Selecting the End Point in High Frequency Titrations C ompa r is0 n of Differential and Ordinary Procedures W. J. BLAEDEL AND H. V. MALMSTADT' C'niversity of Wisconsin, Madison, Wis. Yarious forms of curvature in the ordinary titration curves often cause considerable error in location of the end point. However, such curvature is often unimportant when the end point is located by a differential method, using a uniform flow buret and a differential frequency meter. The advantages of the differential method are discussed for cases where instrument and solution phenomena cause nonlinear response during a titration, and for resolving two end points lying close together. The errors resulting from indiscriminate use of the ordinary method in such cases are given. A buret specially adapted for the differential titration procedure can be used either in the normal manner or to deliver solution at a uniform and low rate of addition. The differential method is more rapid and usually more accurate than the ordinary one for titrations with fairly sharp end points. The conclusions are also applicable to conductometric titrations.
Figure 1,A, is the titration of 20 ml. of 0.01 M sodium chloride with 0.00920 M mercuric perchlorate, which contains a slight excess of perchloric acid (0.006 M ) to prevent hydrolysis. At the end point the total ionic strength of the solution is such that the sensitivity of the high frequency instrument is gradually decreasing. This results in a curved line rather than a straight line after the end point, as shown in Figure 1,A. In the extrapolation of curved segments, there is considerable opportunity for errors of judgment, especially if a minimum number of points is used to establish the curve. This is true even for titrations with sharp end points, and this procedure does not utilize the advantages of a sharp end point. The error of the extrapolation may be reduced somewhat by making measurements more frequently in the region of the end point in order better to delineate the curve; but even so, there is still ambiguity in selecting the point of intersection.
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tivity is sufficiently great so that titrations may be carried out differentially with an increase in accuracy and a decrease in effort, as compared to the ordinary method of titration. A differential procedure has been described briefly (1,d ) , but has not been critically compared to other titration procedures. This paper describes and illustrates the various titration procedures that may be used with high frequency titration instruments ( 1 , S), compares them critically, and states the circumstances under which each may be used to greatest advantage. Three different titration procedures are described, all of which are in use in this laboratory. Each has its own advantages. These procedures are compared by using each with a precipitation, a soluble complex, and an acid-base titration. All examples given here were carried out with the 30-Mc. high frequency titration apparatus ( 1 ) .
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ORDINARY HIGH FREQUENCY TITRATION PROCEDURE
The ordinary, or nondifferential, titration procedure consists of plotting the response from the high frequency titration instru-
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ment against volume of standard solution. A titration curve results which is much like a conductometric titration curve, and the end point is estimated by extrapolating the "straight-line" portions of the curve until they intersect. Several disadvantages of this titration procedure are illustrated in Figures 1,A, 2,A, and 3 and discussed below. The straight-line portions of the curve are very often not straight lines. The causes of such curvature fall predominantly into two classes: those due to the instrument and those due to solution phenomena. Change in the sensitivity of the high frequency titration apparatus with concentration of the solution ( 1 , 3 ) often causes curvature in the titration curve. As the titration proceeds, changes in the ionic strength and activity coe$cient, temperature drifts, and side reactions may also cause curvature.
FIGURE 1 8 VOLUME STANOARO MERCURIC
PERCHLORATE(MLJ
Figure 1. Titration of Sodium Chloride with '/IOMercuric Perchlorate
In Figure 2,A, is shown the titration of 5 ml. of 0.02 N potassium sulfocyanide with 0.02 N silver nitrate, in an end-point volume of about 75 ml. According to ordinary standards, a "very good" titration curve is obtained by taking measurements a t about 0.5-ml. intervals and extrapolating the straight-line portions. In so doing, the end point is in error by 0.1 ml., or 2%. By taking readings a t 0.1-ml. intervals in the region of the end point, it may be seen that the curvature is unsymmetrical, which accounts for the large error. Even by extrapolation of the curved line parts, the error is not greatly reduced, for errors as large as 0.05 ml. (1%) may occur, depending how the extrapolation is made. These errors need not exist, and are due solely to the
Present address, Department of Chemistry, University of Illinois, Urbana, Ill.
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extrapolation method used to locate the end point, for this end point is a very sharp one, as attested by the differential titration curves in Figure 2,B and C. Another example in which the ordinary titration procedure is unsatisfactory is illustrated in Figure 3. In this titration 25 ml. of approximately 0.02 iL’ sodium hydroxide containing a small amount of sodium carbonate are titrated with 0.025 N hydrochloric acid. This titration is a very important one, as it is difficult in practice to prepare and keep standard base that is absolutely carbonate-free. The base used in this experiment is representative of a student preparation of “carbonate-free” base.
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not sharp-end points the ordinary procedure is sometimes superior to the others described in this paper. DROP DIFFERENTIAL TITRATION PROCEDURE
In this procedure, the change in beat frequency per drop of standard solution is plotted against volume of standard solution as the titration progresses, .An S-shaped titration curve results, which is the first derivative of the ordinary titration curve. The drop differential procedure and its application to various titrations have been described ( 1 , 2 ) . This method has several advantages over the ordinary one, but these are mostly the same as in the continuous flow differential method. The drop differential method gives a differential titration curve without any auxiliary equipment such as the constant flow buret and differential frequency meter. Only the regular direct-reading frequency meter is required. If the change in beat frequency per drop undergoes a great change a t the end point, or especially if it reverses, only a rough plot on an expanded scale in the region of the end point is necessary. With experience, and for titrations with sharp end points, end points may be located without plotting to a precision of 1 to 2 drops of standard solution. For greater precision plotting is necessary and the procedure becomes tedious and time-consuming. There are other disadvantages to the procedure. Obviously, the change in beat fre uency per drop must be measurable. If larger portions must %e added, precision is lost in establishing the end point. The change in beat frequency per drop of standard solution is not precisely measurable. Variations in drop size as,hi h as 10 to 20% are noted, and this causes scattering of poinLs wkch determine the titration curve. This scattering causes no appreciable error, providing a good-sized change is involved a t the end point. Considerable manipulation is involved in making the dropwise additions of standard solution and noting the change in frequency
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Figure 2. Titration of Potassium Sulfocyanide with Silver Nitrate End-point volume about 75 ml.
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This titration is illustrative of a type where two substances are titrated, one in great excess of the other. Because the major constituent is titrated first, there should be two distinct end points very close together. However, as shown by the circled points in Figure 3, even when points are taken every 0.5 ml. there is only one point that does not fall very close to the two wings of the curve. This gives little indication of a double end point. If points are taken every 0.1 ml. there is some indication of a double end point, but because there is only a short curved approach to the bicarbonate end point, the break is rather indefinite, and it is practically impossible to establish the first and even the second end point with any degree of precision. Extrapolation of the two long wings until they intersect gives one end point which has no quantitative significance, as there should be two. There are many cases where the ordinary method of titration gives erroneous results, even where the wings of the titration curve seem to be straight. When curved, extrapolation of the wings may be a very uncertain process. However, this method is accurate when used with discrimination, as in the titration of thorium versus oxalate (4). Also, in titrations with poor-i.e.,
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Ordinary Procedure for Titration of Sodium Hydroxide with Hydrochloric Acid About 1 mole % sodium carbonate
V O L U M E 24, NO. 3, M A R C H 1 9 5 2
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meter readings. The use of the special buret described below greatly increases the ease of making the dropwise additions. As little warning is given of approach to the end point in some titrations, there is the likelihood of missing the end point. For these titrations, it is of advantage to carry out a rapid preliminary titration on the sample to locate the end point approximately.
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tinuous flow differential procedure and recording of the titration curves. The maximum deviation of all nine samples was 0.02 ml. and the average deviation less than 0.01 ml. from the theoretical d u e . The curvature of the wings which is so troublesome by the ordinary method is of no consequence by the differential method. This reaction is considered as typical of many titration reactionsthat is, it is definitely not an ideal case because the end point is only moderately sharp and the change a t the end point is only moderately great, with no reversal in direction. Figure 2,B and C, represents the titration of 5 ml. of 0.02 N potassium sulfocyanide with 0.02 N silver nitrate. From these figures it is seen that the curvature which is so serious by the ordinary method (Figure 2 4 ) is of no significance by the differential method. The titration shown in Figure 2,B, was performed using a titration cell with about four times the sensitivity of that shown in Figure 2,C. The time constant of the RC differentiator (6) for the titration shown in Figure 2,B, is 0.5 second and in Figure 2,C, 1 second. For five titrations of this sort, the maximum deviation was 0.01 ml. Figure 4 is for the differential high frequency titration of 25 ml. of approximately 0.02 N sodium hydroxide containing a small amount of sodium carbonate with 0.025 N hydrochloric
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Figure 4. pH and Differential Procedures for Titration of Sodium Hydroxide with Hydrochloric Acid About 1 mole YQsodium carbonate
The drop differential method, where it is applicable, is superior to the ordinary procedure. I t is definitely outmoded by the continuous flow differential method, however, and should be used only when a uniform flow buret nnd differential frequency meter ( 5 ) are not available. UNIFORM FLOW DIFFERENTIAL TITRATION PROCEDURE
A special, but simple, buret (Figure 5 ) is used to add the standard reagent to the titration vessel a t a uniform rate. Consequently, the frequency of the high frequency titration instrument changes a t a fairly constant rate until the end point is reached; then the rate of change of frequency rapidly changes to a new value. Therefore, by using the differential frequency meter ( 5 ) in conjunction with a recording milliammeter, a continuous titration curve is recorded which is the first derivative of the ordinary titration curve. Traces of typical recorded differential high frequency titration curves are given in Figures 1,B, 2,B and 2,C, and 4. Figure 1,B, is for the titration of 20 ml. of 0.01 M sodium chloride with 0.00920 M mercuric perchlorate. The marker pips were put on the titration curve every 0.1 ml. by the marker device described with the differential frequency meter ( 5 ) . The insertion of so many marker pips is not necessarily a recommended procedure, but is used here to illustrate the results obtained by use of such a marker device. Nine identical samples of 0.01 M sodium chloride were titrated on different days with different sensitivities with 0.00920 M mercuric perchlorate using the con-
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Uniform Flow Buret
acid. This titration curve shows how well two end points which are close together can be resolved by the differential titration procedure. Figure 4,A, shows a pH titration curve for an identical sample of base with acid. Five samples were titrated by the differential high frequency method. For both the first and second end points no two results deviated by more than 0.03 ml. Five identical samples were also titrated with a p H meter, and results deviated by as much as 0.07 ml. for the first end point and 0.05 ml. for the second end point. For an identical sample carried out by the ordinary high frequency procedure as shown in Figure
ANALYTICAL CHEMISTRY
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3, it is difficult to establish either the first or second end point without personal prejudice and to within 0.1 or 0.2 ml. In order to increase the magnitude of the change a t the first end point with respect to that a t the second, the RC diffikentiator time constant is kept a t 1 second until shortly after the first end point is reached; then the time constant is switched to 0.5 second. This reduces the sensitivity to one half, and reduces the magnitude of the change a t the second end point to one half of what it would have been had the highest sensitivity been maintained as for the first end point. The point where the time constant is changed is indicated in Figure 4. Because the rate of change of frequency is very close to zero when the time constant is switched, there is only a slight jog in the curve. CONSTANT FLOW BURET
A three-way stopcock (1- to 2-mm. bore) is sealed to an ordinary buret as shown in Figure 5. One outlet is shaped to a bent
tip, and the buret is used in the ordinary manner with this outlet. The other outlet passes through an obstructed tube and into the bent tip to give a low rate of delivery. A switch from normal delivery to the low rate, or vice versa, may be made simply by turning the stopcock through 180'. The obstructed tube consists of a short segment of borosilicate glass tubing filled with ground soft glass. The ground glass is held in place in the glass tube by sintering the ends of the groundglass section. By varying the diameter of the borosilicate glass tubing, the length of the ground-glass plug, and the particle size of the ground lass, any desired flow rate may be obtained. For a flow rate of a t o u t 0.01 ml. per second a t a head of about 80 cm. of water (corresponding to the zero mark on a 50-ml. buret), a 4-cm. length of ground glass (100- to 200-mesh) in a borosilicate glass tube 2-mm. in inside diameter is suitable. Rather than sinter the end of this section, it is better to add 1 or 2 cm. of coarser ground glass (ca. 60- to SO-mesh) and sinter.this to-hold the plu in place. T%e obstructed tube is held in place in the buret with gum rubber connections to facilitate removal for cleaning, though this is seldom necessary. This arrangement is far superior to a capillary obstruction, which has a tendency to plug and requires frequent cleaning. To clean, the obstructed tube is removed and cleaning solution is sucked through in the direction opposite to that followed by the standard solution in the buret. A similar tube is still in use without cleaning after several hundred titrations in this laboratory over a 12-month period. The flow rate of this buret is not constant, but varies with head of liquid in the buret. However, in the region of an end point, where the head changes uniformly and negligibly the variation is slight and uniform. This might introduce a slight slope to the titration curve over many milliliters, which does no harm with the differential method. Uniformity of rate of addition is shown by the very smooth recorded rate curves obtained by adding sodium chloride solution from the constant flow buret into a more dilute sodium chloride solution contained in the high frequency titration cell, where no chemical reaction takes place. The after parts of the curves in Figures 2,B, and 4 show larger irregularities than do the same rates of change of frequency when no chemical reaction is in. volved. In the titration shown in Figure 2,B, the precipitate coagulated into very large particles, and these particles apparently cause larger random frequency fluctuations because of a nonuniform field. However, even though the precipitate coagulates, the fluctuations are very small compared to the end-point break, and there is no difficulty in obtaining the end point. The larger fluctuations in the after part of Figure 4 are characteristic of all acid-base samples titrated and are probably due to evolution of carbon dioxide, which causes random frequency fluctuations. PROCEDURE FOR CONSTANT FLOW DIFFERENTIAL TITRATION
The constant flow buret shown in Figure 5 is mounted so that the tip is a t least 0.125 inch below the surface of solution in the titration vessel. In order to ensure uniform and rapid mixing of the standard reagent, it is desirable to have the buret tip no further than 0.125 inch from the shaft of the borosilicate glass
stirrer, which is powered by a constant speed motor. With this arrangement mixing is rapid and uniform, and the random frer e n c y fluctuations are very small, which also results in smooth ifferential titration curves as seen by the recorded curves in Figures 1,B, 2,B and C, and 4. The sample is placed in the titration vessel and diluted nith water to bring the solution level above the grounded ring ( I ) . The direct-reading frequency meter (6)is set on the 50,000 cycles per second scale and a beat frequency is obtained by adjusting the reference oscillator of the high frequency instrument. A11 titrations presented in this paper were performed with the 30Mc. instrument ( I ) . If the end point is known approximately, standard reagent is added rapidly to within a few milliliters of the end point. Then the beat frequency is again set by adjusting the reference oscillator, the differential frequency output indicator (recording milliammeter a t zero center scale) is switched into the circuit, and the stopcock on the buret is set to deliver standard reagent a t a uniform rate. The chart speed on the recorder is usually set a t either 0.75 or 1.5 inches per minute. The differential titration curve is automatically recorded and it is merely necessary to use one of several methods to note the buret reading a t the end point. One possible method, which is effective but probably more work than necessary, is illustrated in Figure 1,B. In the region shortly before the end point, marker pips are inserted every 0.1 ml. starting with a given reading on the buret. Interpolation between two pips immediately gives the end point. Actually, as the delivery rate of the buret changes so slightly over a few milliliters, it is only necessary to introduce a pip mark about every 0.5 ml. in the region of the end point and to interpolate between two of them for the end point.
If the end point is very sharp, as for titrations shown in Figure 2,B and C, and the fist end point of Figure 4,it is only necessary to observe the buret reading quickly when there is a sudden break in the titration curve. For such sharp end points the panel differential output meter in conjunction with this method can be used instead of the recording milliammeter. If a marker device is not convenient to use, it is just as easy to read the buret every time it crosses a 1-minute or 0.5-minute division on the chart. To obtain the end-point reading it is only necessary to interpolate between two of the minute divisions. This is the procedure most frequently used in this laboratory. For nonroutine work, it is believed that the simplicity of construction and use of the buret described in the above section as compared to an absolutely constant flow type warrants the disadvantage of having to take R few buret readings during the titration. With but little experience end points can be obtained rapidly and precisely. Even titrations with only moderately sharp end points, as in Figure 1,B, can be obtained rapidly with a precision of 0.01 to 0.02 ml. For routine work, it would probably be better to use a constant-flow buret ( 6 ) synchronized with chart drive so that chart lengths represent volumes, and/or so that flow is stopped a t the end point. For this work, no effort was applied in this direction. The pen on the type recorder used travels on an arc. However, it is possible to estimate the end point by the usual manner of taking the mid-point of the steep break if the sensitivity of the differential frequency meter is set so that the recorder pen travels only about one or two scale divisions (0.1 to 0.2 ma.) a t the end point. By this method, the distance traversed by the pen is essentially a straight line and no significant curvature is introduced because of nonlinear coordinates. SURI\IARY
Three different titration techniques using recently described high frequency titration instruments ( I , 3 ) are compared. The differential and ordinary titration methods are complementary, each having its own advantages; which method is best for a particular type of titration depends upon several circumstances. Generally, the uniform flow differential method using a uniform flow buret and differential frequency meter is more rapid than the ordinary method. Various forms of curvature of the titration curves which often cause serious error by the ordinary method are of no consequence by the differential method. Also, the differential method is superior to the ordinary method for resolving two
V O L U M E 2 4 , NO. 3, M A R C H 1 9 5 2 end points which are close together, When conditions are such that the wings of the curve are actually straight lines, but the end point is very poor, the ordinary method is sometimes superior to the differential method. For slow reactions, the differential method may be inapplicable. A buret is described which can be used either in the normal manner or to deliver solution a t a uniform and low rate of addition. ACKNOWLEDGMENT
This work was supported in part by the Wisconsin Alumni Re-
459 search Foundation and in part by grants-in-aid from E. I. du Pont de Nemours & Co. and the Atomic Energy Commission. LITERATURE CITED
(1) Blaedel, W. J., and Malmstadt, H. V., ANAL. CHEM.,22, 734 (1950). (2) rbid., P. 1410. (3) rbid., p. 1413. (4) Ibid., 23, 471 (1951). (5) Ibid., 24, 450 (1952). (6) Lingane, J. J., Ibid., 20, 285 (1948). RECEIVED for review June 16, 1951.
Accepted September 25, 1951.
Precipitation from Homogeneous Solution LOUIS GORDON, Syracuse University, Syracuse, iV. Y. The conventional or heterogeneous process of precipitation, in which a solution of the precipitant is added to the reaction medium, often results in excebsive coprecipitation due to localized concentration effects. In the process of precipitation from homogeneous solution, the precipitant is generated uniformly throughout the entire reaction region. This procedure thus avoids the concentration gradients which characterize the ordinary mode of precipitation. A comparison of the precipitates formed in the two processes indicates the superiority of the homogeneous method because there is much
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less coprecipitation of interfering elements. The precipitate is also more readily filtered and washed, owing to its dense and compact character. The technique of precipitation from homogeneous solution can be an efficient method of separating a substance from other constituents, and therefore it should find further application in gravimetric analysis. The number of fractionation steps in certain precipitation processes can be reduced. Distribution studies of trace materials on carriers formed under slow precipitation conditions closely approximating equilibrium conditions are also possible.
ess, are reduced through the use of dilute solutions there will be some improvement. A much higher degree of improvement will be obtained when the precipitant is generated uniformly and slowly, so that the liquid phase remains homogeneous. The precipitate resulting from such a process will show much less coprecipitation of interfering ions. It can also be readily filtered and washed because of its more compact character. A visual comparison of the precipitates obtained by the two methods is shown in Figures 1 and 2. The apparent volumes of such precipitates and some typical data are compared in Tables I and 11. Because of the unsatisfactory results obtained whenever gelatTable I. Apparent-Volume Ratios of Precipitates inous precipitates similar t o hydrous ferric oxide were encounTime of Settling tered, it is obvious why so much early attention was focused on Months’ Ratio4 this class of substances. In a recent paper Willard (28) reviewed 2 20 Fer] IC oxide the early methods used t o effect a homogeneous change in the pH Thorium oxide 2 9 Stannic oxide 2 20 of a solution. The basic acetate and similar methods (18) and hlasnesium oxalate 0 . 5 hour8 51 2 . 5 hours 34 procedures which employ sodium thiosulfate (11) or hexamethyl4 5 hours 29 17 hours 20 enetetramine (20)are examples. With these methods a gelatinous 2 months 11 precipitate is still obtained, although there is usually a reduction Ratio of volume of ppt. produced by heterogeneous method to that produced by homogeneous method. by about one half in the a p - parent volume as compared to the ammonia precipitate. Table 11. Comparative Separations It was first demonstrated by Element Other Substance Coprecipitated Present, TT’illard and Tang (37) that Substance Pptd. Method of Precipitation Mg. Gram Error, mg. more than just a homogeneous Mn 1 .O Ammonia (24) A1 0 . 1 Aluminum oxide change in pH is required to preMn 1.0 Urea-succinate (36) A1 0 . 1 Ammonia, two pptn. (84) Fe 0 . 1 Co 0 . 0 5 Ferric oxide cipitate hydrous oxides or basic Urea-formate (36) Fe 0.112 c o 1.0 Urea-formate, 2-stage pptn. (36) Fe 0.112 co 1.0 salts so that there is minimum Oxalate-ammonia ( 9 8 ) Ca 0.0503 Mg 0 1 Calcium oxalate Urea (31) Ca 0.0503 .Mg 0.1 coprecipitation of interfering Methyl oxalate (16) Ca 0.0503 M g 0.1 ions. I n an experiment in which Li 0.1 Ammonium oxalate (6) hlagnesium oxalate Mg 0.0100 Ethyl oxalate (14) Li 0 . 1 Mg 0.0112 a solution containing alum and a Other substance determined by analysis of precipitate. urea was heated to boiling, they b Data obtained by Gordon and Wroczynski (18). obtained adense aluminum pre-
MOSG the desirable properties of a substance to be separated from a solution by a precipitation process are minimum coprecipitation and maximum filterability. When a precipitate is produced by the conventional method of adding a reagent directly to a solution, the degree t o which these properties are attained often leaves considerable room for improvement. Such is the case when ammonium hydroxide is used to precipitate hydrous ferric oxide in a gelatinous form a t pH 2 in the presence of manganous ion which precipitates a t p H 8.5 (29). When the concentration effects, characteristic of this heterogeneous proc-
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