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Understanding Complexometric Titrations of Metal Cations with Aminopolycarboxylic Acids (EDTA and Analogs) within the frame of the Notion of Reactions between Groups of Chemical Species

In spite of the apparent technical simplicity with which visual complexometric titrations of metal cations with aminopolycarboxylic acids titrants are performed, a complex chemistry takes place in the titrated solution during the titration, due to the chemical environment and to the insuppressible chemical properties of metal cations, aminopolycarboxylic acids and metallochromic indicators. This chemical complexity makes rigorous exposition and evaluations of complexometric titrations arduous. Nonetheless, by the introduction of the notions of groups of chemical species and reactions between groups of chemical species (with the connected concept of conditional formation constant), a frame is created within which complexometric titrations with aminopolycarboxylic acids can be collocated and which allows a reasonably simple presentation and evaluation of the analytically relevant aspects of this type of titrations. In a sense, the well known concept of conditional complex formation constant is updated in a way that facilitates understanding and use.


Introduction
Aminopolycarboxylic acids are a class of multidentate chelating ligands which are very effective in complexing metal cations through oxygen and nitrogen donor atoms.
Because of this capability, they are employed in a variety of ways in chemistry and in other fields.
A large number of aminopolycarboxylic acids have been synthesized and examined for their properties in complexing metal cations, but only a small number of them are in common use [1].
The most popular of all the aminopolycarboxylic acids is ethylendiaminetetraacetic acid (H 4 EDTA) which corresponds to the structure [2,3]: Diethylenetriaminepentaacetic acid (H 5 DPA) and diaminocyclohexanetetraacetic acid (H 4 DCTA), which correspond to the following structures, are also very common and in many aspects equivalent to EDTA: Probably, the most popular application of aminopolycarboxylic acids in chemistry is their use as titrants in complexometric titrations for the determination of a variety of metal cations [4].
Visual direct complexometric titrations are very rapid and convenient and are performed in a way that is technically very similar to other types of titrations (e.g., visual Acid-Base titrations) [2,3]. From equations (2) and (3) we see that, while the variable t l V is increased during the titration, the analytical concentration of the aminopolycarboxylic acid (which is initially zero) increases, while the analytical concentration of the metal cation decreases. Since M APCA C increases and M Me C decreases during the titration, there will be a point when they become equal and relation (4) is verified: This point is, by definition, the equivalence point. The aim of any analytical complexometric titration is to identify the value of t l V for which relation (4) holds in the titrated solution.
The above definition of the equivalence point is justified from the fact that the fundamental reaction which takes place between the target metal cation, Me z+ , and an amino polycarboxylate anion, APCA nhas a very simple 1: 1 stoichiometry shown in (5): The equilibrium constant of reaction (5) is generally denoted by β 1 : For instance, if Me z+ = Zn 2+ and APCA n-= EDTA 4 ⁻ we have reaction (7), which would take place during the titration of a solution of Zn 2+ with standard EDTA solution: 2 4 2 Zn EDTA ZnEDTA 2 16.5 1 2 4 ZnEDTA 10 Zn EDTA Please note that the simple stoichiometry of complexes and the large values of the corresponding formation constants are at the basis of the usefulness of this class of ligands [4,5].
The metallochromic indicator, In n-, which is responsible for the colour change of the titrated solution when the titration is stopped, is itself a chelating agent which reacts with the target metal cation according to the simple reaction (9): In MeIn However, for the present purposes, it is useful to note that In n ⁻ is a base which may exist as several protonated species, H h In (n-h) ⁻, which differ in the degree of protonation, and that reaction (9) can be rewritten in the form: The outstanding property of a metallochromic indicator is the fact that the free indicator, H h In (n-h) ⁻, has a colour which differs from the colour of the complex MeIn (n-z) ⁻.
At the beginning of the complexometric titration, because of the excess of Me z+ in the titrated solution, we see the colour of the complex MeIn (n-z) ⁻. As the titration advances, Me z+ progressively disappears from the solution because of reaction (5), until a point is reached when the MeIn (n-z) ⁻ complex is decomposed to the free indicator. At this point, the colour of the solution changes from that of MeIn (n-z) ⁻ to that of H h In (n-h) ⁻. As it happens, the colour of the H h In (n-h) ⁻ species depends on the number of bonded protons.
As a consequence, the colour observed when the MeIn (n-z) ⁻ complex is decomposed depends on the prevailing H h In (n-h) ⁻ species formed; which, in turn, depends on the pH of the titrated solution. In order to fix the colour of the solution after the equivalence point the pH of the titrated solution is buffered to a suitable value by adding a pH buffer [2][3][4][5][6]. This is unavoidable because it often happens that one or more of the H h In (n-h) ⁻ species have the same colour as the complex MeIn (n-z) ⁻. Obviously, if a H h In (n-h) ⁻ species, which has the same colour as MeIn (n-z) ⁻, is formed, no change in colour will be observed.
In other words, a metallochromic indicator is such only in a restricted range of pH in which the prevailing form of the free indicator has a colour which differs from the colour of its complex with the target metal cation. For instance, one of the most popular metallochromic indicators is Eriochrome Black T (EBT) which corresponds to the structure: The full protonated species of EBT is H 2 EBT⁻ which has a red colour and is the prevailing species up to pH ≈ 6.5. The dissociation of a proton from H 2 EBT⁻ produces HEBT 2 ⁻ which has a blue colour and is the prevailing H h EBT (3-h) ⁻ specie in the pH range ~ 7 < pH < ~ 11. The full deprotonated species EBT 3 ⁻ has an orange colour and prevails at the highest pH. Now, since the MeEBT (3-z) ⁻ complexes have a red colour (regardless of the nature of the metal), it is quite evident that Eriochrome Black T is not a metallochromic indicator at pH lower than about 7.
From the above, it may also be deduced that a metallochromic indicator is in primis an Acid-Base indicator, so much so that a change in colour may be observed in the titrated solution because of an unintended pH change following the addition of the titrant, even in a point which is far from the equivalence point of the complexometric titration. This event is also prevented by pH buffering the titrated solution.

The Chemical Complexity of Complexometric Titrations
In spite of the apparent operational simplicity of a direct complexometric titration, a very complex chemistry unfolds in the titrated solution as the titration progresses.
This chemistry is not of direct interest to the experimentalist, but must be considered in detail in order to state the conditions which must be met if accurate results must be derived from a complexometric titration.
To give an example, consider the titration of a Zn 2+ solution, pH buffered with an NH 3 /NH 4 + buffer, with a standard solution of EDTA in presence of EBT as indicator.
In abstract, all the reactions listed in Table 1 take place in the titrated solution as the titration progresses. This long list of reactions derives from the fact that a metal cation in water is a Brönsted acid which forms hydroxo complexes (see reactions from 3 to 6 in Table 1) and the amino carboxylate anion and the indicator are bases which can be protonated (see reactions from 7 to 10 and reactions 11 and 12 in Table 1). Furthermore, the presence of the NH 4 + /NH 3 pH buffer introduces an auxiliary ligand, i.e., NH 3 , which competes with the primary ligand, i.e., EDTA 4 ⁻, for the metal cation (see reactions from 15 to 18 in Table 1). Finally, according to reaction 13 and 14 in Table 1, the primary complex ZnEDTA 2 ⁻ may react with H + to form the acid complex ZnHEDTA⁻; or, with OH⁻ to form a basic complex Zn(OH)EDTA 3 ⁻.
Since, in abstract, any reaction taking place in the titrated solution produces a new species, it follows that a large number of different species are generally present in the titrated solution during the titration.
This point is stressed in Figure 1 for the particular case of a 0.01 M Zn 2+ + 0.01 M EDTA 4 ⁻ + 0.25 M NH 3 + 0.25 M NH 4 + + 10 -6 M Eriochrome Black T solution. This state of affairs is replicated with obvious variations which depend on the specific nature of the metal cation, of the titrating aminopolycarboxylic acid, of the auxiliary ligand and of the indicator in any complexometric titration [4,5,8,9].
Protonation of the primary ligand

Acid and Basic Complexes
Protonation of the indicator Acid and Basic Complexes  Table 1 for examples of each type of reaction Figure 2 presents a general scheme of the reactions which typically occur during a complexometric titration.
In Figure 2, the reactions which take place in the titrated solution are classified as primary reactions, which are functional to the titration, and side reactions which are unintended reactions due to insuppressible chemical properties of reactants and products of the primary reactions. Side reactions are classified as hydrolysis reactions and complex formation reactions of the metal cation with the auxiliary ligand (symbolized by Q n -). Then, there are in general side reactions of protonation of the primary ligand (APCA n -) and of the indicator (In n -). Finally, the reactions of the primary complexes, MeAPCA (n-z)and MeIn (n-z) -, to form acid and basic complexes, may occur.
Please note that, in Figure 2, a special symbol for the equilibrium constant of each type of reaction (which will be used in the following) is introduced. Please refer to Table 1 for specific examples of the general reactions in Figure 2 and for typical values of the corresponding equilibrium constants.
Evidently, there it is a conspicuous number of reactions in the titrated solution that are unnecessary for the titration, but which cannot be suppressed and must, then, be considered because they have an effect on the yield of the primary reactions and can dramatically change the outcome of the titration.

Overcoming Side Reactions
Side reactions which take place during a complexometric titration enormously complicate evaluations and discussions concerning this type of titrations.
By the way of example, suppose that you want to To this end, one may start by writing the mass balances for the metal and aminopolycarboxylic acid in the solution.
The mass balance of the metal contains a long list of species: Please note that many of the species bear a charge. For brevity, when appropriate, two dots (i.e., ··) will be used as placeholders respectively for the size and sign indicating the charge. For instance, APCA·· ≣ APCA n ⁻, The mass balances (equation (12) and (13)) and the condition (14) at the equivalence point can be written in a World Journal of Chemical Education 9 more compact and convenient form by introducing the notion of group of (chemical) species. A group of species is simply a collection of species existing in the titrated solution.
From equations (12), (13) and (14) we see that the most convenient choice is to define a first group of species, which we shall indicate with Me' (i.e., Me prime), collecting all the species containing the metal cation, Me z+ , but which do not contain the aminopolycarboxylic acid. That is, we define: Analogously, it is convenient to define a second group of species, indicated by APCA' (i.e., APCA prime), collecting all the species in the solution which contain the aminopolycarboxylic acid ligand, but do not contain the metal cation. That is: By definition, the concentration of the APCA' group is: Finally, we define a complex group of species, MeAPCA', containing the primary complex MeAPCA ·· plus the basic and acid complexes. We have: With definitions (15) (16) and (17), the mass balances of the metal cation and of the aminopolycarboxylic acid, and the condition at the equivalence point assume the very compact form: Thus, the analytical concentration of the metal is given, in each titration point, by the sum of the concentrations of the Me' and MeAPCA' groups; the analytical concentration of the aminopolycarboxylic acid is the sum of the concentrations of the APCA' and MeAPCA' groups of species; finally, the equivalence point of a complexometric titration is reached when the concentrations of the Me' group is equal to that of the APCA' group.
If we now meditate briefly on the above description of the complexometric titration using groups of species and on equations (12 1 ), (13 1 ) and (14 1 ), we realize that this is precisely the same language we would use and relations we would write if in the titrated solution only the single hypothetical reaction (18) took place: Reaction (18) is a rather bizarre reaction which exposes as reactants and products groups of species, rather than, as usual, real chemical species.
Nevertheless, we can attribute significance to this strange reaction by convening that it represents any reaction in which any of the species of the Me' group reacts with any of the species belonging to the APCA' group producing any of the species that make up the complex MeAPCA' group of species.
In other words, reaction (18) describes the overall degree of conversion of the metal cation to a complex which contains the aminopolycarboxylic ligand (regardless of the reactants from which this is accomplished and posing no restrictions on the formed complex, except that it must contain the target metal cation and the amino polycarboxylate ligand). Reaction (18) will be called a reaction between groups of species.
We can extend this vision to the reactions of the indicator by simply defining two additional groups of species In' and MeIn'.
The group of species In-is defined in analogy with group APCA-as:

In
In HIn H In = + + + and its concentration is given by equation (19): Group MeIn-is analogous to group MeAPCA' and includes, beside the pure complex MeIn ·· , the acid MeHIn ·· and basic Me(OH)In ·· complexes, if they are formed.
We have: These definitions of the groups In' and MeIn' allow us to define in a very simple way the transition point of the metallochromic indicator.
In fact, the indicator transition point coincides with the point of the titration when equal concentrations of In' and MeIn' groups are present in the titrated solution and it is then verified relation (21): If we meditate about equation (21), which defines the transition point of the indicator, we see that this is exactly the expression we would have written if in the titrated solution the only reaction of the indicator was: Reaction (22) is a second reaction between groups of species which, during a complexometric titration, takes place simultaneously with reaction (18).
Finally, we have now developed a formalism which allows a very simple but rigorous description of the chemistry behind a complexometric titration.
During a complexometric titration, only the two reactions between groups of species Me' + APCA' ⇌ MeAPCA' and Me' + In' ⇌ MeIn' take place, which exposes five groups of species.
The As a general guide, for the accuracy of analytical results derived from the titration, it is necessary that a point of the ≈ . This is as to say that the change in colour of the indicator must take place as close as possible to the equivalence point. An indicator which changes colour much before the to have low sensitivity for the given titration (and will produce an error in defect on the amount of metal cation determined from the titration). Analogously, it is understood that an indicator with too high sensitivity for the titration at hand changes colour much after the ) and it will produce an error in excess on the amount of metal cation determined from the titration. Although the sensitivity of the indicator is not the only factor which must be considered, it follows that, for accurate results, at least an indicator of appropriate sensitivity for the planned titration must be at hand.

Calculation of the Concentrations of Groups of Chemical Species
As we have shown in the previous paragraph, one may think that during the titration of a metal cation with an aminopolycarboxylic ligand are formed invariably only five groups of species (i.e., Me', APCA', MeAPCA', In' and MeIn') by two reactions between groups of species.
On the contrary, a variable and relatively large number of individual chemical species (which depends on the nature of the metal cation, on the nature of the aminopolycarboxylic ligand and on the nature of the pH buffer and indicator) are present in the titrated solution.
As it will become apparent in the following, it is a worth undertaking to extend this picture by evaluating the concentrations of the five groups of species.
In abstract, there are at least two procedures by which the concentrations of the five groups of species in the titrated solution can be evaluated.  Figure 3 represents the outcome of this procedure when applied to the solution presented in Figure 1.
Each bar in Figure 3 (except bars with labels H + , OH⁻, NH 3 , and NH 4 + ) collects several bars of Figure 1 and this is the reason because Figure 3 is much simpler than Figure  1. Nevertheless, Figure 3 gives a great deal of information on the fundamental chemical features of the represented solution.
For instance, we can easily deduce that the overall yield of the primary complex formation reaction is very high, simply because the concentration of the complex group ZnEDTA' is several orders of magnitude larger than the Zn' group concentration. Analogously, we see that the free form of the indicator is largely prevailing on the zincindicator complexes, so that the represented solution exposes the colour of the free indicator.
Furthermore, by confronting Figure 1 with Figure 3, one can get the intriguing idea that they represent, respectively, a high resolution and low resolution view of the chemistry in the considered solution.
The key point is that the low resolution view of the titrated solution is perfectly adequate for practical purposes and, most importantly, can be achieved without employing great means and it is much easier to present. Please remember that primed symbols represent groups of chemical species, while symbols bearing double dot (indicating, respectively, size and sign of the charge) represent individual chemical species. If we indicate these conversion factors, according to a common use [5,8], respectively by α Me , α APCA , α MeAPCA we have: Analogously we have: Obviously, at first sight this procedure does not appear convenient. In fact, from one side it requires that Nevertheless, although at present it may not appear clear as the five alfa conversion factors can be evaluated, assume that for a described titration all the five alfa coefficients have a definite value which is the same for all the titration points; in other words, assume that each of the alfa coefficients is constant for a given titration.
Then we can elaborate the equilibrium constants of reactions (5) and (9): in the following way. The equilibrium constant of reaction (5) is: From equations (23), (24) and (25) we have: By substituting equations (23 1 ), (24 1 ) and (25 1 ) in the expression of the equilibrium constant (6) and rearranging we have relation (28): By assumption, the alfa coefficients are constant for a given titration, so that the left side member of equation (28) is constant during a particular titration.
If we define the parameter β 1 ' (i.e., β 1 prime) as in equation (29): equation (28) can be written in the abridged form: When the expression of the β 1 ' parameter is confronted with the reaction between groups of species (18): we see immediately that expression (30) is precisely what we would have written by applying the law of mass action to the reaction between groups (18). Then the parameter β 1 ' defined by equation (29) may be understood simply as the equilibrium constant of the reaction between groups Me' + APCA' ⇌ MeAPCA'.
In order to avoid confusion, we shall call β 1 (which is an equilibrium constant in the usual sense) absolute equilibrium (or formation) constant, while, β 1 ' will be indicated (following a widespread usage [2,3,5,8]) as conditional equilibrium (or formation) constant.
Analogously, by elaborating the equilibrium constant of reaction (9) on the basis of equations (26)  MeIn ' +  (that is, the conditional formation constant of the metal-indicator complex).
The usefulness of the above elaborations depends on the fact that, if, for a given titration, constant values for the conditional constants exist and if we are able to evaluate them, then we can calculate the group concentrations precisely as if they were single species concentrations and treat the reactions between groups of species as if they were standard reactions between species.
In fact, if constants values of the conditional constants can be secured, we can write as many relations between the groups of species in the titrated solution as the number of groups of species, and solve the ensuing system of equations to obtain the concentrations of the groups.
The system of equations for calculating the groups concentrations [Me'], [APCA'] and [MeAPCA'] is made up by the two mass balance equations (12 1 ) and (13 1 ) and by the expression (30) of the conditional formation constant of the reaction between groups Me' +APCA' ⇌ MeAPCA'. We have: The mass balances (12 2 ) and (13 2 ) have been derived from the mass balances (12 1 ) and (13 1 ) by using relations (2)  The form of this system is always the same for all titrations, although, of course, the values of the various parameters and the symbols representing the metal and the aminopolycarboxylic acid ligand may change from titration to titration.
In §3.2 are presented directives on how a MS Excel sheet, for solving the above system of equation, can be developed.
Please note that we have not considered the problem of calculating [In'] and [MeIn'] since this is in practice irrelevant.
In fact, in order to evaluate the sensitivity of the indicator, it is only required the conditional constant for the reaction between groups Me' + In' ⇌ MeIn' (i.e., β 1 In ').
In the next paragraph we face in detail the problem of evaluating, for a described titration, the β 1 ' and β 1 In ' conditional formation constants.

Evaluation of the Conditional Formation
Constants, β 1 ' and β 1

In '
The evaluation of the conditional formation constant, 1 ' β , which governs the reaction between groups of species Me' + APCA' ⇌ MeAPCA' is based on the evaluation of the α Me , α APCA , α MeAPCA coefficients whose values are then substituted into equation (29): From this, it should be clear that the absolute formation constant, 1 β , must be known.
As a matter of fact, in order to evaluate the alfa coefficients, all the reactions which take place in the titrated solution (and the corresponding equilibrium constants) must be known.
For example, for the titration of the Zn 2+ cation with EDTA in an ammoniacal buffer a list of reactions and equilibrium constants as the one presented in Table 1 must be at hand.
In order to develop general relations which allow the evaluation of the alfa coefficients, it is useful to write the relevant side reactions in a general form (see Figure 2).
The ubiquitous hydroxo complexes of the metal cation are supposed to be formed according to the general reaction (32): For specific examples of reaction (32) when Me z+ = Zn 2+ , see Table 1.
In order to calculate the conditional constant for the formation of the metal-indicator complexes from relation (31), only the alfa coefficients α In and α MeIn must be evaluated (since α Me is already expressed from equation (45)).
α In is very similar to α APCA and depends on the protonation constants of the indicator according to relation (48) ( ) Although the above derivation may appear tedious, all it is necessary to understand it is that the five alfa conversion factors are expressed by relations (45), (46), (47), (48) and (50), which are collected in Table 2, where are also rewritten the relations which connect the absolute formation constants, β 1 and β 1 In , to the corresponding conditional constants, β 1 ' and β 1 In ', through the five alfa conversion factors.
As a matter of fact, Table 2 contains all the necessary relations for understanding and using the notions of groups of species and reactions between groups of species, which allow a simple but rigorous description of a complexometric titration.
Even a superficial inspection of Table 2 will show that the fundamental requirement for the alfa coefficients to be constant during a titration is that the pH of the titrated solution be constant throughout the titration. However, this does not represent an additional constraint for the execution of the complexometric titration since, as we have anticipated in the introduction, a constant pH of the titrated solution must be maintained in any case to control the behaviour of the metallochromic indicator.
A further condition is that, to keep constant α Me , the equilibrium concentration of the auxiliary ligand, [Q··], also must not appreciably change during the titration. This condition is generally not satisfied during a practical titration. In fact the addition of the titrant will dilute the components of the pH buffer. Although this, in general, has no effect on the pH of the titrated solution, [Q··] steadily decreases during the titration. This implies that α Me decreases and the conditional formation constants increase somewhat during a practical titration (see relation (45), (29) and (31) in Table 1). This effect largely depends on the conditions of a particular titrations and will be neglected in the following since experience shows that it has, in most practical cases, minor effects on predictions. In general, the initial value of [Q··] can be used for evaluations.
Another important point to be stressed is the fact that all the alfa coefficients are ≧ 1. α Me invariably increases by increasing the pH and the concentration of the auxiliary ligand because this favours the side reactions of the metal cation. This increase of α Me will produce a corresponding decrease of the conditional formation constants.
On the contrary, α APCA and α In will always increase by decreasing the pH and this effect will tend to decrease the value of the conditional constants as the pH of the titrated solution is decreased.
α MeAPCA and α MeIn will be in general very close to one because acid and basic complexes are only formed at extreme pH, while practical complexometric titrations are performed in the range ~ 4 < pH < ~ 12. This is necessary in order to avoid the unfavourable effects due to the drop of β 1 ' and of β 1 In ' that generally takes place at very low and very high pH.

Use of Group Concentrations and Reactions between Groups of Species
After the elaborations in the previous paragraph, we are now in a position to perform all the necessary evaluations concerning any complexometric titration by completely ignoring (after the alfa conversion factors and the values of the two conditional constants have been evaluated) the plethora of reactions and species that are present in the titrated solution.
In fact, in order to expose the analytically relevant aspects of the complexometric titration, we only need to assume that in the titrated solution are present the five groups of species which are governed by the two conditional equilibrium constants β 1 ' and β 1 In ' (whose values have been evaluated as described in the previous paragraph).

Complexometric Titration Curves
In the framework of the model of groups of species and reactions between groups of species, the titration curve for a complexometric titration is the plot representing log After <MyTitrationMatrix> has been compiled, it is a simple matter to draw a plot exposing curves log[Me']→V t (ml), log[APCA']→V t (ml) (and eventually log[MeAPCA']→V t (ml)).
Obviously, at first sight, this procedure appears rather tedious, because one needs to solve the system of equations (12 2 ), (13 2 ) and (30) for each of the selected values of V t (ml). However, in the next paragraph, it is shown how to develop an MS Excel file which will create <MyTitrationMatrix> in a matter of minutes.
Aside from that, in Figure 4 it is presented such a plot drawn assuming the following typical representative  It must be realized that, in the frame of the model of groups of species and reactions between groups of species, the specific nature of the metal cation and of the aminopolycarboxylic acid ligand (and indicator) are completely immaterial: the titration of 50 ml of a 0.01 M solution, of whatsoever metal cation, with a 0.02 M standard solution of any aminopolycarboxylic acid ligand, under conditions that happen to produce a value of the conditional constant β 1 ' = 10 8 , will be represented by Figure 4 (with obvious variations in the symbols representing the metal cation and the aminopolycarboxylic ligand).
From Figure 4 we can easily see that the equivalent volume is V eq = 25.00 ml, corresponding to the crossing point between curves representing APCA' and Me' (V eq can be read precisely from <MyTitrationMatrix> in correspondence of the line where the relation log[APCA'] = log[Me'] is verified). Moreover, at the equivalence point log[Me'] eq = log[APCA'] eq = -5.1.
In the plot, two additional horizontal lines, whose ordinates are -log(β 1 In ') and -log(β 1 ') respectively, have been drawn. Please note that, in Figure 4 The logarithmic form of equation (31)   In ') -1). In the case of Figure 4 this implies that the titration will be stopped when log[Me'] ≈ -4 -1 = -5.
Evidently, an indicator for which β 1 In ' = 10 4 is almost ideal for the titration of Figure 4.
As can be judged from the value of β 1 ' = 10 8 (and from the position of the line labelled, -logβ 1 '(APCA) in Figure  4) the indicator appropriate for a given complexometric titration must have, under the conditions of the titration, β 1 In ' << β 1 '. This is obvious, since the aminopolycarboxylic ligand must be able to attack and decompose the MeIn' complex through the reaction between groups: 1 In 1 ' MeIn ' APCA ' MeAPCA ' In ' ' K β β + + → =  and this will happen only if β 1 ' ≫ β 1 In '. From Figure 4 we see that, under typical conditions, a metallochromic indicator for which log(β 1 In ') ≈ 1/2 log(β 1 ') has an appropriate sensitivity (i.e., it changes colour close to the equivalence point). Please note that, since both β 1 In ' and β 1 ' will change, especially by changing the pH and the nature and concentration of the buffer in the titrated solution, a metallochromic indicator which is suitable for a given titration under certain conditions, can become completely unsatisfying if, for instance, the pH of the titrated solution is modified.
The accuracy of results derived from a complexometric titration also fundamentally depends on the value of the conditional constant, β 1 ', of the reaction between groups Me' + APCA' ⇌ MeAPCA'. This is because, hardly, if ever, it will be possible to select a metallochromic indicator which changes colour exactly at the equivalence point, i.e., when [Me'] = [Me'] eq . This implies that the volume, V arr (ml), which we read on the burette when the titration is ended (and which is used instead of V eq (ml) in all evaluations) does, as a matter of fact, differ from V eq (ml). A reasonable standard of accuracy is to pretend that |V eq -V arr | ≦ 0.1 ml, which is about the precision of volumes delivered by a burette.
If  Figure 4, we can stop the titration anywhere in the range -5.64 < log[Me'] < -4.54, without the difference |V eq -V arr | exceeding 0.1 ml. Then, we can meet our standard of accuracy using whatever metallochromic indicator which has, under the condition of the titration, ~ 3.54 < log(β 1 In ') < ~ 4.64.
From this we deduce that, around the equivalent point, there is an interval which has the following property: if the titration is stopped in this interval, then, the error on the volume measured from the titration does not exceed 0.1 ml. The fact is that the amplitude of this interval depends on the value of β 1 '. This interval is already somewhat narrow when, as in Figure 4, β 1 ' = 10 8 (about one log unit) and becomes increasingly narrower as β 1 ' decreases. As a consequence, if the β 1 ' value drops under the threshold of about 10 8 , it may be extremely difficult or impossible to find an appropriate indicator which allows the set standard of accuracy to be met. This is especially true in view of the fact that the number of metallochromic indicators available is limited.
For instance, Figure 5 presents a complexometric titration performed with the same parameters as titration in Figure 4, except that β 1 ' = 10 6 has been assumed.
This titration hardly will provide accurate results. By following the Me' curve in Figure 5, we see that at the equivalence point (at This may be impossible to realize in practice. An additional practical inconvenient of this titration is the slow decrease of [Me'] around the equivalence point which will cause a very gradual change in colour of the indicator and, by consequence, uncertainty and irreproducibility in locating the moment at which the titration has to be ended. In any case, from Figure 5 we can judge that an indicator having β 1 In ' = 10 4 is too sensitive for this titration (as can be seen in the blink of an eye from the position of the horizontal line representing -logβ 1 In '). On the contrary, Figure 6 presents titration curves for a titration performed with the same parameters as titrations in Figure 4 and Figure 5, except that β 1 ' = 10 10 has been assumed.
This titration will give very easily accurate and reproducible results.  horizontal lines of ordinate -log(β1 In ') = -4 and -log(β1') = -10 (whose significance is explained in the text) are also shown By following the Me' curve in Figure 6, we see that at the equivalence point (at V t = 25.0 ml) log[Me'] eq = -6.1. On the other side, from the plot we read log[Me'] -0.1 = -4.6 and log[Me'] +0.1 = -7.6. In order for this titration to meet our standard of accuracy, it is only necessary an indicator which changes colour in the wide interval ~ -4.6 < log[Me'] < ~ -7.6.
It will be relatively easy to find an indicator which has, under the conditions of the titration, a conditional constant higher than about 10 3.6 and lower than about 10 6.6 .
For instance, the indicator specified in Figure 6, which has β 1 In ' = 10 4 , has a convenient sensitivity for this titration (as can be judged from the position of line with label -logβ 1 '(In)).
Furthermore, because of the rapid variation of [Me'] around the equivalence point, the change in colour will be very abrupt, making the location of the moment at which the titration has to be ended very simple and reproducible.
It can be noted at this point that most of the metal cations complexes with aminopolycarboxylic acids have absolute formation constants which largely exceed 10 10 .
However, most of the complexometric titrations in practical use are performed with conditional formation constants which generally are under about 10 10 , because of the drop of the conditional formation constants due to the side reactions, mainly of the amino polycarboxylate anion in acidic solutions and of the metal cation with auxiliary ligands which are introduced for a variety of reasons.

Using MS Excel for Drawing Complexometric Titration Curves
The data necessary to draw complexometric titration curves (like the ones presented above) for specified values of the titration parameters (i.e., M t C , M 0 C , eq l V , β 1 ' and β 1 In '), can be generated in a matter of seconds after a MS Excel sheet, which we shall call <ArrayGenerator> has been developed.
Data produced by <ArrayGenerator> are then copied and pasted as values to a second empty MS Excel sheet which we shall call <MyTitrationMatrix>. <MyTitrationMatrix> sheet exposes a matrix of numbers with four column and 502 rows. Each row (except the first row used for labels indicating the contents of each column) is an array with four elements (corresponding, respectively, to values of V t (ml), In order to develop the <ArrayGenerator> sheet in a short time, consider Figure 7 which represents the first 12 rows of the corresponding MS Excel sheet.
Cells with a pale yellow background contain labels while cells with dashed background are left empty.
As a first step enter the following labels, <···>, in the indicated cells of <ArrayGenerator> sheet: <Burette→Vt(ml)> cell A1; In cell J2 enter the default value for the molar concentration of the standard aminopolycarboxylic acid solution (i.e., 0.02); in cell J3 enter the default value for the molar concentration of the metal cation in the titrated solution (i.e., 0.010); in cell J4 enter the default value for the volume in ml of the titrated solution (i.e., 50.00); finally in cell J5 enter the default value for the β 1 '(APCA) (i.e., 10 8 ).
Column A of the <ArrayGenerator> sheet is labelled <Burette→V t (ml)> and must be intended as representing the burette which delivers the titrant solution. As a real burette, it can deliver up to 50 ml of titrant and it is graduated each 0.1 ml. Then, using standard Excel tools, generate, in cells from A2 to A502, 501 V t (ml) values from zero up to 50 spaced 0.1 (see Figure 7).
In cells indicated in the following Table 3, enter the corresponding code (you can copy the code and paste it in the appropriate cell). After this, extend with the usual Excel tools, the code in cells B2, C2, D2, E2, F3 and G3 up to cells B502, C502, D502, E502, F502 and G502.
This will fill columns of the <ArrayGenerator> sheet with numbers whose significance is specified in the label identifying each column.
Finally, cells are formatted as shown in Figure 7. Naturally, the default titration parameters, in cells J2, J3, J4 and J5, are modified to simulate any planned titration; the contents of the <ArrayGenerator> sheet are automatically updated on changing the titration parameters. In cell J5 of <ArrayGenerator> enter the value of the conditional equilibrium constant for reaction between groups Zn' + EDTA' ⇌ ZnEDTA'. From the list of reactions in Table 1 and expressions for the alfa coefficients in Table 2 we evaluate β 1 ' = 3.45·10 8 (numerical details of this calculation can be seen in Table  4).
After this, select and copy columns A, E, F and G of <ArrayGenerator> and paste them as values in an empty <MyTitrationMatrix> sheet.
It is useful to update labels in row 1, by substituting Finally, select columns A, B and C of <MyTitrationMatrix> and insert a line graph, with the standard Excel procedure.
After formatting, the plot should appear like the one in Figure 8.
From Figure 8 one can see an equivalence point at 25.20 ml which is reached when the [Me'] group concentration is 10 -5.48 (actually, these values are read from <MyTitrationMatrix> sheet).
In order to keep the difference |V eq -V arr | ≦ 0.1 ml, an indicator is required which changes colour in the interval -4.8 < log[Me'] < -6.2 (these values can be read in <MyTitrationMatrix> sheet in correspondence of V t = 25.1 ml and 25.3 ml, respectively).
Is Eriochrome Black T (see §1) such an indicator?
In order to answer the above question we need to calculate the conditional constant for reaction between groups of species Zn' + EBT' ⇌ ZnEBT'. This is easily done on the basis of the list of reactions in Table 1 and relations in Table 2 (see Table 4 for details). We calculate β 1 '(EBT) = 5.9·10 3 .  From this value we deduce that equal concentrations of the red ZnEBT' complex and the blue EBT' indicator are present when, during the titration, log[Zn'] has dropped to -3.8 (= -log(β 1 '(EBT)). The pure blue colour of EBT' will appear somewhat further during the titration when log[Zn'] has fallen to about -4.8 (= -log(β 1 '(EBT)) -1).
From this, it is clear that the titration will be stopped somewhat before the equivalence point (i.e., EBT sensitivity is a little low for this titration) but in any case the difference |V eq -V arr | does not exceed ~ 0.1 ml (which is the set standard of accuracy).
The horizontal line in Figure 8, representing the transition point of the indicator (log[Zn'] = -log(β 1 '(EBT))) helps to understand the above considerations. Table 5 presents a list of reactions and of the corresponding equilibrium constants which will be the basis of discussion in this paragraph.

The Special Meaning of the Complexometric Titration of Mg 2+ Cation with EDTA
The complexometric titration of magnesium with EDTA in presence of Eriochrome Black T as indicator is obviously based on the reactions between groups: Supposing that the sample contains several mg/l of magnesium, a 100 ml volume is transferred to a titration flask and, according to a standard procedure [4,6], buffered by adding 10 ml of the pH ≈ 10 ammoniacal buffer (10 M NH 3 + 1.3 M NH 4 Cl). Then sufficient Eriochrome Back T indicator powder is added to obtain a clearly visible red colour. Finally, the solution is titrated with standard EDTA solution until a pure blue colour appears.
A representative case of this titration is presented in Figure 9.
As can be seen from Figure 9, under the described conditions, β 1 '(EDTA Mg ) = 8.3·10 7 and β 1 '(EBT Mg ) = 1.4·10 5 are calculated from relations in Table 2 and reactions in Table 5. The full blue colour of the free Eriochrome Black T indicator is predicted to develop somewhat after the equivalence point (i.e., the sensitivity of Eriochrome Black T for magnesium is somewhat higher than required) at log[Mg'] ≈ (log(β 1 '(EBT)) -1) = -6.14. However, the titration is stopped within the interval -6.2 < log[Mg'] < -4.74, which is necessary to keep the absolute accuracy within ± 0.1 ml.
Calcium 2+ ion is the most common interferent in the determination of magnesium. In fact, the presence of calcium during the determination of magnesium, under the conditions described above, introduces two additional reactions between groups. From data in Table 5 and relations in Table 2, we have:  Figure 10 represents the complexometric titration of magnesium under the same conditions of Figure 9, except that an equal concentration of calcium has been supposed to be present in the titrated solution.
As can be seen, curve representing log[Ca'] and line representing -log β 1 '(EBT Ca ) have been added. Simulation has been performed with MS Windows Forms application attached to [9] The red colour of the solution at the beginning of this titration is essentially due to the complexes of the group MgEBT'. This can be seen at the blink of an eye from Figure 10, since line representing -logβ 1 '(EBT Mg ) lies much under line representing -logβ 1 '(EBT Ca ) (which implies that MgEBT' is much more stable than CaEBT'). As a consequence, in order for the indicator to change colour, MgEBT' must be decomposed. As in Figure 9, this will happen when log[Mg'] ≈ (-log(β 1 '(EBT Mg )) -1) = -6.14.
The volume of 0.02 M EDTA standard solution consumed, that, in abstract, we read on the burette when log[Mg'] ≈ -6.14 is calculated to be V arr = 11.08 ml.
Evidently, this volume is, to a very good approximation, twice the volume we would have determined if only magnesium was present. This is because an equal volume of EDTA is consumed to produce the complex CaEDTA' from Ca', whose concentration, when the indicator changes colour, has dropped to very low value (≲ 10 -8.2 M as judged from Figure 10).
It is easy to see that the product M t arr · l C V (= 0.02·0.01108 = 2.21·10 -4 moles) does not correspond to the number of moles of magnesium but, within the accuracy of the titration, it represents the sum of the number of moles of calcium and magnesium.
Because of this, in general, the determination of magnesium, according to the above procedure, is best interpreted on the basis of relation (57) Only in the case one can assure that no calcium is present in the titrated solution, relation (57) will give the amount of magnesium in the titrated solution.
It must be understood that titrations presented in Figure  9 and Figure 10, although they may appear at first sight very different from each other, are practically indistinguishable, that is they appear to perform exactly in the same way. This is a demonstration of the fact that titrations with aminopolycarboxylic acids hardly, if ever, allow the determination of a single metal cation in presence of others, (that is, complexometric titrations with aminopolycarboxylic acids are not selective).
Although this lack of selectivity is in general considered as a limitation of the technique, it is, nonetheless, useful from a practical point of view.
In fact, for instance, total water hardness is obtained essentially by performing the above titration on a water sample.
From equation (57) one gets the idea that if magnesium is absent from the titrated solution, the above titration would allow the determination of calcium.
In abstract this is true, although the sensitivity of Eriochrome Black T for calcium is somewhat low.
However, in practice, calcium cannot be determined by the above titration procedure because of the slowness of the reaction CaEBT' + EDTA' ⇌ CaEDTA' + EBT' which is responsible for the change of colour of the titrated solution.
In fact, if magnesium is absent, the initial red colour of the solution is obviously due to the complex CaEBT' and (as a consequence of the slow decomposition of CaEBT' by EDTA') the solution will turn blue very slowly. Since visual titrations are performed by continually adding the titrant until the colour change is observed, it is easy to see that when the blue colour finally slowly appears in the solution and the titration is stopped, the volume that we read on the burette is, presumably, much larger than the equivalence volume. This is what it is meant when we say that Ca 2+ blocks Eriochrome Black T. Then, another indicator must be used for the complexometric titration of calcium.
However, there are a number of intriguing ways in which this obstacle can be overcome.
The most used chemical device for determining calcium using exactly the same conditions described above for the determination of magnesium, rests on the fact that the MgEDTA 2 ⁻ complex can be prepared in a pure solid form as a sodium salt (i.e., Na 2 Mg(EDTA)).
By adding to the calcium solution a controlled amount of the Na 2 Mg(EDTA) salt, the titration of calcium will change into the titration of magnesium without altering the volume of EDTA consumed in the titration. This is because the reaction MgEDTA' + Ca' ⇌ CaEDTA' + Mg' takes place with a very high yield under the described conditions, since the CaEDTA' complex (β 1 '(EDTA Ca ) = 1.0·10 10 ) is much more stable than the corresponding MgEDTA' complex (β 1 '(EDTA Mg ) = 8.3·10 7 ).
This type of titration is usually called a replacement titration.
As a matter of fact, for many metal cations, replacement reactions of the type MgEDTA' + Me' ⇌ MeEDTA' + Mg' proceed with an high yield, so that the complexometric titration of a variety of metals can be reduced to the determination of magnesium (i.e., Mg 2+ can be substituted and titrated in place of the target Me z+ cation).
Even more intriguing is the fact that the titration of magnesium with EDTA in presence of EBT can be inverted (that is, EDTA can be titrated with a standard solution of Mg 2+ , still using EBT as an indicator). This inverted titration is usually called a back-titration. Figure 11 gives an example of the back-titration of an EDTA solution with a standard 0.02 M solution of MgSO 4 . It is supposed that the back-titration is performed under the same conditions as the direct titration (i.e., the EDTA solution is buffered with about 1 ml of pH ≈ 10 ammoniacal buffer each 10 ml of titrated solution), so that the reactions between groups and the corresponding conditional constants are exactly the same as during the direct titration (as can be judged confronting Figure 11 with Figure 9). As a matter of fact Figure 11 is generated with <ArrayGenerator> simply by entering the concentration of the magnesium standard solution in place of that of the EDTA standard solution and vice versa. Then, data are transferred, as usually, to <MyTitrationMatrix> sheet and labels for EDTA' and Mg' are exchanged.
By observing Figure 11, it is clear that Eriochrome Black T, which is a suitable indicator for the direct titration, is not appropriate for the back-titration.
In fact, obviously, during the back-titration, the change of colour is reversed and the titration is stopped when the initial blue colour turns to red.
This will happen when log[Mg'] ≈ -log(β 1 '(EBT Mg )) +1 = -4.14 and then much after the equivalence point (producing an enormous error in excess on the amount or concentration of EDTA).
Then, once again, we should change Eriochrome Black T with an indicator of higher sensitivity.
However we can circumvent this problem in a very ingenious way, if a second burette containing the 0.02 M EDTA standard solution used for direct titrations is at hand.
In this case, once we have, during the back-titration surpassed the equivalence point and the solution has