which convert vanadium(V) to lower valence states, and ions such as Hg(I), Ag(I), Tl(I), Pb(II), and tungstate which yield precipitates under the experimental conditions employed interfered with the normal determination of vanadium. These interferences were, however, easily eliminated by standard methods. ACKNOWLEDGMENT
The authors thank Umadas Mukherjee and Sameer Bose for the generous provision of facilities.
LITERATURE CITED
(1) Das Gupta, A. K., Singh, M. hi., J . Sci. I n d . Research ( I n d i a ) 11B, 268 (1952). (2) Dyrssen, David, Acta. Chent. Scand. 10,353 (1956).
(3) Hillebrand, W.F., Lundell, G. E. F., Bright, H. A,, Hoffman, J. I., “Applied Inorganic Analysis,” 2nd ed., p. 458, Wiley, Xew York, 1953. (4) Meloan, C. E., Holkeboer, P., Brandt, W.W.,ANAL. CHEM.32,791 (1960). (5) Ryan, D. E., Lutwick, 0. D., Can. J .
83, 97, 925, 3rd ed., Interscience, New York, 1959. (7) Shome, S. C., Analyst 75, 27 (1950). (8) Shome. S. C., ANAL. CHEX 23, 1186 (19>1). (9) lalvitie, N. A4., Ibid., 25,604 (1953). (10) Wallace, G. W., hfellon, Li. G., Ibid., 32, 204 (1960). (11) West, P. R., J . Chem. Educ. 18, 528 (1941). ANAL. (12) Wise, W.M., Brandt, W.W., CHEM.27, 1392 (1955). (13) Wright, E. R., Mellon, M. G., IND.EKG.CHEM.,ANAL.ED. 9, 251
(1937).
Chem. 31, 9 (1953). (6) Sandell, E. B., “Colorimetric Determination of Traces of Metals,” pp.
RECEIVED for review August 26, 1960. Accepted November 8, 1960.
Spectrophotometric Titration of Weak Acids or Bases in Aqueous Solution Using the Type II Plot STANLEY BRUCKENSTEIN and D. C. NELSON’ School o f Chemistry, University o f Minnesota, Minneapolis 7 4, Minn. The Type II plot of Higuchi, Rehm, and Barnstein for the spectrophotometric titration of weak acids or bases has been tested in aqueous solutions. Accurate results are obtained provided the ionic strength i s held constant during titration. A theoretical analysis of the range of application of the Type II plot i s considered for solutions of weak acids or bases. Experimentally, it i s found that acids (or bases) with pK 6 9.0 can be determined with an accuracy of 0.2y0 in 0.1M concentration, and 0.07M sodium carbonate can be titrated to bicarbonate with 0.2% accuracy. The Type II plot can be used to determine the end point of weaker acids (or bases) than i s possible using more conventional methods of end point detection.
T
use of an optical device as a n aid in following a neutralization reaction, either by observing the color change in the compound titrated or through the use of a n indicator, is not a recent development. Tingle (IO) used a pocket spectroscope in 1918 to observe the desired colored constituent and detected the end point visually, while Mika (8) used a spectrophotometer to follow the color change of a n indicator a t the end point. Excellent reviews of photometric titrations have been given by Goddu and Hume (3) and by Malinstadt (7). Goddu and Hume (4) described the spectrophotometric titration of various HE
1 Present address, Scientific and Process Instruments Division, Beckman Instruments, Inc., 10609 N.E. 18th St., Bellevue, Wash.
438
ANALYTICAL CHEMISTRY
substituted phenols ip aqueous solutions, detecting the end point by a plot of absorbance us. volume of base added. As the acid became veaker or more dilute, the line obtained exhibited more curvature in the vicinity of the end point. Since the end point detection is based upon extrapolation of the straight line portions obtained before and after the equivalence point in such a titration. it is necessary that a straight line region exist over a significant fraction of the titration for this method to be applicable. Grunwald (5) has proposed a mathematical method for determining the end point in spectrophotometric titrations with appreciable curvature in their absorbance-volume plots. On the other hand, Higuchi et al. ( 2 , 6 , 9 ) proposed four different ways of treating the absorbance data obtained in a spectrophotometric titration which avoid some of the problems inherent in the conventional absorbance-volume plots. This work is concerned with a detailed study of the principles and applications of one of the methods first proposed by Higuchi, Rehm, and Barnstein (6). Consider a weak base, B, and its conjugate acid, BH. (Designation of the charge type of the base or its conjugate acid nil1 be avoided until it is relevant to the discussion.) The equilibrium constant for the acid dissociation reaction
the activities and activity coefficients of the species indicated by the subscript. Defining the concentration con~, stant, ( K B H )as
where a and f represent, respectively,
Higuchi, Rehm, and Barnstein
(KBH),= K B H ~ B H / ~ H ~ B (3)
one obtains Equation 4 on substituting Equation 3 into Equation 2. (4)
If a n indicator base, I, is added t o a solution containing B and BH, the hydrogen ion concentration of this solution can be determined from the ratio of the two forms of the indicator using Equation 5, (5)
where (KIH)cis the concentration constant for the equilibrium IH=H++I
(6)
Elimicating the hydrogen ion concentration from Equation 4 by substituting from Equation 5, one obtains
Higuchi, Rehm, and Barnstein (6) assume that in the titration of B with a strong acid the reaction to form B H is quantitative. Therefore, if S milliliters of standard acid are required to reach the equivalence point and only X milliliters of standard acid have been added,
(6)
0
I
2
3
Figure 1. Titration of acetic acid using bromocresol green as indicator
-In 1.OM sodium chloride - - - No sodium chloride combined Equation 7 with Equation 8 to obtain Equation 9.
Thus, a plot of the reciprocal volume of standard acid added to the solution against the spectrophotometrically determined ratio, [I]/[IH], should be a straight line with an intercept equal to the reciprocal of the volume of the standard acid needed just to neutralize the weak base. Equation 9 differs from the result obtained by Higuchi et al. (6) in that it distinguishes betn.een concentration and thermodynamic equilibrium constants, while their result did not. It is easily shown that if a weak acid, BH, is titrated with a strong base in the presence of an indicator base, I, the expression analogous to Equation 9 is Equation 10. In Equation 10, X and S refer to the volume of standard base added during the course of the titration.
Higuchi et al. (6) did not apply Equations 9 or 10 in aqueous solutions, but did show that Equation 9 could be used in glacial acetic acid as a solvent. In applying Equations 9 or 10 to aqueous solution one must consider the two assumptions which have been made in deriving Equations 9 and 10
Figure 2. Titration of acetic acid using p-nitrophenol as indicator
-In 1.OM sodium chloride - - - No sodium chloride and BH are bases of different charge types, K , varies with the ionic strength. Consider the situation in Iyhich BH has a charge of 0 and I H a charge of THEORY - l-e.g., the titration of acetic acid Effect of Ionic Strength on ( K I H ) ~ / with sodium hydroxide using bromo(KBH)c. The titration of a n-eak acid cresol green (BCG) as indicator. Ap(or base) with a strong base (or acid) plying the limiting law to Equation 11, generally results in a change in ionic Equation 12 is obtained strength during the course of the titration. The only exceptions are the titration of a singly charged anionic weak base with strong acid where the subscripts on the activity and the titration of a singly charged coefficients indicate the charge of the cationic weak acid with strong base. ion. In the initial stages of the titraTherefore, the effect of ionic strength tion of a weak uncharged acid with on (KIH)c/(KBH)c,which was assumed sodium hydroxide using a singly charged to be negligible in the initial derivaanionic indicator acid, the value of tion, must be considered. In addition, i t K , is very nearly equal to K ~ K B H was assumed that the reaction of B with because the ionic strength is low and strong acid (or BH with strong base) f- nearly unity. In the later stages of is quantitative throughout the titrathe titration, K , is greater than I * i d tion and does not depend on K B (or KsH since the value of 1If- * increases KBH) with ionic strength. Therefore, the The effect of ionic strength on the plot according to Equation 10 is not ratio of the two concentration conlinear and the plot 1/X us. [IH]/[I] stants may be predicted using Equation is concave don-nv-ardsince K , decreases 11 LL. as 1/X increases. This has been found to be the case in the titration of acetic acid with sodium hydroxide using bromocresol green as indicator (Figure 1). Nearly all practical nnalyticaI iyork No attempt is made to calculate the involves such high concentrations of deviations from linearity because the electrolyte that the Debye-I-luckel limassumptions concerning activity coefiting law may not be used to estimate ficients are only approximate. This quantitatively the activity coefficients is clearly demonstrated nhen one conof ions. In addition, there is no good siders the case of the titration of unway to predict the activity roefficient charged acid with sodium hydroxide of an uncharged species. However, in the presence of an uncharged inqualitative conclusions can be drawn dicator acid-e.g., acetic acid using from Equation 11 if it is assumed that p-nitrophenol as indieator. From Equathe limiting law is qualitatively correct, tion 11 and the limiting la\T, it appears and that the activity coefficients of unthat all coefficients cancel and that charged species are unity. When I H and the conditions under which these assumptions hold.
VOL. 33, NO. 3, MARCH 1961
439
Table I.
Titration Ranges for Application of Equation 10 to End Point of Weak Acid. Concentration of HB, A' 0.1 0.01 0.001
p1CBIi
fmin
4 00 5 00 G 00 7 00 8 00 9 00 9 40 9 70 10 00
0.62 0.27 0.09 0.03 0.01 0.003 0.002
fmsr
fmin
fmar
fmin
fmar
0.999999 0.99999 0.9999 0,999 0.99 0.90 0.75 0.50 0.01
0.92 0.62 0.27 0.09 0.03 0.01 0. O O i
0.99999 0.9399 0.999 0.99 0.90 0,047
0.99 0.92 0.62 0.26 0.067
0.9999 0.999 0.99 0.90 0.19
.fmlD= fraction titrated Ti-hichmust be exceeded for e 5 0.001; which must not be exceeded for e 2 -0.001.
K , = KIH/KBH throughout the titration when IH and B H are of the same charge type. -4plot of the results of the titration of acetic acid using sodium hydroxide and p-nitrophenol as indicator according to Equation 10 should be a straight line. +4 straight line is actually obtained in practice. However, the value of the intercept, l/S, found is not correct (Figure 2). Obviously, specific ionic interactions must be considered in predicting the quantitative variation of K,. In the example discussed, the fact that IH and B H nere of the same charge type merely decreased the variation in K , with ionic strength to the point where the nonlinearity was not visibly obvious. There is still enough change in K , during the course of the titration to give incorrect end points. -4 practical solution to the ionic strength problem is to perform the titrations under conditions of constant ionic strength by adding a sufficient quantity of inert electrolyte (sodium chloride) to the unknon-n acid (or base) so as to obtain a n ionic strength of 1 on dilution. This solution is then titrated with 1JP sodium hydroxide (or hydrochloric acid), Effect of KBH on Accuracy of Extrapolation. I n the application of Higuchi's method t o aqueous solutions two interrelated questions arise: For what values of KB and K B H do Equations 10 and 11 apply? Over what titration ranges do Equations 10 and 11 hold? In the case of the titration of a weak base with strong acid, Equation 13
X
- ([H']
fma,
=
S - 1 (14) - [OH])(V/.V)
The difference between the approximate expression, Equation 8, and the exact expression, Equation 14, is the - [OH-]} ( V / N ) in term ([Hf] the denominator. Substituting Equation 14 into Equation 7, the exact expression, Equation 15, resultq. 1
X - ([H'] - [OH-]) V/.V
-
In the case of the titration of a weak acid with a strong base it is easily proved that 1
+ { [H+] - [OH])lT/AV-In Equation 16, X , S, and N refer to the standard strong base. Equation 15 is the counterpart of Equation 9 and Equation 16 that of Equation 10. Comparing Equations 9 and 15 or 10 and 16, it is seen that the left hand side of Equation 17 represents the error ( e ) introduced by using the approximate Equations 9 and 10 instead of the exact Equations 15 and 16.
X i ( [H"] - [OH-] } V/L\~X (17)
440 *
ANALYTICAL CHEMISTRY
eCf
=
{[H+]- [OH- ;
(18)
where f reprwents the fraction titrated, C is the sum of [B] ,BH]. and the volume i.; held constant throughout the titration. For small values of P, /H-j = ICeK h f f therefore ,
+
fraction titrated
I & ( [ H + I - [ O H - ] I V / . ' / ( N x X=/ I & e
is a n exact statement of the equilibrium ratio of the base to its conjugate acid in terms of the analytical concentrations of this acid-base couple and the equilibrium concentration of hydrogen and hydroxyl ion. Defining X and S as previously, V as the total volume after X milliliters of standard acid of normality N have been added to the weak base, Equation 14 folloirs directly from Equation 13.
calculate from Equation 1; the range of values n ithin IT hich K B H must lie for e 5 0.001 by making the substitutions C = S X S 1- and f = S / S to obtain Equation 1s
The minus sign in Equation 17 refers to the titration of a rveak base 11-ith strong acid and the plus sign to the titration of a weak acid with strong base. Assuming that the relative error in volume measurement a t any point during the titration is O.l%, no error greater than the random buret error will be introduced by using Equations 9 and 10 in place of Equations 15 and 16 when e _< 0.001. I t is possible to
\\here K , = [H+][OH-1. Table I was calculatcd ironi Equation 19 assuming e = =0.001. The positive value of e give- the minimum value of the fraction titrated ( f m > J of a neak acid vith dissociation constant K B H n hich must be exceeded if Equation 10 is not to tliffcr significantly from Equation 16. The negative value of e gives the ma\iinuni value of the fraction titrated (fmn,) \T hich cannot be exceeded under the same conditions. Table I Ehons that Equatiw 10 is valid over a considerable titration range for acids of different strength and concentration. ;It a fixed concentration of weak acid, C, decreamg KBH results in a decrease in both j,n.,and fmln, until ultimately there is only a very narroiv titration range for n hich Equation 10 holds. In order to specify juqt lion iwak a n acid may become and -till be determinable by the Type I1 plot. it is necessary to consider briefly the effect of range of extrapolation on the uncertainty of 1/S. Assuming that the same random error exists in tTTo sets of data obtained over two different titration ranges, the set of data requiring the least extrapolation will give the best values of S. It will be arbitrarily assumed that fmax must be equal to or exceed 0.5 in order for the extrapolatioii according to Equation 10 to yield a u-elldefined value of S . Under these conditions, if the weak acid concentration is 0.1M, S can be determined with equal accuracy for all acids for which KBE 2 2 X Obviously, there is an upper limit to KBE, since as KBH increases, fm,n increases (Table I). This increase in fmln poses no serious problems because even n hen KBH = 10-4, Equation 10 is follo\\ed in the range 0.63 5 f 5 0.999999. It is apparent from Table I that the Type I1 plot is ne11 suited to the deterniination of weak acids and bases in aqueous solutions. Recently, Connors and Higuchi ( 2 ) developed a n experimental technique which corrects for the incompleteness of the titration reaction and demon-
strated its usefulness in glacial acetic acid as a solveiit. Their technique should also 1%ork in aqueous ,elutions. Titration of Mixture of Two Weak Acids. FIRS^ ESD POIXT. I n a mivture of two neak acids, C molar B H and C‘ molar B’H. !There KBH>> KB“), if S milliliters of standard base are required to titrate I3H quantitativelyl Equation 20
Table II.
Titration Ranges for Application of Equation 10 to First and Second End Points of Mixture of Two Weak Acids.
log KBH K B’H ~
3.00
PI~BH
4 00 5 00 A 00 7 00 8 00 9 00
fmin
f”,,, First End Point* 0.58 0.25 0.09 0.03 0.01 0.003
0.5; 0.35 0.19 ...
0 44 0.16
0.07 , . .
... ...
5.00
4.00 fmin
fmnx
...
...
f“,S
,fI”:”
0.61 0.27 0.0tl 0.03
0.91 0 90 0.90 0 90 0.89 0.80
0.99
0.90 0.9;)
0.003
0.99 0 . 98 0 . 89
0.99999!)
0.62
0.99I19YY
0 , $,99o!l
0.82 0.13 0.10 0 . os 0 . 09
0.99099 0 . 99B9 0 . ‘J9Y 0 . !I9 0.!I0
0.01
Fecond End Point< ~KB‘H is obtained as the exact equivalent of Equation 10, nhere X 5 S.and the other symbols have their earlier significance. If the left hand side of Equation 20 is dir ided into the left hand side of Eqiiatioii 10, Equation 21 i- obtained,
1
+ e - e’
(21)
I\ here e 13 defined bj Equation 18 and e’ by Equation 22
e ‘ ( ’ >, ,f
=
[B’]
(2%)
I t is seen from Equation 21 that the error in the titration of a neak acid in the presence of a. neaker acid consists of two additive errors. One part of the error, e , is the same as that found in the absence of the second weak acid, while the second part, e’, arises from the partial neutralization of B’H. Making the wbstitution [B’]/C’ = KB”/([H+] KB~H),and assuming that [ H T ] = K B H (-~ f)/f >> KB’H, Equation 23 is obtained.
+
1 - f ( e - e ’ Y = KBHT-
f
Equation 23 differs from the expression for a solution of a pure weak acid by in the parenthe quantity C’KB~~/KaH theses of the negative term and results in a decrease in fmax as compared to the pure solution of BH. f m a X may be calculated approiimately from the expression (1 - fin,,) = -C’KB(H,’ ( e - e’ ) CKBH. If C’ = C, and ( e e’) = - 0.001, fmax 0.9 and 0.99 when KBH/KB,H = 101 and 106, respectively. This approximate expression is not accurate when KBH/KB“ = lo3 and E:quatioii 23 must be used t o calculate finaY and f m l n accurately. Table I1 was calculated from Equation 23 assuming that C = C’ = 0.1; pKBti = 4.00, 5.00, 6.00, 7.00, 8.00, and 9.00; and log KBHIKB” = 3.00, 4.00, and 5.00. It is obvious that no titration is possible \>
(e’‘’
[H+], Equation 2 i is obtained.
+ e”)C’
The error e” arises because a t the first end point all of [BH] has not been coiiverted into [B] and, as additional qtandard base is added in excess of S , part of the base will be used to conwrt [BH] into [B] simultaneousl? nith the titration of B‘H. The effect of the additional term i n the parenthesis of Equation 27 as compared to Equation 19 is to increaqe the value of fmln. The value of f m , n may be calculated approximately from the e”) expression (1 - f’)/f’* = ( e C’KBa, assuming KglH 6 0. Table I1 chons that there is a sufficiently vide titmtion range for the second end point t o be detected in a mivture of t n o neak acids when C = C’ = 0.1, if 4 5 PKB” 5 9, and log KBH/KB” 5 3. Using the approvimate expression for f m i n given above, it is found that there is insufficient titration range when C’ = C, KBH/KBfH = 100, and (e e”) =
+
0.001.
-
Polybasic Acids. The extiapolation method will apply only if the VOL. 33, NO. 3, MARCH 1961
441
successive dissociation constants differ by a t least lo3 to lo4. Under these conditions, the results found for an equimolar mixture of two weak acids can be used without a significant error in a solution of a polybasic acid. Titration of Bases. Equation 19 may be used in the case of the titration of a n-eak base with strong acid provided KB is substituted for KBH. Similarly, Equations 23 and 27 may be used for calculation of the errors in the titration of a mixture of two n-eak bases if K Band K B p are substituted for K B , and K B ' H I respectively. Accuracy Limits. I n estimating the range of application of Equations 9 and 10, it has been assumed above that a given value of 1/X does not differ from the appropriate evact expression (Equation 17, 20, or 24) by more than 0.1%. The Type I1 plot can be used over a wider range of concentration and pH values than shon-n by Tables I and I1 if less accuracy is acceptable. In the case of a solution of a weak acid, relaxing the accuracy limits permits the titration of weaker acids for a specified concentration of acid or more dilute acid solutions for specified K B H . For example, the titration of 0.01M weak acid of p K B H = 9.00 is not feasible if e is to be kept nithin the limits of 0.001, since Table I s h o w that f m j n = 0.01 and fmak = 0.047 for this case. However, if one desires to keep e within the wider limits of +0.01, one calculates from Equation 19 that f m . n = 0.003 and fmax = 0.90. The feasibility of titrating a dibasic acid to its first end point depends upon K1, K 2 , C, and the permissible deviation of Equation 9 from Equation 23. In the case of a 0.111.1 solution of a dibasic acid, pK1 = 5.00 and pK2 = 8.00, if the permissible deviation is only =t 0.001 Table I1 shows that fm,. = 0.16 and fmsx = 0.35, and the titration is marginal. If one is satisfied with less accuracy and is willing to accept deviations within the limits of =t0.01, f m l n = 0.09 and fmax = 0.90, and the titration is feasible. Similar improvements in the range of applicability of the graphical method occur in the detection of the second end point of a dibasic acid or an equimolar mixture of two acids. Treatment of Data. Values of [ I H ] / [ I ] were calculated from the absorbance data obtained during titration using the previously determined molar absorptivities of the various indicators. The method of least squares was used to calculate the conof Equastants, 8, and (KIH)J(RBH)~ tion 28
*
442
ANALYTICAL CHEMISTRY
calibrated by weighing the mercury displaced. I n all experiments, the volume of the titrant delivered was known with a relative accuracy of a t least =tO.lyoof full capacity of the buret. TITRATIONPROCEDURES. The various reagents were added to the titration cell and distilled water was used to dilute t o the calibration marking on the cell neck. Titrant was added from a microburet, the titration cell was stoppered, and its contents mere thoroughly mixed by repeated inversion. The absorbance was then determined and the entire titration n-as carried out in this fashion. There is a very slight amount of capillary leakage through the ground glass joint of the cell, but this leakage can be completely avoided if Teflon standard taper stoppers (Kontes Glass Co.) are substituted for the ordinary standard taper stopper. The use of this special cell in conjunction with microburets permits the performance of spectrophotometric titrations without modification of either the Beckman DU or Model B spectrophotometers. Scribing the mark on the cell neck permitted the preparation of solutions of known concentration directly in the cell--e.g., all Beer's law calibration curves for the various indicators used in this work mere determined by filling the titration cell to the mark with buffer of the desired pH and making successive additions of a concentrated stock solution from a 0.1-ml. microburet. Chemicals. BROMOCRESOL GREEN (BCG). A stock solution 1.30 X 10-4M in BCG was prepared and added to the titration vessel using a microburet. The indicator has a yellow acid form and a blue basic form and was EXPERIMENTAL used as a two-color indicator. Beer's law is obeyed a t the acid maximum, Apparatus. SPECTROPHOTOMETRIC 450 mp, and a t 650 mp for the basic hfEASUREhlEXTS. A battery-powered form; 650 mp is not the basic abBeckman D U spectrophotometer, sorption maximum but was chosen equipped with a IO-cm. cell compartfor convenience. The molar absorpment, was used. tivity of the acid form a t 450 mp is TITRATION CELL. A cylindrical boro5.2 X lo3 liters per mole-cm. and is silicate glass cell, 5 em. long, was connegligible a t 650 mp. The molar abstructed from borosilicate glass tubing, sorptivity of the basic form is 6.44 X optical flats, and a 10/18 standard taper lo3 liters per mole-em. a t 650 mp and joint. This cell differed from the con6.41 X lo2 liters per mole-cm. a t 450 ventional cylindrical ground glass-stopmp. [IH]/[I] was calculated allowpered, 5-em. cell supplied by Beckman ing for the absorbance of the basic in that a neck about 50 mm. long sepaform of the indicator a t the acid maxrated the ground joint and the cell body. imum from the expression A line was scribed around this neck a t a point about 15 mm. away from the cell body. The volume of water necessary to fill the cell to the marking on the neck was determined by conventional where A represents the absorbance and volumetric calibration techniques (15.19 E the molar absorptivity of the species ml. in the cme of the cell used in this indicated by the superscript a t the wave work). The actual cell length was length used as subscript. determined by comparison with a cell THYMOL BLUE (TB). A 1.25 X of certified length using the method of 10-4M stock solution was prepared differential spectrophotometry. from the uncharged acid form by adding VOLUMETRICEQUIPMENT. All voluan equimolar amount of sodium hymetric equipment was calibrated using droxide. This indicator was used as a conventional techniques. Gilmont one-color indicator, mith all values of ultramicroburets (mercury displacement type with dial micrometers) of [IH]/ [I] calculated from the absorbance of the blue alkaline form a t 595 1.0-, 0.1-, and 0.01- ml. total capacity mp and the total concentration of inwere used to deliver titrant solutions to dicator. Beer's law is obeyed a t 595 the titration cell. These burets were rather than Equation 9 or 10, because the uncertainty in [I]/[IH] is much larger than 1/X-Le., it was assumed that all the error resided in the determination of [I]/[IH]. The positive exponents of Equation 28 are used in the titration of a weak base with a strong acid and the negative exponents are used in the titration of a weak acid with a strong base. Estimation of Concentration Constants. In the previous discussion of the range of applicability of Higuchi's technique all calculations are based on the concentration constant, K,, not on the thermodynamic constant. Quite generally, the concentration constant for uncharged and anionic acids increases with ionic strength and reaches a maximum value in the vicinity of an ionic strength of approximately 1, while there is only a slight effect noted for singly charged cationic acids. If one uses the data available for citric acid ( I ) to estimate these changes, 0.30, 0.70, and 1.3 pK units should be subtracted from the thermodynamic pII values of uncharged, -1, and -2 acids, respectively. The concentration nutoprotoIysis constant of water i. 0.28 pK unit smaller in 1M sodium chloride solution than a t zero ionic strength ; therefore, corrections for bases of various charge types can be calculated from the expression K B B K B =K,. While these calculations are only approximate, they do increase the reliability of predictions based upon Equations 19, 23, and 27 and have been used below.
mp, and the molar absorptivity is 1.6 X lo4liters per mole-cm. p-NITROPHENOL. A 0.02M stock solution was prepared and added to the titration vessel with a microburet. This substance was used as a one-color indicator, with all values of [IH]/[I] calculated from the absorbance of the yellow, singly charged, anionic alkaline form a t 430 mp and the total indicator concentration. Beer's law is obeyed a t this wave length and the molar absorptivity is 9.16 X lo3 liters per mole-cm. ACETICACID. Reagent grade glacial acetic acid was diluted with distilled water to approximately 0.1M and standardized against standard 1M sodium hydroxide using phenolphthalein as indicator. HYDROCHLORIC ACID. Concentrated reagent grade acid was diluted with distilled water to approximately 1.OM. The acid was standardized against sodium carbonate using bromocresol green as indicator and boiling the solution to remove carbon dioxide just before the end point. ~-HYDROXYRENZOIC ACID. An approximately 0.034 solution of acid was made up in 1Jf sodium chloride and standardized by potentiometric titration using a glass electrode. PHEiTTLAL4NINE. Aliquots Of an aqueous phenylalanine solution were evaporated to dryness on a steam bath, and the residue was dissolved in glacial acetic acid and titrated with perchloric acid in acetic acid using crystal violet as indicator. POTASSIUM ACETATE.Reagent grade material was analyzed in the same manner as phenylalanine. POTASSIUM DIHYDROGEN PHOSPH 4TE. Analytical reagent grade salt mas recrystallized from water. The recrgstallized material was dried a t 80" C. in a vacuum oven for 1 week. SOLUTIONS. Sodium Carbonate. Analytical reagent grade material was heated a t 280" C. for 0.5 hour. The material was cooled in a desiccator and weighed out as needed. Sodium Hydroxide. A solution (40 grams per 100 ml. of water) of reagent grade material was filtered through a fine glass filter and diluted with boiled distilled water. The base was standardized against potassium biphthalate using phenolphthalein as indicator. Pcriodic testing with barium chloride solution verified the absence of carbonate. RESULTS AND DISCUSSION
Acetic Acid. Figure 1 contains plots according t o Equation 10 of the titration of -0.1M acetic acid using as titrant 1.OM sodium hydroxide in the presence and absence of 1.OM sodium chloride using bromocresol green as indicator. The line obtained in the absence of 1.OM sodium chloride is concave downward as was predicted from Equation 12. Performing the titration in 1.OM sodium chloride solution yields a straight line. The ana-
lytical results in 1.OM sodium chloride solution are given in Table I11 (experiments 1, a and b) and are satisfactory. Figure 2 contains plots according to Equation 10 of the titration of -0.1M acetic acid with 1.OM sodium hydroxide using p-nitrophenol as indicator in the presence and absence of 1,OM sodium chloride. In both cases straight lines are obtained. However, comparison of the results obtained as given in Table 111, experiments 1, c and d, 1, g and h, show that about 1.7% more acetic acid is found than is actually present if the sodium chloride is omitted. In the presence of 1.044 sodium chloride, an accuracy and precision of about 0.1% is obtained. Experiments 1, e and f of Table I11 test the effect of obtaining data in the early stages of a titration and extrapolating a considerable distance to obtain the end point. In experiment 1, e, there is a larger error than was noted in the other titrations (0.7%). A similar cxperiment involving the titration of potassium acetate with hydrochloric acid is described below. In Table 111, the values of el and eh, the error terms of Equation 19, for the actual lowest and highest fraction were calculated using pKHAc = 4.76 and the ionic strength corrections given under Estimation of Concentration Constants. Potassium Acetate. The results of the titration of -0.06M hydrochloric acid using bromocresol green as indicator are given in Table 111, experiments 2, a-c. For a titration range of -20 to ~ 9 0 %the~ data indicate A 0.2%. Experan accuracy of iments 2, g, h, and i involve extrapolation from 50% titrated and yield the poorest results. The accuracy in these A 0.570. experiments is Experiments 2, j, k, and 1 &howthat the titration of 0.007M potassium acetate can be performed with an acA 0.5%. curacy of e l and ehwere calculated from Equation 19 (suitably modified for the titration of a weak base). Phenylalanine. Phenylalanine (pK = 9.13) is a cationic acid. I t s acid dissociation constant may be considered to be independent of ionic strength as a first approuimation. Experiments 3, a, b, and c give the results obtained for 0.05 and 0.07-1.1 solutions using thymol blue as indicator. The accuracy is f. 0.6%. e l and eh were calculated from Equation 19. p-Hydroxybenzoic Acid. This is a dibasic acid with pK1 = 4.48 and pK2 = 9.40. In 1 . O X sodium chloride it is estimated that (pKI), = 4.2 and (pK2), = 8.7. Therefore, it is possible to titrate this acid with sodium hydroxide as a mono- or dibasic acid using the graphical technique. The results
-
-
-
obtained are given in Table 111, experiments 4, a-e. The first end point was detected using p-nitrophenol as indicator and the second end point was detected using thymol blue as indicator. The error in detecting the first end point is negligible while that in the second end point is f 0.5%. el and e h were calculated from Equations 23 and 27 for the first and second end points, respectively. Potassium Dihydrogen Phosphate. This compound was titrated with 1.OM sodium hydroxide t o HPOa-2 a t two concentration levels, 0.06 and 0.01M. The results are given in Table 111, experiments 5 , a-c and 5, d-f, respectively. The accuracy and precision a t both concentration levels are f 0.2%. Thymol blue was the indicator used. The values (pK2), and (pK3), were estimated as described earlier, using pK2 = 7.21 and pK3 = 12.37, and were used to calculate el and eh from Equation 23. Sodium Bicarbonate. hpproximately 0.07M and 0.007M sodium carbonate was titrated to HCO3using 1.OM hydrochloric acid and thymol blue as indicator. The results obtained are given in Table 111, experiments 6, a-e. The accuracy is &0.2% a t 0.07M and f.0.3% a t 0.0076f sodium carbonate. el and eh were calculated from Equation 23 (suitably modified for the titration of a diacid base) using (KI), = 1 0 - 6 0 5 and = 10-966 for the first and second concentration dissociation constants of carbonic acid.
-
CONCLUSION
The performance of a Type I1 titration is no lengthier than a conventional photometric titration in the case where absorbance measurements are made a t only one wave length. Slightly more time is required if absorbance measurements must be made a t two wave lengths. Ordinarily, absorbance measurements are made a t 5 to 8 points before the equivalence point. The calculation time required to process the data using the method of least squares and a desk calculating machine is 15 to 20 minutes, and about 5 minutes using an IBM 610 computer. In the latter case only about 90 seconds of operator attention is necessary. In most cases, choosing I H such that KIa = KBH is satisfactory, provided the restrictions imposed by Equations 19, 23, and 27 are kept in mind. By suitable choice of indicator concentration and wave length it would be possible to use an indicator which is 100 times weaker or stronger than BH. In practice, bromocresol green, p-nitrophenol, and thymol blue permit the performance of nearly all titrations of interest. VOL 33, NO. 3, MARCH 1961
443
Table 111.
Expt , 1, a
Determination of Acids and Bases by Photometric Titration in 1 M Sodium Chloride
Concn., M
Titration Range, 73
0.066 0.066 0.006 0,006 0.006 0.006 0.006 0,006
20-90 20-90 82-94 82-94 20-50 20-50 82-94 82-94
BCG BCG PNP
0.065 0.065 0.065 0.053 0.053 0.053 0,065 0.065 0.065 0.0066 0.0066 0,0066
Phenylalanined
p-hydroxy benzoic acidd
Compound Xcetic a
d d
IJ
c
d
e
f
ge
he 2, a b
Potassium acetate/
C
d
e
f
E 1
j
k 1
3, a
b C
4,
b C
d
e
5, R b C
d
e
f
ti, a
?;anCOsj
I3 C
d
e a
Indicators
hlilliequivalents Taken Found
Relative Error,
%
el
X
BCG BCG PXP PNjP
0 0 0 0 1 1 0 0
997 997 996 988 137 137 988 988
0.998 0,996 0.996 0.987 1.144 1,138 1.004 1.006
+0 -0 -0 -0 0 0 1 1
25-90 25-90 35-90 20-85 20-85 20-85 20-50 20-50 20-50 20-85 20-85 20-85
BCG BCG BCG BCG BCG BCG BCG BCG BCG BCG BCG BCG
0 0 0 0 0 0 0 0 0 0 0 0
9i6
976 976 800 800 800 976 976 976 0985 0985 0985
0.976 0.977 0.975 0.801 0.799 0.797 0.973 0.977 0.964 0,0980 0,0982 0.0992
0 0 +0 1 -0 1 0 1 -0 1 -0 4 0 3 0.1 -1 2 -0 5 -0 3 0 7
-0.4 -0.4 -0.4 -0.3 -0.3 -0.3 -0.2 -0.2 -0.2 -2.0 -2.0 -2,o
0.048 0.073 0.073
28-47 18-65 18-55
TB TB TB
0 731 1 096 1 096
0.735 1 ,098 1,086
0 6 0 2 -0 9
-0.4 -0.4 -0.4
0.029 0,029 0,040 0.040 0,040
75-96 -12-96 (1)75-(1)95 (1)75-( 1)95 (1)75-( 1)95
Ph-P PSP TB TB
0 0 0 0 0
439 439 607 607 607
0,439 0.439 0,602 0.610 0.606
0 0 -0 0 0
0 0 8
0.063 0.063 0.063 0.010 0,010 0.010
79-88 79-88 79-88 79-88 79-88 79-88
TB TB TR ’rB
0 0 0 0 0 0
954 954 954 1577 1577 1577
0.958 0.956 0.955 0.1573 0.1573 0.1576
0 0 0 -0 -0
0.065 0.0G5 0.065 0,0065 0 . 0065
82-90 52-06 82-96 82-96 82-96
TU
0 0 0 0 0
091 a91 991 0991 OW1
0.986 0.992 0,990 0,0988 0.0993
PNP
TB
m
TB
TB
TB TB TB
-1)
1 1 1 1 7 1 6 8
1% 12 0.1 0.1 12 12 0.1 0.1
f,
x
103~
0.07 0.07 0.04 0.04 1.4 1.4 0.04 0.04 -2,1
-2.1 -2.1 -1.4 -1.4 -1.4
-0.4 -0.4 -0.4 12 12 12
-0.6 -0.9 -0.7
5 2
0.5 0.5 0.02 0.02 0.02
0.02 0.02 I).002 0.002 0.002
4 2 1 3 3 1
0.01 0.01 0.01 0.3 0.3 0 .3
0.06 0.06 0.06
-0 *5 0 1 -0 1
-0 3 0 2
-1.5 -1 5 -1.5 1 1
0.7 0.7 0 .7 - ti -6 -6 -4 -4
BCG = bromocresol green: P S P = p-iiitrophrnol, T U = thymol hl\ie. ei = deviation from Equations 9 or 10 calculation at lowest value of titrutioii r:iiige using Equation 19, 2 3 , or 27. = deviation from Equation 9 or 10 calcxilated a t highrst v a h c of titration range using Equation 1‘3, 2 3 , or 27. Titrant used was 1.11 sodium hydroxide. Titration performed in pure water. KOsodium chlorido ;itided to hold ionic strtwgth constant. 4h
e
Titrant used was 1.11 hl-drochloric acid.
It is apparent that the linear extrapolation method of Higuchi may be used in aqueous solution provided the ionic strength is held constant during the titration. The accuracy of the method is limited by three factors: deviations of the approsimate relation used to make the graphical plot from the exact relations, photometric errors, and extrapolation errors. Under ideal conditions-Le., titrations for u-hich no significant deviation from Equations 9 or 10 occurs-the photometric and extrapolation error determines the ultimate accuracy. -4ccuracies of -0.2% for titrations are estimated n hen it is permissible to include data obtained up to 9Oy0 titrated. Extrapolation of data from a maximum titration value of 50% is a less reliable procedure, and 444
e
ANALYTICAL CHEMISTRY
accuracies of the ordcr of 0.5% are to be expected. T h e predictions of Table; I and I1 are in subhintial agreement with the experimental results and calenrly define the range of applicability of the Type I1 plot in aqueous solution. ACKNOWLEDGMENT
Tlic author5 thank I. 11. KolthofY for hi, comments during the ryperimental portion of thiq n o ~ k . This nork u a s zponsored, in part, by the Office of Ordnance Research. LITERATURE CITED
( 1 ) Bruckenstein, S.,Iiolthoff, I. AI., in “Treatise on -1nalytical Chemistry,”
I. 11. Iiolthoff and. P. J. Elviiig. eds., Chap. 12, Interscience, SPK Tork, 1959. ( 2 ) Connors, K. -I., Higuchi. T.. AXAL. CHEM.3 2 , 9 3 (1960). (3) Godtlu, R. F., Hume. D T.,Zbid., 26, 1679 (1954). ( 4 ) I b i d . , p. 1740. ( 5 ) Grunn-ald, E., Ibid., 28, 1112 (19.56\. (ti) Higuchi, T., Rehm, C.> Barnstein, C., I b i d . , 28, 1.506 (1956).
( 7 ) llalmstadt, H., Rec. C R e v l . Progress 17, 1 (1956). (8) lIika, J., 2. anal. C h c m . 128, 159
(194FI’I. ( 9 ) Reiim, C’., Higuchi, T.. . 4 s i ~ CHEX. . 29, 367 (1957). (10) Tingle, -A,) J . :lm. Chrni. SOC.40, 873 (1918).
KECEI~EII for review July 27, 1960. Accepted October 21, 1960. Taken in part from a thesis submitted hv D. C. Selson in partial fulfilment of the requirements for the degree of master of science.
