Here u1, UZ, us, uls, and ups are the mobilities in the membrane phase of anion 1, anion 2, the dissociated resin cation, the undissociated ion pair of the resin cation and anion 1, and the undissociated ion pair of the resin cation and anion 2, respectively. kl and k2 are constants related to the standard chemical potentials of anion 1 and anion 2, respectively. In the external solution and membrane phase, Kl and Kz are dissociation constants of ion pairs of the exchanger cation with anion 1 and anion 2, respectively,and
Equation 11 differs from the empirical Equation 1 in that the former consists of two logarithmic terms, each having the same form as the single term in Equation 1. However, if 7 is very nearly unity or zero, one of the two terms in Equation 11 becomes negligible and Equation 11 reduces to the same form as Equation 1. Alternatively, as has been suggested by Sandblom, Eisenman, and Walker (14), Equation 11 can be replaced with good approximation by an equation consisting of a single logarithmic term, by introducing “average ionic selectivities.” Hence, the selectivity ratio appearing in Equation 1 may be regarded only as an average value and, therefore, may show the kind of variation we found experimentally. Further, the selectivity ratio in the second logarithmic term of Equation 11 has been equated to the ion-exchange equilibrium constant since it is reasonable to assume that uls = uzs (14). However, in the derivation of Equation 11, the dissociation constants KI and Kz have been defined only in terms of concentrations of the various species in the membrane phase and not in terms of their activities. Con-
5
sequently, the quantity ! ! is only the ion-exchange kl Kz selectivity coefficient and not the ion-exchange equilibrium constant. It is interesting to note that the ion-exchange selectivity coefficients of certain anions in the case of solid anion-exchangers show a decrease at low concentrations of the ion of interest in the external solution (18, 19). Thus, the decrease in selectivity ratios at low concentrations, observed in this study, might be related to a similar decrease in the ionexchange selectivity coefficient, itself. While the specific origin of this special effect can only be surmised at this point, it is nevertheless clear that the selectivities of the liquid membrane electrodes we studied are largely consistent with current theories of “mobile-site’’ type membrane electrodes. Furthermore, the adoption of our graphical procedure permits the evaluation of selectivity ratio values which are both internally consistent and applicable to a range of solution conditions. This finding is highly encouraging because it demonstrates that meaningful comparisons can be made between individual electrodes, as well as classes of electrodes, used under different experimental conditions and in different laboratories. RECEIVED for review April 7, 1969. Accepted June 12, 1969. Financial support of the Office of Saline Water and the University Program for Scientific Measurement and Instrumentation is gratefully acknowledged. (18) B. Chu, D. C. Whitney, and R. M. Diamond, J . Inorg. Nucl. Chem., 24, 1405 (1962). (19) H. Levin, W. J. Diamond, and B. J. Brown, Ind. Eng. Chem., 51, 313 (1959).
Difference between the Inflection Point and the Equivalence Point in Coulometric Titrations of Weak Acids George Marinenko and Charles E. Champion Institute .for Materials Research, National Bureau of Standards, Washington, D . C . 20234 Mannitoboric acids of varying pKa were successfully used to test independently the validity of the Roller equation for evaluation of the error due to noncoincidence of the inflection point and the equivalence point of acid base titration. The use of mannitoboric acid in media of different mannitol concentrations has provided the first test of the Roller equation under conditions where both parameters, c and K, are varied for a single acid of known stoichiometry. Moreover, the applicability of the Roller equation to complex systems such as boric acid-mannitol has not been established prior to this investigation.
IT HAS BEEN KNOWN for many years that as the acids become progressively weaker and the hydrolysis of their salts more pronounced, the inflection point of a potentiometric acid-base titration curve occurs prior to the stoichiometric equivalence point. The subject has been treated theoretically in great detail in the early part of this century (1-11). According to Roller ( 4 ) the difference between pH at the equivalence point (ep) and the pH at the inflection point (ip) can be represented by the following equation: (pH)ep 1208
- (pH)ip
= 0.65 dKW/cK,
ANALYTICAL CHEMISTRY
where K, is the ion-product constant for water (K, = at 25 “C), cis the concentration of the salt of the weak acid at the equivalence point (moles/liter), and K, is the dissociation constant of the weak acid. It is apparent from this equation that (pH),, - (PH)~,in an unsymmetrical titration of weak acid with strong base becomes larger as the concentration of salt at the equivalence point is decreased and as the acid becomes weaker. Converting the above equation to reflect the error in the titer as a function of cK,, Roller arrived at essentially the following relationship : E. D. Eastman, J . Amer. Chem. Soc., 47,332 (1925). E. D. Eastman, ibid., 50, 418 (1928). E. D. Eastman, ibid., 56,2646 (1934). P. S. Roller, ibid., 50, l(1928). P. S . Roller, ibid., 54,’3485 (1932). P. S . Roller, ibid., 57,98 (1935). S . Kilpi, 2.Phys. Chem., Leipzig, 173, 223 (1935). (8) Ibid., p 427. (9) Ibid., p 435. (10) Ibid., p 239. (1 1) D. A. MacTnnes, “The Principles of Electrochemistry,” Reinhold Publishing Corp., New York, N. Y.,1939, pp 307-312.
(1) (2) (3) (4) (5) (6) (7)
A
=
-3
X lo-'' (cK,)-'
where A is the percentage error in the titer due to the [(PH),~(pH)i,] difference. Several methods have been used to study, experimentally, the dependence of the inflection point on cK,. Valik, in collaboration with MacInnes, conducted differential potentiometric titration of the dibasic aspartic acid, in which K,, = 1.5 X and K,, = 2.5 X 10-lo. However, these investigators could not find any difference between the titer to the first inflection point and that from the first to the second inflection point. MacInnes pointed out that the difference between the stoichiometric and potentiometric end points increased as the acid or base became weaker, but the difficulty of locating the end point also increased. He stated that "it may be safely concluded that within the accuracy to which the potentiometric end point of a titration can be established, it is identical with the stoichiometric end point" (11). Bates and Wichers (12), in their precise intercomparison of acids by differential potentiometric titration with hydrogen electrode, were also concerned about the difference between the inflection point and the equivalence point. It was pointed out that a study of the course of titration curves adjacent to and on both sides of the inflection point was made by Bates and Canham (12). The inflection point determined experimentally appeared at pH lower than the equivalence point computed from the dissociation constant of the acid and the ion-product constant for water, along with the estimated values of the activity coefficients. Moreover, the differences were somewhat greater than the uncertainties in the calculations and larger than those estimated by the Roller equation. In the course of the coulometric investigation of boric acidmannitol system in our laboratory, pK, of boric acid was measured in media containing different amounts of mannitol. The results of this study indicate that by merely varying the concentration of mannitol one can have a whole spectrum of acids, varying smoothly in pK,. The continuum of acids thus produced offers a unique possibility for the study of the relationship between the inflection point and the equivalence point. EXPERIMENTAL
The measurements were conducted using the previouslydescribed instrumentation and apparatus (13). Because, in coulometric acidimetry, it is advantageous to employ 1M KCI as the supporting electrolyte, all measurements reported here were made in aqueous 1M KC1 at 25 "C. No attempt was made to control the temperature but on the basis of other measurements in this laboratory, it is known that the temperature of the environment does not differ from 25 "C by more than a degree. In the determination of pK, of boric acid as a function of mannitol concentration, two concentrations of boric acid solutions were used. They were prepared by weighing appropriate amounts of borax (NBS pH standard), addition of a precalculated weight of mannitol to produce the desired final concentration of mannitol, and dilution to volume with 1M KC1. Nitrogen was passed through the solutions to remove Con, after which the pH was measured. Because this borax is very close to stoichiometric Na2B407. 10H20, or in the resulting solution
--
is close to 1, the
CE3803
(12) R . G. Bates and E. Wichers, J. Res. Nut. Bur. Stand., A , 59, 9 (1957). (13) G. Marinenko and J. K. Taylor, ANAL.CHEM., 40,1645 (1968).
0*51 0.3
T
0.2
W
t 0
z
e
J
0.05
z z 0.03
b z
0.02
0
gE
0.01
u
z
8
0,005 0.003
0.002
0.000
& +& - I
I/,
3
4
5
7
6
d
8
9
PK
Figure 1. Effect of mannitol concentration on the apparent ionization constant of boric acid in 1MKCl at 25 "C 0 5 X 10-3M boric acid 0 3 X 10-2M boric acid A 2.5 X 10-*MKH2P044- 2.5 X 10-*MNa2HPO4 0 1 X 10-2M potassium acid phthalate (pKJ
value of pH measured, was taken to be equal to pK,, where pK, is the apparent dissociation constant of the boric acid. A plot of the dependence of pK, of boric acid in media containing different amounts of mannitol is shown in Figure 1. The effect of mannitol on the ionic equilibria of phthalate and phosphate buffers was also measured. These two bulTers were used in the role of controls and the appropriate curves for them are also plotted on the same figure. The values of pK, (or pH) plotted on the abscissa axis were obtained by measurements of the same parameters in the absence of mannitol and the horizontal arrows indicate the apparent change of these parameters caused by the presence of 1M KCl. A number of different size samples of virtually stoichiometric boric acid were titrated coulometrically to a differential potentiometric inflection point in 1M KCl containing various amounts sf mannitol. In the case of a very weak acid (K, N 10-9 to lo-'), it was difficult to obtain an accurate inflection point by a single titration. Consequently, in these cases, after the accurate potentiometric curve was obtained, a small amount of HC1 (10-15 peq) was added to the titration cell to neutralize the excess KOH which was generated beyond the end point and to regenerate some of the boric acid, returning the system to the region preceding the inflection point. Subsequently, the sample was again retitrated. After several such repeated titrations, a more reliable value of (pH)i, was established. Figure 2 shows data for two such titrations. Both the upper and the lower curves show titration of approximately the same size sample of boric acid. The upper titration ( A ) was carried out in 1M KCI, while the lower curve ( B ) corresponds to a titration carried out in 1M KC1 and 0.25M mannitol. The parameters calculated for the systems are VOL. 41, NO. 10,AUGUST 1969
1209
10.0
PH
EQUIVALENCE
WlNT
9.6
9.4 9.0
-
8.6
-
8.2
-
7.8
-
PH
I
9.4
7.4 -
t-
7.0 I
1
I
1
I
MH3B03; K , = 3.98 X loT6;cK, = 2.40 X
shown in the legend of the figure. The presentation of these titration curves is somewhat unconventional and for that reason it merits some explanation. It is well known that a relatively smooth potentiometric titration curve, when plotted as a first derivative, may show a significantly large amount of scatter. Accordingly, while the potentiometric curve itself may appear to be very precisely reproduced under a given set of conditions, the random scatter of points in the differential plot would impair precise determination of the inflection point. When the acid is as strong as B, in the lower part of Figure 2, the pH difference measured for a reaction increment is sufficiently large that the uncertainty of the measurement itself does not affect significantly the differential titration curve. On the other hand, for boric acid titrated in noncomplexed form in 1M KCl, shown in the upper part of Figure 2, the p H differences measured per microequivalent of hydrogen ion reduced amount to only a few hundredths of a pH unit and therefore the random errors of measurement affect significantly the differential plot. For this reason, the multiple retitration method was used in order to identify more precisely, the pH of the inflection point. After the sample is titrated beyond the inflection point, a few hundredths of a ml of strong acid such as hydrochloric acid are delivered into the titration cell thus regenerating some of the boric acid, after which a differential potentiometric titration is carried out. The composite curve resulting from 3 such repetitive titrations is shown as A of Figure 2. It is seen that for titration A , where the cK, value is 7.27. 10-13, the potentiometric inflection point occurs approximately 3 prior to the calculated equivalence point, whereas for cK, value of 2.40. in B,it is practically coincident with the equivalence point. The calculated values of the fraction titrated are based on the assumption that assay of the same material in an electrolyte containing higher concentrations of mannitol and boric acid is accurate. The validity of this assumption is substantiated by titration of a number of large samples in an electrolyte containing concentrations of mannitol as high as 1M. It was found that the assay of boric acid between cK, of 10-8 and lop4 is 99.999,% and it is independent of cK, (within 0.003 %). 1210
ANALYTICAL CHEMISTRY
i 1
u
Figure 2. End points of titrations of 0.06 meq samples of H3BOaat different apparent values of K, A . C = 6.32 X 10-4 MH3B03; K, = 1.5 X cK, = 7.27 X 10 -13 B. C = 6.02 X 10 -9
\
Figure 3. Error encountered in coulometric titrations of mannitoboric acid at different cK, values due to the difference between potentiometric inflection point and the equivalence point Titrations conducted in: 0 lMKCl ( K , = 1.2 X 0 1M KCl 0.047M mannitol (K, = 2.4 x 10-7) 0 1M KCI 0.25M mannitol ( K , = 4.0 X loW6) Solid line represents the error calculated by Roller equation
+ +
RESULTS AND DISCUSSION
On the basis of data shown in Figure 1, it is evident that in mannitol concentration range from 2 x to l M , pKa of boric acid is a linear function of the log C ,, where C, designates the molar concentration of mannitol. In this range of mannitol concentration,
Because neither the phthalate nor phosphate hydrolytic equilibria are affected by the same concentration of mannitol as evidenced by the constancy of pH of these two buffers, it seems reasonable to conclude that the changes in pK, of boric acid result from the mannitol-borate interaction (complexation). The elucidation of the hydrolytic equilibria in this system is under the investigation in our laboratories and will be described elsewhere. The data obtained in titrations of boric acid for several values of cK, are summarized in Figure 3. In the figure, experimental points for error in the assay of boric acid titrated in different media are plotted against the appropriate CK,. For CK, = 2.5 X the point on a graph represents an average value because some titration results were greater than 100% and the error function for these results could not be represented on this log graph. The experimental value of the error of this point is 0.002 =t0.010 absolute per cent, where the uncertainty figure represents the standard deviation of the mean based on 6 degrees of freedom. The solid line represents the plot of the titration error predicted by the Roller equation. It can be seen that the errors encountered due to the difference between the inflection point
and the equivalence point, determined in these titrations, are in an excellent agreement with the errors predicted by the Roller equation over 5 orders of magnitude of the cKa values. The use of mannitoboric acid in media of different mannitol concentrations has provided the first test of the Roller equation under conditions where both parameters, c and K,, can be varied for a single acid of known stoichiometry. Moreover,
the applicability of the Roller equation to hydrolytic equilibria involving a complex acid in a medium which is not simply water has been verified, a fact which was not known until the present investigation. RECEIVED for review February 2 5 , 1969. Accepted May 14, 1969.
High-Frequency Titrimetric Determination of Total Base Number of Lubricating Oils Brian P. Caughley Wellington Polytechnic, Private Bag, Wellington, New Zealand Michael V. Joblin British Petroleum (New Zealand) Ltd., Waiwhetu, Wellington, New Zealand One of the tests performed on lubricating oils is the measurement of their “Total Base Number” (TBN). The ASTM and IP method of measuring TBN involves titrating an alcoholic solution of the oil with alcoholic hydrochloric acid, and plotting a graph of pH reading against the volume of acid added. This method is rather slow, and the results are not very reproducible. Another disadvantage of this potentiometric method is that the graph obtained does not always show a satisfactory end point. Also, during the determination, carbon particles in used oils tend to adhere to the electrodes, and this contamination reduces the reproducibility of the result. An alternative method of determining TBN values has been developed using high-frequency measurements. This method aims at speeding up the determination of TBN, eliminating the possibility of contamination of the electrodes, always obtaining a sharp break in the graph at the end point, and obtaining a more reproducible and more meaningful result.
ONEOF THE TESTS performed on new and used lubricating oils is the measurement of the “Total Base Number” of the oil. The basicity of an oil is due to organic and inorganic bases in the oil, such as amino compounds, salts of weak acids (soaps), salts of heavy metals, and addition agents such as inhibitors and detergents. Basicity is an important property of oils because the bases offset acidity (and hence corrosion of metal) which arises from oxidation of the oil during its use. The Total Base Number (TBN) of an oil is defined as: “The quantity of acid, expressed in terms of the equivalent number of milligrams of potassium hydroxide, that is required to neutralize all basic constituents in one gram of sample” ( I ) . The ASTM and IP method ( I ) of measuring TBN involves titrating an alcoholic solution (isopropanol/toluene) of the oil with standardized alcoholic (isopropanol) hydrochloric acid, and plotting a graph of pH reading (measured on a pH meter) against the number of milliliters of hydrochloric acid added. This method is rather slow because of the slow adjustment of the potential and the large number of points which must be recorded; a single determination may take at least 30 minutes. Also, the range of results is fairly wide. Table I shows the (1) “1966 Book of ASTM Standards” (Part 17), p 239, ASTM Designation D664-58, Philadelphia, Pa., 1966. “I. P. Standards for Petroleum and Its Products,” 26th Ed., Part 1, p 762, I. P.
approximate range of results for oils of various TBN values, determined potentiometrically. It is difficult to assess the reliability of these results, particularly for used oils, unless a more accurate and meaningful method of measuring TBN can be found. Another difficulty in the potentiometric titration method is that the graph obtained frequently does not show a satisfactory end point (see Figure 1). To overcome this problem (which is particularly marked when dealing with used oils of low TBN) a standard buffer solution, of 2:4:6 trimethyl pyridine (gamma-collidine) and hydrochloric acid in isopropanol is used. The apparent pH of this buffer solution (as indicated by pH meter) is measured accurately, and the end point in the titration of the oil solution with alcoholic hydrochloric acid is taken at the apparent pH of this nonaqueous buffer (approximately pH = 4). Whether the result so obtained is very meaningful, and actually measures TBN as previously defined, is questionable. Still another problem with this method is that carbon particles which occur in used oils tend to adhere to the electrodes during the experiment, and this contamination may also reduce the accuracy of the result. Careful, regular cleaning of the electrodes in cold chromic acid solution is thus essential. A lubricating oil is normally replaced when its TBN falls below a certain level (which is, of course, determined by several different factors), but as can be seen from Figure 2, the time for replacement may be difficult to determine because of these errors in measuring TBN. An alternative method of determining TBN values has been developed using high-frequency measurements. This method Table I. Range of TBN Values Obtained Potentiometrically for Oils Which Give Sharp End Points and Poor End Points in Titration Curves. TBN 1-5
Sharp end point 0.2
Poor end point At least 0 . 3 5-20 1 At least 1.5 20-100 4 At least 6 100-250 10 At least 15 The range is greater for oils which do not have sharp breaks in the titration curve, and which thus require a buffer solution pH to be used.
Designation 177164, London, England, 1967. VOL. 41,
NO. 10,AUGUST 1969
1211
