ANALYTICAL CHEMISTRY
1226 of the vanadyl-metavanadate couple a t room temperatme.
LITERATURE CITED
Upon heating to 80” C. the derivative voltage decreases because the vanadyl-metavanadate couple is much more nearly reversible in hot solution, and also because of the increase in ionic mobility at higher temperatures. The second rise is attributed to the eshaust’ionof vanadyl ions and the rapid fall after the end point is due t,o the presence of the more nearly reversible cerous-ceric couple. In this latter case, the most satisfactory end point is the first sharp decrease due to the presence of the cerous-ceric couple. A ratio of ferrous t,o vanadyl of 10 t’o 10 ml. gave a check between potentiometric and derivative methods by 0.02 and 0.02 ml., respectively. A ratio of 20 ml. of ferrous to 1 ml. of vanadyl gave checks of 0.01 and 0.02 nil., respectively. Thr fvrrous was apprusimately 0.09 N a n d the vanadate 0.2 S. In a11 cases, the end-point readings become stable in 5 to 30 secon&, and in one case (ceric-vanadyl titration) Ptability was achieved five timei! faster with the derivative method. The approach of the end point is easily anticipated with practice. The advantages of thie unit are the simplicity, the sharper breaks afforded (as in iodine-thiosulfitte titrations), the elimination of plotting, the rapid at3tninmentof stable readings, the elimination of a reference half-cell (esp!cially advantageous for yniall volumes and high t e m p m t u r w l , and ability to detect a succwpion of c.nd point9 simply.
(1) Baker, H. H., and sftiller, R. H., Tram. Electmchet?~.Soc., 76, 75 (1939). (2) Cox,D. C., J . Am. C h e m Soc., 47, 2138 (1925). (3) Delahay, P.,AXAL.CHEhr., 20, 1212 (1948). (4) Delahay, P., Aiial. Chim.Acta, 1, 19 (1947) (5) Ibid., 4,636 (1950). (6) Foulk, C. W., and Bawden, 4.T., J . d m Chem. Soc , 48, 2045 (1926). (7) Furman, S. H., and Wallace, J. H., Jr., I M . , 53, 1283 (1931).
(8) Glasstone, S., and Hickling, A,, “Electrolytic Oxidation and Reduction,’’ p. 123, S e w York, D. Van Nostrand Co., 1936. (9) Kolthoff, I. M., and Lingane, J. J., “Polarography,” S e w York,
Interscience Publishers, 1941. (10) XIacInnes, D. H., and Jones. P. T., J . -4m. C h .SOL, 48, 2831 (1926). (11) Muller, ’Erioh, “Eleknonietrische AIassanalyae,” 7th ed., Dresden, T. Steinkopff,1942. (12) Muller, R . H., A N A I .CHEM.,22, 72 (1950). (13) Myers, R. .J.. and Pwift, E. H.. ,J. A m . C h m . Soc.,70, 1047 (1 948). (14) Van Name, H. C,., and Fenwick, F., Ibid.. 47, 9 (1925). (15) Ibid., p. 19. (16) Watt, G. W.,and Otto, J. B., Jr.. J . Elecbodi~na.Soc., 98, 1 (1951). (17) Willard, €I. H., a n d Fenwick, F., (1922). (18) Ibid., p. 2516.
J. 87n. C 7 w n
Soc.,
44,2604
RECEIVEB February 28. 1951.
Three-Dimensional Model for Interpreting Electrometric Processes CHARLES 5. REILLEY, W. DONALD COOKE’, AND N. HOWELL FIiRMlN Princeton Unirersity, Princeton, N. J . Recent work on coulonietric procedures prompted inquiry into the common basis for many electrochemical methods. A description is given of a surface, plotted in three dimensions with current, per cent oxidized, and voltage as coordinates. This surface is presented as a unified basis for explaining and relating polarography,amperometry,potentiometry, and polarized end-point phenomena such as the “dead-stop” method. The equation for the surface under proper conditions is shown to give a quan-
I
S SONE recent work on coulometric titrations, the need arose
for more sensitive electronietric end-point, procedures, because the conventional methods did not, have the desired sensitivity for use in the microgram and submicrogram regions. In the search for applicable procedures the interrelationship of various electrometric procedures was studied. A relationship capable of unifying potentiometric, amperometric, polarographic. and other electrochemical methods was found. As s result of this study, t x o tv-o new end-point procedures have been developed: :t differential method (10)and an amperometric t.itration of high sensitivitp ( 8 ) . It is hoped that this unified explanation will prove advantageous in the clarification, development, and application of electronietric procedures to new situations. When a platinum electrode is placed in a solution containing a reversible couple such as ferric-ferrous or a dropping mercury electrode in hydroquinone-quinone, the properties concerning their interaction may be depicted on a graph of three dimensions: current, voltage, and per cent of the couple in the oxidized st’ate 1
Present address, Cornell University. Ithacit, S . Y.
titative explanation for these electrochemical procedures. This study has been instrumental In the development of a derivative polarographic end point and a sensitive end-point procedure for &metric microtitrations. This surface provides a fundamental picture of the relationship between the various electrochemical methods and should prove valuable in the application and understanding of existing techniques as well as in the development and extension of newer methods of electrometric analysis.
(see Figure 1 ) . The equation for the surface of this figure may be written for various stages of concentration polarization:
where z is the fraction of the total concentration, CO,in the reduced state and the other terms have their usual significance (7).
E Ea R T n F fred
k,,, fox
kred
=
voltage of electrode
= standard oxidation-reduction potential = gas constant = absolute temperature =
electron change between oxidized and reduced forms
= faraday (96,500 coulombs) = activity coefficient of reduced state = diffusion constant for oxidized state such that
=
ido
k,, =
CO(J) current (positive for cathodic current and negative for anodic current) --red
i
idc =
CO(1 - 2) = activity coefficient of oxidized state = diffusion constant for reduced state such that
V O L U M E 23, NO. 9, S E P T E M B E R 1 9 5 1 k,, and kFed may be calculated from the IlkoviE equation for the case using B dropping mercury electrode, whereas for platinum electrodes in unstirred solutions k,, and kred would be a function of time. For the cases of platinum electrodes in stirred solutionR, k,, and ked are determined empirically, as factors such as rate of stirring, electrode shape, and size are difficult to determine.
1227 in Equation 1 the terms for a fixed per cent oxidation, one obtains the more familiar formula for a polarographic wave:
E
=
where
-id, - RT - In i-
nF
idc
(2?
-i
+
El/ 2 is E when i equals ' / a ( z d o idc) ZCO(negative by convention) - Z)CO
i d a equals -kred i d s equals kox(l
This equation has been treated elsewhere (7). CONSTANT CURRENT PLANE
If the condition for a potentiometric titration-namely, zero current in the indicator circuit-is placrd in Equation 1, it reduces to the Sernst equation:
Figure 1.
Three-Dimensional Model for Interpreting Electrometric Processes
This figure and equation are used to explain several electrometric techniques by drawing planeti in the solid figure and placing certain restrictions on the equations
vo
CONSTANT PER CENT OXIDIZED P L 4 N E
Figure 2, -4, illustrates a plane in the solid figure where the per cent osidized is constant near 50y0 B shows the two-dimenBional curve taken from -4, which is seen to be the polarogram expected for a reversible system consisting of a mixture containing equal quantities of oxidized and reductd forms By substituting
V O L T I G E
I
A
i
B
V O L T A O E
Z O X I D I 2 ED
A
Figure 3.
Zero Current Plane with Resulting Potentiometric Curve
Figure 3 shows how the plane in the solid figure appears in t h c usual two-dimensional graph. The dotted line, C, illustrates the condition when another couple is added in excess, such as cericcerous to the ferric-ferrous. I t may be noted from Equation 1 and the diagram in Figure 3, A , that by applying a current, the potentiometric curve is shifted so that an early or late end point will result, depending upon the direction of the current. Van Kame and Fenwick ( 1 2 ) noticed this effect upon polarizing a platinum indicator electrode. The shape of a differential potentiometric titration may be obtained by differentiating Equation 3:
!
I
x!
dx Figure 2.
I
\ I
Constant Per Cent Oxidized Plane with Resulting Polarographic Wave
(4)
As the end point is approached, 5 will approach either zero or
ANALYTICAL CHEMISTRY
1228 unity for an oxidation or a reduction, respectively. limiting cases, when
x + 0, dE -+ dx
For these
m
x + l , - -dE +-m dx dE x equals 0.5, - = -:4RT/nF
dx
In practice, dx is held approximately constant through addition of equal increments of titrant, a small sample of the solution being temporarily withdrawn for a reference before each addition. Thus the potential difference, dE, between this reference and the main body of the solution becomes a measure of the derivative dE/dx.
VOLT A O E
A
CONSTANT VOLTAGE PLANE
Equation 1 can be rearranged to:
This equation has the linear form of i ( a ) = b - cz, where a, b, and c are constants for a given voltage. The slope is given by -c/a and the intercept by b/a. When E is much larger than EO the equation will reduce to:
i
= -kredC@
(6)
When E is much smaller than EO:
i
= koxCo(l
XI
100
0
%OX ID1Z E D
Figure 4.
Q
Constant Voltage Plane with Resulting Linear Relationship
Basis for quantitative polarography and amperometric titrations
- 2)
(7)
Equations 6 and 7 show the relationship a t voltages removed from EO and are the basis for quantitative polarography and amperometric titrations. The amperometric titration of an oxidation-reduction system is similar to a scheme of Muller (9) in which the titration is carried out to a predetermined voltage. .4t this point there is zero current in the indicator circuit, since the potential of the solution is equal to the predetermined potential impressed upon the indicator electrode. The selection of this voltage for the amperometric method should take into account the same considerations as are applied to potentiometric end points (6). Figure 4 shows how the plane a t a fixed voltage appears in two dimensions. The dotted line would be the result of the addition of another couple like ceric-cerous to the ferricferrous after the end point in an amperometric titration.
mid-point, z = 0.5. If the two k values may be assumed equal to one another, a simplified expression is obtained for the slope a t the mid-point:
dE - = - di
4RT nFkCo
(9)
If a series of polarograms is taken a t a platinum electrode in a stirred solution of a polarographically reversible system, as in the titration of ferrous with ceric, results are obtained similar to those in Figure 1 (10). Figure 1, A , is typical of the original solution of ferrous iron, B when half titrated with ceric ion, C a t the end point, and D a t some point past the end point. The slope of the polarographic curve, as it crosses the zero current axis, varies considerably during the titration. The expression for this variation in slope may be obtained on differentiation of Equation 1 with respect to x and where the current, i, equals zero:
The variation of this slope is similar t o the form of that for the differential potentiometric titration (Equation 4). Use has been made of this fact for a derivative polarographic titration (10). I n this derivative titration, a small, constant current, d z , is placed across two electrodes, and the resulting dE is measured, since, as for the differential potentiometric titration, dE is a measure of the derivative, dE/di. (This derivative refers to a polarographic wave, as the derivative is taken with respect to current, whereas in the differential potentiometric titration, the derivative was taken with respect to the fraction in the reduced state.) The so-called “dead-stop” end point of Foulk and Bawden ( 4 ) has been explained in several papers (1, 3,4,8, 11). The functioning of the dead-stop end point is also of a derivative nature [see Figure 2 (IO)]. The potential across the two electrodes is held constant (dET)and the resulting current, d i , is measured with a sensitive galvanometer. The applied voltage distributes itself between the two electrodes [dEl and dE2 of Figure 2 ( I O ) ] such that the net potential dE, plus dE2 is the same as that applied dET. The resulting current is a function of di/dE and varies inversely as the dE/di of the derivative polarographic titration described above. Thus,
This shows that the slope initially, when x + 1, approaches m, and a t the end point, where x -+ 0, approaches - m . At the
For a reversible system, then, the current initially and a t the end point should approach zero, and elsewhere reach a value de-
POLARIZED SYSTEMS USING PLATINUM ELECTRODES
-
V O L U M E 23, NO. 9, S E P T E M B E R 1 9 5 1
1229 current immediately past the end point in accordance with the results of Foulk and Bawden (4). COMBINATION OF TWO SYSTEMS I
If two reversible systems such as cericcerous and ferric-ferrous are placed with proper alignment of their respective voltage axes, a three-dimensional plot results as in Figure 5 . A potentiometric titration is seen t o follow curve ABCDE. B represents the point where the first system is 50% oxidized and D represents the point where the second system is 50% V O L T A G E oxidized. The voltages a t these points Figure 5. Three-Dimensional View, Showing Interaction of Two Systems correspond t o respective Eo’s if activities are neglected. The end point ocABCDE. Potentiometric curve JKC-CLM. Amperometric titration curve curs in region C. If a polarogram is run FGCHI. Polarogram of solution at endpoint a t the end point, a curve such as FGCHI would result. The FG wave represents the oxidation current for the higher voltage couple (cericpendent upon 2. The shape of this end point has been reported cerous) and the H I wave represents the reduction wave of the for the titration of metavanadate by ferrous ( 5 ) . If the k values lower voltage couple (ferric-ferrous). The amperometric titraare identical, the mid-point of the titration should give a value of tion end point is illustrated by the intersection of two straight da/dE = -nFkCo/4RT. As the previous equations were based lines, JKC and CLM. The coincidence of the potentiometric upon a rapid, reversible system, other approaches must be used and amperometric end points is seen a t C. for irreversible systems. The dead-stop end point has been applied for the most part to irreversible systems titrated iodometriLITERATURE CITED cally. Figure 3 (IO)shows an irreversible system such as arsenite Bottger, W., and Forsche, H. E., Mikrochemie, 30, 138 (1942). Cooke, W. D., Rellley, C. K.,and Furman, N. H., ANAL.CHEW, when titrated with iodine. In the initial solution, A , reduction 23, in press. of H1O+and oxidation of OH- may be the only reactions that Delahay, P., Anal. Chim. Acta, 4, 635 (1950). could occur to any appreciable extent. As such a small voltage Foulk, C. W., and Bawden, A. T., J. Am. Chem. Soc., 48, 2045 (about 10 to 15 mv.) is usually applied, no current will flow due (1926). Gale, R . H., and Mosher, E., ANAL.CHEM.,22, 942 (1950). to the hydroxyl or hydronium ion discharge. Any irreversible Kolthoff, I. M., and Furman, N. H., “Potentiometric TitraRubstances present which are oxidizable or reducible a t the elections,” Kew York, John Wiley & Sons, 1926. trodes will give a curve as shown by the dotted line(s) in A. ThioKolthoff, I. hl., and Lingane, J. J., “Polarography,” Kew York, sulfate, for example, has been shown to be electrolytically ouiInterscience Publishers, 1941. Mitchell, J., Jr., and Smith, D. hl., “Aquametry,” p. 86, Kew dizable ( I ) . Thus, as the slope is about zero initially, and reYork, Interscience Publishers, 1948. mains so up to the end point [Figures 3 A and B ( I O ) ] , the current, RZiiller, Erich, “Electrometrische Rfassanalyse,” Dresden, T. d,, will also be about zero in this region. Any impurity, or traces Steinknopff, 1942. of iodine to couple with iodide, will increase the slope and give Reilley, C. N., Cooke, W. D., and Furman, N.H., ANAL.CHEM., 23, 1223 (1951). rise to a small current. Immediately past the end point [Figure Stock, J. T., Metallurgia, 37, 220-3 (1948). 3, C ( I O ) ] , the slope suddenly increases owing to the presence of Van Kame, R. G., and Fenwick, F., J . Am. Chem. Soc., 47, 19 excess iodine, and the current, di, increases rapidly. Thus, up (1925). to and a t the end point, the current is small, with a sudden rise in RECEIVED hIarch 30, 1951
+--
Variability in the Beckman Spectrophotometer W. 0. CASTER‘ Nutritional Chemistry Laboratory, Nutrition Branch, U.S . Public Health Service, Washington, D . C.
A
M J M B E R of approaches have been used in describing spectrophotometric error. Some workers ( 3 , 8, I S ) have concerned themselves primarily with estimating the highest degree of precision obtainable with an instrument under certain given conditions. Others ( I O , 20, 23) have been more interested in locating and estimating the magnitude of the errors introduced into their analytical data as a result of using a spectrophotometer. The latter approach can be criticized, in that it may not distinguish between errors traceable to the instrument and errors inherent in the technique. However, this approach yields over-all error values which are usually of more practical interest to the analytical chemist. As these approaches are accompanied by different testing conditions, it is not surprising that certain differences of opinion have ’Present address, Physiological Chemistry Department, University of Minnesota. Minneapolis, Minn.
arisen concerning the accuracy expected from a given instrument. It is reported that, under certain conditions, the Beckman quartz spectrophotometer is capable of yielding transmittance readings with a precision of 0.1% (6, 8), 3~0.07%(ZO), or 0.02% ( 5 ) transmittance. Ewing and Parsons ( I O ) reported that although individual Beckman spectrophotometers may give highly consistent results, there is a spread of several per cent between the analytical results obtained from a series of different instruments. In a collaborative assay ( 1 5 ) differences of as much as 10% were observed in the standardization of a series of Beckman spectrophotometers. By special techniques, Bastian (4,6) reported analytical errors smaller than 0.1% with the Beckman spectrophotometer. These widely differing results pose a real problem for the analyst. Under what conditions is it safe to report such values as 1742 ( I I ) , 27,450 (IQ), 4954 ( 2 5 ) ,and 14,704 (@), which imply
