Chemical Education A CHIMIA Column Teaching Acid-Base Equilibria by Using Log-log Diagrams

: The purpose of this column consists in illustrating the utility of log-log diagrams for the graphical determination of the pH value of acid-base systems in dilute aqueous solution. This methodology allows to tackle systems going well beyond the case of mono- or diprotic acids, providing fast and accurate estimations of the pH. It is also argued that the use of so-called Sillén diagrams affords a better understanding of complex acid-base systems.

Acid-base equilibria, viewed in terms of the Brønsted-Lowry definition, [1] constitute a classical topic in general chemistry. They are crucial not only because of the quintessential importance of acids [2] but also because they provide an ideal playground for the discussion and understanding of chemical equilibrium in (mostly) dilute solutions. Consequently, virtually all textbooks on general chemistry will feature a specific chapter devoted to acid-base equilibria. However, the depth of the corresponding mathematical treatment is very often affected by approximations. This usually leads to a set of simple equations used for the calculation of the pH value for solutions of strong or weak acid and bases depending on their concentration and dissociation constants. Correspondingly, students will often focus on learning these equations by heart, thereby missing the chance of a more thorough understanding of the chemistry governing acid-base equilibria. To a certain degree, this is reasonable, given the relative complexity of a rigorous treatment of even the simple dissociation equilibrium of a monoprotic weak acid HA, such as e.g. acetic acid. In fact, in order to exactly compute [H + ] for a corresponding solution, one needs a cubic equation (eq. 1) in [H + ], whereas a diprotic acid H 2 A will lead to a fourth-order polynomial in [H + ] (eq. 2). [3] (1) For polyprotic acids the situation is even more complex and clearly out of reach for manual solutions and calling for computational ones. [4] On the other hand, graphical representations, e.g. in form of distribution diagrams may help in visualizing which species deriving from a specific dissociation of a polyprotic acid will be present in relevant concentrations with respect to the total concentration of the acid at a defined pH. Fig.  1 shows such a diagram for the triprotic acid H 3 PO 4 .
The curves in such a diagram represent the mole fraction α i of a certain species i as a function of pH and are derived from expressions such as e.g.: This kind of distribution (or speciation) diagram is clearly useful when it comes to judging, for example, which dissociation equilibrium will be relevant at which pH. So for the case of H 3 PO 4 , a relatively strong acid with pK a1 = 2.16, it is evident that the second and third dissociation will be relevant at pH values above ca. 5 and ca. 10, respectively, meaning that H 3 PO 4 in a not too dilute solution essentially behaves as a monoprotic acid and can be quantitatively treated as such.
By expressing the mole fractions α i in logarithmic form, one obtains, e.g. for the concentration of A 2-in the case of a diprotic acid H 2 A:

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Expressions of this kind are now the basis for plotting logC (log[species]) as a function of pH, thereby obtaining a log-log diagram of the type illustrated in Fig. 2 for the triprotic system of aspartic acid as a specific example. Such a diagram is often called a Sillén diagram honouring the name of the Swedish chemist Lars Gunnar Sillén who contributed in popularizing log-log diagrams. [6] It can be shown that the curves in a Sillén diagram may be approximated by linear segments of slope 0, +1, -1, +2, -2, etc., as recognizable in Fig. 2. Hence, a Sillén diagram may be constructed "by hand". Note that the greater deviations from the calculated curves occur at pH values close to the corresponding pK a values, where also slope changes occur, the maximum error being 0.3 on the logarithmic scale, i.e. a factor of 2 in concentration at pH= pK a . The diagram can now be used for the -mostly quite accurate -estimation of the pH of a solution, not only of a simple monoprotic acid, but also of more complex systems, including mixtures of different acids, given that the total concentrations of the involved species are known. In order to do this, one needs to formulate the so-called proton condition (in German: Protonenherkunftsgleichung, PHG). This accounts for all contributions to [H + ] (equilibrium concentration) deriving from deprotonations/protonations of all involved species including water. With the proton condition in hand, one can identify in the diagram the intersection of two dominant species (dominant intersection), i.e. by neglecting all other species occurring there in lower concentrations, thus allowing for an estimation of the pH value of the corresponding solution. In the case of a solution of e.g. aspartic acid (H 2 Asp), or sodium aspartate (NaHAsp), respectively, the proton conditions and the dominant intersections read as follows: The estimated pH values of 2.9-3.0 and 6.9-7.0, respectively, correspond quite well to the calculated ones of 3.02 and 6.95. [4] The slightly underestimated pH value in the former case is mainly due to the rather close proximity of the H + line to the dominant intersection, meaning that [H + ] cannot be totally neglected.
Let us now determine the pH of a solution prepared from both sodium aspartate and acetic acid, being their total concentrations [Asp] tot =0.03 M and [OAc] tot =0.001 M. The original Sillén diagram for the aspartic acid system must be completed by the estimated curves for acetic acid (blue dotted lines in Fig. 2). Next, the proton condition for this new system must be formulated and the dominant intersection must be identified: The exactly computed pH value for such a solution is 5.45, [4] again confirming the accuracy of the graphical approach, which usually lies within at most ±0.1 pH units.
The didactic value of the graphical methodology resides in the visualization of even rather complex acid-base systems at equilibrium. Theformulationoftheprotoncondition anditsuseforthedetermination of the dominant intersection, and hence the pH value, requires the ability to first locate an approximate pH range pertinent to the solution under scrutiny. This, in turn, must be based on a proficiency in comparing and judging the relative relevance of specific equilibria in the concert of a number of interdependent ones. Thus, the systematic use of Sillén diagrams greatly contributes in acquiring these skills.