AN AID IN THE CONSTRUCTION OF TITRATION GRAPHS GRANT W. SMITH University of North Dakota, Grand Forks, North Dakota
TmRamoN graphs are commonly used in elementary courses in quantitative analysis to demonstrate the change in concentration of an ion during the course of the titration. A neutralization curve, for example, represents the pH values continuously from the start of the titration to the end point or well beyond the end point. Titration graphs are extremely useful to the teacher of analytical chemistry as a device for showing variations in the sharpness of end point with changes in concentration of reagents, type of reagents used, and nature of the indicator. This graphical method of presentation is applicable to nearly all types of titration systems, including neutralization, precipitation, oxidation-reduction and potentiometric titrations. The general usage of this device is emphasized by the fact that practically all modem textbooks of quantitative analysis employ it, although there is considerable variati6n in the degree of detail with which it is discussed. It is common practice, also, to present the various methods of computation of ion concentrations necessary for the constrnction of titration graphs. These methods involve the use of the Law of Mass Action and chemical equilibrium and include the more common cases of dissociation of weak electrolytes, common ion effects, buffer solukions, hydrolysis of various types of salts, solubility product and oxidation-reduction equilibria. Probably every experienced teacher of quantitative analysis fully realizes the difficulty which the average student experiences in attempting to construct even the simple titration graphs due to the complexity of the subject. The fundamental difficulty is the failure to f o m a clear picture of the nature of the titration system a t various stages of completion. If one does not realize clearly that a t certain stages a neutralization system may consist of a simple solution of an acid, while a t other stages it may be a buffer solution composed of a weak acid and its salt, and a t still another stage it may be simply a solution of a salt, it is obvious that he cannot properly apply the appropriate mathematical formulations for the constrnction of the titration curve. In order to aid students in the elementary analysis courses who experience this common difficulty, the writer recently devised a graphical method of presentation to help them form a clear picture of the nature of the system they are dealing with. The use of this method in the classroom met with such marked success and favorable reaction from the students that it was
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3
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b*
F
E
1:
0 Figum 1.
20
40
60
80
G
B
100 120
140
.
160
PER CENT NEUTRALIZED The Titration of m AcLcid with BConc.n*ation
180
200
of th- S mm.
felt that others might also like to consider it for their use in similar situations. The following presentation of the method is intended merely as an illustration of its application to two cases: simple neutralization and simple precipitation reactions. It is felt that the teacher or student may readily adapt the procedure to the more complex cases as the need arises, since the method is in itself quite simple and direct. Since this device is intended only to show graphically the nature of the system a t any given stage of the reaction, the student must still have a sound knowledge of the principles of chemical equilibrium in order to apply the information to the actual construction of a titration graph. SIMPLE NEUTRALIZATION SYSTEMS
The case of the titration of a monobasic acid, HA, with a base, BOH, is represented in Figure 1. The reaction is HA
Acid
+
BOH Base
-
BA Salt
+
H20 Water
The ordinates in Figure 1 represent equivalents of solute and also total volume of solution as the titration progresses. The abscissae represent per cent neutrali-
1E18
APRIL, 1949
nation: point B represents 100% neutralization, i. e., the end point of the reaction; points on BC show the amount of excess base added after the end point is reached. At C, "200% neutralization," an amount of excess base equal to the amount of original acid present has been added. In order to show the change in volume, the line V oto Vzwhas been dravn to illustrate the case in which the acid and base are of equal concentrations. If the concentrations of acid and base differ, this volume line will have a different slope, but is always straight. VOis the original volume of the sample. I t is readily seen that the amount of acid decreases linearly to a value of zero a t the end point, and that the number of equivalents of acid remaining a t any intermediate point in the titration is given by points on EB. Simultaneously, the amount of salt formed increases from zero a t the start of the titration to the value B D a t the end point. Beyond the end point, as the addition of base is continued, the amount of salt remains constant, while the amount of excess base increases linearly from zero to the value, CF, which is equal to the amount of salt and also the original amount of acid. Concentrations are easily determined by taking into account the actual equivalents of acid, salt, or base present a t any stage of the titration and observing the value of the corresponding volume given by points on the volume line. An example will serve to illustrate the general method of use of the graph. In the titration of 100 ml. of 0.1 N acid, HA, with 0.1 N base, BOH, the ordinate A E represents 0.01 equivalent of acid in the original sample and VOequals 100 ml. At 60% neutralization, wx and x y equal the number of equivalents of acid and salt, respectively, and the volume increase is equal to yz. Since wx represents four-tenths of the original acid and the volume has increased to 160 ml., the concentration of acid is now
regard the hydrogen-ion concentration as equal to the total ooneentrstion, i. e.,
Similarly, the concentration of salt is
Reyion 1. Using the equilibrium expression for the ionization of a weak acid, we solve far the hydrogen-ion concentration in the usual manner. I n most eases we can use the simple expression
(0.6)(0.1)(100/160) = 0.0375 N
More simply, since the ratio of salt to acid is seen to be 3: 2, the concentration of salt equals (3/2) (0.025 N) = 0.0375 N Four types of neutralization reactions will now be discussed in detail. For convenience in the following discussions, let us divide the titration process into four parts: first, the sample at-the beginning of the titration, consisting of a solution of an acid; second, the addition of base up to, but not including, the end point (region AB in Figure 1); third, the endpoint (point B), a t which we have only a solution of a salt; and fourth, the region beyond the end point, during which interval base is being added in excess (region BC in Figure 1). Type I : Strong Acid Tilrated with Strong Base. Consider the titration of 100 ml. of 0.1 N solution of strong acid with strong base of the same strength. Region 1 (Point A). Sinoe the sample is a strong acid, we m i y
[Ht]
=
C, = 0.1 M
Region 8 (Region AR). I n the mixture of the strong aoid and a salt, the hydrogen-ion eonoent,retion is considered to hc thc same as the concentration of bhe acid:
[H+] = C. At 60% ncutrslization, for examplc,
[Ht]
=
(0.4)(0.1)(100/160) = 0.025 M
I t is well to point out that [HC]is not equal to C. in the region very close to the end point, since the ionization of water itself hecomes significant in this region. However, for the purpose of constructing the simple titration curve, i t is generally unnecessary t,o consider points in this very limited region. In the example under dkouqsion, [H+1 may be considered prrteticslly equal to C , s t least down to u,ithin about 0.02% of the end point. This limitation is somewhat greater when more dilute solutions are undor consideration. Region 9 (End Point, B ) . Here we have a salt of the NaCl t.vpv, whose concentration C. = (1)(0.1)(100/200) = 0.05 AT For a solution of this type of salt, since hydrolysis is negligible, the hydrogen-ion concentration is t,he same it8 that for. p u x water, 1 0 ' (room temperature). Region 4 (Regim RC). Here the situation is analogous t,o that in region AB, except that 8. strong base instead of strong acid is present with the salt. We consider the hydroxyl-ion coneentration as equal to the concentration of base; the effect of thc salt is negligible for our purpose. At 140y0 neutralized, since the total volume is 240 ml.,
[OH-]
=
Ca = (0.4)(0.1)(100/240) = 0.0167 121
Then
whom K , is the ion product constant for water.
Type 2: Weak Acid Tilrated with Strong Base. An example of this type would be the titration of acetic acid with sodium hydroxide.
[Ht]
=
dm.
as an accrptable approximation, u.here C. represents the concen-
tration of the acid and K,is the ioniaation constant. Regzon 8. As soon as base is added to the acid, a mixture of the wcak acid and its salt is formed. This is a buffer mixture, and tho hydrogen-ion concentration for such a solution is
where C,/C, is the ratio of the concentration of the acid to that of the salt. Sinoe it is not necessary t o know the actual concentrations to evaluate this ratio, the value is easily read directly from the graph. For example, a t 60 per cent neutralization, the ratio hence, C./C. ii 0.4/0.6 or
[H+l = It m y be desirable to demonstrate that the evaluation of the ratio C./C, does not require a. knowledge of the actual concentration values. This is easily done by calculating the actual values to show that the same factor occurs in both numerator and denominator, i. e.,
JOURNAL OF CHEMICAL EDUCATION
190
Region 4. As in Region 2 preceding the end point, in Region 4 we have a buffer solution. This buffer solution is composed of the excess of weak base added and the salt formed during the nentralization. The amount of salt remains constant, but the amount of bsse increases linearly. For such a buffer
[OH-] =
c* Ks -. C,
Since only the ratio of the concentrations, Co/C., is involved, the relative amounts of bsse and salt obtained directly from the g a p h can he used without actually calculating the concentrstions. At 140% neutralized we have 0.4 [OH-] = -.Ks = 0.4Ks 1;o From the value of [OH-] the hydrogen-ion concentration is easily found.
Type 4: Weak Base Titrated with Strong Acid. If the "acid" and "base" regions in Figure 1 are interchanged, the graph of the titration of a weak base with a strong acid is obtained. It is obvious that this is the counternart of T v ~ 2e and is treated similarlv. rsgulaa.
PER CENT REACTION COMPLETED ~ h . ~ i t - t i ~.f ~ N.CI ~ i t hA ~ N O . of the same concentration
In the region AB close to A, the simple formula for a bu5er cannot be applied, since C 8 does not approximate the true anion concentration satisfactorily. One should avoid the use of the simple bu5er formula until approximately 10 per cent of the acid has been neutrdiaed. Immediately adjacent to the end point, B, it is necessary to use discretion in applying this simple calculation, also, as noted in the case of Type 1, h g i o n 2. Region 3. At the end point of this resction we have a solution of thesalt of the weak acid and strong base. Thissalt hydrolyzes, and the hydrogen-ion concentration is given by the expression
The value of Cs,the concentration of the salt, is seen to be half that of the original acid sample, since the volume has been doubled and the total equivalents of salt equal the original equivalents of acid. Region 4. Since t h r bnw pw.;rnt i r ~this raw is astrongonc, the anmt m+tlwd of caleulnrim is u r d a - in rhc corrrspnnd~ngregion of Type 1above.
Type S Weak Acid Titrated with Weak Base. The titration of acetic acid with ammonium hydroxide is an example of this type. Region 1. The hydrogen-ion concentration of the sample is found in the same manner as in the preceding type. ) Region 2. Again we have a bu5er solution, as in Type 2, so the treatment is the same as it was in that case. Region 5. The salt present a t this end point is a salt of a weak acid and weak base,~andwill be appreciably hydrolyzed. However, if the ionieation constants K . and K Oof the acid and base, respectively, are about equal, the pH of the solution will be practicallv that of m r e water. i . e.. about 7 . In eeneral the hvdroeen-
".
Region I .
The sample is a weak bsse, so [OH-] = d ? Z ,appmximately
Region B. The solutionin this region is a buffer consisting of the weak base and its salt. I t is treated as in Type 3, Region 4. The same limitations to the utility of the simple hu5er formula pointed out under Type 2, Region 2, apply here, also. Region 5. The salt solution a t the end point is farmed from the weak base and strong acid; hence,
Region 4. The buffer solution in this region is of the same type and is treated in the same way as that in Type 2, Region 2, or Type 3, Region 2.
Extension of the treatment to cases involving polyacid bases and polybasic acids, if desired, may be carried out easily.
SIMPLE PRECIPITATION SYSTEMS Figure 2 shows the graph of the titration of a soluble chloride with silver nitrate. It is similar in form to that for the neutralization system. As an example, consider the titration of 100 ml. of 0.1 N NaCl with 0.1 N AgNOs. As before, we shall divide the process into four regions. Since we are dealing with solubilities, we must use the solubility product relationship in our mathematical analysis: [Ag+l[Clj = KS.P Region 1. Our original sample is a solution of NaCl and since it is completely ionized, the concentration of chloride ion,
[CI-] = C. = 0.1 M where C . is the concentration of the NaCI. Region 2. As AgN03is added from the buret, AgCl is precipitated and the amount of C1- remaining in solution decreases linearly. At any stage of the reaction in this region the vslue of [CI-] is readily determined from the graph by considering the number of equivalents remaining and also the total volume of the solution. For example, when 80% of the chloride has been precii)itat,ed (80 ml. of AgNOa added)
APRIL. 1949
since 0.2 of the original salt r e d s unpreeipitated and the volume of the solution has increased from 100 ml. to 180 ml. In the region uerg close to the end point, the method illustrated shove cannot he applied, since the AgCl itself produces an appreciable quantity of Agf and C1- ions In the example cited, assum, um of the simple method ing a value of 1.1 X 10-'ofor K s . ~ .the of calculation begins to yield significantly erroneous values for [CI-] when more than 99.9 ml. of AgNOs have been added. More accurate values may he readily obtained in this region, if desired, by the proper consideration of the ions from both the soluble chloride and the slightly soluble AgCI. In actual practice, however, it is rarely necessary to carry the calculation into this very limited region in order to determine the course of the titration curve. I t is easily shown that this region hecomes smaller and less significant the more insoluble the salt formed in the titration, and it increases with decreasing concentration of solutions used. Region 3. At the end point the chloride is present as the silver salt, and the only chloride ions in solution are those which result from the slight solubility of silver chloride. Thus,
Regim 4. Beyond the end point we are adding AgN03 in excess, and this remains in the ~ d u t i o ncausing the concentration
191
of silver ions to increase linearly. The Ag+ concentration is the same as that of the AgNOj added in excess. I t is calculated by consideration of the number of equivalents of AgNOn present at any specified time, as shown by the graph, and also the volume of solution. At 160%, the Agf concentration, [AgC] = (0.6)(0.1)(100/260) = 0.023 M The C1- concentration is then readily dotcrmined from the soluhility product relationship.
In conclusion, a graphical method has been outlined for the classroom oresentation of background material necessary for the thorough understanding of the constmction of titration graphs. Detailed discussion of two simple types of titration, neutralization, and precipitation, has been given. The method js easily capable of extension to most of the other types of titration reactions, and should help the student form a clear mental picture of the processes involved. The author would he pleased to hear from teachers who may try the method in their classes, and would welcome suggestions as to its improvement or modification.
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