JULY 1979 The Analyst Vol. 104 No. 1239 Simplification of the Mathematical Evaluation of Titration Results by Regarding Complexes of the Type A,B as n Complexes of the Type AB Sten Johansson Department of Analytical Chemistry, The Royal Institute of Technology, S-100 44 Stochholnz 70, Sweden When titration results are evaluated mathematically, the calculations can be simplified and generalised by regarding complexes of the type A,B as n com- plexes of the type AB. For example, a diprotic acid can be replaced by two monoprotic acids, each of the same molarity as the diprotic acid but with other equilibrium constants. The conditions to be fulfilled if such simplifica- tions are to give correct results are discussed in detail. Keywords : Titration results ; inathematical evaluation Titrations may be carried out for different purposes.A physical chemist may perform a titration in order to determine equilibrium constants. He takes great pains to obtain the correct values for the constants, and titrations of this type may take a long time. An analytical chemist, on the other hand, performs a titration in order to determine the con- centration of one or several components in a sample. The equilibrium constants are often known, or they are determined by a separate titration. The exact value of the constant is of subordinate significance, the important thing being to obtain a correct value for the equivalence volume rapidly. The stepwise addition of equal volumes of titrant has proved to be a very useful method for obtaining quick and accurate results., A titration of this type cannot be terminated at the equivalence point, so the equivalence volume must be calculated from the results of the titration (e.g., millilitres of sodium hydroxide solution veysus pH values).In this department a computer program TITRA2s3 has long been in use for the evaluation of such acid - base titrations. In order to give the program a more general application, and to avoid the complications connected with chained equilibria, all acids are considered to be monoprotic. For example, a diprotic acid is replaced by two monoprotic acids, each of the same molarity as the diprotic acid but with different equilibrium constants. Certain conditions must be fulfilled if this simplified method of calculating equilibria is to be accurate, and these will be discussed in detail here.Simms’ Titration Constants His argument was more or less as follows. Suppose we have one solution containing a diprotic acid with the stability constants K, = [HA]/([H][A]) and K , = [H,A]/([H][HA]) and another solution containing the same concentrations of each of two monoprotic acids with the stability constants G, and G,. (The signs of charge are omitted in this paper except where necessary for clarity.) If we now titrate each solution with standard sodium hydroxide solution and after each addition measure the pH value, then we can draw up two series of paired values (millilitres of sodium hydroxide solution and pH). For these series to be identical at every point, the conditions are: Simms4 is apparently the first person to have dealt with this problem.K, = G, + G, .. . . . . * * (1) K, x K , = Gl x G, .. .. .. * (2) 1/K2 = l/Gl + 1/G2 . . * . . . - * (3) from which it follows that The above indicates that G, and G, are true constants. Simms calls the G constants 593594 JOHANSSON : SIMPLIFICATION OF THE MATHEMATICAL Analyst, VoZ. 104 titration constants. Although he refers to dissociation constants, the same term will be used in this paper for stability constants from now on. Simms continues his argument to prove that an n-protic acid with the constants K,, G,. K,, .... K , can be replaced by n monoprotic acids with the titration constants G,, G,, .... In this instance the following equations apply: n K, = 2 Gi i = l n-1 n K,K, = 2 2 GiGj n-2 n-11 n K1K2K3, ....K , = G,G,G,,. ... G, Kankare5 derived the same equations using a different approach. Klas6 used Simms’ formula in order to facilitate the calculation of the hydrogen-ion concentration of aqueous solutions of several polyprotic acids. Calculation of Titration Constants from the K values. calculate K , and K,. are known, is not easy to carry out. equations (1) and (3) G, m K, and G, m K2. constants are equal to the stability constants defined in the usual way. equations (1) and (2) yield G, = +Kl and G, == 2K, = +Kl. The more protons that are in the acid, the more difficult it is to calculate the G values If we take a diprotic acid, where G, and G, are known, it is easy to The converse calculation, i.e., to calculate G, and G, if K , and K , If G, is much greater than G, we obtain from the In this instance, therefore, the titration If G, = G, Thus, the titration constants will vary between - * (4) .... .. K, < G1, < K, . . * * (5) K, Q G , < 2K, .. .. .. .. If the program used for the evaluation of a. titration requires an approximate value for the titration constants, then it is usually sufficient to estimate this according to equations (4) and (5). It may be convenient to use some kind of standard procedure and the mid-point of the interval can then be taken as the initial value of the constant. The simple equations (4) and (5) are not applicable to acids with more than two protons. In those instances Table I can be used for the calculation of approximate log G values from known log K TABLE I VALUES OF ai FOR CONVERTING LOG Xi TO APPROXIMATE VALUES OF LOG Gi log Gi = log K, + a$.Number of protons in the acid a1 a2 a3 a 4 a5 a6 2 -0.12 +0.18 3 -0.18 +0.10 +0.30 4 -0.20 $0.07 t 0 . 2 4 +0.40 5 -0.22 +0.05 +0.22 +0.35 +0.48 6 -0.23 +0.04 +0.21 $0.34 +0.44 $0.54July, 1979 EVALUATION O F TITRATION RESULTS 596; values. Table I shows the values of ai to be added to the log K , values in order to obtain log G,. at the mid-point of the interval, according to the equation log Gi (mid-point) = log K, + a, If necessary, the values can be calculated more exactly by solving G1 and G, from equations (1) and (2) : G,,, = Kl/2 & $VK; - 4K1K2 The data on stability constants in the available literature, however, are often so unreliable that not much is gained from a calculation of this kind as a rule.The best results are obtained when the programs are so designed as to allow for a certain amount of error in the G values without causing an unacceptable degree of inaccuracy in the concentration deter- minations. It is also evident from equation (6) that G, and G, do not have red values if K , is less than 4K,. A titration of a diprotic acid with the quotient K,/K, < 4 or (log K, - log K,) < 0.60 cannot therefore be treated as a titration of two monoprotic acids. Significance of Titration Constants of a diprotic acid HB,-B2H can be illustrated as follows, mainly according to Adams7: Simms’ titration constants have a simple physical significance. The protolysis equilibrium .. .. * * (7) .. The constants k,, k2, k, and k, are stability constants defined by the following equations: k, [HI [-B,-B,] = [HB,-B2-] k, [HI [-B,-B,] = [-B,-B,H] k3 [HI [-B,-B,H] = [HBI-BZH] k, [HI [HB,-B2-] = [HB,-B,H] If K , and K , are the ordinarily defined stability constants then the following is valid: K , = k, + k, If the addition of a proton to the group B,- is independent of the presence or absence of a proton at the B2- group then k, = k3.In the same way k , = k , is obtained with the addition of a proton to the B,- group. In these circumstances we obtain Kl = k, + k , .. .. .. .. * - (8) Kl x K , = k, x k , . . .. .. .. .. (9) If one compares the equations (1) and (2) with (8) and (S), it can be seen that k, = G, and k, = G,. Simms’ titration constants can therefore be expressed as the intrinsic constants of the individual groups (also called microscopic constants) of the diprotic acid if the additions of hydrogen ions to the two groups are independent of each other.Similar reasoning can be applied to polyprotic acids. In his paper, Adams7 says that as a result of these statistical considerations KJK, > 4 is valid for the quotient between the two stability constants of a diprotic acid. The value 4596 JOHANSSON : SIMPLIFICATION OF THE MATHEMATICAL Analyst, VoZ. 104 is obtained if the acid is symmetrical and if the addition of hydrogen ions to both groups takes place independently of each other, i.e., k , = k , = k , = k,. For other polyprotic acids values for these statistical quotients are reported.* If Adams' argument was correct, it would mean that all n-protic acids could bt: replaced by n monoprotic acids with the same concentration.However, Bjerrum,S~lO Simms,4~11 Ricci,l2 Meites* and Kankare5 have pointed out that the real conditions cannot be expressed as simply as by these statistical quotients. To clarify this point, Adams' argument can be expanded and generalised. Development and Generalisation of Adarns' Ideas Consider the reaction A + IB = AB with the stability constant defined by the concentrations according to CAB]/( [A] [B]). In the solution there are ionic or molecular fornis of AB, A, B and other charged or uncharged molecules originating from reactions betwee:n these three and the solvent or other species present in the solvent. The probability of the reaction in question determines the extent to which the reaction takes place ; at equilibrium, certain definite proportions exist between the different species in the solution, and these proportions can be said to be controlled by the equilibrium constants.Each one of AB, A and B therefore occur in the solution in many different forms. When we determine a constant our methods of measurement are not precise enough to measure each separate constant, so that some forms are actually measured in the constant while other forms cannot be measured, for example, because they are affected by electrostatic forces in the solution. If the different measurable forms of AB are designated (AB),, (AB),, . . ., (AB), and the different measurable forms of A and B are designated (A)l, (A),, . . ., (A)m and (B)l, (B),, .. ., (B)n, then the stability constant has the following appearance : Reactions may occur between all of these molecules. 1 If we now especially consider the constants describing the reactions of (AB), to other measurable forms of AB etc., we obtain the following expressic n : where k , 1. In the same way we obtain, for A and BJuly, 1979 EVALUATION OF TITRATION RESULTS m 597 n u = l We then obtain the following expression for the constant K : If the reaction (A), + (B)l = W)l is the main reaction, then the constants kA, k, and k , should not be much greater than 1 in practice. These constants can be regarded as a sort of side-reaction coefficient (a-coefficient)13 that is used to adjust the main reaction constant in order to obtain a value for the constant being measured.It can be seen that if the transition (AB), to (AB),, etc., means that (AB), reacts with another ion, L, in the solution, then K,, has the same appearance as a normal a- coefficient : k,, = 1 + [L] X k:; f [LIZ x kzA + . . . + [LIZ-‘ x k S 1 ) The factor k , is still constant if the concentration of L is constant. This type of side reaction will probably have an effect at higher ionic strengths. The non-measurable forms of AB, A and B can be expressed, in the same way as the measurable forms, as (AB),, (A), and (B)l, respectively. The non-measurable forms that are thought to be measured in the constant are of special interest. They are designated by (AB),,, to (AB),, (A)m+l to (A), and (B),+l to (B)T, and can then be expressed as follows: w=Z+l The ratio between the measured species and the species that are thought to be measured is consequently k ~ ~ / ( 1 + g ’ A B ) .In the following, ( 1 + g’m) will be designated by gAB (gAB > 1 ) . This treatment gives rise to further correction factors for the main reaction constant. The concentrations (AB),, (A), and (B), must be divided by g-, gA and gB, respectively, so that the constant has the following appearance :598 JOHANSSON : SIMPLIFICATION OF THE MATHEMATICAL Analyst, Vol. 104 kaB . . g A gB The coefficients k=/gm, kA/gA and kB/gB should. be close to 1, but they can be both greater or less than 1. If the reaction (A), + (B), = (AB), is the main reaction for which the constant is to be measured, then they should be, in reality, a comprehensive expression for the activity coefficients and side-reaction coefficients for the side reactions that are not otherwise taken into account.They will be called $ from now on. These +-coefficients are constant if the conditions for the reaction are kept constant. The side reactions for which the constants are known, and for which it is also known that the reaction products are not calculated within the constant K, can be considered by ordinary a-coefficients. If, then, (AB), has a side reaction with L it is considered by CC(AB)~(L) so that instead of +D we obtain a- = $AB + ~ ( A B ) , ( L ) . - 1. The conditional constant now has the following appearance : The above suggests that all equilibrium constants can be regarded as concentration constants but as conditional13 concentration constants. As Simmsll and Ricci12 have shown, it still holds that polyprotic acids can be replaced by monoprotic acids.The condition for this is, however, that the +-coefficients and the or-coefficients are kept constant, and that the ratio between the constants is greater than the statistical quotient. If this reasoning is applied to Adams' diagratm [equation (7)], given the condition that the reaction via HB,-B,- is the main reaction (and that no reactions other than those in Adams' diagram occur) , the following expression is obtained for the constants (excluding the ionic charges) : where and [HBI-B;] = [HBI-BJ + [B,-B,H] If the reaction via B,-B,H occurs to an equally large extent as the reaction via HB,-B,, then k, = k, and $HBl-B2 = 2.If the reaction occurs to a lesser extent, &B1-B2 has a value between 1 and 2. The following is then valid for the quotient of the constants: where The size of K, and k, is of decisive importance for the size of the quotient K,/K2. then the maximum value of the quotient K1/K2 will be 4. 1 <+AB1--B2 <4 If k , = k ,July, 1979 EVALUATION OF TITRATION RESULTS 599 It is difficult to comment on the size of the quotient k,/k, other than to say that as a rule it should hold that k, > k,. However, Campbell and Meites,14 in a paper on titration graphs for acids with very small ratios of successive dissociation constants, have indicated a possible structure for such abnormal acids in which K , is greater than K,. An indicator that reacts like an abnormal acid has been discussed by Schwarzenbach.15 It is also conceivable that side reactions take place to such a large extent that the ratio between the conditional constants falls below the statistical quotients.This instance is dealt with more fully in the examples below. Examples of the Influence of Side Reactions Titration of adipic acid These problems occurred when we were titrating adipic acid. We often perform our acid - base titrations at an ionic strength of 0.5, using barium chloride as the ionic medium. The following constants are stated at p = 0 for adipic acid, H2A16J7: H + A =HA log Kl = 5.41 H + HA = H,A log K , = 4.42 Ba + A =BaA log KBaA = 1.85 Thus, when adipic acid is titrated in the barium chloride medium, a conditional constant Ki is valid for K, if the side reaction of A with Ba is taken into account.Taking these constants as a starting point, rough calculations indicate that Alog K = log Ki - log K , should be approximately 0.7 in the barium chloride medium when p = 0.5. Thus, we have come close to the critical value Alog K = 0.60 where, theoretically, the evaluation program TITRA no longer functions. These titrations were evaluated with regard to the constants in order to obtain the true value for Alog K. In this instance the computer program LETAGROP~* was used. The values obtained were log K; = 4.74 and log K , = 4.09, ie., Alog K = 0.65. Therefore, the evaluation of the concentrations by TITRA functioned normally and the relative error was less than 0.1 yo. Titrdion of the disodiwn salt of ethylenediaminetetraacetic acid (EDTA) In the previous example, the complexation with the ionic medium was assumed to be very weak.A stronger complexation results in more pronounced changes. This is true for, for example, EDTA. Titration of the disodium salt of EDTA with sodium hydroxide solution yields two steps on the titration graph, the second step being very small. At an ionic strength of 0.5 the following values for the stability constants can be assumed provided that no side reactions occur: H + Y = H Y H + HY = H,Y log K, = 10.2 log K , = 6.1 The two protons dissociate one after the other and as a result first HY3- and then Y4- are formed. If, however, 0.133 M barium chloride and 0.100 M sodium chloride solutions are used as the ionic medium the ionic strength will still be 0.5 but the conditions for the titration will be radically changed.The reasons for this are the reactions between barium chloride and EDTA according to the following formulae : Ba + Y = BaY log KBay = 7.4 Ba + HY = BaHY log KBaHP = 1.7 (The values of the constants given above are based upon those given by Schwarzenbach et al.19) The conditional acid constant values that can be calculated by using these values are log K; = 4.6 and log Ki = 5.2. It is evident that Ki has now become greater than K;, which means that both protons will react almost simultaneously during the titration.,O600 JOHANSSON Consequently, there will be only one step in the titration graph but this will not, however, be identical with the titration graph obtained for a monoprotic acid of twice the concentra- tion.14 (At the beginning of the titration giraph there is, of course, another difference, depending on the fact that Na,H,Y is an ampholyte.) In this instance the computer program TITRA~ will not work in its usual form.Non-acid Complexes The reasoning principally used for acids, i.e., proton complexes, can also be applied to other types of complexes. This has been partly carried out for metal complexes by Bjerrum,21 Fronaeus,, and K l a ~ . , ~ These workers have not, however, considered the applica- tion of their ideas within analytical chemistry. Even if the above principles can theoretically be applied to complexation titrations, in practice their application is limited. If in a titration of a metal ion M with a complexing agent L, [MI is measured by a metal ion sensitive indicator electrode, reasoning identical with the above can be used if only the complexes ML, M,L, etc., are formed.Usually, however, complexes of the type ML, ML,, etc., are formed. A solution con- taining only these complexes, by analogy with the above, can be replaced by another solution with complexes of the type ML. If these two solutions are titrated, the concentration of L must be the same in both solutions. If the titration is performed potentiometrically then the indicator electrode should respond to the L concentration. Such ligand-responsive electrodes are, however, rare. In these instances L must absorb light, or else an indicator must be used. Most of the indi- cators in use are metallochromic, and are therefore not suitable for studying variations in the ligand concentration.Similarly, it is rather unusual for such titrations to be performed photometrically. I am indebted to Professor Folke Ingman for valuable discussions and to Mrs. Alison Holmstrom for translating the manuscript. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. References Johansson, A., Analyst, 1970, 95, 535. Ingman, F., Johansson, A., Johansson, S., and Karlsson, R., Analytica Chim. Acta, 1973, 64, 113 Johansson, A., and Johansson, S., Analyst, 1979, 104, 601. Simms, H. S., J . A m . Chem. SOC., 1926, 48, 1239. Kankare, J . J., Talanta, 1975, 22, 1005. Klas, J., Analytica Chim. Acta, 1968, 41, 549. Adams, E. Q., J . A m . Chem. SOC., 1916, 38, 1503. Meites, L., J . Chem. Educ., 1972, 49, 682. Bjerrum, N., Z. Phys. Chem., 1923, 106, 219. Bjerrum, N., Ergedn. Exakt. Naturw., 1926, 48, 125. Simms, H. S., J . A m . Chem. SOC., 1926, 48, 12511. Ricci, J . E., “Hydrogen Ion Concentration,” Princeton University Press, Princeton, N. J., 1952. Ringbom, A., “Complexation in Analytical Chemistry,” Interscience, New York, 1963. Campbell, B. H., and Meites, L., Talanta, 1974, 21, 117. Schwarzenbach, G., Helv. Chim. Acta, 1943, 26, 418. Gana, R., and Ingold, C. K., J . Chem. SOC., 1931, 2153. Topp, N. E., and Davies, C. W., J . Chem. SOC., 1940, 87. Phyllis, B., SillCn, L. G., and Whiteker, R., Ark. Kemi, 1969, 31, 365. Schwarzenbach, G., and Ackermann, H., Helv. Chim. Acta, 1947, 30, 1798. Schwarzenbach, G., and Sulzberger, R., Helv. Chim. A d a , 1943, 26, 453. Bjerrum, J ., “Metal Ammine Formation in Aqueous Solution,” Haase, Copenhagen, 1941. Fronaeus, S., “Komplexsystem hos koppar,” Gleerupska Universitets, Bokhandeln, Lund, 1948. Klas, J., Analytica Chim. Acta, 1975, 74, 220. Received May 22nd, 1978 Amended September 1 lth, 1978 Accepted December 14th, 1978