. 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. 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. 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. 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. 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. 26. Applications (Titration & Buffer Intensity) Part 3
27. 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. 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. 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. 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. 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. 32. pK1 pK2 pK2 pK3pK1 HA (acetic acid) H2A (carbonic acid) H3A (phosphoric acid) H3A (citric acid) pHpH CT = 1 mM
33. 33. pK1 pK2 pK2 pK2 pK3pK1 HA (acetic acid) H2A (carbonic acid) H3A (phosphoric acid) H3A (citric acid) pHpH CT = 10 mM
34. 34. pK1 pK2 pK2 pK2 pK3pK1 HA (acetic acid) H2A (carbonic acid) H3A (phosphoric acid) H3A (citric acid) pHpH CT = 100 mM pK1
35. 35. pK1 pK2 pK1 pK2 pK3 pK1 pK2 pK3pK1 HA (acetic acid) H2A (carbonic acid) H3A (phosphoric acid) H3A (citric acid) pHpH CT  
36. 36. Ref www.aqion.de/file/acid-base-systems.pdf www.aqion.de/site/38 (EN) www.aqion.de/site/34 (DE)

#### Description

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.

#### Transcript

1. 1. Acid-Base Systems Simple Analytical Formulas aqion.de updated 2017-08-30
2. 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. 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. 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. 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. 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. 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. 8. The Elegance of Ionization Fractions aj (as the smallest Building Blocks) pK2 pK3 H3A (citric acid) pH pK1 pK2 pK3pK1 H3A (phosphoric acid)
10. 10. Polyprotic Acid HNA (The complete Set of Equations) Part 1 the general case (N = 1, 2, 3, ...)
11. 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. 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. 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. 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. 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. 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. 17. Solving the Set of Equations (Notation & Assumptions) Part 2
18. 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. 19. Assumptions Replace Activities {..}  Concentrations [..] This is legitimate for: – small ionic strengths, I0 (dilute systems) – or conditional equilibrium constants
20. 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. 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. 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. 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. 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. 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. 26. Applications (Titration & Buffer Intensity) Part 3
27. 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. 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. 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. 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. 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. 32. pK1 pK2 pK2 pK3pK1 HA (acetic acid) H2A (carbonic acid) H3A (phosphoric acid) H3A (citric acid) pHpH CT = 1 mM
33. 33. pK1 pK2 pK2 pK2 pK3pK1 HA (acetic acid) H2A (carbonic acid) H3A (phosphoric acid) H3A (citric acid) pHpH CT = 10 mM
34. 34. pK1 pK2 pK2 pK2 pK3pK1 HA (acetic acid) H2A (carbonic acid) H3A (phosphoric acid) H3A (citric acid) pHpH CT = 100 mM pK1
35. 35. pK1 pK2 pK1 pK2 pK3 pK1 pK2 pK3pK1 HA (acetic acid) H2A (carbonic acid) H3A (phosphoric acid) H3A (citric acid) pHpH CT  
36. 36. 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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