ANALYTICAL CHEMISTRY, VOL. 51, NO. 4, APRIL 1979
585
high purity commercial solvents could be achieved. IO 0
'QE I
6 0
20
-
I0
m
c
0
x
=2
:
-
05 03
02
CONCLUSION Solvent scatter and luminescence have been shown to be major factors controlling the limits of detection in the conventional fluorescence technique. Solvent luminescence can frequently be reduced by orders of magnitude with careful selection or purification. Temporal resolution would also be valuable for isolating any sample emission long lived in comparison to the fluorescent impurities of the solvent. Spectral resolution is of little value, however, unless the emission is greatly red-shifted from the excitation energy. In contrast, the interference caused by scattered emission can frequently be attenuated with either spectral or time resolution, because of its sharp spectral and fast temporal characteristics. ACKNOWLEDGMENT We thank J. M. Harris for the initial work on the low cost cell design and the laser system.
01
LITERATURE CITED C
350
400
450
550
500
600
Wavelength ( n d
Figure 5. Methylcyclohexane purification. (0-0-0) (O-.O-.O) Purified
Unpurified.
mixtures but can only be purchased in less than spectral grade quality. T h e luminescence of the best of 4 equal fractions collected from a packed Vigreux column (-10 theoretical plates) is presented in Figure 5 . Although significant improvement was made, much more elaborate purification procedures would be necessary before results competitive with
(1) R. E. Brown, K. D. Legg, M. W . Wolf, and L. A. Singer, Anal. Chem., 46( 12). 1690 (1974). (2) F. E. Lytle and M. S. Kelsey, Anal. Chem., 46(7), 855 (1974). (3) R . N. Zare and M. R. Berman, Anal. Chem., 47(7), 1200 (1975). (4) J. H. Richardson and S. M. George, Anal. Chem., 50(4), 616 (1978). (5) C. A. Parker, "Photoluminescence of Solutions", Elsevier, New York, 1968. (6) C. Reichardt, and K. Dimroth, Fortschr. Chem., Forsch., 11, 1-69 (1968). (7) F. J. Smith, J. K . Smith, and S.P. McGlynn, Rev. Sci. Instrum., 33(12), 1367 (1962).
RECEIVED for review November 16, 1978. Accepted January 18, 1979. This research was supported in part through funds provided by the National Science Foundation under Grants MPS75-05907 and CHE77-24312.
Algorithm for the Determination of Decay Rate Constants by Reversal Current Chronopotentiometry Donald A. Tryk and Su-Moon Park" Department of Chemistry, The University of New Mexico, Albuquerque, New Mexico 8713 1
Current-reversal chronopotentiometry (CRCP) is a technique which can complement cyclic voltammetry by providing quantitative kinetic data for slow following reactions investigated qualitatively by cyclic voltammetry. CRCP has perhaps been under-utilized because of the difficulties of its data treatment. We wish to report a convenient algorithm for the computation of pseudo-first-order decay rate constants of electrogenerated species using the method of current-reversa1 chronopotentiometry. For the case where the forward electrolysis current, if, equals the reversal current, i,, the diffusion equation yields the analytical solution ( I ) . 2 erf
6= erf d
m
(1)
where t is the forward electrolysis time, T is the time elapsed between the current polarity switching time and the transition time for the electrogenerated species, and h is the pseudofirst-order decay rate constant. For the case in which i, # if, the solution is (2) (u
+ 1) erf 6 = erf d
m
(2)
where u = irjifi 0003-2700/79/0351-0585$01 O O / O
The usual method for the computation of k has been to read values of the dimensionless quantity ht from a graph or table of k t vs. the experimental dimensionless quantity, T / t (I). Herman has outlined a method for computer-generating such a table ( 3 ) . In order to increase the precision of h , the experiment may be run a t various values o f t to obtain a series of values of kt. These may be plotted against t , the slope of the line being k ( 1 ) . In terms of computer programming, however, a method requiring the use of a stored table and interpolation therefrom suffers from defects in storage economy, speed, and precision. Another approach is to recast Equation 1in an explicit form, giving the quantity t l r as a function of k r : 1
t / 7 = -[erf-1(2
kr
erf
\/G)I*1 ~
(3)
where e r r ' is the inverse of the error function. T h e inverse error function is readily implemented as a subroutine using the Newton method ( 4 ) , since the derivative of the error function is available ( 5 ) , and the error function itself is commonly available as a subroutine. The geometric form of Equation 3 is shown in Figure 1. At large values of t / 7 , k r 1979 American Chemical Soclety
586
ANALYTICAL CHEMISTRY, VOL. 51, NO. 4, APRIL 1979
t/r
o
v 0
, t,s
10
15
Experimental data with computer-generated M values. Each point is averaged from three experiments (See text for experimental details) Figure 2.
kr
Theoretical curve for CRCP based on Equation 3. The thinner lines demonstrate the operation of the False Position algorithm. T h e asymptote lies at k T = 0 . 2 2 7 5 Figure 1.
becomes insensitive to changes in t / 7, but this characteristic is inherent in Equation 1. The problem then becomes one of finding roots of Equation 3 so that values of k i can be computed for input values of t / T . Rather than attempting to derive a derivative of t / T with respect to k T , which would be necessary in order to employ the Newton method, t h e False Position method ( 4 ) may be used. This method does not require a derivative to be known and is nearly as efficient as the Newton method. Also, the False Position method required less than half the average number of iterations required by a simple stepping search when used on a sample data set. Equation 3 was fitted to several different differentiable functional forms in order that the Newton method could be used directly, but a good fit could not be obtained. In its simplest form, the False Position method may be set up using the origin (point 2) and a point on the curve in Figure 1 arbitrarily close to the asymptote, h i = 0.22 (point l), as t h e starting points. In geometric terms, a line is drawn between the two points, and the intersection of this line with the horizontal line corresponding to an experimental t / +value is found. The 127 coordinate of this point is used to compute a value of t / T using Equation 3. This coordinate pair corresponds to a point on the curve (point 3) from which a second line is drawn to point 1. Again the intersection with the horizontal t / T line is found, and the k T value used to compute the t / T coordinate of a new point on the curve (point 4). The process is continued until the desired accuracy is obtained. T h e iteration formula is then:
where ( t / T ) d is the desired experimental value. I n order to achieve the efficiency indicated above for this algorithm, the curve was divided into two parts, and t / r values below 5.6970 were treated using (0.1600, 5.6970) as the point of false position. Values of t / T above this were treated using (0.2200, 12.6524). This treatment can easily be extended to encompass situations in which i, # if by using Equation 2. Having now a value for h i for an experimental value of t / i , one computes k t and plots values of k t vs. t as in Ref. 1. Figure 2 shows a plot using computer-generated values for an experiment similar to one run by Testa and Reinmuth ( I ) on the oxidation of p-aminophenol (PAP). Twice-recrystallized PAP was 1.13 mM in 0.0965 M H2S04,run at a current of f l WAusing a Pt disk electrode of area 0.021 cm2. The k value was 0.132 f 0.008 5-l a t the 90% confidence level. This compares well with the value of Testa and Reinmuth ( I ) , which was 0.103 f 0.003 s-l for 0.102 M H2S04and with that of Herman and Bard ( 6 ) det,ermined by cyclic chronopotentiometry, which was 0.115 s-' for 0.10 M H2S04. All of these values are a t 30.0 "C. A print-out of a FORTRAN IV program using the False Position method with least-squares and plotting subrout,ines is available from the authors. LITERATURE CITED (1) A. C. Testa and W. H. Reinmuth, Anal. Chem., 32, 1512 (1960); C. Furhni and G. Morpurgo, J . Electroanal. Chem., 1, 351 (1959-60). (2) 0. Fischer and 0. Dracka, J . Electroanal. Chem., 75, 301 (1977); 0. Dracka, Collect. Czech. Chem. Comm , 25, 338 (1960). (3) H. B. Herman in "Electrochemistry: Calculations, Simulation and Instrumentation", J. S. Mattson, H. B. Mark, and H. S. MacDonald, Jr., Ed., Marcel Dekker, New York, 1972, p 63. (4) F. S. Acton, "Numerical Methods That Work", Harper and Row, New Y w k . 1972, p 52. (5) M. Abramowitz and I. A. Stegun, Ed., "Handbook of Mathematical Functions", NBS Applied Mathematics Series, No. 55, U.S. Government Printing Office, 1965, p 311. (6) H. B. Herman and A. J. Bard, Anal. Chem., 36, 510 (1964).
RECEIVED for review July 17, 1978. Accepted December 19, 1978.
Digestion Tube Diffusion and Collection of Ammonia for Nitrogen-15 and Total Nitrogen Determination William A. O'Deen" and L. K. Porter USDA, Science & Education Administration, Agricultural Research,
Total N in seeds, plants, soils, manures, sludges, organic residues, and organic compounds has traditionally been determined by Kjeldahl analysis. The Kjeldahl digestion
P. 0. Box
E, Fort Collins, Colorado 80522
converts the organic r\; to NH4+. Use of digestion tubes arrayed in an aluminum heating block allows greater numbers of samples to be digested simultaneously ( 1 ) . T h e NI&+
This article not subject to U.S. Copyright. Published 1979 by the American Chemical Society
