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

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- 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 (I0) 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

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