Pseudo-Three-Dimensional Presentation of Reaction Rate Data Stanley N . Derning Department of Chemistry, Emory University, Atlanta, Ga. 30322 DEVELOPMENT of a reliable analytical method requires a study of the effects of many independent variables upon the measurable response parameter (e.g., absorbance, reaction rate, titrant volume, etc.). If there are n o interactions among variables, several simple graphs, each presenting experimentally-measured values of the response parameter as a function of one of the variables, are sufficient for a meaningful presentation of the data. However, if interactions among variables occur, then to show these interactions severai curves must be plotted o n each graph, all curves plotted as response parameter cs. one variable but each curve representing a different level of a second variable. Such plots are in effect the projection of information from three-space to twospace. An alternate approach to this problem of presenting a n experimentally-measured dependent variable as a function of two independent variables is to use pseudo-three-dimensional plots. This method of plotting has been used for the presentation of various types of experimental data ( I , 2). Kinetic methods of analysis have become established as useful techniques for the analytical chemist (3-5). The development of a kinetic method requires a knowledge of the effects of several variables upon the reaction rate. The use of reaction-rate methods to analyze mixtures of two or more catalytic species is aided by pseudo-three-dimensional graphs. The functions relating reaction rate to the concentrations of (usually) two reactants may be different for each of the catalysts in the mixture and therefore optimum conditions may be chosen for carrying out the analysis. All calculations were done using a PDP-9 computer (Digital Equipment Corporation, Maynard, Mass.) with 8K of 18-bit words and a I-psec memory cycle time. FORT R A N was used as the main programming language and called a n assembly language subroutine for output of X-Y values through a dual 12-bit digital-to-analog converter interface to a n analog X-Y recorder (Model 7004A, HewlettPackard, Palo Alto, Calif.). The digital output correr sponding to the least significant bit of the X-axis signal was used to control a relay closure to ground for use as a pen-lift signal to the recorder. The plotting procedure is in effect essentially the same as that described by Bordass and Linnett (6). Those portions of a surface that would not be visible from the perspective of the viewer are suppressed. Provision is made within the program to specify the ratio of height-to-width of the drawing and still retain the desired angles between the X, Y , and Z axes in the conventional isometric plots. The variable sensitivity of the X and Y inputs of the recorder can be used to vary the dimensions of he drawing if desired. Each plot takes from four to about 15 minutes, depending upon the complexity of the calculations, the number of grid lines (1) R. D. Sacks and J. P. Walters, ANAL.CHEM., 42, 61 (1970). (2) B. G. Willis, W. H. Woodruff, J. R. Frysinger, D. W. Margerum, and H. L. Pardue, ibid., p 1350. (3) G. G. Guilbault, ibid., p 334R. (4) H. B. Mark, Jr., and G. A. Rechnitz, “Kinetics in Analytical Chemistrq,” Interscience, New York, N. Y., 1968. ( 5 ) G. A. Rechnitz, ANAL.CHEM., 40,455R (1968). (6) W. T. Bordass and J. W. Linnett, J. CIiem. Educ., 47, 672 (1970). 1726
Figure 1. Rate coefficient for osmium catalysis as a function of arsenic(II1) and cerium(1V) concentrations As
Ce R
= = = =
[As(III)] from O . O M to 0.lM [Ce(IV)]from 0.01M to O . 1 M rate coefficient for osmium catalysis 1.04 X lo4 sec-l at maximum arsenic and cerium concentrations
drawn, etc. Copies of the program are available upon request. Figures 1 , 2, and 3 are pseudo-three-dimensional plots of rate coefficient (vertical axis) as a function of the concentrations of cerium(1V) (distal axis) and arsenic(II1) (proximal axis) for the catalysis by osmium ( 7 ) , iodine (8), and ruthenium (9), respectively. Rate coefficient is defined here as f([Ce(IV)], [As(III)]) in the rate expression R = f([Ce(IV)], [As(III)]) [Catalyst]. The qualitative differences in the mechanisms of catalysis are readily apparent. Quantitative differences could be obtained for any pair of cerium and arsenic concentrations by measuring the distance from the cerium-arsenic plane to the reaction surface; however, numerical computation from the mathematical rate equations is easier and preferable for quantitative work. The figures show not only conditions of cerium and arsenic concentrations for obtaining the greatest rate of reaction, but also those areas that are most sensitive to small uncertainties in reactant concentration. For some analyses, the analyst may prefer to sacrifice detection limits in order to gain insensitivity to reactant concentrations. Further, (7) R . L. Habig, H. L. Pardue, and 3. B. Worthington, ANAL. CHEM., 39,600 (1967). (8) P. A. Rodriguez and H. L. Pardue, ibid., 41, 1376 (1969). (9) 1. B. Worthington and H. L. Pardue, ibid., 42, 1157 (1970).
ANALYTICAL CHEMISTRY, VOL. 43, NO. 12, OCTOBER 1971
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Figure 2. Rate coefficient for iodine catalysis as a function of arsenic(II1) and cerium(1V) concentrations As
Ce R
= =
= =
[As(III)] from O . O M to 0.1.44 [Ce(IV)] from O.OM to 0.1M rate coefficient for iodine catalysis 1.03 X lo5 sec-l at maximum arsenic and cerium concentrations
Figure 4. Ratio of rate coefficient for ruthenium catalysis to rate coefficient for osmium catalysis as a function of arsenic(II1) and cerium(1V) concentrations As = [As(III)J from 0.0714 to 0.1M Ce = [Ce(IV)J from O.OM to 0.1M R J R , = ratio of rate coefficient for ruthenium catalysis to
rate coefficient for osmium catalysis maximum arsenic and cerium concentrations (ratios greater than 10.0 set equal to 10.0)
= 10.0 at
the extent t o which the reaction is not pseudo-zero-order in cerium and arsenic can be estimated by locating o n the reaction surface a line segment originating at the initial concentrations of cerium a n d arsenic, of slope equal t o 2/1 when projected to the cerium-arsenic plane (A[Ce(IV)]/ A[As(III)I = 2/1 as required by the stoichiometry of the reaction), and of length equal to the corresponding extent of reaction. Information of this type can be obtained mathematically by differentiation, optimization, and other means, but the results are difficult to interpret without the understanding of the total reaction surface afforded by pseudo-three-dimensional plots. Mixtures of catalytic species may be determined using reaction-rate techniques if at least two analyses are carried out at different reactant concentrations and simultaneous equations are solved. Thus,
,r
a
+ ~12x2 a?ixi +
ailxi
=
022x2 =
bl
(11
b?
(2)
where x j = concentration of catalyst j
V
Figure 3. Rate coefficient for ruthenium catalysis as a function of arsenic(II1) and cerium(1V) concentrations As
Ce R
= = = =
[As(III)] from O.OM to 0.1M [Ce(IV)Jfrom O.OM to 0.1M
aij = rate coefficient for catalyst j under reaction conditions i br = total rate for reaction conditions i Solving gives
rate coefficient for ruthenium catalysis 3.20 X lo6 sec-l at maximum arsenic and cerium concentrations
XI =
blaaz a11a22
- alzbz - u21u12
ANALYTICAL CHEMISTRY, VOL. 43, NO. 12, OCTOBER 1971
(3) 1727
x2 =
u11u2? mb2
bla21
(4)
a21a12
If it is assumed that there is no error in pression for the error analysis becomes
ai3, the
+ /3jC(Abq)2
AX,' = aj2(Abi)2
general ex(5)
The error coefficients are
To minimize the error coefficients, the term u11u22should be very large with respect to the term mlal?. Thus, the ratios should be ull/a12should be very large and the ratio u2ir'u2? very small. Phrased differently, for minimal error in the determination of the catalyst concentrations, it is desirable to carry out one of the analyses at conditions of reactant
1728
concentrations such that the contribution to the total rate is large for one catalyst and small for the other, and to carry out the other analysis at conditions such that the relative contributions of the catalysts to the total rate are reversed. A pseudo-three-dimensional presentation of a surface representing the ratio of the two rate coefficient surfaces should be useful for determining regions of reactant concentrations that meet the above criteria. Figure 4 is a plot of the ratio of the rate coefficient of ruthenium catalysis to the rate coefficient of osmium catalysis. The surface is truncated for ratios greater than ten. There is a broad area of cerium-arsenic concentrations over which the ratio exceeds the arbitrary value of 10. The region over which the ratio is less than the arbitrary value of 0.1 is less extensive. A qualitative understanding of the relationships between the three variables is possible that cannot be obtained from strictly two-dimensional projections or from mathematical expressions. Reasonable conditions for optimum analytical conditions can be chosen. ACKNOWLEDGMEhT
The computer interface was constructed by Lee H. Altmayer. M. Kent Burel prepared some of the plots. RECEIVED
ANALYTICAL CHEMISTRY, VOL. 43. NO. 12, OCTOBER 1971
for review April 14, 1971. Accepted June 21, 1971.
