the titration to the second end point, which is designated Vea. Therefore, for practical purposes, (5) 17DC
\;C HC1 = 70 HKOo
3 65( VI - VDC)SI
-1_
?V
6.30(Vi
= ~
(6)
- V D_ C)S~ _ _ (7)
TV
Equation 4 can be rewritten as shown in Equation 8 and then simplified as given by Equation 9. V D C = 5/-D
- T/o
s ’VI
- (1.000 -
x) s,
= V D - ‘ V O - (1.000
-s -2)
V2A
(9)
TiDc can be calculated rapidly by Equation 9 and used in Equations 5 and 6 to determine percentages of sulfuric and hydrochloric acids or in Equations 5 and 7 to determine percentages of sulfuric and nitric acids. ACKNOWLEDGMENT
v2
(8)
For practical titrations V 0 will be small compared to Vn, and Arz/Arl is nearly unity for a good base, so the second term in Equation 8 is essentially Vo. Also, for a good base (1.000 A7z/A7J, is small, usually about 0.005. It is not necessary to multiply by exactly VZ,but only by the closest whole number of milliliters from the start of
The authors wish to thank Ralph Magin for help in performing some of the titrations.
Kon-Aqueous Solvents,” G. F. Smith Chemical Co., Columbus, Ohio, 1952. (5) Fritz, J. S., Yamamura, S. S., ANAL. CHEM.29,1079 (1957). ( 6 ) Harlow, G. A., Wyld, G. E. A., Ibid., 30,69-73 (1958). ( 7 ) Hine, J., Hine, RI., J. Am. Chem. SOC. 74, 5266 (1952). (8) Malmstadt, H. V., AKAL.CHEY. 29, 1901 (1957). (9) Malmstadt, H. V., Fett, E. R., Ibid., 27, 1757 (1955). (10) Malmstadt, H. V., Roberts, C. B., Zhid., 28, 1408 (1956). 111) Zbid.. 28. 1412 (19561. (12) Malmst$dt, H.‘ V., Vassallo, D. A,, Anal. Chim. Acta 16, 455 (1957). (13) E. H. Sargent Co., Scientffic Apparatus and Methods 7 , KO. 2 (1955). (14) Seidell, A., LISolubilities‘of Inorganic
and Metal Organic Compounds,” Vol. I, p. 233, Van Nostrand, Xew York, 1953. -__-.
LITERATURE CITED
(15) Van der Hiedje, B. H., Anal. Chim. Acta 16,392 (1957). (16) Rolff, J. P., Zbid., 1, 90 (1947).
(1) Bruss, D. B., Wyld, G. E. -4., ANAL. CHEM.2 9 , 2 3 2 (1957). (2) Critchfield, F. E., Johnson, J. B., Zbid., 26, 1803 (1954). (3) CundifT, R. H., Markunas, P. C., Zbid., 2 8 , 7 9 2 (1956). (4) Fritz, J. S., “Acid-Base Titrations in
RECEIVED for review February 3,1958. Accepted October 30, 1958. Work sponsored by Standard Oil of Indiana, U. S. Air Force under contract A F 18 (603)-137 monitored by the Air Force Office of Scientific Research, Air Research and Development Command.
Thermometric Titration of Weak Monoprotic Acids JOSEPH JORDAN and WM. H. DUMBAUGH, Jr. Department o f Chemistry, The Pennsylvania Stafe Universify, University Park, Pa.
b The applicability of thermoanalytical titrimetry to the determination of weak acids in dilute aqueous solutions has been investigated. Four acids, whose ionization constants covered a range of nine orders of magnitude, were studied systematically a t concentrations between 1 0-3M and 1 0-’ M. Sodium hydroxide served as the alkalimetric titrant. Kinetic effects were eliminated by using an automatic procedure which maintained the rate of addition of the titrant one millionth that of prevailing rates of neutralization. The thermometric end points were precise and accurate to = t l % for concentrations of 3 X 10F3Mand higher. Within the range of acids studied, ionization constants had no effect on precision and accuracy, because the heats of neutralization were approximately equal. Potentialities and limitations are evaluated of “enthalpy-dependent’’ methods vs. classical “free energy-dependent” alkalimetric procedures.
F
a pioneering study by Linde, Rogers, and Hume (6) it is apparent that the feasibility of thermometric titrations (and the precision and ROM
210
ANALYTICAL CHEMISTRY
accuracy which can be attained) may depend in principle on the prevailing concentrations as well as on the equilibrium constants, enthalpies, and kinetics of the processes involved. Kinetic considerations may conveniently be eliminated whenever the rate of additioq of the titrant (“titration rate,” R , expressed in moles per second) is slow relative to the rate of the relevant reaction. In the automatic procedure described previously (5) the rate of delivery of titrant was of the order of 0.02 ml. per second. Assuming that a 50-ml. sample of monoprotic acid ( l o + to 10-1M) is titrated with sodium hydroxide of suitable concentration (0.1 to 5M depending on the concentration of the titrate), R t is readily estimated to be of the following order of magnitude. 2 X 10-6
< Rt < 10-4 mole per second (1)
1 2. 3. 4.
Borica Acetic Monochloroacetic Trichloroacetic
This compares Kith a rate of neutralization, R,, of lo2 < R, < lo3 moles per second ( 2 ) which can be computed from pertinent neutralization constants ( = 10” liters per mole-second) (3). It is evident from a comparison of relationships 1 and 2 that (RtlRn) 5 (3) which is totally negligible n-ithin an experimental error of AI%. K i t h the remaining criteria in mind, a systematic study was undertaken to investigate the applicability of thermometric titrations to the determination of weak acids covering the widest possible range of acid strength, but having comparable neutralization enthalpies. As a convenient “test series” the folloning acids (listed with their heats of neutralization a i t h a strong base, and pertinent ionization constants a t 25” C.)
K , = 5.8 x K , = 1.8 x K , = 1.6 x K, = 3 x
4Hn0 = - 10.6 kcal. per mole 10-6; AH,O = -13.5 kcal. per mole 10-3; AH,’ = - 14.4 kcal. per mole 10-I; AH,’ = - 13.4 kcal. per mole 10-10;
a Under the experimental conditions prevailing in this investigation, boric acid behaves as a monoprotic acid.
were selected for a critical analytical study in dilute aqueous solutions (2). The choice was based on the consideration that in conventional-e.g. , potentiometric-acid-base titrations the main drawback is inherent in the difficulty of determining end points in dilute aqueous solutions, whenever the ionization constants are small. The significance of thermometric titrations from the view point of quantitative analysis is consequently contingent on whether they may shon- less dependence on ionization constants than classical procedures. I n 0.01111 solution acids 1 t o 4 J ielded, 1%ith sodium h j droxide, thei~nometric titration curves of similar shape and quantitative results of equal precision and accuracy. I n contradistinction to this behavior, in comparable potentiometric titrations trichloroacetic acid Yields an excellent inflection a t the end point, while in the case of 0.01M boric acid, no nell-defined inflection is discernible. The information obtained in this study is of general significance, because the relative ionizatiort constant and neutralization enthalpy values of acids 1 to 4 appear to be generally typical of weak acids: They ill appear to have comparable heats of neutralization, R hile their ionization constants vary enormously ( 4 ) . EXPERIMENTAL
Naterials. Reagent grade chemicals 11ere used throughout. Apparatus and procedure nere generally the same as described previously (6). Jordan and Alleman have reported in their thermometric titration curves an anomaly which appeared as, a short horizontal portion between the end point and the teiminal ‘‘exce!jS reagent line.” It has now been ascertained that this was due to failure to maintain isothermal conditions throughout the titrant. I n the n-ork of Jordan and Alleman, prior to titration, a fraction of the titrant (0.5 to 0.6 ml.) in the tip of the constant flow buret was allowed to attain thermal equilibrium within the Dewar titration flask, while the rest of the titrant remained a t room temperature. The latter was as a rule slightly different from the temperature prevsiling in the Den-ar. This condition did not affect the validity of the analytical results leported by Jordan and Alleman, involving the chelatimetric determination of divalent cations with (ethylenetlinitril0)tetraacetate. However, the chelation enthalpies tabulated in the previous paper ( 6 ) may require correction. To cscertain the corrected enthalpy values, it is planned to repeat in thef,e laboratories the relevant titrations, under accurately controlled isothermal conditions throughout the titrant. Khen definitive data are available, the corrected chelation heats xi11 be reported.
In the course of the investigations described here, isothermal conditions in the titrant were maintained by stoppering the Dewar and allowing temperature equilibrium to be attained before the buret tip was inserted in the solution, Heat exchange between the buret with the titrant and the solution in the Dewar n.as negligible during the short period of actual titration (about 1 minute). Satisfactory results can also be obtained if a suitable capillary buret tip is used to minimize the corresponding volume of titrant. To facilitate accurate estimation of the end point, it was found advantageous to maintain the titrant slightly colder than the solution t o be titrated. This yielded a conveniently acute angle between the two branches of the titration curve preceding and follon ing the end point. TT’hen the titrant nas warmer than the soluticn titrated, the angle was obtuse and the break a t the end point was not sufficiently sharp to attain optimum accuracy. The abscissa scale was calibrated for volume by weighing the water delivered (in a given time interval) by the buret which Tvas driven by a synchronous motor. The recorder chart (on which the temperature change was recorded) was also driven by a synchronous motor and a length of chart was measured over a given time interval. Combining this latter speed with the volume delivered per unit time, a calibration was obtained in milliliters of titrant per millimeter of chart. A chart speed of 50.8 mm. per minute was employed, yielding a calibration of 0.0243 ml. of titrant per mm. of chart. The accuracy of such a calibration is evidently dependent on the constancy of the speeds of the two synchronous motors. A series of measurements of the speeds of the eynchionous motors showed that any error introduced by transient variations was negligible. The dimensions of the chart paper (E. H. Sargent and Co., No. S-29349) were found to be within experimental error, independent of changes in relative humidity, variations fiom chart to chart, and variations in different sections of the same chart roll. I n each experiment, 50 ml. of acid were titrated with standard sodium hydroxide in a carbon dioxide-free system. I n order to give the abscissa a convenient length for measurement and keep the time of titration relatively short (to facilitate maintenance of adiabatic conditions), the concentration of sodium hydroxide was adjusted to be 50 to 100 times greater than that of the acid titrated. Solutions used in this study were standardized independently by classical methods to provide reference concentrations. These are considered accurate to ~ k 0 . 3 or 7 ~better. All titrations were carried out a t 25” C.
RESULTS
A trace of an automatically recorded typical therinometric titration curve of 0.01M boric acid is shown in Figure 1, curve I. Curves similar in shape were obtained a t other concentrations in a range betneen and 10-lM. The thermometric titration curves of the other monoprotic acids in the series studied were also similar. Even a strong acid, hydrochloric acid, in 0.01M solution gave an almost identical titration curve, as is illustrated in Figure 1, curve 11. Portions A B on the titration curves in r!gure 1 represent “temperature-time blanks” with no titrant added. The actual titrations u-ere started at points B. Sections C D represent the excess reagent line. The end points, C, ivere e\trapolated linearly as indicated in dotted lines. The titration curves exhibited fluctuations on the blank portions and on the excess reagmt lines. These fluctuations had an amplitude of the order of 10+ mv. (Corresponding to a temperature of loF4“C.) and a frequency of 1 cycle per second. Such fluctuations were conspicuous by their almost complete absence from the ascending portions of the curves, BC. Titrations nere performed in a range of concentrations betv een 0.0008 and 0.1M. The precision of the titrations mas in general the same a t all concentrations between 0.003.1f and 0.1X. The corresponding standard deviations are plotted in Figure 2. The average standard deviation of a single measurement in a range of concentrations between 0.003 and 0.1,IZ was 0.9%. Below 0.003V the precision dropped to 12%. The accuracy of the alkalimetric determinations is shown in Figure 3. A semilogarithmic type of plot was used n hich reflects with equal fidelity any deviations a t IOK as well as a t high concentrations (8). Only results a t concentrations higher than 0.003M were included in Figure 3, because the precision a t smaller concentrations was too poor t o be significant in terms of accuraq-. The mean accuracy of ail titrations plotted in Figure 3 was &I%. DISCUSSION
The fluctuations obserT ed on specific portions of the thermometric titration curves (Figure 1, curves I and 11) can reasonably be ascribed to thermistor “noise.” I n accordance n-ith recent experience in the Department of Colloid Science, Cambridge, England, a noise of comparable order of magnitude appears to be gencrall? common t o thermistors and represents their main limitation for use in temperature measurements (1). The fact that the noise was barely evident in the ascending part of the titration curves is accounted for as follows. For a noise signal to be noticeVOL. 31, NO. 2, FEBRUARY 1959
* 21 1
TIME
10
LOG C
Figure 2.
Precision of thermometric titrations
C.
True concentration, mmole/liter Standard deviation of single determination, per cent 1. Trichloroacetic acid II. Monochloroacetic acid 111. Acetic acid IV. Boric acid u.
VOLUME
OF TITRANT
Figure 1. Experimental and theoretical thermometric titration curves Experimental curve of 0.01M boric acid titrated with 1M sodium hydroxide II. Experimental curve of 0.01 M hydrochloric acid titrated with 1 M sodium hydroxide Ill. Calculated theoretical curve for 0.01M boric acid titrated with 1M sodium hydroxide AB. Temperature-time blank B. Start of titration C. End point CD. Excess reagent line Note: Titrant temperature was 1.1 2’ C. below initial reactant temperature
1.
able, its amplitude must be appreciable compared to that of the “main signal.” On the ascending portions (BC in Figure 1) of the thermometric titration curves obtained in this investigation, the change in the main signal was of the order of 1 mv.-Le., a hundred times greater than the noise amplitude. However, on the horizontal portions, A B , of the curves the main signal was constant, Consequently, the noise yielded distinctly visible fluctuations on the temperature-time blanks. The fluctuations in the excess reagent line regions, CD, can obviously be explained by similar considerations. The over-all ‘situation regarding the appearance of fluctuations on the thermometric titration curves is analogous to what is commonly observed in polarograms obtained a t the dropping mercury electrode, where the periodic current fluctuations are much more prominent on the diffusion current regions than on the ascending parts of the wave (7). The experimentally obtained thermometric titration curves of weak acids were in satisfactory agreement with the calculated curves, everything else being equal, As an illustration, the theoretical curve corresponding to Figure 1, curve I, has been plotted as curve I11 in the same figure. Figure 3, curve 212
ANALYTICAL CHEMISTRY
103I02
I’
-0 0
101
-
=
100
099’ 098-
0971
1
1
1
1
I
I
4
I
I
0200 0400 0600 0800 1000 1200 1400 IWJ 1800 2000 2
b
0
C Accuracy of thermometric titrations LOG
Figure 3.
Cdet. Determined concentration C, I, 11, 111, IV. Same as in Figure 2
111, has been computed by rigorous mathematical procedures for the titration of 0.01M boric acid with 1M sodium hydroxide on the basis of the following assumptions: AH = AHo = -10.6 kcal. per mole; K, = 5.8 X 10-10; the activity coefficient for the protonated form of boric acid was assumed to be unity; activity coefficients for all charged species were evaluated on the basis of applicable As Debye-Huckel approximations. can be seen in Figure 1, curve 111, the theoretical curve exhibits a small, but definite rounding in the end point region corresponding to an incompleteness a t the equivalence point of 4.1%. The rounding is somewhat blurred in the experimental curve (Figure 1,curve I) by the noise fluctuations superimposed on the excess reagent line. However, it is evident from a comparison of curves I and 111in Figure 1that the experimental and theoretical curves are virtually superimposable except for the fluctuations. Theoretical calculations similar to those used for computing Figure 3, curve 111,indicated that in order to determine a thermometric titration end point Kith a given accuracy, it is necessary to consider the completeness of the reaction a t the “farthest point of the excess reagent line” (points D in Figure 1)
from which the end point is back-extrapolated. For instance, if it is desired that the end point error should not exceed 1% and the extrapolation is performed from a point corresponding to 100% excess reagent, it is necessary and sufficient that K. 2 ( 4) Khenever this minimum equilibrium requirement is met, the precision and accuracy of thermometric titrations are solely a function of the unknown concentrations and of the heats of neutralization involved. I n this study, a t concentrations exceeding a critical minimum of 0.003M the precision and accuracy of the alkalimetric determinations were indeed the same for all acids studied, because the corresponding enthalpies of neutralization varied only between 10 and 15 kcal. per mole. The poor precision and accuracy a t concentrations lower than 0.003M are accounted for by the fact that under the experimental conditions the ascending part of the titration curves, BC, was less than 40 mm. whenever the concentration of the unknown was less than 0.003M. A minimum length of 40 mm. was required for linear extrapolation along BC to an end point m-ith a precision to 1%.
Advantages a n d Limitations of Thermometric Titrations. It is evident that the information obtained in this study is of apprr.ciihlp analytical significance for t h e uetermination of weak acids at lorn concentrations. For example, boric acid may be determined with a n accuracy t o 1% in the millimolar concentration range. It is believed that this is the only method ~1 hich permits the direct alkalimetric titration with comparable accuracy of an acid with a n ionization constant as small as 10-'0 a t concentrations of the order of 10-3 and l O - * X in aqueous solutions. Because the heat of ncutralization of most monoprotic acids is of the order of 10 to 15 kcal. per mole, thermochemical titrations are applicable to a number of analytically important organic and inorganic acids the ionization constants of which meet requirement (4). Some of these acids with their ionization constant at 25" C. are: salicylic 1 x nitrous 5 X formic 2 X lop4, glycolic 1.5 X lactic 1.4 X benzoic 7 X lo+, phenylacetic 5.4 X 10-6, hydrazoic 1.5 X 10-6, propionic 1.4 X uric 1.5 X 10-6, o-nitrophenol 5.6 X 10-8, hypochlorous 4 X 10-8, m-nitrophenol 3.9 X 10-9, hydrocyanic 1 X p-chlorophenol 6 X 10-'0, arsenious 5.5 X 10-'0, and phenol 1 x 10-10. An "enthalpimetric sensitivity index (ESI)" conveniently characterizes the limitations inherent in a given set of experimental conditions. I n a previous paper (5) the ESI was defined by the equation PI%= CmtnAH" (5) nhere P,, denotes the E S I for 1% accuracy in calories per liter, C,,,. (expressed in millimoles per liter) is the minimum concentration of the unknown nhich can be determined with a n accuracy of = k l % , and AH" is the heat of neutralization expressed in kilocalories per mole. Results obtained in the present study yielded for P I , a value of 40 cal. per liter. Khile the ESI has no exact significance, it might be used as a convenient empirical parameter for characterizing the sensitivity of a given titration setup. End point extrapolation in thermometric titrations depends on discrete breaks between linear portions of a titration curve. This situation engrnders a n almost prohibitive disad-
vantage in the application of direct alkalimetric titrations of mixtures of acids. Because the heats of neutralization of-Fost known acids differ little, 'uirect alkalimetric titrations of free acids cannot be used per se for determining any one component in binary (or more complex) mixtures. The most important potentialities and limitations of alkalimetric thermochemical titrations are due to the fact that the heats of neutralization of various acids happen to differ by 50% or less. Compared with the prevailing extensive differences in ionization constants, this is equivalent to a remarkable invariance. Its energetic significance is apparent from the following considerations. Generally, the heat of neutralization (with a strong base) of an acid can be additively equated to its heat of ionization, AH,", plus the heat of neutralization of strong acids (- 13.5 kcal. per mole) : 10 kcal. per mole < -AH,' = (-AH," 13.5) < 15 kcal. per mole (6) I n turn, the enthalpy of ionization is a n additive function of a free energy and a n entropy term:
+
AH,'
=
AFi"
+ TASi"
=
+
- R T In K i TAS," (7) A comparison of Relations 6 and 7 makes it evident that for a given acid, A F i o is energetically compensated by the T A S , " term, yielding a AH," value which varies considerably less from acid to acid than either ASi" or AF,". This situation is illustrated in Table I, in which are listed the relevant thermodynamic data for the four acids investigated in the present study. Table 1. Data a t
ACKNOWLEDGMENT
The authors wish to thank the Research Corp. for a grant in support of this work. LITERATURE CITED
(1) Berger, R. L.. private communication. (2) Dumbaugh, W. H.,Jr., thesis, Penn-
sylvania State L niversity, January 1959. f3'1 Einen. 11..Maever. J.. Xaturwissenschu&k 14, 1 (1955). ' (4) Harned, H. S.. Owen, B. B., "Physical Chemistry of Electrolytic Solutions,'J 3rd ed.. I). 667, Reinhold, Yew York, 1958. (5) Jordan, J., Alleman, T. G., ANAL. CHEhI. 29, 9 (1957). (6) Linde, H. ITr.$Rogers, L. B., Hume, D. Y . ,Zbid., 2 5 , 404 (1953). (7) .Meit::, L., "Polarographic Techniques, p. 43, Figure 4 l a , Interscience, Xew York, 1955. (8) Ibid., p. 158. \ - I
I
_
RECEIVED for review August 7 , 1957. Accepted October 2 , 1958. Division of Analytical Chemistry, 132nd Meeting, ACS, New York, September 1957. Work based on the doctoral dissertation of William H. Dumbaugh, Jr.
Free Energy and Entropy
25' C. for Selected Monoprotic Acids (2)
Acid Boric Acetic Monochloroacetic Trichloroacetic
acetic, monochloroacetic, and trichloroacetic acids yield potentiometric titration curves which differ appreciably from each other. However, the corresponding thermometric titration curves obtained in this study were similar, because for each acid the AF," values in column 1 and the T A S i " values in column 3 are comparable. An interpretation of this remarkable compensation effect between free energy and entropy terms (which appears to be general for weak acids, a t least a t 25" C.) will be published elsewhere.
AFi", Kcal./ Mole 12.6 6.5
-22& 1
TASi", Kca1.l Mole -9.6 -6.6
3.9
-18h 1
-5.4
0.9
ASi", E.U. -32A 1
-2*5
Relative Acidities of Organic Acids in Pyridine and WaterCorrection I n the article on "Relative Acidities of Organic Acids in Pyridine and Water" [ANAL.CHFM. 30, 1978 (1958)l Equation 2 should read: pK. (H,O) = 0.0100 X AHSP
-1h2
+ 3.61
The footnotes in Table I1 should read: Calculated from Equation 3. * Calculated from Equation 2. C. 4.STREULI a
Because the free energies listed in column 1 vary by a factor of 15, boric,
VOL. 31, NO. 2, FEBRUARY 1959
213
