It is reasonable to believe that the accuracy and precision of this instrument for integration of a n input signal can be increased by a factor of two by the use of precision-matched capacitors, a chopper stabilizer on the follower amplifier, or high quality soIid state amplifiers. These possibilities will be the subject of future investigations. Applications of this instrument for densitometry in paper and thin layer chromatography and electrophoresis may also be found. The work of Seligson, et al. (6) illustrates the application of a similar system to electrophoresis. The system described in that work requires a separate amplifier for each chromatographic peak being integrated. I n addition, each integrator is operating throughout the entire separation and measurement operation. I n
the system described here, the integrating capacitors are switched into the amplifier circuit only during the period of the peak to be integrated. As a result, only one integrating amplifier is required irrespective of the number of peaks being integrated. I n addition, integration error resulting from amplifier offset is limited to that occurring during the period of the peak. ACKNOWLEDGMENT
LITERATURE CITED
(1) Consolidated Electrodynamics Corp., Pasadena, Calif., Bull. 34210-1, Nov.
1963. (2) Dal Nogare, S., Juvet, R. S. Jr., “Gas Liquid Chromatography,” p. 198, Interscience, New York, 1962. (3) Disc Instruments Inc., Santa Ana, Calif., Bull. 200. (4)Hoffman, E. G., ANAL. CHEY. 34, 1216 (1962). (5) Infotronics Corp., Houston, Texas, Product Data Sheet CRS-11. (6) Lotito, L. A., XcKay, D. K., Seligson, D., Clin. Chem. 11, (3) 386 (1965).
(7) Malmstadt, H. V., Enke, C. G., Toren,
The authors acknowledge discussions with Willard Faulkner of the Cleveland Clinic which provided inspiration for the initiation of this work. We also acknowledge the assistance of Jonathan University Amy of the Purdue Chemistry Department during the conduct of this research.
E. C.,“Electronics forscientists,”~.356, W. A. Benjamin, Inc., New York, 1962. (8) Pardue, H. L., Dahl, W. E., J. Electroanal. Chem. 8 , 268 (1964). (9) Perkin-Elmer Corp., Wilton, Conn., Bull. 990-9089. (10) Sawyer, D. T., Barr, J. K., ANAL.
CHEM.34, 1213 (1962).
(11) Self-organizing Systems, Inc., Dallas, Texas, Bull. 105, Sept. 1963.
A Simple Graphical Method for locating the End Point of a pH or a Potentiometric Titration Stephen R. Cohen, Department of Neurology, College of Physicians and Surgeons, New York, N. Y.
T
10032
method for finding the end point of a p H or a potentiometric titration is to plot ApH/AV or AE/AV as a function of V , the volume of titrant, and to take the value of V at the maximum as the end point. This graph consists of two parts, one for volumes less than the end point and one for greater volumes, which are extrapolated to intersect at the end point. Because these branches are curved and increase rapidly as the end point is approached, the intersection is poorly defined and, even with precise data, the end point is uncertain by several times the reading error of the buret. I n addition, because p H or E changes very rapidly with V near the end point the values of ApH/AV or AE/AV in the critical region around the end point often have scatter which greatly increases the uncertainty of the end point. The following graphical procedure can be used t o find the end point precisely even with moderate experimental scatter. It is easier to use than Gram’s methods ( I ) . Unlike them, it can also be applied to differential titrations where ApH/AV or A E / A V , rather than p H or E , is measured as a function of V . Plot the values of ApH/AV or AE/AV against V as usual. Draw the separate curves through the data points before and after the end point. These curves should be smooth, but need not be fitted to the data with more than reasonable care. Curves drawn by eye are adequate. It is not necessary to extend HE MOST COMMON
158
ANALYTICAL CHEMISTRY
them beyond the data points to find their intersection. Select several values of pH/AV or AE/AV, and read the corresponding volumes from these two curves. Call them L for the volume taken from the left-hand curves (before the end point), and R for the volume taken from the right-hand curve (after the end point). Compute M = l/* ( L R ) ,the average of these two values; and D = R - L , the difference between these two values. Graph 211 as a function of D and extrapolate to D = 0 to find the end point. With reasonable care i t will be located to within the reading error of the buret. This graph should be a well defined, straight line. Inaccuracies in drawing two curves of the ApH/AV us. V graph will produce a slight scatter in the values of M, which will have a negligible effect on the end point. The above procedure can be derived theoretically. For titrations such as acidbase titrations, which have a titration curve that is symmetrical about the end point, M is theoretically independent of D . I n practice some slight dependence may sometimes be found, but the slope of the graph 111 us. D will be small If the titration curve is not symmetrical about the end point, M will still vary linearly with D , but the slope will not be zero. This will not introduce a n y systematic error, but may reduce the precision of the extrapolated end point. This slight difficulty with unsymmetrical titration curves can be removed if
+
desired, and M made independent of D, by using a weighted average of L and R for 111. Let A be the reacting chemical species being titrated-Le., the species present in excess before the end pointand let B be the reacting chemical species being added; let n, and nb be the number of equivalents per mole of A and B , respectively, for the reaction
1
-A 120
+ nb-1 B
+
Products
at the endpoint. Compute D as before, but use the weighted average M =L nbD/(n, nb) for -11. [M may be seen to be a weighted average by rewriting this expression as M = (n,L+nbR)/(n, nb).] -4s before, graph this modified value of J1 as a function of D and extrapolate to D = 0 to find the end point. The slope of this graph will be essentially zero. When this weighted average is used for X , it is important that na and ?& be taken from the stoichiometry for the end point reaction and not from the stoichiometry of the overall titration. For example when Na2C0g is titrated with HC1 to the CO, end point, the end point reaction is HCOa- H+-, COz HzO,and, therefore, n.ya2COI = 1 and not 2.
+
+
+
+
+
LITERATURE CITED
(1) Gram, Gunnar, Acta Chem. Scand. 4, 559 (1950); Analyst 77, 661 (1952).
