Information • Textbooks • Media • Resources
Effect of Dissolved CO2 on Gran Plots Motomichi Inoue CIPM, University of Sonora, Hermosillo, Sonora, Mexico Quintus Fernando* Department of Chemistry, University of Arizona, Tucson, AZ 85721-0041;
[email protected] Most widely used undergraduate texts on quantitative analysis, even today, recommend the use of a first-derivative plot of d(pH)/dVb versus Vb to locate the equivalence point of a sigmoid-shaped titration curve of pH versus volume Vb of strong base added. Even instrument manufacturers produce automated titration devices that use derivative plots to locate the equivalence points in titration curves. More than 40 years ago Gunnar Gran showed that the equivalence point in such titrations could be conveniently located by the use of a linear plot (Gran plot) that, upon extrapolation, intersected the volume axis at the equivalence point, provided that the assumptions made in obtaining the linear plot were valid (1). Despite the publication of several articles in this Journal on the advantages of using Gran plots (2–8), this method has largely been ignored by textbook writers; only recently have a few scattered reports about the use of Gran plots appeared in analytical chemistry texts (9, 10). Even in these discussions, the merits of using Gran plots are not clearly stated. The rapidly rising segment of any titration curve is subject to the highest experimental errors and from this point of view, is the least desirable segment to employ for locating the equivalence points. For example, in an acid–base titration, this segment has the lowest buffer capacity and is therefore subject to large errors caused by extraneous acids or bases—for example, CO2 absorbed from air. Gran plots have proved to be useful not only in locating equivalence point but also in determining equilibrium constants and their variation as a function of the ionic strength of the solution. The slope of the Gran plot, Vb[H+] versus Vb, gives the acid dissociation constant Ka of a weak acid that is being titrated with a strong base at a constant ionic strength. Students find that the variation of the slope Ka with ionic strength is a convincing experimental demonstration that the K a value varies predictably with the ionic strength, I, of the solution. In fact, the variation of Ka with I can be experimentally determined to show which form of the Debye–Hückel equation or Davies equation will fit the experimental results (11). The influence of CO2 dissolved in strong base solutions is the most bothersome problem in the determination of acid dissociation constants. In all the published discussions mentioned above, however, the effect of absorbed CO2 on Gran plots has been ignored. It is important to address the question, if the concentration of strong base, Cb, is not used in Gran plots, will the presence of H2CO3, HCO3, and CO32 affect the Gran plot and to what extent will the plot’s linearity be affected? Is it necessary, therefore, to prepare KOH solutions that are completely free of K2CO3? The preparation of CO2-free solutions is important and time-consuming, and the storage of these CO2-free solutions out of contact with CO2 from air is troublesome.
1132
The main objective of this paper is to discuss the effect of dissolved CO2. Dissolved CO2 does not give a large error in Gran plots; for this reason, time-consuming steps, such as boiling distilled water followed by cooling under nitrogen gas flow, can be omitted for the determination of the acid dissociation constants in a time-limited undergraduate laboratory experiment when Gran plots are employed. Gran Plots and Determination of Acid Dissociation Constants of Weak Acids The acid dissociation constant of a weak monobasic acid AH is defined by
H+ A– AH
Ka =
(1)
where all terms in brackets are concentrations in mol/L. When a potentiometric titration of a weak acid is carried out with a strong base such as KOH, the following charge balance equation is valid at any point in the titration: [K+] + [H+] = [A] + [OH ]
(2)
+
The value of [K ] is given by
K+ =
V bC b
(3)
V0 + Vb
where V0 is the initial volume of the titration solution, Cb the concentration of the base, and Vb the added volume of the base solution. Since titrations of weak acids are normally carried out in a pH range of 3–7, [OH ] and [H+] are negligibly small in comparison with [A] and [K+] when a certain amount of the KOH solution is added. In the Vb region where this assumption is valid, eq 2 is rewritten as
A– = K+ =
V bC b V0 + Vb
(4)
The total moles of AH in the system is equal to the moles at the equivalence point, VeqCb and the total molar concentration of AH at any time during titration is given by [AH]total = VeqCb(Vo + Vb)1. Therefore, [AH] is given by
AH = AH
total
– A– =
V eq – V b C b V0 + Vb
(5)
Substitution of eqs 4 and 5 in eq 1 gives
Ka =
V b H+ V eq – V b
Journal of Chemical Education • Vol. 78 No. 8 August 2001 • JChemEd.chem.wisc.edu
(6)
Information • Textbooks • Media • Resources
This can be rewritten as +
V b H = K a V eq – V b
AH =
(7)
The plot of Vb[H+] versus Vb, (i.e., the Gran plot), gives a straight line in a certain pH range where [H+] and [OH ] are negligible, and the slope of this straight line gives the Ka value of the weak acid. The value of Veq is the value of Vb at Vb[H+] = 0. Effect of CO2 on Gran Plots
Ka =
HCO3
1
2
CO3 + H
+
(8)
2
(pK a ,CO = 9.88)
(9)
2
where the pK a values at an ionic strength of 0.1 are shown in the parentheses (12). At pH < 7, the only significant charged species is [HCO3 ], which should be added to eq 2: [K+] + [H+] = [A] + [OH] + [HCO3 ]
(10)
+
In the pH region where [H ] and [OH ] can be ignored, the charge balance is given by [K+] = [A] + [HCO3]
(11)
where
HCO3– =
K a ,CO H2CO3 1
2
(12)
H+
Almost all CO2 involved in a titration system is provided by CO2 dissolved in KOH solution. When the CO2 concentration in the hydroxide solution is zCb (where z is a fractional number), the concentration of total CO2 in the titration system is zCbVb(V0 + Vb)1:
zC bV b V0 + Vb
= H2CO3 + HCO3– + CO32
(13)
Because [CO 32] is negligible at pH < pK a ,CO , the combination of eqs 12 and 13 leads to 1
HCO3– =
H+ 1 – α1z V b
α1 zC bV b
V0 + Vb
V b H+ =
(14)
K aV eq 1 – α1z
1
2
K a ,CO + H+ 1
(15)
which shows the fraction of carbonic acid in the form HCO3 . Substitution for [K+] and [HCO3] in eq 11 gives –
A =
1 – α1z C bV b V0 + Vb
and [AH] given by eq 5 is modified as
2
Determination of Concentration of Dissolved CO2 In an extension of the above discussion, the concentration of CO2 dissolved in the KOH solution used can be determined by extending the titration, through the end point, to high pH values. This method has been published by some authors (2, 8, 13), but its theoretical basis has not been described. The principle of this method is therefore explained in this section. In a pH region beyond the end point, the charge balance is given by [K+] = [A] + [OH ] + 2[CO32] + [HCO3]
(20)
+
[K ] is given by eq 3. Since [HA] is negligible at pH values much higher than the pH at the end point of the titration, [A] is given by total
=
V eqC b
(21)
V0 + Vb
Since [H2CO3] can be neglected at pH > pK a ,CO , 2
2
(19)
1
A– = AH K a ,CO
– K aV b
When α1z is negligible, eq 19 is identical with eq 7, and a Gran plot gives the correct values of Veq and Ka. When z has a certain value, the Vb[H+] versus Vb plot systematically deviates from a straight line that is extrapolated from the region α1z [H+] so that α2 ≈ 1, this equation can be rewritten as 2
V0 + Vb +
H
=
Cb
2
V b – 2zV b – V eq
Kw
(27)
Hence the plot of (V0 + Vb)[H+]1 versus Vb gives a straight line when z is not very large. At (V0 + Vb)[H+]1 = 0, (1 – 2z)Vb,0 = Veq
(28)
where Vb,0 is the intercept on the volume axis on the basic side. Obviously, the value Vb,0 deviates from the intercept Veq on the acidic side when CO2 is dissolved in the solution being titrated. From eq 28, the proportion z of CO2 dissolved in the KOH solution can be determined by
z=
V b,0 – V eq 2V b,0
(29)
For a more reliable determination of CO2 concentration dissolved in KOH solutions, it is preferable to titrate a strong acid such as HCl with KOH as described by Martell and Motekaitis (13). In such a titration system of a strong acid HCl versus a strong base KOH, the acid can be assumed to be completely dissociated, and the charge balance equation and [Cl ] in the acidic region are given by [K+] + [H+] = [Cl ]
Cl– = HCl total =
C bV eq V0 + Vb
(30) (31)
Substitution of eqs 3 and 31 into equation 30 gives (V0 + Vb)[H+] = Cb(Veq – Vb) +
(32)
The plot of (V0 + Vb)[H ] versus Vb gives a straight line throughout the acidic region, and the intercept on the volume axis gives the volume Veq at the equivalence point. The z value can be obtained on the basis of eq 29 from the difference between the intercept Veq of the (V0 + Vb)[H+] versus Vb plot in the acidic region and the intercept Vb,0 of the (V0 + Vb)[H+]1 versus Vb plot in the basic region (eq 27). The advantage of the use of a strong acid is that the linearity of eq 32 is valid throughout the acidic region. On the other hand, the linearity
1134
Figure 1. Titration of benzoic acid with a “CO2-free” KOH solution, and Gran plot; the ordinate Fa of the Gran plot for the acidic region is 105Vb[H+]; Fb on the basic region is 1012(V0 + Vb)[H+]1. The quantity of benzoic acid is 0.2054 mmol; [KOH] = 0.1076 M; V0 = 51.0 mL; ionic strength = 0.1 (KCl); T = 25 °C; CO2 content z in the KOH solution is 0.003, determined by titration with HCl under a nitrogen atmosphere. [H+] was determined from pH meter readings calibrated by assuming complete dissociation of HCl in 0.1 M KCl. Solid line is the titration curve calculated with pKa = 4.00. The slope of the Gran plot in the acidic region gives pKa = 4.02.
of the plot in the basic region is valid only when z is small, as predicted from eq 27; this method is not applicable when z is large. The value of (Vb,0 – Veq) and hence the z value has a large error, even when Vb,0 and Veq are determined with an error of 0.1%; the detection limit of z is at best 0.002. Examples of Experiments
Example 1: Determination of pKa of Benzoic Acid Figure 1 shows the titration of benzoic acid with a “CO2free” KOH solution; the z value was determined to be 0.003 by titration with HCl. The experimental conditions are described in the legend to the figure. When a pH electrode is calibrated with standard buffers, the pH meter readings give the logarithmic values of the reciprocal hydrogen-ion activities, log aH, and an appropriate form of the Debye– Hückel expression is required for the conversion of aH to [H+] (i.e., a hydrogen ion concentration given by mol/L). In this experiment, the pH electrode was calibrated by assuming the complete dissociation of HCl in 0.1 M KCl, and [H+] values were determined directly from the pH meter readings; this procedure is described in detail by Martell and Motekaitis (13). The solid line in Figure 1 shows a simulation curve obtained by minimizing the deviation factor (13), Σwi {log[H+]i,obs – log[H+]i, calc}2, where wi = 1/{log[H+]i+1 – log[H+]i–1], and the pKa value for the best fit is 4.00, which agrees with the reported value 4.01 ± 0.02 (12). As shown in the same figure, the Gran plot, Vb[H+] versus Vb, is linear in the region of log[H+] = 4–6. The slope gives a pKa value of 4.02, which agrees well with the above values.
Journal of Chemical Education • Vol. 78 No. 8 August 2001 • JChemEd.chem.wisc.edu
Information • Textbooks • Media • Resources
Figure 2. Titration of benzoic acid with KOH containing CO2. The experimental conditions were the same as for the titration shown in Figure 1, except z = 0.04. The solid line is the titration curve calculated with pKa = 3.91; the slope of the Gran plot in the acidic region gives pKa = 4.03.
Figure 3. Gran plots for titration of HCl with (䊊) the “CO2-free” KOH solution (Example 1) and (×) the CO2-containing KOH solution (Example 2). Only the region close to the equivalence points is shown so that the intercepts are readily located. The ordinate φ is (V0 + Vb)[H+] for the acidic region and 1014(Vo + Vb)[H+]1 for the basic region. The quantity of HCl is 0.516 mmol; V0 = 55.0 mL; other conditions the same as those for the titration shown in Figure 1. Vertical dotted lines show the intercepts Veq and vertical solid lines show Vb,0.
Example 2: Effect of Dissolved CO2
advantages, no time-consuming processes are required for the preparation and storage of strong base solutions used for Gran plots. The determination of the pKa value of a weak acid, therefore, can be completed in a single 3-hour undergraduate laboratory. In instruction to students, it is highly recommended to show that the validity of Gran plot (consequently the reliability of Veq and pKa values) should be always checked by the value of α1z. Obviously, the use of “CO2-free” titrants is required for titrations in research.
Figure 2 shows an example of the titration in which the titrant contains CO2 with z = 0.04. The solid line shows a simulation curve for the minimum deviation factor. The agreement between the observed and calculated curves is poor, and the pKa value, 3.91, obtained for the best fit differs from the value obtained for the titration with the “CO2-free” KOH solution (Fig. 1). Thus, the presence of CO2 causes a large error. On the other hand, the Vb[H+] versus Vb plot gives a straight line between log[H+] = 4 and 6 as shown in Figure 2, and the slope gives pKa = 4.03; the acid dissociation constant is still correctly determined within experimental error although the z value is as high as 0.04.
Example 3: Determination of CO2 Concentration Figure 3 shows (V0 + Vb)[H+] versus Vb and (V0 + Vb)[H+]1 versus Vb plots for the titration of an HCl solution with the “CO 2-free” and CO2-containing KOH solutions used in Examples 1 and 2, respectively; the experimental conditions are shown in the legend. The differences between the Veq and Vb,0 gave the z values described above.
Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Conclusion These examples of titrations clearly show the advantages of Gran plots: (i) the exact concentrations of the titrant and the titrand are not required for the determination of pKa values, and, more importantly, (ii) the presence of CO2 does not cause a serious error in pKa values even when the z value is as high as 0.04 (such a high CO2 content is not permissible for titration unless a Gran plot is used). Owing to these
11. 12.
13.
Gran, G. Analyst 1952, 77, 661. Rossotti, F. J. C.; Rossotti, H. J. Chem. Educ. 1965, 42, 375. West, K.; Flath, P. J. Chem. Educ. 1981, 58, 555. Barnhard, R. J. J. Chem. Educ. 1983, 60, 679. Boiani, J. A. J. Chem. Educ. 1986, 63, 725. Schwartz, L. M. J. Chem. Educ. 1987, 64, 947. Chau, F. T.; Tse, H. K.; Cheng, F. L. J. Chem. Educ. 1990, 67, A8. Vesala, A. J. Chem. Educ. 1992, 69, 577. Harris, D. C. Quantitative Chemical Analysis, 5th ed.; Freeman: New York, 1997; p 281. Butler, J. N.; Cogley, D. R. Ionic Equilibrium, Solubility and pH Calculations; Wiley: New York, 1998; p 84. Seymour M. D.; Fernando, Q. J. Chem. Educ. 1977, 54, 225. Martell, A. E.; Smith, R. M.; Motekaitis, R. J. NIST Critical Stability Constants of Metal Complexes Database; NIST Standard Reference Data; National Institute of Standards and Technology: Gaithersburg, MD, 1993. Martell, A. E.; Motekaitis, R. J. Determination and Use of Stability Constants; VCH: New York, 1992; pp 30, 48–50; a program for calculation is given in this book.
JChemEd.chem.wisc.edu • Vol. 78 No. 8 August 2001 • Journal of Chemical Education
1135
