Equivalence Points

The concept of equivalence points (EP) runs like a golden thread through acid-base theory and applications. There are different types of equivalence points. This article provides a classification of EPs and semi-EPs. This is done for the general case of N-protic acids (based on simple mathematical equations).

1. Equivalence Points (EP) Systematics & Classification aqion.de updated 2017-08-29

2. The set of Equivalence Points (EP & semi-EP ) is the unique barcode of an acid-base system. Motivation

3. An Equivalence Point is a special equilibrium state at which chemical equivalent quantities of acids and bases are mixed: Definition EP: [acid] = [base] square brackets indicate molar concentrations

4. acid (proton donor) conj. base (proton acceptor) conjugate pair HA = H + + A -

5. HA + H 2 O = A - + H 3 O + conjugate pair conjugate pair acid base conj. base conj. acid

6. Diprotic Acid H 2 A = H + + HA - HA - = H + + A -2 conj. acid base acid conj. base 1 st dissociation step: 2 nd dissociation step: HA - is the conj. base of acid H 2 A , and HA - is the conj. acid of base A -2

7. HA - is the conj. base of acid H 2 A , and HA - is the conj. acid of base A -2 EP of HA - : [H 2 A] = [A -2 ] Diprotic Acid Notation: The diprotic acid has 3 species: [H 2 A], [HA - ] and [A -2 ]. They add up to the total amount : C T = [H 2 A] + [HA - ] + [A -2 ]  [ H 2 A] T

8. Triprotic Acid (H 3 A) EP of H 2 A - : [H 3 A] = [ H A -2 ] EP of HA -2 : [HA -2 ] = [A -3 ] In addition, there are much more types of equivalence points. Let‘s systematize it.

9. Further Procedure Triprotic Acid H 3 A Part 1a Part 1b Part 2 N-protic Acid H N A General Approach: H N A + H 2 O “Pure-Acid” Approach simple & nice equations (but without self-ionization of H 2 O)  fails for very dilute acids with C T < 10 -3 M

10. Equivalence & Semi-Equivalence Points of H 3 A Part 1a Triprotic Acid

11. Triprotic Acid (H 3 A) The 3 -protic acid dissolves into 3 +1 species: H 3 A H 2 A -1 HA -2 A -3 There are several ways/equations to define equivalence points (including semi -equivalence points ). a t least there are 2  3 +1 EPs: EP n with n = 0, ½, 1, ... 3 Note: We abbreviate the dissolved, neutral acid-species H 3 A also by H 3 A.

12. Triprotic Acid (H 3 A) EP [ H + ] = [ H 2 A - ] EP 1/2 [H 3 A] = [ H 2 A - ] EP 1 [ H 3 A ] = [ HA -2 ] EP 3/2 [ H 2 A - ] = [ HA -2 ] EP 2 [H 2 A - ] = [ A -3 ] EP 5/2 [ HA -2 ] = [ A -3 ] EP 3 [HA -2 ] = [ OH - ] H + H 3 A H 2 A - HA -2 A -3 OH - EP EP 2 EP 1 EP 3 Equivalence Points H 3 A H 2 A - HA -2 A -3 EP 1/2 EP 5 /2 EP 3/2 Semi-EPs

13. Triprotic Acid 1 st diss. step: H 3 A = H + + H 2 A - K 1 = [H + ] [ H 2 A - ] / [ H 3 A] 2 nd diss. step: H 2 A - = H + + HA -2 K 2 = [H + ] [HA -2 ] / [H 2 A - ] 3 rd diss. step: HA -2 = H + + A -3 K 3 = [H + ] [HA -3 ] / [HA -2 ] stepwise equilibrium constants K 1 = K 1 = [H + ]  pH = pK 1 [H 2 A - ] = [ H 3 A] (semi-EP) [H + ] [ H 2 A - ] [H 3 A ] K 1 K 2 = K 1 K 2 = [H + ] 2  pH = ½ ( pK 1 + pK 2 ) [ HA -2 ] = [ H 3 A] (EP) [H + ] 2 [ HA -2 ] [H 3 A ] Each EP (or semi-EP) is characterized by one specific pH value that relies directly on the acidity constants K: EP  pH  pK pH =  lg [H + ] pK =  lg K

14. EP [ H + ] = [ H 2 A - ] EP 1/2 [H 3 A] = [ H 2 A - ]  pH 1/2 = pK 1 EP 1 [ H 3 A ] = [ HA -2 ]  pH 1 = ½ (pK 1 + pK 2 ) EP 3/2 [ H 2 A - ] = [ HA -2 ]  pH 3/2 = pK 2 EP 2 [H 2 A - ] = [ A -3 ]  pH 2 = ½ ( pK 2 + pK 3 ) EP 5/2 [ HA -2 ] = [ A -3 ]  pH 5/2 = pK 3 EP 3 [HA -2 ] = [ OH - ] Each EP (or semi-EP) is characterized by one pH value that relies directly on the acidity constants K: EP  pH  pK The two “external EPs” EP and EP 3 are non-trivial; they depend, in addition, on the total amount of acid, C T .

15. EP 1/2 EP 1 EP 3/2 EP 2 EP 5/2 pH 1/2 pH 1 pH 3/2 pH 2 pH 5/2 phosphoric acid 2.15 4.68 7.21 9.78 12.35 citric acid 3.13 3.94 4.76 5.58 6.4 pK 1 pK 2 pK 3 phosphoric acid 2.15 7.21 12.35 citric acid 3.13 4.76 6.4 pK 1 pK 2 EP 1/2 EP 1 pK 3 pH EP 5 /2 EP 2 EP 3 /2 (midpoint) (midpoint) pH 1 pH 2 EP EP 3 Examples

16. Equivalence & Semi-Equivalence Points of H N A Part 1b N-protic Acid (N = 1, 2, 3, ...)

17. EP [ H + ] = [ 1] EP 1/2 [ ] = [ 1]  pH 1/2 = pK 1 EP 1 [0] = [ 2]  pH 1 = ½ (pK 1 + pK 2 ) EP 3/2 [ 1 ] = [ 2 ]  pH 3/2 = pK 2 EP 2 [1] = [ 3 ]  pH 2 = ½ ( pK 2 + pK 3 ) ⁞ EP N- ½ [N-1] = [ N]  pH N- ½ = pK N EP N [N-1] = [ OH - ] An N-protic acid H N A has internal EPs two external EPs and 2N+1 EPs: N+1 acid-species : [j]  [ H N-j A -j ] with j = 0,1, ... N

18. N Acid H N A EP 1/2 EP 1 EP 3/2 EP 2 EP 5/2 pH 1/2 pH 1 pH 3/2 pH 2 pH 5/2 1 acetic acid 4.76         2 (composite) carbonic acid 6.35 8.34 10.33     3 phosphoric acid 2.15 4.68 7.21 9.78 12.35 3 citric acid 3.13 3.94 4.76 5.58 6.4 Acid Type pK 1 pK 2 pK 3 acetic acid HA 4.76     ( composite) carbonic acid H 2 A 6.35 10.33   phosphoric acid H 3 A 2.15 7.21 12.35 citric acid H 3 A 3.13 4.76 6.4 Examples for I nternal EPs

19. Unified Notation for I nternal EPs EP n   pH n = ½ (pK n + pK n+1 )  pH n = pK n+1/2 [n-1] = [n+1] [n-½] = [n+½] integer n = j (n = 1, 2, ... N-1) half-integer n = j- ½ (n = ½, ³/ ₂ , ... N- ½) semi-EPs: Note: j is integer and indicates the acid-species [j] and acidity constants K j ; n is integer and half-integer and labels the EPs and semi-EPs. t he true, chemical meaning of n becomes clear later in the text

20. External EPs EP N [N-1] = [ OH - ]  pH N  14 EP [ H + ] = [ 1 ]  pH  0 There is no simple relationship between EP  pH . pH of EP and EP N depend on K 1 to K N and on C T (= total amount of acid). However, for C T   the following asymptotic behavior exists:

21. pH C T [M] C T [M] n=0 n=0.5 n=1.5 n=1 n=2 n=0 n=0.5 n=1.5 n=1 n=2 n=2.5 n=3 H 3 A (phosphoric acid) H 2 A (carbonic acid) Examples Internal EPs: red lines (independent of C T ) External EPs: blue and green lines (C T -dependent )

22. Caution: This approach is valid only for sufficiently high-concentrated acids H N A with C T > 10 -3 M These restrictions & limits are removed in Part 2 (because self-ionization of H 2 O and charge balance is ignored)

23. General Approach: H N A + H 2 O Part 2

24. General Relationship EP n  pH n pH n (Equivalent Fraction) pH 1/2 pH 3/2 pH pH 1 pH 2 EP 1/2 EP 3/2 EP EP 1 EP 2 Titration Curve of Diprotic Acid (Example: 100 mM H 2 CO 3 ) n = Y 1 ( pH ) + w( pH ) C T

25. Basic Equation for H N A n = Y 1 ( x ) + w( x ) C T self-ionization H 2 O i onization f ractions (j = 0, 1, ... N) with c umulative equilibrium constants: k 1 = K 1 , k 2 = K 1 K 2 , ... total amount of acid x  [H + ] = 10 -pH Ref: www.aqion.de/file/acid-base-systems.pdf

26. Plots of EP n in pH-C T Diagrams f or an integer or half-integer n you get from one curve C T = C T (n,x) with x = 10 -pH n Given: N-protic acid H N A  2N+1 curves Note: You can perform the calculations by Excel, for example.

27. n=0 n=0.5 n=1 n=0 n=0.5 n=1 n=1.5 n=2 HA (acetic acid) H 2 A (carbonic acid) C T [M] n=0 0.5 1.5 1 2 2.5 n=3 H 3 A (phosphoric acid) n=0 0.5 1.5 1 2 2.5 n=3 H 3 A (citric acid) pH C T [M] pH EP & semi-EP internal EP external EP external EP N dashed lines refer to the “pure-acid case” in Part 1

28. n = Y 1 ( x ) + w( x ) C T General Case n = Y 1 ( x ) “Pure-Acid” Case either w = 0 (ignoring self-ionization) or C T  (high amount of acid) EP n  pH n = ½ (pK n + pK n+1 ) for integer n pK n+1/2 for half-integer n

29. comprises two subsystems (as limiting cases): pure H 2 O defined by w( x ) = 0 for C T = 0 pure H N A defined by n – Y 1 ( x ) = 0 for C T   n = Y 1 ( x ) + w( x ) C T The general equation for H 2 O + H N A Alternative Interpretation ... and each subsystem has its own EPs

30. The general equation for H 2 O + H N A decouples into two subsystems: C T = C T   pure H 2 O pure H N A with one single EP at pH 7 with EP n at pH n = ½ (pK n + pK n+1 ) and semi-EP n at pH n = pK n+1/2 w = 0 n – Y 1 = 0 (poles of C T ) EP n 

31. Example H 3 PO 4 C T [M] pure acid (H 3 A) pure H 2 O two uncoupled subsystems pH pH = 7 EP 1/2 EP 3/2 EP 1 EP 2 EP 5/2 EP EP 3

32. Example H 3 PO 4 C T [M] acid + H 2 O pH EP 1/2 EP 3/2 EP 1 EP 2 EP 5/2 EP EP 3 C oupling of two Subsystems

33. (B + = Na + , K + , NH 4 + , ...) n = = C B amount of base C T amount of acid H N A BOH + n Titration of weak a cid H N A with s trong base BOH Meaning of n

34. EP 1  1 = C B / C T EP ½  ½ = C B / C T EP 1 : [ acid] T = [base] T Generalization to other EPs: EP n  n = C B / C T ⁞ C T C B T = total amount

35. General Approach vs. Pure-Acid Limit Summary

36. „Pure-Acid“ Approach H N A Subsystem General Approach H N A + H 2 O valid for large C T only (> 10 -3 M) definition based on acid species EP n : [n-1] = [n+1] semi-EP n : [n- ½ ] = [n+ ½ ] outcome ½ (pK n + pK n+1 )  EP pK n+1/2  semi-EP pH n = C T   (n Y 1 = 0) n based on total amount of compounds EP n : n = = [H N A] T [strong base] T C T C B

37. Titration & Buffer Intensity EP n ( integer n)  minimum buffer intensity semi-EP n ( half-integer n)  maximum buffer intensity pH titration curve buffer intensity β EP 1/2 EP 3/2 EP EP 1 EP 2 pH 1/2 pH 3/2 pH pH 1 pH 2 Example: Diprotic Acid (100 mM H 2 CO 3 )

38. Ref www.aqion.de/file/acid-base-systems.pdf www.aqion.de/site/68 (EN) www.aqion.de/site/34 (DE)