# Utilization of α-Values

**2. Calculate the concentration of free calcium(II) in a solution initially prepared with 0.020 M calcium and 0.10 M total EDTA. The solution is buffered at a pH of 2.**

*What affect will pH have on the ligand?*

Students should realize that EDTA^{4-} is a base and that in the presence of an acid the protonated species will form. After groups have discussed this, state this to the class as a whole.

*How do we expect the formation of the calcium-EDTA complex to change based on this pH?*

Students should recognize that with increased amounts of protonated EDTA, according to LeChatlier’s principle, the amount of calcium-EDTA that can form is reduced. After groups have discussed this, state this conclusion to the class as a whole.

*Can we now solve for the amount of uncomplexed calcium ion?*

I use a set of equilibrium constant tables that has an additional table that contains a listing of the fraction of EDTA that exists as EDTA^{4- } (α-values) reported for each half a pH unit. This table is on the same page as the Ka values for EDTA and Kf values for metals with EDTA.

Students see this table and invariably take the α-value for pH 2 and multiply it the total concentration of EDTA to arrive at a concentration of EDTA^{4-}. They then tend to think that, since Kf is large, they can use the initial amount of calcium(II), calculated amount of EDTA^{4-}, and set up a table for a reaction that goes to completion. In the line of the table for the reaction at completion, they will list the complexed calcium as equivalent to the initial concentration of EDTA^{4-} they just calculated, and show that EDTA^{4-} goes to zero. At this time, it is important to point out to the entire class that the concentration of EDTA^{4-} cannot go to zero, since there are large quantities of the protonated forms in solution that must redistribute to produce some finate amount of EDTA^{4-}. With this being the case, the students can appreciate that the amount of complexed calcium must be larger than they calculated, but are stumped about how to actually do the calculation. Some are now tempted to think that the presence of a buffer with a pH of 2 is insignificant because the large *K _{f}* will shift the equilibrium towards the formation of the complex.

*At this point the concept of an α-value should be introduced. *

I spend about an hour introducing the concept of α-values. The first thing I point out is that having a table of α-values for EDTA^{4-} implies that the fraction of EDTA that exists in solution as EDTA^{4-} only depends on the pH and has no dependence on the total concentration of EDTA, and that we need to show that this is the case.

*I ask the groups to write an expression for the fraction of EDTA that exists in solution as EDTA ^{4-}.*

Most groups are able to write this expression without any difficulty. I write it on the board and then instruct them to take the reciprocal of this expression and examine how the equation now consists of a series of ratios of species. Write the reciprocal expression on the board.

*I now ask them if they can evaluate terms for each ratio that will only depend on the pH (or concentration of hydronium ion).*

Some groups realize that they will need to use the Ka expressions for EDTA to evaluate terms for the ratios. Others need to be prompted by asking them what else we know about the system or about EDTA. I also write the four dissociation reactions for EDTA on the board as a way of getting them to think about how we might evaluate these terms. Once they realize that the Ka expressions will need to be used, it can be pointed out that the ratio of HEDTA^{3-} to EDTA^{4-} can be evaluated using only Ka4. I ask them to use appropriate Ka expressions to evaluate that ratio as well as the ratio of H_{2}EDTA^{2-} to EDTA^{4-} and to then see if a pattern is emerging such that they can guess the forms of the final two ratios. They usually can guess the final two terms after seeing the pattern for the earlier two. We can then examine how the only variable in the expression is the concentration of hydronium ion. I then point out how we could do a similar process for any other of the protonated EDTA species in solution, arriving at a set of α-value expressions for each species observed in the dissociation of EDTA. I then draw the axes of a graph on the board that has the value of α on the y-axis and pH from 0-14 on the x-axis.

*I then ask the groups to draw a plot for the α-values for each species observed in a triprotic acid (H _{3}A, H_{2}A^{-}, HA^{2-}, A^{3-}).*

Remind them that the sum of the α-values for all four species must be 1 at every pH value. It also helps to have them think about the two extreme species (H3A and A3-) first and to think about what the plot of their α-values would look like as a function of pH. Most groups are able to reason out that there should be a lot of H3A and not much A3- at highly acidic pH and the reverse of this at highly basic pH. Once they see that, they can usually rationalize that the intermediate species must have an α-value that is low at highly acidic and highly basic pH with some maximum point at an intermediate pH. At this point, I show them examples of plots of α-values for different systems and we look at where the buffers are on the plots, how the plots vary with different species depending on the Ka values for the system, etc.

*Now we can ask how the α-value can be used to calculate the concentration of EDTA ^{4-} at a pH of 2?*

Groups can usually substitute the α-value expression into the Kf expression for the reaction. I then show them how the α-value, as a constant for the particular pH, can be brought up onto the side with Kf to create a conditional constant, and how the magnitude of the conditional constant is then used to determine whether or not the reaction goes to completion and what strategy we will use to actually calculate the equilibrium concentrations. We also write the reaction that is described by the conditional constant. We can now construct a table of initial and equilibrium values under the conditional reaction and calculate the amount of calcium complex that forms.

*Are there any other calcium complexes that can form with the ions present? Are they significant?*

The students will probably not think of hydroxide as a ligand, but remind them that since metals will exist in solution as cations, hydroxide can often form metal complexes. Have them look up the *K _{f}* for calcium hydroxide and assess whether or not this is significant. Remind them that the system is buffered at a pH of 2.

*Evaluate a second conditional constant that incorporates in the complexation of calcium by hydroxide ion.*

I indicate that this will probably be a process analogous to what we just did for the protonation of EDTA and ask whether they think it might be possible to calculate an α-value for the calcium ion. I also ask them what might be the variable in this expression and most realize it should likely be the concentration of competing ligand.

*Derive an expression for the α-value for the calcium ion as a function of hydroxide. *

*Incorporate the α-value into Kf and calculate a second conditional constant that accounts for both protonation of the EDTA and complexation of calcium by hydroxide. *

*What is the concentration of calcium EDTA complex in the solution?*

*What would a plot of complexation of calcium with EDTA look like as a function of pH?*

Students will often think that one of the extremes leads to maximized complexation, but challenge them to consider what would happen at an acidic pH and a basic pH. I show them a page in the accompanying text that has a compilation of the conditional constants for the calcium complex with EDTA to illustrate this trend.

*What if we had a metal that forms multiple hydroxide complexes such as cadmium? How do we factor hydroxide complexation?*

Students should realize that if they had an α-value for Cd^{2+}, they could find a conditional constant. Allow them time to derive the expression for αCd^{2+} and find its value. Suggest also that they start with the terms that include Cd(OH)+ and Cd(OH)2 and see if they can see a pattern that would allow them to guess the final two terms. They may need help getting started but, as with the α-values for ligands, allow them to simplify the individual terms. I then show them a table from the text that shows the conditional constants for the cadmium complex with EDTA as a function of pH to illustrate the trend in complexation with pH.

*What if there were more competing ligands?*

Discuss how a total α-value could be written with separate sets of terms for all competing ligands, emphasizing how we must know the concentration of uncomplexes ligand in the final solution to be able to calculate an α-value.