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An Attractive Way of Developing the Concept of Systematic Titration Error of Visual Acid-Base Titrations (on the Basis of Logarithmic Acid-Base Diagrams)

Francesco Salvatore*, Maria Michela Salvatore

Dipartimento di Scienze Chimiche, Università degli Studi di Napoli “Federico II”, Via Cintia, Napoli, Italy

Abstract: For any titration, the titration error is, by definition, the difference between the volume of titrant added to reach the end point and the volume of titrant necessary to reach a stoichiometrically defined equivalent point. A graphical approach is presented, which allows a smooth and far reaching quantitative discussion of the systematic titration error of Acid-Base titrations, on the basis of the Logarithmic Acid-Base Diagram representing the titrated solution. Considerations and relations are developed which connect this diagram to the titration error. Examples are fully developed, which show that the procedure suggested unavoidably goes beyond the technical or practical topic of evaluating the systematic titration error, and it can be especially rewarding from an educational point of view. Finally, for the reader’s convenience, algebraic expressions and brief instructions to draw Logarithmic Acid-Base Diagrams (also known as Sillén’s diagrams), by using a spreadsheet, are provided.

Keywords: acid base titration, systematic error, acid base diagrams, amino acid, carbonate, bicarbonate

Received March 05, 2014; Revised March 17, 2014; Accepted March 22, 2014

Citation: Francesco Salvatore, Maria Michela Salvatore, An Attractive Way of Developing the Concept of Systematic Titration Error of Visual Acid-Base Titrations (on the Basis of Logarithmic Acid-Base Diagrams). World Journal of Chemical Education, 2 (1). 8-20. January 2014. doi:10.12691/wjce-2-1-3

1. Introduction

Logarithmic Acid-Base Diagrams (in the following called, for brevity, LABD) were introduced in a pre computer era, [1], as an alternative to algebraic calculations, for the evaluation of pH and equilibrium concentrations of all species present in a specified (aqueous) solution of acids and/or bases.

In the computer era, there is no need for plots to calculate the equilibrium concentrations of species present in whatever solution (if the pertinent data are available).

However, graphical equilibrium calculations based on LABDs are still popular and very useful because of their recognized capabilities of transferring chemical knowledge and enhancing understanding of concepts related to Acid-Base reactions. Books are available which deal with this topic, [2,3,4]. Please note that, in reference [4], Logarithmic Acid-Base Diagrams are called Sillén’s diagrams.

Then, we shall go beyond the subject of using LABDs for graphical pH calculations, in order to focus on the topic of systematic errors which can bias analytical results derived from Acid-Base titrations (especially visual Acid-Base titrations).

We shall discuss how LABD of the titrated solution can assist in the process of introducing, detecting, evaluating and, eventually, avoiding the systematic titration error for a specified Acid-Base titration, because we believe that this argument, if presented in an attractive way, is eminently suitable to promote a deep understanding of a broad range of concepts related to Acid-Base reactions and titrations.

In dealing with chemical equilibrium is fundamental to distinguish analytical concentrations of substances dissolved to prepare any given solution from equilibrium concentrations of species present in the solution.

Although there is an agreed and convenient symbol to indicate molar equilibrium concentrations (i.e., [X], where X symbolizes any species in the considered solution), there is no established symbol to indicate analytical molar concentrations. Therefore, we shall use the custom symbol [[Y]], to indicate the molar analytical concentration of any substance Y which has been dissolved to prepare the considered solution. By way of example, we can prepare a solution in which [[HCl]] = 0.1 M; however, because HCl is fully dissociated, in this solution [HCl] ≈ 0 and [Cl–] = 0.1 M.

Furthermore, for the dissociation constants of weak acids, which are necessary to achieve our aims, we shall consistently use throughout the values at 25°C and zero ionic strength. These can, eventually, be substituted with practical or custom dissociation constants. The same applies to transition intervals of Acid-Base indicators.

2. Acid-Base Titrations, Logarithmic Acid-Base Diagrams and Systematic Titration Error

2.1. Acid-Base Titrations and Logarithmic Acid-Base Diagrams

Whatsoever solution of acids and/or bases can be represented by a Logarithmic Acid-Base Diagram (LABD), which is constructed by drawing in the plane log[j]→pH a curve for each species, j, present in the considered solution at the equilibrium concentration [j].

Instructions and algebraic expressions for developing a MS Excel file designed for drawing any required LABD are postponed, for convenience, to §4. In addition, a Windows Forms application for drawing appealing LABDs is supplied on a CD attached to reference [2].

For the present, it is sufficient to assume that the LABD representing any given solution is readily available.

LABD of pure water is the simplest one, since it is constituted by only two curves (actually two lines) representing log[H⁺] and log[OH–], respectively (see Figure 1). In Figure 1, line identified with label H has slope -1 and represents log[H⁺] (= -pH); line with label OH has slope +1 and represents log[OH–] (= - pKw + pH). pKw is the ion product of water. The two lines intersect at pH = 7.0, which is the pH of pure water at 25°C.

Figure 1. Logarithmic Acid-Base diagram of pure water at 25 °C. Line identified with label H represents log[H⁺] and line with label OH represents log[OH–] as functions of pH

As we add to water solutes having Acid-Base properties, additional curves must be added in order to represent the new species which are formed in the solution.

For instance, if we add a monoprotic acid, HA, of specified pKa and analytical concentration [[HA]], two new curves will appear in the LABD representing the solution. These, curves represent log[HA] and log[A–], respectively, and the LABD will expose four curves identified by labels H, OH, HA and A–.

For instance, Figure 2 is a LABD which represents a solution 0.100 M of acetic acid. This means that, for Figure 2, [[CH₃COOH]] = 0.100 M. Curve with label CH₃COOH represents log[CH₃COOH] and curve with label CH₃COO– represents log[CH₃COO–]. The CH₃COOH curve meets CH₃COO– curve at pH = pKa = 4.75 (pKa is the dissociation constant of acetic acid). The horizontal asymptote to the curves of CH₃COOH (or of CH₃COO–) intersects the log[j] axis at log[[CH₃COOH]]. This diagram embodies the acetic acid dissociation equilibrium and mass balance, but not the charge balance. This implies that Figure 2. also represents a 0.100 M solution of acetate (i.e., [[CH₃COO–]] = 0.100 M) or any other solution in which [CH₃COOH] + [CH₃COO–] = 0.100 M.

Figure 2. Logarithmic Acid-Base diagram of 0.100 M acetic acid, at 25°C. Curve with label CH₃COOH represents log[CH₃COOH] and curve with label CH₃COO– represents log[CH₃COO–] as a function of pH

In general, a LABD exposes as many curves as the number of species in the solution which it represents, becoming, by consequence, more and more crowded as the complexity of the considered solution increases.

In the following, we shall assume that the reader has a general knowledge of the significance of Logarithmic Acid-Base Diagrams as presented in references [1,2,3,4].

On the other side, the solution represented in a LABD can be titrated with the standard solution of a strong base (e.g., NaOH → alkalimetric titrations) or with the standard solution of a strong acid (e.g., HCl → acidimetric titrations).

As a matter of fact, a titration amounts, in a figurative way, to traversing the LABD of the titrated solution from its initial pH, pHin, to a final pH, pHend.

In practice, the evolution of the pH during an Acid-Base titration is caused by the stepwise addition of increasing volumes, , of the titrant solution to an initial volume, , of the titrated solution.

The target of an analytical Acid-Base titration is the determination of the equivalent volume, , which, by definition, is the volume of titrant solution which contains a number of moles exactly equal (or stoichiometrically equivalent) to the number of moles of the titrated substance in the titrated solution. In general, is used to derive the analytical concentration or number of moles of the titrated substance by means of simple stoichiometric calculations presented in any standard general or analytical chemistry textbook, [5].

By definition, analytical results derived by using the equivalent volume are absolutely accurate and free from systematic error.

In addition, by definition, when a volume of titrant exactly equal to has been added, the titrated solution’s pH assumes a value which is the equivalent point pH, pHeq.

However, during Visual Acid-Base Titrations (VABT), the colour change of an Acid-Base indicator is used as a signal to stop the titration. When a VABT is discontinued, the pH of the solution, indicated with pHend, coincides with either pH value delimiting the transition interval of the indicator.

To be more specific, for an alkalimetric titration, pHend coincides with the upper limit of the transition interval of the indicator. For an acidimetric titration, pHend coincides with the lower limit of the transition interval.

Standard transition intervals for common Acid-Base indicators are readily available from analytical chemistry textbooks, [5], or from the Web.

The volume of titrant solution consumed to reach pHend is the end point volume, , which can be read from the burette when the Acid-Base titration is ended.

It must be understood that, once we have specified the indicator, pHend is automatically set and that, if not by chance, pHend is different from pHeq.

Of course, we can change pHend by selecting a different indicator, but we have only a limited or no control on pHeq, which is fixed by the analytical composition of the titrated solution.

2.2. Systematic Titration Error

Suppose that 25.00 ml of the 0.100 M solution of acetic acid represented in Figure 2 are titrated with a standard solution 0.1000 M of NaOH (i.e., [[NaOH]] = 0.1000 M), in presence of phenolphthalein as an indicator (transition interval {8.2, 9.5}).

This titration starts at pHin ≈ 2.9, which is the pH of a solution [[CH₃COOH]] = 0.1 M. This pH is easily read from Figure 2, because it corresponds to the pH at the intersection point between curves representing log[H⁺] and log[CH₃COO–]. By the way of example, this is a simple demonstration of how a LABD can help in pH calculations. From this point, we move towards higher pHs and we meet pHeq ≈ 8.9, which is the pH corresponding to the intersection point between the curve representing log[CH₃COOH] and the curve representing log[OH–]. This is because, by neglecting the small change in volume of the titrated solution, pHeq is, for our purposes, not significantly different from the pH of a 0.1 M solution of sodium acetate. By the way, changes in volume of the titrated solutions will be neglected throughout.

We should then stop the titration when pHeq is reached and read from the burette.

However, during the VABD with phenolphthalein, we do not stop at pHeq, but continue to pHend ≈ 9.5, which is the upper limit of the phenolphthalein transition interval. After that, what we read from the burette is , which of course is, in abstract, larger than .

For any specified VABT, we define the variables and according to equations (1) and (2):

[picture] | (1)

[picture] | (2)

In general, for whatever VABT, both, and , will be different from zero.

is easily evaluated for a planned titration. In fact, is set by the chosen indicator and, on the other side, the evaluation of is a standard pH calculation which is most readily performed graphically using the LABD of the titrated solution (as we have done above) or algebraically (as described in standard textbooks, [4,5]).

Although the fact that is the primordial origin of the systematic titration error, the accuracy of a planned titration cannot be assessed simply on the basis of the value of .

Obviously, a lower value is not worse than a higher value, but not always it is better or necessary.

On the contrary, the most direct and physically significant measure of the titration accuracy is .

In fact, in the following, we shall identify with the systematic titration error and our target will be to show how it can be evaluated.

Obviously, the primordial systematic error, , will propagate to the analytical results derived from the titration data.

Once has been evaluated, it will be very easy to assess its effects on analytical results.

To this end, we note that to perform a titration we use a burette to deliver the titrant. Whatever burette has a limited precision which we shall indicate with.

This implies that is affected by an uncertainty (casual error) which is at least . For instance, a typical Class A burette might have , so that whatever volume read from this burette carries an uncertainty which is not less than ±0.05 ml.

The casual error on will then propagates as a casual error on the analytical results.

The key point is that we need to worry about the systematic error only in the case . In fact, in the opposite case, i.e., if , the effects of the systematic error will be embedded in the casual error of the burette and will not produce any practical effect on analytical results.

An obvious corollary of this is that it is of no use to make efforts to reduce to the absolute zero; in fact, it is only necessary, in practice, to reduce, if it is possible, to a value lower than .

2.2.1. Evaluation of the Systematic Titration Error Expressed as

Figure 3, which is the simulated titration curve of the above described titration of 25.00 ml of the acetic acid solution represented in Figure 2 (performed with a 0.1000 M standard solution of NaOH), shows the characteristic shape of a simple titration curve.

Figure 3. Alkalimetric titration curve of 25.00 ml of a solution 0.100 M acetic acid with a standard 0.1000 M solution of NaOH

In general, a titration curve has a slope which depends on the pH, and which we shall indicate with . If the titration curve has been drawn in the plane, carries the dimensions of ml⁻¹.

For any specified titration, there is a relation between and , which involves the slope of the titration curve and which, for the present purposes, can be formulated, in an approximate but very simple and useful way, as in equation (3):

[picture] | (3)

In practice, , which is, by definition, the slope of the titration curve at the titration end point, can be regarded as a conversion factor, which converts the readily evaluated in the corresponding (which is a measure of the systematic error).

Obviously, for the use of equation (3), we need to evaluate and, for this purpose, we shall use the LABD of the titrated solution.

To this end, it is necessary to select in the LABD representing the titrated solution the curve representing a particular species, which we shall call the Slope Determining Species (SDS).

At each selected pH, the curve representing the SDS, in a LABD, is the one that lays highest in the diagram and which at the selected pH has a non zero slope.

For instance, with reference to Figure 2, the SDS at pH = 1, 4, 8 and 10 coincides, in the order, with H⁺, CH₃COO–, CH₃COOH and OH–. It is evident that the SDS will always be H⁺, at sufficient low pH, and OH– at sufficient high pH.

Actually, what we need for the use of equation (3) it is to pinpoint the SDS at the end point of the titration, that is to say when . After that, is readily read from the LABD of the titrated solution.

Finally, can be evaluated from relation (4):

[picture] | (4)

In equation (4), [[TitrantConc]] stands for the known titrant analytical concentration (e.g., [[NaOH]] for alkalimetric and [[HCl]] for acidimetric titrations) and , is the initial volume of the titrated solution (usually between 10 and 50 ml).

Equation (4) holds both for alkalimetric and acidimetric titrations. However, since acidimetric titration curves have negative slope, must be, in such a case, interpreted as the absolute value of the slope. In other words, for an acidimetric titration, the sign of , calculated from equation (4), must be inverted.

For instance, for the above considered titration of 25.00 ml of [[CH₃COOH]] = 0.100 M with [[NaOH]] = 0.100 M, performed using phenolphthalein as an indicator, we need to pinpoint, from Figure 2, the SDS at pHend ≈ 9.5.

We see that and, by consequence:

Using this value and other titration data in equation (4), we calculate :

And finally:

Since, from equation (1), we have:

the systematic titration error of this titration is, from equation (3), evaluated to be:

The interpretation of this result is straightforward. If the titration is performed with a burette which can measure volumes within ± 0.05 ml, then the systematic titration error is negligible. The uncertainty on the concentration or number of moles of acetic acid calculated from the titration data is governed by the precision of the burette and is calculated from relation (5):

[picture] | (5)

For the current case relation (5) yields:

In contrast with the above result, consider the titration of 25.00 ml of 0.100 M hypochlorous acid (i.e., [[HClO]] = 0.100 M) with [[NaOH]] = 0.100 M in presence of phenolphthalein as an indicator (pHend ≈ 9.5, as it was before).

The LABD of the titrated solution is presented in Figure 4, from which we can see that pHeq ≈ 10.3 (this is the pH corresponding to the intersection point of the curves representing log[HClO] and log[OH–]).

Figure 4. Logarithmic Acid-Base diagram of 0.100 M hypochlorous acid (pKa = 7.53) at 25°C.

Therefore, , which means that the titration is ended before the equivalent point. In principle, we add less NaOH than required and this is an omen for a systematic error in defect on the concentration or number of moles of HClO determined from the titration.

However, as we have demonstrated in the previous example, this error will be practically important only if its absolute value exceeds .

For this reason we repeat the above procedure to evaluate the systematic error, .

From Figure 4, we can smoothly see that the SDS at pH ≈ 9.5 is HClO and that log[HClO] = -3.0. Then, for we have:

and finally:

It is obvious that this titration is affected by a systematic error which largely exceeds the uncertainty introduced by the precision of the burette.

In this case, the uncertainty introduced by the limited precision of the burette is irrelevant and the % systematic error on the concentration or number of moles of hypochlorous acid is calculated from relation (6):

[picture] | (6)

Equation (6), for the current case, yields:

In practice, if we titrate, in the way we have described above, 25.00 ml of a solution in which [[HClO]] = 0.100 M, our experimental result will be: [[HClO]] ≈ 0.098 M.

2.2.2. The Exceptional Case of StrongAcid ↔ StrongBase Titrations

In the following we shall consider NaOH as a prototype for strong bases and HCl as a prototype for strong acids. This is made only for convenience, since whichever strong base can be substituted for NaOH and, analogously, any strong acid for HCl.

Then, under the label StrongAcid ↔ StrongBase titrations are included alkalimetric titrations of HCl solutions with standard solutions of NaOH, and, vice versa, acidimetric titrations of NaOH solutions with a standard solution of HCl.

Within the frame developed above for the systematic titration error, the LABD of the titrated solution serves both qualitative and quantitative purposes.

In a qualitative way, by observing the LABD at pHend, the nature of the SDS can be established.

Then, the value of , to be used in equation (4), can promptly be read from the LABD.

Furthermore, the LABD of the titrated solution is used for the graphical evaluation of pHeq (although this is outside the subject of this paper) which is needed to establish .

In general, pHeq depends on the composition (i.e., the concentration of acid and/or base) in the titrated solution and so does . In addition, the LABD of the titrated solution and, by consequence, depend on the composition of the titrated solution. The outcome is that, in general, the systematic titration error cannot be evaluated a priori, without an assumption on the analytical composition of the titrated solution.

The sense of the discussion on the systematic titration error is to show that the titration of a given acid or base can be performed with a negligible systematic error at least for a number of assumed values of its analytical concentration, so that one can be confident about the titration error when an actual sample, of unknown concentration, is titrated. Nevertheless, titrations of the same substance at different concentrations will have different systematic errors, even if all other titration data are kept unchanged.

Now, please consider the LABD in Figure 5, which represents a 0.05 M solution of HCl. The horizontal line at log[[HCl]] = log0.05 = -1.3 represents log[Cl–]. The simplicity of the LABD of HCl depends on the fact that, being HCl a strong acid, [HCl] never assumes significant values (at least when we consider diluted solutions of the acid).

Analogously, the LABD of a NaOH solution exposes only a horizontal line, located at log[[NaOH]], which represents log[Na⁺], since, [NaOH] is insignificant in any diluted solution of sodium hydroxide.

From this, we see that the plot of Figure 5 also represents a solution in which [[NaOH]] = 0.05 M, if the horizontal line is interpreted as representing log[Na⁺].

Figure 5. Logarithmic Acid-Base diagram of 0.05 M HCl (or 0.05 M NaOH) at 25°C

Many of the very special features of StrongAcid ↔ StrongBase titrations depend on the simplicity of the plot of the titrated solution.

As it is apparent from Figure 5, the SDS during the StrongAcid ↔ StrongBase titrations is either H⁺ or OH–. This is for the simple reason that there are no curves (or lines) with non zero slope in the LABD of the titrated solution, except lines representing log[H⁺] and log[OH–].

Then, SDS ≡ OH―, and , if the StrongAcid ↔ StrongBase titration is ended a pH > 7.

By inserting this constraint in equation (4) we have:

[picture] | (7)

On the other side, when the StrongAcid ↔ StrongBase titration is ended at pH < 7, then SDS ≡ H⁺. In such a case and equation (4) becomes:

[picture] | (8)

From equations (7) and (8) we see that the slope of the StrongAcid ↔ StrongBase titration curve at is always the same, regardless of the analytical concentrations of the titrated StrongAcid or StrongBase, (e.g., [[HCl]] or [[NaOH]] in the titrated solution).

In practice, equation (7) applies to all titrations which are ended at alkaline pH, whenever the SDS coincides with OH–. This happens to be the case for a number of alkalimetric titrations of weak acids, e.g., for the alkalimetric acetic acid titration presented in the previous paragraph (but not for the alkalimetric titration of hypochlorous acid).

Analogously, equation (8) applies to all titrations which are ended in the acidic range, whenever the SDS coincides with H⁺. This often happens for acidimetric titrations of weak bases.

Nevertheless, StrongAcid ↔ StrongBase titrations have a unique feature which, strictly, is never found in titrations of weak acids or bases. We allude to the fact that StrongAcid ↔ StrongBase titrations have an invariant pHeq ≈ 7, independent on the concentration of the titrated StrongAcid or StrongBase. Within our framework, the consequence of this is that equation (1) becomes:

When all this (i.e., equations (3), (7), (8) and (9)) is considered, we deduce that any StrongAcid ↔ StrongBase titration has a constant systematic error, , which is independent on the concentration of the titrated StrongAcid or StrongBase. only depends on variables that are under the full control of the operator, i.e., the chosen indicator, the titrant concentration and the initial volume of titrated solution. The implication of this is that, unlikely all other cases, the systematic titration error of StrongAcid ↔ StrongBase titrations can be evaluated a priori, without knowledge of the amount of StrongAcid or StrongBase in the titrated solutions.

Figure 6. Plot of the systematic titration error for StrongAcid ↔ StrongBase titrations calculated for = 50 ml and = 0.1000 M. symbolizes [[NaOH]] in alkalimetric titrations and [[HCl]] in acidimetric titrations

Plots of as a function of for alkalimetric and acidimetric StrongAcid ↔ StrongBase titrations are presented in Figure 6 (which has been calculated from the above relations assuming = 0.1000 M and = 50 ml, which are typical values).

If we assume for the burette a precision of ± 0.05 ml, which is a very common case, from Figure 6 we see that StrongAcid ↔ StrongBase titrations can be ended (under the assumed conditions) anywhere in the range without the systematic error exceeding the precision of the burette.

This implies that almost all the common indicators (except a few) are suitable for the StrongAcid ↔ StrongBase titration.

For instance, whichever indicator with an upper limit of its transition interval falling between pH ≈ 4.7 ad pH ≈ 9.3 (e.g., bromocresol green, methyl red, bromothymol blue, phenol red, phenolphthalein, etc..) is a suitable indicator for the alkalimetric titration of a StrongAcid.

Phenolphthalein, which strictly corresponds to , is on the border, since, as can be easily evaluated, yields ≈ + 0.09 ml. This is a small systematic error which can be tolerated for many purposes and which, in practice, can altogether be avoided by stopping the titration in correspondence of the legendary phenolphthalein’s pale red.

Although, strictly, the value of depends on the particular indicator chosen, from the practical viewpoint, all the mentioned indicators will produce the same results, which are dominated by the uncertainty introduced by the burette limited precision.

3. Attractive Examples

3.1. Acid-Base Titrations of Amino Acids

In this paragraph, we consider the problem of evaluating, within the framework developed above for the systematic titration error, the utility of Acid-Base titrations as a method for determining accurately the amount of an amino acid in a given solution.

Although this may appear a matter of mere academic interest, it offers, on the one side, a beautiful chance to expose a number of general questions which can arise (or may intentionally be raised) while coping with Acid-Base reactions and titrations. On the other side, it is a way to penetrate deeply in the Acid-Base chemistry of amino acids, which is a worthy undertaking because amino acids are ubiquitous in biology and biochemistry.

As it is well known, natural amino acids are described by the general formula H₂NCHRCOOH, in which R represents the amino acid side chain bonded to the alpha carbon atom, which also bears the basic amino group (―NH₂) and the acid carboxylic group (―COOH).

Simple amino acids which do not bear in the side chain additional functional groups with Acid-Base properties will be called GlycineType amino acids (this group includes several natural amino acids, e.g., Alanine, Valine, Leucine, etc.). Natural amino acids bearing an additional acidic function in the side chain are classified below as AsparticAcidType amino acids (for instance, Glutamic acid). Finally, natural amino acids with a basic group in the side chain are classified as LysineType amino acids (for instance, Ornithine).

3.1.1. GlycineType Amino Acids

Two dissociation constants (respectively, pKa1 and pKa2) are associated with GlycineType amino acids. pKa1 and pKa2 connect three different species according to the scheme in Figure 7.

Figure 7. Schematic presentation of the Acid-Base properties of GlycineType amino acids

For instance, for Alanine, pKa1 = 2.31 and pKa2 = 9.7. pKa1 and pKa2 are much the same for all natural GlycineType amino acids.

From the above scheme, it must result that, in water, the neutral species, i.e., H₂NCHRCOOH ≡ ⁺H₂NCHRCOO―, is amphiprotic and can react both with acids or bases.

In the context of Acid-Base titrations, this must be interpreted to signify that, in abstract, an amino acid can be titrated either alkalimetrically or acidimetrically. The question is: of the two possible titration modes, which one would provide more accurate results?

Of course, the answer is incorporated in the pKa1 and pKa2 values characterizing the Acid-Base behaviour of the amino acid, but it requires an extra bit of thinking.

We take Alanine as representative of GlycineType natural amino acids and consider a solution containing 0.05 M Alanine (i.e., [[Alanine]] = 0.05 M).

LABD of this solution is exposed in Figure 8. The neutral amino acid is denoted HAla, to show that Alanine has an acidic proton which can be dissociated. The dissociation of the proton from Alanine produces the conjugated base, Ala–. H₂Ala⁺ is the conjugated acid of HAla formed upon protonation of the neutral species. In Figure 8, the curves representing H₂Ala⁺ and HAla intersect at pH = pKa1 = 2.31, while curves of HAla and Ala– intersect at pH = pKa2 = 9.7.

To solve the above dilemma, we observe that the initial pH of the Alanine solution is about 6, which graphically corresponds to the point where the curve of H₂Ala⁺ meets the curve of Ala–.

An acidimetric titration amounts to traversing the Alanine LABD, from pHin ≈ 6, towards lower pH. In this journey we meet pHeq1 ≈ 1.9 (which graphically corresponds to the crossing point between curves representing H⁺ and HAla).

Symmetrically, an alkalimetric titration starts at pHin ≈ 6 and reaches the equivalent point at pHeq2 ≈ 11.2 (which graphically corresponds to the crossing point between curves representing HAla and OH–).

Even a superficial consideration of these statements, within the frame of Figure 8, will show that an alkalimetric titration is, of the two, the more convenient.

In fact, from the LABD in Figure 8, we see at a glance that, if our target is the acidimetric equivalent point, the SDS must be selected between H⁺, HAla and H₂Ala⁺. On the other side, if our target is the alkalimetric equivalent point, the SDS will be either OH– or HAla. Since any possible SDS around the acidimetric equivalent point has an higher concentration than that of any possible SDS around the alkalimetric equivalent point, will be higher, and the systematic error lower, if we select an end point around pHeq2 ≈ 11.2 rather than around pHeq1 ≈ 1.9.

Furthermore, we can easily predict that even the alkalimetric titration (although certainly more favourable than the acidimetric) will not produce very accurate results because of the rather large concentrations of HAla and OH– (which are the potential SDS around pHeq2). This prognosis can easily be verified.

Figure 8. Logarithmic Acid-Base diagram of 0.05 M Alanine, at 25 °C. Label HAla stands for the neutral amino acid

To this end, the most appropriate indicator (among common Acid-Base indicators), is thymolphthalein (transition interval {9.4, 10.6}).

This choice amounts to fixing pHend ≈ 10.6 and 10.6 - 11.2 = - 0.6 (which is an omen for a systematic error in defect).

In any case, by taking for the NaOH titrating solution [[NaOH]] = 0.1000 M and = 50 ml for the initial volume of titrated solution, calculations are as it follows:

Since in the titration are consumed about 25 ml of NaOH, this amounts to a % systematic error, on the concentration or number of moles of Alanine, of about -13%.

The limitations of visual Acid-Base titrations with respect to the determination of GlycineType amino acids (e.g., Glycine, Valine, Leucine, etc.) are quite evident.

However, the above described Alanine titration is eminently suitable for a laboratory experiment aimed to demonstrate impressively the effects of systematic error.

Suppose that we weight accurately, at the analytical balance, an amount of pure Alanine (which is easily available) close to ≈ 0.0025 moles (for the evaluations below we assume a weight of 0.2227 g, which correspond exactly to 0.0025 moles of Alanine). After that, we transfer the weighted sample to a titration flask with about 50 ml distilled water. After adding the thymolphthalein indicator, the Alanine solution is titrated, with standard 0.1000 M NaOH, until the full blue colour of the indicator has been developed.

The stoichiometric equivalent volume corresponding to 0.0025 moles of Alanine is calculated to be 25.00 ml. However, on the basis of the above evaluations, we shall measure a smaller end point volume: ≈ 22 ml. By using 22 ml to calculate the number of moles of Alanine we obtain ≈ 10⁻³×22×[[NaOH]] = 0.0022 moles (instead of 0.0025). Finally, the molecular weight of Alanine, MWAlanine, is calculated by dividing its weight (0.2227 g) by ≈ 0.0022 moles. The result will be: MWAlanine ≈ 0.2227/0.0022 = 101 u.m.a.. When this value is compared to the real one (i.e., MWAlanine = 89.09 u.m.a.), one is memorably impressed that the difference between the two amounts about to adding a carbon atom to the Alanine formula.

3.1.2. AsparticAcidType Amino Acids

However, as it happens, thinks may change if the amino acid side chain contains an additional acidic functional group, which implies that the neutral molecule bears two dissociable protons according to the scheme in Figure 9.

Figure 9. Schematic presentation of the Acid-Base properties of AsparticAcidType amino acids.

In this case, four species, which differ in the number of bonded protons, are produced in the solution.

Aspartic acid itself (pKa1 = 1.94, pKa2 = 3.7 and pKa3 = 9.62) is a good representative of this class of amino acids.

In abstract, a solution of the neutral amino acid not only could be titrated acidimetrically or alkalimetrically, as in the case of Alanine (in fact, Aspartic acid is, as all the amino acids, an amphiprotic substance). But a further question can be raised which attains to the fact that Aspartic acid is de facto a diprotic acid. Then, in principle, an alkalimetric titration of Aspartic acid can be performed up to a first stoichiometric equivalent point or continued to a second stoichiometric equivalent point (this dilemma arises with all multiprotic acids).

After excluding the acidimetric titration (on the basis of what we have seen for Alanine), the question is: an alkalimetric Aspartic acid titration yields more accurate results if targeted at the first or at the second equivalent point?

For evaluations below, we assume the usual typical values (i.e., [[AsparticAcid]] = 0.05 M, [[NaOH]] = 0.1000 M, = 50 ml).

LAMD of the titrated solution is presented in Figure 10, in which H₂Asp symbolizes the neutral amino acid. By consequence, the conjugated acid of H₂Asp is H₃Asp⁺, while the bases derived from the stepwise dissociation of the two acidic protons from the neutral species are HAsp– and Asp2–. The curves representing H₃Asp⁺ and H₂Asp intersect at pH = pKa1 = 1.94; curves of H₂Asp and HAsp– intersect at pH = pKa2 = 3.7; curves of HAsp– and Asp2– intersect at pH = pKa3 = 9.62.

In order to discuss the raised question, we begin locating in the LABD the initial pH of the titrated solution, pHin.

Figure 10. Logarithmic Acid-Base diagram of 0.05 M Aspartic acid, at 25 °C. Label H₂Asp stands for the neutral amino acid

At a glance we obtain pHin ≈ 2.8, corresponding to the crossing point between curves representing H₃Asp⁺ and HAsp–.

By traversing the LABD from pHin towards higher pH we must meet two stoichiometric equivalent points.

The first equivalent point corresponds to the conversion H₂Asp→HAsp– and is reached at pHeq1 ≈ 6.7 (corresponding to the crossing point between curves representing H₂Asp and Asp²–).

The second equivalent point corresponds to the conversion H₂Asp→Asp²– and is reached at pHeq2 ≈ 11.1 (corresponding to the crossing point between curves representing HAsp– and OH–).

Then, if we target the titration at the first equivalent point, an indicator must be chosen with an upper limit of its transition interval close to pHeq1. Otherwise, an indicator with its upper limit of its transition interval around pHeq2 must be employed.

From Figure 10, it is seen at a glance that the alkalimetric titration of Aspartic acid must be targeted at pHeq1 ≈ 6.7, simply because, around pHeq1, any possible SDS (i.e., H₂Asp and Asp²–) has a much lower concentration than that of any possible SDS (i.e., HAsp– and OH–) around pHeq2 ≈ 11.1.

Then, the answer to the above question is that Aspartic acid is best titrated as a monoprotic acid fixing pHend as close as possible to pHeq1 ≈ 6.7.

A suitable indicator for this titration is methyl red (transition interval: {4.4, 6.3}). This implies pHend ≈ 6.3 and ≈ 6.3 – 6.7 = - 0.4 (which is a warning for a systematic error in defect).

However, from Figure 10, evaluations are as it follows:

From this we can smoothly see that, unlike Alanine and congeners, Aspartic acid can be determined alkalimetrically, practically without the systematic error exceeding the precision of the burette.

The above titration has another favourable feature, which relates to the fact that, for a selected indicator, is independent on the concentration of Aspartic acid and, by consequence, can be calculated a priori. This is because pHeq1, in a broad range of concentrations, is independent on the concentration of Aspartic acid.

For instance pHeq1 would still be ≈ 6.7 for the alkalimetric titration of a 0.005 M solution of Aspartic acid.

This can be seen considering that the LABD of a solution, in which [[AsparticAcid]] ≠ 0.05 M, can be generated from the one in Figure 10, by translating, as a single group, the curves representing H₃Asp⁺, H₂Asp, HAsp– and Asp²– either upward or downward. Obviously, whatever simultaneous vertical translation of these curves does not modify the abscissa of the intersection point between the curves of H₂Asp⁺ and Asp2―, which determines pHeq1.

It is also useful to mention that chemically, the independence of pHeq1 from the Aspartic acid concentration is due to the fact that HAsp–, which is produced at the first equivalent point, is an amphiprotic species. This rule can be extended to all titrations in which the species produced by the Acid-Base reaction between the titrant and the titrated substance is amphiprotic.

3.1.3. LysineType Amino Acids

Acid-Base properties of LysineType amino acids are described by scheme in Figure 11.

Figure 11. Schematic presentation of the Acid-Base properties of LysineType amino acids

We take Lysine as representative of its own class. The neutral species will be indicated by HLys to expose its acidic dissociable proton. Dissociation of the HLys proton produces the singly charged base Lys–. HLys can accept up to two protons producing the acidic species H₂Lys⁺ and then the fully protonated and double charged species H₃Lys²⁺.

With respect to Acid-Base titrations, we can raise for Lysine questions analogous to those raised above for GlycineType and AsparticAcidType amino acids.

They are formulated as it follows:

Must we titrate a Lysine solution acidimetrically or alkalimetrically?

If, for a given choice of the titration mode, it exists more than one stoichiometric equivalent point, which one should we target at?

For evaluations below, we assume as usual the typical value, i.e., [[Lysine]] = 0.05 M, for the amino acid solution to be titrated. LABD of this solution is presented in Figure 12. Curves representing H₃Lys²⁺ and H₂Lys⁺ intersect at pH = pKa1 = 2.19; curves of H₂Lys⁺ and HLys intersect at pH = pKa2 = 9.12; curves of HLys and Lys– intersect at pH = pKa3 = 10.68.

Figure 12. Logarithmic Acid-Base diagram of 0.05 M Lysine, at 25 °C. Label HLys stands for the neutral amino acid

We start by observing that a lysine solution has an alkaline pH, which can easily be found graphically since it corresponds to the intersection point between curves of H₂Lys⁺ and Lys–. From the diagram we read pHin ≈ 9.9, and this high initial pH, per se, excludes the alkalimetric titration. Then, we consider the acidimetric titration and find that there are, in abstract, two stoichiometric equivalent points corresponding to the conversions HLys→H₂Lys⁺ and HLys→H₃Lys²⁺. When the first equivalent point has been reached HLys has been stoichiometrically converted to H₂Lys⁺. So pHeq1 is the pH of an H₂Lys⁺ solution, which graphically corresponds to the point where the curve of H₃Lys²⁺ intersects the curve of HLys (i.e., pHeq1 ≈ 5.7). pHeq1 does not depend on the concentration of Lysine since species H₂Lys⁺ is amphiprotic.

The second equivalent point is reached when the initial HLys has been stoichiometrically converted to the fully protonated H₃Lys²⁺ acid and it corresponds to the intersection point between curves of H₂Lys⁺ and H⁺ (i.e., pHeq2 ≈ 1.8).

From LABD in Figure 12, we see at a glance that the acidimetric titration of Lysine is by far more accurate if stopped around pHeq1. This is because, around the first acidimetric equivalent point (pHeq1 ≈ 5.7), the SDS is either H₃Lys²⁺ or HLys; while around the second acidimetric point (pHeq2 ≈ 1.8) the SDS is either H₂Lys⁺ or H⁺; then it can be seen that the concentration of the SDS will be lowest if we select an end point around pHeq1 ≈ 5.7. Furthermore, around pH ≈ 5.7, [H₃Lys²⁺] and [HLys] are so low that we can predict very accurate results.

Between the common indicators, bromothymol blue (transition interval: {6.0, 7.0}) is an appropriate choice. In fact, this corresponds to pHend ≈ 6 and ≈ 6.0 – 5.7 = 0.3. At pH ≈ 6.0, SDS ≡ HLys and evaluations are as it follows:

There is no doubt that this titration is capable of great accuracy and that the uncertainty on the results will be fully controlled by the precision of the burette.

This conclusion extrapolates to all LysineType natural amino acids.

Please note that Lysine and its congeners are very often sold in the form of the corresponding hydrochloride salts.

What we have demonstrated above does not apply to a solution of Lysine×HCl (Lysine hydrochloride).

A 0.05 M solution of Lysine×HCl is still represented by the LABD of Figure 12, but the key point is that pHin ≈ 5.7. Then, from Figure 12, we see that a Lysine hydrochloride solution cannot be accurately titrated neither acidimetrically nor alkalimetrically.

3.2. Titrations of Bicarbonate and Carbonate Mixtures with the “Two Indicators Method”

The Acid-Base titration of solutions containing mixtures of carbonate and bicarbonate is a classical experiment suggested in many analytical textbooks.

The very elegant two indicators method for the determination of carbonate and bicarbonate consists in the acidimetric titration of their mixtures (usually with an HCl standard solution).

First, phenolphthalein is added to the titrated solution, which at the beginning appears intensely red, and the acidimetric titration is interrupted when the red colour disappears. After reading the volume of titrant delivered (i.e., ), the solution is coloured in blue by adding bromocresol green and the titration continued until the blue colour turns yellow. The volume of the HCl solution consumed to reach the bromocresol end point is .

Consequently, two end points volumes are elegantly collected in a single titration, which are then used, with other titration data, to evaluate the concentrations (i.e., [[CO₃2–]] and [[HCO₃–]]) or the number of moles of carbonate and bicarbonate in the titrated solution.

The moles of HCl consumed to reach the phenolphthalein end point (pHend1 ≈ 8.2) are assumed to convert the carbonate to bicarbonate. Then we have:

[picture] | (10)

At the bromocresol end point ((pHend2 ≈ 4), carbonate and bicarbonate have been converted to carbonic acid. Then, the number of moles of HCl consumed is related to the number of moles of carbonate and bicarbonate by the obvious relation (11):

[picture] | (11)

By combining relations (10) and (11), one obtains relation (12), from which the number of moles of bicarbonate can be calculated:

[picture] | (12)

Although the two indicators method appears very simple and elegant, it is not the method of choice and various authors, [5], suggest alternative methods (which require more manipulations and reactants). The question is: what is the problem with the two indicator method?

In the following we shall answer within the framework of the systematic titration error.

For convenience, for evaluations below, we assume that 50.0 ml of a solution containing [[CO₃2–]] = [[HCO₃–]] = 0.025 M, are titrated with standard 0.1000 M HCl.

LABD representing the assumed carbonate + bicarbonate solution is presented in Figure 13.

Figure 13. Logarithmic Acid-Base diagram of 0.05 M carbonic acid, at 25°C. Curve representing carbonic acid (H₂CO₃) intersects curve representing bicarbonate (HCO₃–) at pH = pKa1 = 6.35; bicarbonate curve and carbonate (CO₃2–) curve intersect at pH = pKa2 = 10.33

On the basis of the titration data and of Figure 13, we deduce the following stoichiometric values:

→First stoichiometric equivalent point↓:

pHeq1 ≈ 8.4 (→crossing point between curves of H₂CO3 and CO₃2–);

→→Second stoichiometric equivalent point↓:

pHeq₂ ≈ 3.8 (→crossing point between curves of HCO₃– and H⁺);

It can be appreciated that the phenolphthalein end point at pHend1 ≈ 8.2, is, in effect, very close to pHeq1 ( = 8.2 - 8.4 = - 0.2), and so does the bromocresol end point at pHend₂ ≈ 4 ( = 3.8 – 4 = - 0.2).

From Figure13, evaluations for are as it follows:

Analogously, evaluations for are as it follows:

Obviously, there is a systematic error on the number of moles of carbonate which, from equation (6), is evaluated at about + 1.2%.

On the other side, the number of moles of bicarbonate depends on . This difference carries a negative systematic error which is about -0.4 ml . This implies that the concentration or number of moles of bicarbonate is affected by a % systematic error of about -3%.

These evaluations are based on Figure 13 (which has been drawn by using values at 25° and zero ionic strength for the dissociation constants of carbonic acid, i.e., pKa1 = 6.35 and pKa2 = 10.33), on standard transition ranges for the indicators and on the specified titration data.

In practice, changes on the assumed values can have effects on the accuracy (in either direction). Nevertheless, it is evident that the original sin of the two indicators method is the systematic error in excess on the carbonate which then propagates unfavourably to the bicarbonate through equation (12).

Some authors, [6], suggest the use of methyl red instead of bromocresol green in the above procedure. This amounts to taking pHend₂ ≈ 4.4. From Figure 2, we see that this choice makes the error on more negative and, by consequence cannot, in general, be justified (unless much lower concentrations of total carbonate are considered than assumed in the present example).

Although bicarbonate cannot be determined very accurately when carbonate is present in a given solution, the acidimetric titration of bicarbonate to the bromocresol end point produces very accurate results. This is easily seen as a special case of the two indicators method, in which the cause of inaccuracy (i.e., the large error on ) is removed since, by definition, = 0, in absence of carbonate.

The same can be said for the determination of carbonate in absence of bicarbonate, since in such a case can be directly connected to the number moles of carbonate through relation (13):

[picture] | (13)

which is obtained from equation (11) when it can be assumed that = 0.

4. Using MS Excel for Drawing Logarithmic Acid-Base Diagrams

Table 1 provides algebraic expressions needed for drawing LABDs for solutions containing monoprotic (HA), diprotic (H₂D) and triprotic (H₃T) acids.

Table 1. Algebraic expressions for drawing Logarithmic Acid-Base Diagrams*

Species which differ only in the number of bonded protons constitute an Acid-Base group. For instance there are four different species in the Acid-Base group of any triprotic acid. The variables , and which appear in Table 1, are analytical group concentrations.

For instance, , would represent the total concentration of carbonic acid Acid-Base group. So, in abstract, if a solution is prepared dissolving specified analytical concentrations of , and , is interpreted as:

However, in most cases an Acid-Base group is introduced in the solution through a single substance, so that only one of the analytical concentrations, i.e., [[...]], in the group is different from zero. For instance, a solution of sodium bicarbonate has: ; but for a mixture of sodium carbonate and bicarbonate (as the one described in the previous paragraph) we have: . In practice, in a given solution, the analytical concentration of an Acid-Base group coincides with the sum of the equilibrium concentrations, [...], of all species belonging to the group. Naturally, analytical group concentrations are deduced from the amount of the various substances employed to prepare the considered solution.

In a given solution, each Acid-Base group is considered independently and its analytical concentration and acid dissociation constants (which connect the species in a group) are fixed. Then, the equilibrium concentration of any species in the considered group can be calculated as a function of [H⁺] (or, which is the same, of pH) from the appropriate expression in Table 1.

If a solution contains several Acid-Base groups, the above procedure is repeated for all Acid-Base groups, since the LABD representing such a solution is simply the superposition of the LABDs of each Acid-Base group.

Then, a MS Excel .xlxs file, which we shall call myLABD.xlxs, for drawing the logarithmic Acid-Base diagram representing any solution containing one or several Acid-Base groups is readily developed using functions in Table1.

One can start with an Excel file exposing four sheets which, for convenience, are renamed: Sheet1 → MonoproticAcid, Sheet2 → DiproticAcid, Sheet3 → TriproticAcid, Sheet4 → myPlot.

In the MonoproticAcid sheet the first row is reserved to labels. Cells from A1 up to I1 are labelled in the order: pH, log[H⁺], [H⁺], log[OH–], [OH–], [HA], log[HA], [A–], log[A–], pKw, pKa, CHA.

Under cell labelled pKw (i.e., cell J2) we input the water self dissociation constant, pKw (e.g., 14 at 25 °C); then, in cell K2, under label pKa, we input the value of pKa (for instance, 9.25 for an ammonia solution). It is useful also to have values for Kw and Ka, which are readily obtained inserting in cells J3 and K3, respectively, the functions 10-pKw (=10^-J2) and 10-pKa (=10^-K2).

In column A (starting with cell A2) of the MonoproticAcid sheet we generate 140 pH values, from 0 to 14, spaced 0.1 pH units. Under label log[H⁺], cell B2, we input the function –pH ( =-A2). In cell C2 (under label [H⁺]) we input the function 10-pH (i.e., =10^-A2). Then, we extend the functions in cells, B2 and C2 up to cells B142 and C142. Under cell log[OH–], cell D2, we input the function pH - pKw (=A2-$J$2); please note the absolute reference to cell J2, written $J$2. Then, we extend the content of cell D2 up to cell D142.

Under the cell [OH–], cell E2, we input the function 10logOH (i.e., = 10^D2) and extend its contents up to cell E142.

In cell F2 (under the label [HA]) we now input the [H⁺] function in row 4 of Table 1. To do so we need a value for the Acid-Base group concentration, which is written in cell L2 under cell labelled CHA. For instance we use CHA = 0.1(→in cell L2).

Specifically, in cell F2 we write: =($L$2*C2)/(C2+$K$3) and extend its content up to cell F142. Then, in cell G2 (under label log[HA]), we input function log₁₀[HA] (=LOG10(F2)) and its content is extended up to cell G142.

In cell H2 (under label [A–]) we input function in row 5 of Table 1 (=($L$2*$K$3)/(C2+$K$3)) and extend the function up to cell H142. Then, under label log[A–] (cell I2), we input the function log₁₀[A–] (=LOG10(H2)) and we extend the function up to cell I142.

Finally, we have in the MonoproticAcid sheet values necessary for drawing curves for log[H⁺], log[OH–], log[HA] and log[A–]. Changing pKa in cell K2 and/or CHA in cell L2 will automatically update all values in the sheet.

In order to visualize the LABD, an empty dispersion plot is inserted in the myPlot sheet. Then, data to be plotted are selected from the MonoproticAcid sheet. Values in the first column of the MonoproticAcid sheet (i.e., pH) represent the abscissa for all the curves to be drawn. Then, for each curve to be drawn we select the appropriate column in the MonoproticAcid sheet, i.e., columns with labels log[H⁺], log[OH–], log [HA], log[A–]. The Excel plot appears like the one in Figure 14.

It represents a solution in which the Acid-Base group myHA/myA–, characterized by pKa = 9.25 has a 0.1 M total analytical concentration.

The coordinates, (pH, log[j]) of any point on any curve in the original LABD, exposed in the myPlot sheet, can be read within 0.1 log units by positioning the mouse pointer on the selected point.

The procedure we have shown above for the monoprotic acid can be easily extended, introducing the appropriate changes, to develop the sheet DiproticAcid, using [H⁺] functions in rows 7, 8 and 9 in Table 1. Finally, one can develop the sheet TriproticAcid on the basis of functions in rows 11, 12, 13 and 14 (of Table 1).

LAMD of any solution of acids and bases are drawn in myPlot sheet selecting the appropriate columns in MonoproticAcid, DiproticAcid and TriproticAcid sheets, with the usual Excel operations.

Obviously, there are infinite ways to personalize plots using the standard Excel tools.

Figure 14. Logarithmic Acid-Base diagram representing the 0.1 M solution of a monoprotic acid (having pKa = 9.25) drawn in MS Excel

5. Conclusions

If an Acid-Base titration is considered within the frame of the Logarithmic Acid-Base Diagram of the titrated solution, the topic of systematic titration error, which is generally treated in a qualitative way, can smoothly be transferred at the quantitative level, without lengthy algebraic manipulations which would otherwise be required.

References

[1]   | Sillén, L.G., Graphical presentation of equilibrium data. Chapter 8 in Kolthoff, I.M., Elving, P.J. and Sandell, E.B. (eds), Treatise on analytical chemistry, © The Interscience Encyclopedia, Inc., New York, 1959.
[2]   | Salvatore, F., Interpretazione grafica del pH e della capacità tamponante di soluzioni acquose di acidi e basi, © Aracne editrice s.r.l., Roma, 2009.
[3]   | Kahlert, H. and Scholtz, F., Acid-Base Diagrams, © Springer-Verlag. Berlin Heidelberg, 2013.
[4]   | Butler, J. N., Ionic equilibrium: solubility and pH calculations, © Wiley-Interscience, New York, 1998.
[5]   | Harris, D.C., Quantitative Chemical Analysis, © W. H. Freeman & Company, New York, 2011.
[6]   | Nollet, L. M. L., Handbook of water analysis, © Marcel Dekker, Inc., New York, 2000.