Frederick D. Tabbutt Reed College
Portland, Oregon
I
I
Titration Curves from Logarithmic: contentration
T h e use of logarithmic concentration diagrams to solve problems in ionic equilibrium has been explained a number of times in the American literature1 since it was first introduced here by Sillen.2 The utility of the method both from a practical and a pedagogical viewpoint is undeniable. The purpose of this paper is to demonstrate a further use of the log c diagrame-one of providing a quick and detailed picture of a titration curve. The method to be described in this paper is an obvious extension of the log c diagram method. It is mathematically very simple--in fact, no calculations are necessary. It is quick. It is versatile. It is as accurate as a titration curve can be read. Most important, it gives the student an intuitive insight into the factors which make a titration feasible. The method is most easily understood by those who are familiar with the way of treating equilibria using log c diagrams.8 The method for extracting the titration curve from the log c diagram is this: draw the log c diagram for the solution using the initial analytical concentrations; then identify the initial point and each equivalence point on the log c diagram using the appropriate proton condition; finally, fill in the intermediate points. It works, despite s~hstantialchanges in the roncrnrmtion of the tirrxnt. hecauw. until rlw finnl eiruivn1mi:e r~c~inr. the titration curve is determined by the analyticai con: centration of the system being titrated, which changes only slightly with dilution. Beyond the last equivalnnre point rhr rurve isdetermined ton good npproxi~nntion by the nndytienl eoncentration of thc titnnt nt the equivalence point. Acid-Base Titrations
As an example, take the case of the titration of a 0.10 M HC1 solution with 0.10 M NaOH. Only the OHH+ lines of the log c diagram shown in Figure 1 are required. The proton condition4 for the initial pH (a%), the equivalence point (loo%), and a stoichiometric amount
of the titrant in excess of the equivalence point (200%) are listed below. The points corresponding to these are noted in Figure 1.
Figure 1.
0% 100% 200%
Log c diagram for strong arid and rtrong base.
+
[HC1 = [OH-] [Cl-I = 0.10 [Hf1 = [OH-] [Cl-I, [Hf1 = [OH-] [Na+l [GI-], [OH-] = 0.10 [Na+] [Ht1 = [OH-]
+ +
++
Dilution is neglected as is usually done in sketching a 'For those not familiar with log c diagrams, briefly, they am constructed in the following manner (see any of the refs. ( 1 3 ) for a more oamplete explanation). By combining the mass halance and equilibrium expressions, it is possible to relate and graph the logarithm of the concentration of all species which can exist in solution as a function of a master variable, e.g., pH for acid-base equilibria. The identification of a system point (e.g., for acid-base systems; pK., log analytical concentration) greatly simplifies the construction of the diagram, because a11 (acid-base) equilibrium systems have the same shape relative to their system point. Finally, to determine the pH of s. solution one uses the proton condition. This is the equation which relates concentre tions which oan exist in solution to maintain the initial proton level. For example, if we have a solution af NaHCO* then from the zero level of HCOI- and H20:
'
FREISER, H., AND FERNANDO, Q., "Ionic Equilibria in A n e lytical Chemistry," John Wiley & Sons, Inc., New York, 1963; BUTLER,J. N., "Ionic Equilibrium, a Mathematical Approach," Addison-Wesley Publishing Co., Reading, Mass., 1964; BUTLER, J. N., "Solubility and pH Calculations," Addison-Wesley Puhlishing Co., Reading, Mass., 1964; ABRAHAMBON, A. W., "Understanding Physical Chemistry," Chapter 14, W. A. Benjamin, Inc., Q.,J. CEEM. New York, 1964; and FREISER,H., AND FERNANDO, Ennc., 42,35 (1965). ' SILLEN, L. G., "Graphical Presentation of Equilibrium Data," in "Treatise on Analytical Chemistry," (Editors: KO> TBOFF, I. M. AND ELVING, P. J.), Part I, Vd. 1, Chapter 8, Interscience (a division of John Wiley & Sons, Ino.), New York, 1959.
" " - L + . -OHThe proton condition is: [HGOal
+ [H'l
=
[OH-]
+ [Coal
The dominant equality (highest intersection on the log c diagram) consistent with the proton condition is that which determines the pH of the solution. For other master variables there are similar considerations. ' For solutions containing only strong acids and bases the pmton condition is the same as the electroneutrality expression.
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May 1966
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titration curve. A correction can he made for this, hut it is not worth the trouble a t this stage. To identify points intermediate to 0 and loo%, the appropriate proton conditions are considered. For example, if CAis the analytical concentration of the acid, [Cl-1, and CB is the analytical concentration of base added, [Na+], then at 20% CB = 0.2 CA. From the proton condition, CB
+ lH+1
+
[OH-I CA [H+] = 0.8 CA or log [H+1 = log CA - 0.097 =
To the accuracy that one can read the log c diagram (or the titration curve), the 20% point falls on the H + line 0.1 log unit below the 0% point. In a similar manner, it can be seen that to the accuracy that is consistent with readability the following table is appropriate.
in Figure 4a. A similar approach is used for weak acids and bases. For exampIe, consider the titration of a dihasic weak base, 0.10 M NaaC03,with HC1. On the log c diagram in Figure 2 are indicated the 0, 100, 200, and 300% points corresponding to zero levels of CO3-, HC03-, 0.10 M H + respectively. As the H&03 and HzCOa titration proceeds from the C03- zero level the proton condition is:
+
[H+l
% 20 30 50 70 90 99 99.9
One simply follows along the H + line marking the percentages relative to the drop from the 0% line until the OH- intersection is about 0.5 units away from the intersection with H+. In this last interval of 0.5 log units the percentage rapidly goes to 100% as the OH- contribution t,o the electroneutrality expression becomes imp~rtant.~ Finally, consider the points intermediate to 100% and 200%. 4 t the 110% point CB = 1.1 CAand substitution into the proton condition gives [OH-] = 0.1 CA which predicts that the 110% point will be one log unit below 200% on the OH- line. With one or two more calculations the symmetry relative to 100% becomes apparent. Thus, two log units down from 200% is 101%, etc. The titration curve is then plotted directly from the log c diagram as % versus pH, or given the initial volume of the HC1 solution, the percentages may he converted to volume of titrant. Such a plot is shown
=
lC1-I
+ [OH-]
and substituting for [Cl-] in terms of the titration, P, and the analytical concentration of carbonate, CB, we have:
For 10%
Log units down from 0% on H + line 0.1 0.15 0.3 0.5 1.0 2.0 3.0
Titration,
+ 2[H&Os] + [HCOs-I
[H']
Figure 2.
+ ZIHnCOal + [HCOI-] = 0.010 + [OH-]
Log c diagram for 0.1
M carbonate.
It can he seen from Figure 2 that the only point consistent with this equation is that where [HCOa-] = 0.010. Moreover, it can be seen, then, that by dropping down from log CB by 1 log unit on the HCOa- line, we have 10%; 0.52 log units on the HCOa- line, 30%; 0.3 log units on the HC03- line, 50%. Beyond 50'%, intersections on the HC0,- line are not so accurately read. However, if one considers it from the point of view of approach to the 100% zero level, i.e., a proton condition relative to HCOa-:
' This last interval can he evaluated, but it is seldom neces-
Relative to the concentration, CO,of H + or OH- at the intersection when CA - CB = 0, the following table relate the vertical distance from the intersection to the point on the descending line. sary.
Multiplicative factor relating ezcess of H + to C,
0.38 0.30 0.23
For example, if h
=
3 Co = [H+] 2 Co 1 Co 0 . 5 CO 0 . 1 CO
0.23 log units, then
lOO(0.19 X 10') 0.1 % titrated = 99.999981
% from equiv. pt.
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then the values of the COas line equal to CB(l - P)/100 determine the per cent up to 100 as shown. Only Cs(l - P)/100 is considered because throughout this region OH-, H E 0 3 and H f are sufficiently low on the log c diagram to he negligible. The 100-200% interval is identifiable in an analogous manner. Beyond 200% the situation is the same as the strong acid-strong base case already described. As a general rule then, any acid-base titration can he followed along the appropriate lines on the log c diagram without the need of deriving the equations mentioned, provided these lines do not come too close to the lines of the other species in the appropriate proton condition. The log c diagram and titration curve for 0.1 M H2S with NaOH are shown in Figures 3 and 4c. Three points are worth mentioning here. First, note that the pH undergoes an abrupt initial change. This is due to
the easily repressed ionization of the weak acid, HzS. In fact, as a general rule, the lower the determining intersection on the log c diagram, the more poorly poised is the solution corresponding to this zero level. 0
-2 log c -4 -6
Redox Titrations
Only minor changes of the above method are required for redox titrations. The major difference results from the way the log e diagram is drawn. As shown in Figure 5, using reduction potential, EOZ,,Cd, as the master variable, the system points are Ei,.,.a, CA for redox couples where both oxidized and reduced form are soluble. These are quite simiiar to the acid-base systems. For those redox couples where one form is insoluble, the system is represented by a straight l i i e . V n the redox log e diagram of Figure 5 are represented the following redox couples: E0
+ Fef++ = Fe++ e + AgCl = Ag + C1-
0.771
e
-8
1
3
5
7 pH
9
1
1
1
5e f MnO4-
3
Log c diagrom for 0.01 M sulfide.
Figure 3.
It is also apparent that the lower the intersection for an equivalence point, the sharper it will be. Second, the equivalence point for the second proton is poorly defined because S- a t this concentration is a stronger base than OH- (system point to the right of the OH- line). Finally, and this speaks more to the log e diagram method, note how the pH for the 100% point, which is equivalent to a 0.1 M HS- solution, is correctly predicted, i.e., it is not the arithmetic mean of the two pK's. 14
12 10 8
"
~~6
4.
2 0 ' 2 0
4 0 b b 2 b ' & ' & ' 0
2
'
4
'
6
rnl NmOH rnl HCI ml NmOH Figure 4. Acid-base titration curves for 2 5 ml oliquots of o.
b. c.
+ 8H+ = Mn++ + 4Hn0
0.222 1.51
The analytical concentration of the Fe+++-Fe++ systemis0.10 M andthat of theMn0.--Mn++is0.02 M. The titration curve can be constructed from this diagram for the titration of 0.10 M Fe++ with MnOl- in one molar acid. The E O z .of, ~the 0% point is uncertain in a redox titration. The iron cannot be entirely Fe++ and this point is usually incalculable. In this case assume that the iron has been treated with a silver at 0% is fixed reductor, in 1M HCl, in which case Eoz.rcd as shown. At the 100% point We++] = 5[MnO,-I
which is represented in Figure 5. The intermediate percentages are filled in relative to the 50 and 200% points just as in the previous cases. These are indicated inFigure 5, and the titration curve is shown in Figure 6a. Now consider the redox titration of a mixture. Figure 7 is the log c diagram of a mixture of 0.1 M vanadium and 0.02 M iron. On it are marked the percentages for the titration of +5 vanadium and +3 iron with a chromous solution. The titration curve for a 25 rnl aliquot is shown in Figure 6b. In the transposition from Figure 7 to Figure 66, each 100% interval is converted to a volume and the intermediate volumes for these percentages are computed for plotting.
0.1 M HCI with 0.1 M NaOH. 0.1 M NozCOa with 0.1 M HCI. 0.01 M H2S with 0.1 M NoOH.
Mixtures of weakly ionized species of differing concentrations can be considered in a manner similar to polybasic acids. An example of a mixture will be treated in a redox titration. It is also worth pointing out that the titration error is estimable from the log c diagram. For instance, if phenolphthalein is used as the indicator for the HClNaOH titration and if the endpoint occurs a t pH = 8.5, it can be seen from Figure 1 that the endpoint error is +0.003%. The calculated value [using the method derived by Butler,' p. 1081 is +0.006%; this takes dilution into account. Neglecting the dilution factor the calculated error is +0.003%. Figure 5.
6 These are simply generalieations which derive from the Nernst equation. See footnote 2.
Eox.red Log c diagram for 0.1 M iron, 0.02 M manganese, ond silver in 1 M HCI.
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Precipitation Titrations
Dilution Effects
A final example is that of a titration involving a precipitation. The log e diagram for three silver halides and silver chromate is shown in Figure 8 using pAg as the master variable. Suppose that one had a solution which was 0.01 M in chloride, bromide and 0.001 M in iodide and that it is titrated with 0.01 M AgN03 solution. The pAg a t Oyois effectively determined by the initial iodide concentration. The initial progress of the titration is followed along the iodide line. As before, 0.3 log units down from 0% is 50%; 0.5 is 70%; 1 is 90%, etc. At a and b in Figure 8, bromide and chloride are precipitated respectively. a, b, and care the potentiometric endpoints, while a', b', and e' are the equivalence points. The estimated endpoint error is -0.16% for the iodide and -0.3% for the bromide endpoint. The titration curve for a 25 ml aliquot with 0.01 M AgN03is shown in Figure 9.
The dilution effects which were previously alluded to as being negligible, now will he examined. From the point of view of the log c diagram, the effect of dilution is to slide the system points vertically downward during the course of the titration. If the titration is carried out with a titraut which is very dilute compared to the solution to be titrated, then the correction will be large. But usually the titrant is as concentrated, if not more concentrated, than the solution. Consider the redox titration in Figure 5 and 6a, and examine the effect of dilution for this titration. The correction a t 50, 100, and 200y0 is zero because since the iron and manganese system points slide the same amount, the intersections occur at the same E o z , r r d . The largest correction for dilution is in the last 100% (100-200% interval). If 0.1 M is the concentration of the initial solutions of Fe++ and Mn04-, then a t 200970 the concentration of each
O.Oia
. . . . . . . 2
4
6
e ' i o '
m l KMn04
-0.5~,
o
2
4
6 rnl
8 i b i 2 1 4 1 6
Cr"
Figure 60. Redox tihotion curves for 2 5 ml aliquot of 0.1 M Fefc with 0.1 M M n 0 . i in 1 M HCl.
Figure 6b. Redox titration curves for 25 ml aliquot of 0.1 M VOiC and 0.02 M Fettt with 0.5 M Crt+.
Of additional interest is a consideration of the Mohr titration which can be made with the log c diagram in Figure 8. For any initial concentration of chloride the equivalence point occurs a t the Ag++-Cl- intersection. The CrOa=concentration required for the formation of Ag,Cr04 at that equivalence point is easily identified at the same pAg of this intersection. Analyses involving rather complicated equilibria, e.g., the Liebig-Deniges titration can be clearly sorted out in a similar fashion. The latter titration involves complex ion formation. But, although the nature of the equilibria may vary, the basic method is applicable.
would be 0.083 M and 0.0167 M respectively. The 200% point would be where the Mn++ and Mn04- lines crossed with a system point at a log c value of -1.78 [log (0.02 X 0.83) ] compared to the uncorrected value of -1.70. This means that a progressive shift downward in log c values between 100 and 220y0 is necessary for the correction. But since the maximum shift of 0.08 in log ccorresponds to less than 0.01 v upward it is hardly worth plotting the corrected value. In the case of the carbonate titration in Figure 3 there would be no correction for the 50, 100, and 150% points. Thc intervals in between would have negligible changes.
Figure 7 lobovel. Log c diagram for 0.1 M vmnodium, 0.02 M iros and 0.2 M chromium in 1 M HCIOe Figure 8 (right). Log c diagram for inrolvble silver rdtr.
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The pH of the 200% point would be shifted upward by 0.24 units [I/* (log 0.33)], the pH at 300% by 0.3 units (log 0.25) 1. In a favorable case such as the first one the dilution error is negligible. I n less favorable cases of several equivalence points the error is not negligible although still minor. For this reason students are not advised to correct for dilution.
estimate endpoint errors. After a preliminary introduction to log c diagrams the student can consider titrations involving sophisticated equilibria and quantitatively understand the basis for their procedure. I n fact, the log c diagram is more useful than the titration curve itself.
Advantages to the Method
A primary advantage to be gained by this method is a rapid and quantitative insight into a titration without being bogged down by algebra. I n terms of saving time the student can soon dispense with the preliminary log c diagram in considering simple titrations. He learns to recognize the poised system, i.e., in the region around pK. or Eo, and recalling his log table simply adds or subtracts the appropriate amount to the master variable for 50% approaching 0 and 100% in the limit. For complicated titrations, by drawing the log c diagram he can identify the well poised regions (corresponding to high concentrations of species on either side of the proton condition) and the unstable regions (corresponding to low concentrations in the proton condition). He can easily determine from the log c diagram whether a titration is feasible. He can determine endpoints and
0
10
2 0 3 0
4 0 5 0
6 0 7 0
ml b N O 3 Titrotion curve for 25 ml oliquot of 0.01 M Br; and 0.001 M I- with 0.01 M AgNOs. Figure 9.
0.01 M CI;
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