Table I. D e t e r m i n a t i o n of V a n a d i u m in BCSe Steels Certified value,
BCS steel
64a 252
Vanadium found, 7a
70
S
1.546, 1.55, 1.56, 1.538, 1.556. Average value 1.55 0.440, 0.460, 0.450, 0.450, 0.455. Average value 0.451
1.57
0.0084
Several analogs of 1,2,3-phenyloxyamidine have been synthesized and characterized by elemental analysis and IR spectra. Their physical properties, analytical applications, and stability constants of their metal complexes have also been studied.
EXPERIMENTAL 0.46
0.0074
BCS, British Chemical Standards, Bureau of Analyzed Samples, Ltd, Newham Hall, Middlesbrough, Yorks.
Preparation of the Reagent. The reagent, 11, is prepared by the condensation of N-phenylbenzimidoyl chloride and N-phenylhydroxylamine in absolute CGHS-N-OH
I
C,H,--C=h'-C,jH, dine. The reagent yields water-insoluble blue-violet precipitate with vanadate ions in 1.0 to 10.OM acetic acid media which is quantitatively extracted in chloroform. The absorbance curve of the colored system shows a broad absorbance maxima a t 550-575 n m when measured against a chloroform blank. The reagent (A,,, 315 nm) and the vanadium complex have well-separated absorption maxima so that the excess of the reagent does not interfere in the photometric determinations. Final determination of vanadium is based on the measurement of the absorbance of the blue-violet extract a t 560 nm. The colored system conforms to Beer's law and the optimum range for color measurement as computed by Sandell's method (7) is 2.4 to 8.8 ppm of vanadium. The Sandell sensitivity (8) of the color reaction is 0.011 pg of vanadium per cm2 at 560 nm. The molar absorptivity is 4.21 x 103 liter mole-l cm-1. The method provides a simple and efficient method for the separation and determination of vanadium in the presence of several diverse ions. The common ions including Fe(III), Mo(VI), Cr(III), Mn(II), Cu(II), Ni(II), Sb(III), A s O ~ ~do- not interfere. Ti(1V) and W042- inhibit the color development and thus seriously interfere with the determination. The vanadium content of the BCS steels, 64a and 252, has been determined precisely and accurately. The results are summarized in Table I. Because of the high sensitivity of the reagent, small sample weights can be used, thus avoiding long dissolution times. The results of detailed investigations will be communicated shortly. "Colorimetric Determination of Traces of Metals," 3rd. ed.. Interscience Publishers, Inc., N e w Y o r k , N . Y . , 1959, p 97. (8) Ref. 7, p 83.
(7) E. B. Sandell,
I1
ether medium. N-Phenylbenzimidoyl chloride can be obtained by several methods (9-11) starting from benzanilide by the action of phosphorus pentachloride or thionyl chloride or by the action of phosphorus pentachloride on benzophenone oxime. Procedure. Twelve grams of N-phenylbenzimidoyl chloride (0.055 mole) were taken in about 150 ml of absolute ether in a 500-ml short-necked, round-bottomed flask and cooled in ice-cold water. To this, N-phenylhydroxylamine 6.1 grams (0.056 mole) in absolute ether was added in portions over a 20-minute period while stirring with a glass rod. A light brown oil slowly separates out. Stirring and shaking were continued till the light-brown oil solidified and separated as shining crystals resembling sodium chloride. This was filtered off and washed with 3 x 20-ml portions of ether. The resulting 1,2,3-phenyloxyamidine hydrochloride weighed 6.5 grams. The hydrochloride salt thus obtained was treated with 25 ml of 1N ammonia solution in a conical flask and the precipitated 1,2,3-phenyloxyamidine free base was filtered off, recrystallized three times from 70% aqueous ethanol to give pale yellow crystals, mp 170"; yield, 4.5 grams. Anal. Calcd for C19H16N20: C, 79.16., H, 5 . 5 5 . , Pi, 9.72. Found: C, 79.06., H, 5.62., N, 9.87. The substance was insoluble in water and soluble in chloroform, ethanol, acetone, benzene, and acetic acid.
ACKNOWLEDGMENT The authors are indebted to R. N. Tandon, Principal, College of Science, Raipur, for providing research facilities and to G. Alfred, Synthetic Drugs Project, Hyderabad-37, for the elemental analysis.
RECEIVEDfor review May 25, 1973. Accepted February 28, 1974. (9) I Ugi F Beck and U Fetzer , Ber (10) 0 Wallach A n n 184, 79 (11) E Beckmann Ann 252, 1
95, 126 (1962)
Finite Difference Methods in the Theory of Chronopotentiometry D. K. McCreary and P. D. Schettler Chemistry Department, Juniata College, Huntingdon, Pa. 16652
Many different methods have been utilized t o solve chronopotentiometric diffusion equations; however, the use of finite differences in conjunction with a computer may be the most universally applicable one. Strangely, however, numerical methods seem t o have been overlooked, in spite of the fact that such techniques may be of the broadest applicability. While most methods work only for linear equations or with certain restrictions as t o parameters, a set of finite difference equations can be constructed for any set of 1610
9
chronopotentiometric equations. Finding a stable solution can be a difficulty, but it would appear that this can be circumvented by a judicious application of theory and experience. In any event, instability is easy t o detect, once it has occurred. Accuracy depends primarily on the computation time expended. We have used the method to solve two chronopotentiometric mechanisms, and we have discovered some distinctive features in their @i us. io plots. T h e error associated with our computations is about 5%.
A N A L Y T I C A L C H E M I S T R Y , VOL. 46, NO. 11, SEPTEMBER 1974
Plots of i o ~ l us. l ~ io where io is the current density and T is the transition time are particularly valuable and have long been used to obtain information about electrode kinetics. This utility is much enhanced by use of a library of theoretical solutions of the appropriate sets of partial differential (chronopotentiometric) equations. In some cases exact solutions to these equations can be found (1-7). Unfortunately, many interesting cases either require approximations which are usually only applicable under certain restricted conditions, or remain unsolved. The use of finite differences does not have such restrictions per se, although the stability of the solution may not be predictable a priori when nonlinear partial differential equations are involved. Nevertheless, an unstable solution is easy to detect as a matter of practice and, by a judicious choice of time and distance increments, solutions may be found to a wide variety of problems not approachable by other means. One catalytic mechanism studied was bf'
A+Y===B
bulk
I
'b
(2 1 where CA and Cg are the concentrations (moles/cm3) of components A and B, respectively, D is t,he diffusion constant of both species, t is the time (sec) and x is the distance from the electrode (cm). For mechanism I, p = 1 and kf = k ; Cy where k ; and k b are the forward and backward rate constants of the bulk reaction. It is assumed that Cy is large in comparison to other concentrations and, hence, l z f is a constant. For mechanism 11, p = 2 and kf and k b are the forward and backward rate constants, respectively. For both systems, the appropriate initial and boundary conditions are as follows: c,(x, 0) = cAO, cB(x, 0) = cBO for all x (3)
lim c A ( x , t ) =
The electrode reaction may be either cathodic or anodic and, hence, we have not included the electron. Delahay (3) had previously considered special cases of this mechanism, in particular the case wherein Y is in excess so that its concentration remains constant, the initial concentration of A is zero, and the backward rate constant of the bulk reaction is zero. In this paper computer results for I are presented where Y is still in excess, but where the initial concentration of A and the backward rate constant are both finite. The other half-cell catalytic mechanism that is described here is kf
2A ==+ B
I1
kb
B +A
+
Y
electrode
Again the electrode process can be either an oxidation or a reduction. When the current density is either very high or very low, the differential equations describing this system have exact solutions, but the techniques used for these special cases are not applicable in the intermediate region. Similar cases were studied by Reinmuth ( 5 ) for the mechanism pY e 0 R, but again under restrictive conditions. These catalytic or self catalyzed reactions have some importance. An example of Mechanism I is the Fe", HzOz system as reported by Kolthoff (8). There have been suggestions of self catalyzed processes such as Mechanism I1 in respect to electrode processes in the metal ammonia system (9).
cB0f o r
all i
(4)
";F3
w=o
=
-
acB(x7t)l
r=O =
ax
>
- i o / n ~ for ~ all i 0 (5) where io is the current density (A/cm2), F is the faraday, and CO is the concentration before electrolysis begins (equilibrium concentration). The equations for mechanism I can be solved by the method of Laplace transforms, the solution being C B ( O , f ) = (kf + kb)-'{kf(CAo f CBO) -
+
[io(kf + kb)'/2/?ZFD'/2] e r f [ ( k f kb)"2t1/2]) (6) which reduces to Delahay's solution (3) when hb and are set equal to zero. The transition time is thus given by 7
bulk
/) =
X-m
e le ctr ode
B - A
cA0,lim c,(x,
X-m
= (kf
+
kb)-'{erf-' [nFD'/2kf(CAo +
CBo)/(kf + kb)'i2iO])2 (7) where erf-' indicates the inverse of the error function. Exact general solutions of the equations for Mechanism I1 remain elusive except in two special limiting cases. In the first, a t very high current densities, where the bulk reaction does not have time to contribute to the concent,ration of the electroactive species, the solution is C,(O,t)
CBO
-
2iof*/2/nF(D~)1/2 (8)
-+
THEORY The diffusion equations of both mechanism I and mechanism I1 have the form
(1) P. Delahay and T. Berzins, J. Amer. Chem. SOC.,75, 2486 (1953). (2)T. Berzins and P. Delahay, J. Amer. Chem. SOC.,75, 4205 (1953). (3)P. Delahay, C. C. Mattax, and T. Berzins, J. Amer. Chem. SOC.,76, 5319 (1954). (4)W. H. Reinmuth, Anal. Chem., 32, 1514 (1960). (5)W. H. Reinmuth, Anal. Chem., 33, 322 (1961). (6)L. Gierst and A . L. Juliard. J. Phys. Chem.,57, 701 (1953). (7)P. Delahay and G. Mamantov, Anal Chem., 27, 478 (1955). (8)I. M. Kolthoff and E. P. Parry, J. Amer. Chem. SOC..7 3 , 3718 (1951)
i,r1/2 I ~ / ~ c , O ~ F ( D T ) ' / ~ (9) In the second, a t very low current densities, the transition time is so long that the species reacting in the bulk solution are essentially in equilibrium a t all times, and the solution becomes CB(0, f ) = CBo + l/?CAo - [ i , ~ " 2 / ? 2 F ( D ~ ) *+ /2] kb/8kf
+
l/{[kb(CBo
+
1/2CAo)/kf]
[iot"2kb/kfnF(DT)i/2] + [kb2/16kf2]}112
(10)
iof'" = [(CBo f 1/CAo)nF(DT)"2]/i, (11) Equations 9 and 11 describe the lower and upper bounds of i o ~ l for l ~ this mechanism. We utilized numerical methods to obtain solutions of the nonlinear case (Mechanism 11) between these bounds. Using a forward difference for the time differential and a central difference for the distance differential, the finite difference approximation (written in computational form) is (9)P. D. Schettler, C. L. Van Antwerp, J. A. Hamilton, J. E. Thilly, and J. D. Spear, in "Electrons in Fluids," N. Kestner and J. Jortner, Ed., SpringerVerlag, New York, N.Y., 1973 p 239.
A N A L Y T I C A L C H E M I S T R Y , VOL. 46, NO. 11, SEPTEMBER 1974
1611
Tt
*
-
?,'=-I, *
, -:=-?
3 3 F -iI
L
ZiF-,
.
.X F - ,
.:CF~-7.
uc
c
-02a
- - -21 +
++iii_t
4
055
+---i
5-
-!NE
Figure 1. G(0,f ) is plotted against time (sec) for mechanism I1 at a current density of 1 A and with the values of k, and kb varied as follows:
sec-l
-
- -)
kf = 1 X i o 5 crn3/rnole sec, k,, = 1 sec-'. (- -) kf = 1 x 109 cm3/ mole sec, kb = 1 X i o 4 sec-'. (-1 kf = 1 x 1 o ' O crn3/mo1esec. kb = 1 x 1o5 sec(-
(A-sec1/'/crn2)IS plotted vs Figure 3. Mechanism I1 Log of log 4 (A/cm2)where k, = 1 X i o 9 cm3/rnole-sec, and kb = 1 x i o 4
'
the limit h and g 0. Utilizing the theoretical work reported for linear equations (10) as a guide, we found that stability was attained under the following conditions:
hD/g2 .xi
g.i,/nFD
5
1
X
5
1.0
X
lo-'
(16)
l o - ' max (C,'); I = A o r B
[hkf(CAo)Pand h k , ~ , ~ ]5 1 x 10-1 max
(17)
(c,'); I
A or B
=
(18)
CALCULATIONS A N D RESULTS In the calculations that were made, D = 1 X low5cm2/ sec, CA' = 1.0 x IO+ mole/cm3, Ceo = 1.0 x 10-5 mole/ cm", and IZ = 1. The parameters hf, k b , and io were varied.
These values produced results within 5% of values obtained by direct calculation (Equations 7, 9, 11) in regions where comparisons could be made. In addition, the results did not change excessively outside this region for certain select,ed points where h was decreased by an order of magnitude. The parameter h was adjusted (within the constraints of Equations 16-18) so as to produce a constant change of concentration a t the first iteration; this means that error in the transition time will be of the same order of magnitude (-6%) from run to run. The effects of changes in the rate constants k f and hb are depicted in Figure 1. As expected from qualitative consideration of the mechanism, faster reaction kinetics produce longer transition times by increasing the amount of electroactive species available a t the electrode. Figures 2 and 3 show logarithm i ~ ~ lus. / *logarithm io plots of mechanisms I and 11, respectively. The two mechanisms differ greatly in the characteristic shapes as indicated. In Figure 2, the curves go off to infinity as low current densities are approached. However, in Figure 3, we find an inflection point and a plateau a t low current densities. It has been pointed out that this is characteristic of pre-kinetic cases. The error due to the approximation for mechanism I is easily obtained as the exact solution is available and our mean error was 5.6%. For mechanism 11, there is no exact solution; however, in the limit of high current density the i g ~ l / ?graph approaches the value given by Equation 9, and such a comparison suggests a mean error of 5% for the computer results here also. Decreasing h by a factor of ten in mechanism 11 gives an error of about 1%,with a resultant large increase in computer time. We see no reason why the method cannot be extended to other nonlinear (or otherwise intractable) mechanisms that cannot be solved by rigorous means but which, nevertheless, are of interest in interpreting laboratory data. The
The parameters h and g were adjusted so as to find regions of stability that reasonably could be expected to extend to
(10) G. E. Forsythe and W. R. Wasow, "Finite Difference Methods for Partial Differential Equations," Wiley, New York. N.Y.. 1960.
.il_ i
+-__+
_---_
._
.L.
ii--i-
~~
.3
1
.,~->-4.-++.++4++
