Simple analog system for simultaneous kinetic analysis

described in references (21, 24, 25) while a voltammetric analytical study on reaction 6 is in preparation. Of course only curve a was present when the concentrations of carbon dioxide and oxygen verified (see Equations 3 and 5 ) the condition ~ [ C O ~ ] D C=O[ ~O~Z ’] ~D O ~ ~ ’ ~ (10) co32-Together. The cathodic anodic current-potential profiles obtained with the three species present together in the melt were in agreement with the results presented in the previous paragraphs. A representative example of the curves (recorded under conditions of excess of O1 on COz at the microelectrode surface) is reported in Figure 9. Curves a and b are separated by a

large potential range which indicates that Equation 2 behaves as voltanimetrically irreversible. From an analytical point of view, it can be noted that voltammetric profiles such as those reported in Figure 9 offer the opportunity of a quantitative determination of all the species present in solution. With reference to Figure 9

Solution Containing COz, 02,and

Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on September 5, 2015 | http://pubs.acs.org Publication Date: April 1, 1972 | doi: 10.1021/ac60312a025

(24) P. G. Zambonin, J. Electrounul. Chern., 24, 365 (1970). (25) P. G . Zambonin, V. L. Cardetta, and G. Signorile, ibid., 28, 237 (1970).

(id, = KI[CO~~-I; ( i d 0 = KZ[COzl;

(id,= KdOd

(11)

Of course the same analytical informations can be obtained (see the previous paragraph) from current-potential curves recorded in the presence of an excess of CO2 or when the bulk concentrations of the two gases are in the ratio required by Equation 10. RECEIVED for review July 19, 1971. Accepted October 5. 1971.

Simple Analog System for Simultaneous Kinetic Analysis James B. Worthingtonl and Harry L. PardueZ Department of Chemistry, Purdue University, Lafayette, Ind. 47907

A system for automatic graphical presentation of simultaneous kinetic analysis data is presented. The system uses the graphical extrapolation method for the determination of the concentration of two component mixtures. A simple analog circuit coupled with an X-Y recorder provides a graphical plot containing linear regions related to component concentration. Extrapolations from these linear regions provide concentration directly. The system is illustrated for the determination of rare earth mixtures using a ligand exchange reaction. Quantitative determinations of lanthanum and neodymium mixtures down to 2 X 10-6Mwith accuracies of 0.3 to 16% are reported. DURING THE PAST DECADE, there has been a sharp increase in interest in kinetic methods of analysis (1-3). One of the primary reasons for this has been the development by several investigators of instrumentation for the automation of the measurement and computational operations involved in these analyses (4-6). A survey of the literature in this area shows that most instrumental developments have been directed toward single component analyses. Until recently, few attempts have been made to automate methods for simultaneous kinetic analyses. It is probable that this analytical approach will be much more attractive if instrumentation is developed which removes the time-consuming and tedious operations from the methodology. Pinkle and Mark ( 7 ) proposed one approach which utilized an analog computer to solve simultaneous equations involved Present address, Diamond Shamrock Corp., Painesville, Ohio. Author to whom correspondenceshould be addressed. (1) K. B. Yatsimirskii, “Kinetic Methods of Analysis,” Pergamon Press, New York, N. Y., 1966. (2) H. B. Mark, Jr., and G. A. Rechnitz, “Kinetics in Analytical Chemistry,” Interscience, New York, N. Y., 1968. (3) G. G. Guilbault, ANAL.CHEM., 42, 334R (1970). (4) S. R. Crouch, ibid., 41,880 (1969). (5) G. E. James and H. L. Pardue, ibid., 40,796 (1968). (6) Zbid., 41, 1618 (1969). (7) D. Pinkle and H. B. Mark, Jr., Tufunta,12, 491 (1965).

in the so-called “method of proportional equations” (2). Their data were restricted to simulated reaction rates with no chemical data being reported. More recently, Margerum, Pardue, and coworkers have reported upon digital computer methods for the collection and processing of such data (8-11). While the digital computer has proved to be a powerful tool for these analyses, it is probable that situations will exist in which it is not possible to dedicate a digital computer to this type problem, and simpler, less expensive instrumentation will be desired. This report describes a simple analog system which simplifies the data collection and processing steps in simultaneous kinetic analyses. The approach taken in this work involves the recording of reaction response curves in such a manner that concentrations are easily read from the graphs without additional computations or replotting of the data. The method is described for the analysis of two component mixtures. Results are reported for the analysis of mixtures of rare earth ions (lanthanum and neodymium) using ligand exchange reactions reported by Margerum and coworkers ( 4 9 ) . The reactions used involve the exchange between Cu(I1) and the rare earth complexes of trans-l,2-diaminocyclohexaneN,N,N‘,N-tetraacetate (MCyDTA). The reaction sequence is represented in Equations 1 and 2.

HCyDTA3-

+ Cu (11)

faat __f

CuCyDTA?

+ &I+ (2)

Under controlled conditions (8, 9), the rate of production of CuCyDTA2- is first order in the concentration of rare ~~~~

~

(8) J. B. Pausch and D. W. Margerum, ANAL.CHEM.,41, 226 (1969). (9) D. W. Margerum, J. B. Pausch, G. A. Nyssen, and G. F. Smith, ibid., p 233. (10) B. G. Willis, J. A. Bittikofer, and H. L. Pardue, ibid., 42, 1340 (1970). (11) B. G. Willis, W. H. Woodruff, J. R. Frysinger, D. W. Margerum, and H. L. Pardue, ibid., p 1350. ANALYTICAL CHEMISTRY, VOL. 44, NO. 4, APRIL 1972

767

Time

If it is assumed that there is no product present at the start of the reaction, then the concentration of the product at any time t is given by Pt

=

A , - At

(6)

Substituting for A t in Equation 6 from Equation 5 yields


P1 = A,(1

- e-k5t)

(7)

This expression, for a single component sample, predicts that a plot of product concentration us. e-kat should be linear. It predicts further that at very long times, when t + 03 and e--kot -+ 0, the intercept is equal to the initial concentration, Ao, of the rate limiting species. Thus, a single component analysis could be performed by plotting product concentration us. e - kat and extrapolating the resulting plot to the zero and infinite time axes. If two species undergo first order reactions simultaneously, and if the two reactions are independent of one another and produce a common product, then Equation 7 is an accurate representation of the time dependent product concentration resulting from each reaction, provided the appropriate rate constant is used for each. The product concentration resulting from both reactions at any time t is the sum of two such expressions and is given by Pt

e-kt

Figure 1. Product concentration computed data

DS.

k , = 4 X 10-2 sec-1 0 = A , = 1.B, = 0 0 = A , = O.B, = 1

exponential functions,

kb = 5 X 0,. = A , =

Sec-'

Bo = 1

earth complexes (MCyDTA-) and is independent of the Cu(I1) concentration. The formation of CuCyDTA2- is monitored by the adsorption of radiant energy at 300 nm by this species. Although results are reported only for this one chemical system, the method should be applicable to any two-component mixture involving first order reactions and an easily monitored reactant or product. Accordingly, the principles of the method are presented for a general situation. GENERAL CONSIDERATIONS

The basis of the method is outlined for the analysis of a twocomponent mixture. The explanation will be based on detection of the product of a reaction. Similar derivations can be presented for the case in which the disappearance of reactants is followed. The following generalized reactions are used to simplify notation. A + R ~ . ' P

(3)

B+R%P

(4) It is assumed that the reaction is first order in A and B and zero order in reactant, R , and product, P, and that the first order rate constants have the relationship, ka > kb. Also, it is assumed that the presence of A or B has n o effect on the rate constant for the other reactant. For the case of a single component A with an initial concentration of A,, the concentration of A at any time t is represented by A! 768

=

Aoe-k't

(5)

ANALYTICAL CHEMISTRY, VOL. 44, NO. 4, APRIL 1972

=

+

A,(1 - e-knt) B,(1 - e-kbt)

(8)

This expression is analytically useful only if there is some difference between k , and kb. To illustrate the utility of this expression it will be assumed that k , is much larger than kb. If this is the case, then there will be an early time period during which e-kbt will still remain close to unity. During this time the quantity 1 - e-kst is close to zero and the form of Equation 8 approaches that of Equation 7, and a plot of Pt us. e-katwill permit an evaluation of the concentration of the faster reacting component by the procedure outlined above. Also, there will be a later time period during which e-kat approaches zero while e-kbt differs from unity but has not yet approached zero. During this time period the product concentration will be represented by Pt

=

A,

+ B,(1

- e-kat)

(9)

Using reasoning similar to that applied above to Equation 7, it follows that a plot of Pt US. e - k s t will yield a straight line with intercepts (at t = 0 and t + respectively) of A, and A, Bo. Thus, Bo can be obtained by difference. It is expected that the two plots would show regions of nonlinearity at intermediate times when both reactions are contributing effectively to changes in Pt. In principle, for markedly different rate constants and no interfering absorbing species, it should be possible to obtain initial concentrations of both species in a two-component mixture from a single plot of product concentration us. the exponential term for the more slowly reacting component. In more practical situations it is desirable to construct two plots, one of P1 us. e-knt and one of Pt us. ,-Izat, and to utilize extrapolated data taken from both plots. The principles, procedures, and some of the practical characteristics of the proposed method can be clarified using the diagrams in Figures 1 and 2. These figures represent plots of computed data for two rate constants with a ratio of apsec-') proximately 1O:l ( k , = 4 X lo-* sec-l, kb = 5 X and two different concentration ratios. The rate constants were selected to be consistent with those of the chemical system to be evaluated experimentally.

+

Time

Figure 2. Product concentration os. exponential functions, computed data k, = 4 X WC-’ 0 = A , = 1, Bo = 0


0 = A , = 0, Bo =

Abscissa a

ka = 5 X SW-’ 0,. = A , = 1, Bo = 0.1

0.1

I.o

I

I

08

I

I

1

0.6

I

0.4

I

I

0.2

I

I

QO

e-kt

Figure 1 represents the situation for equimolar mixtures of the two components. Data for both components are included in this figure, with the circles representing the faster reacting component and the squares representing the slower reacting component. The open circles and squares represent data for single components, and the closed circles and squares represent data for the two component mixture. The abscissa for the upper plot is proportional to e--kntwhile that for the lower plot is proportional to e - k b t . The plots representing mixture data are drawn as they would be plotted by semiautomated instrumentation described below. Each contains both the linear and nonlinear regions expected. The suggested procedure involves the extrapolation (dashed lines) of the linear portions of the curves to the t = 0 and t + m axes to obtain experimental values for A , and A , Bo. Applying the reasoning outlined above using Equations 7 and 8, it follows that extrapolation of the plot for the faster reacting component to t --c m should yield an intercept equal to the initial concentration, A,, of this species. Similarly, extrapolation of the plot for the slower reacting component to the t + axis should yield an intercept equal to the sum of the Bo. Also, extrapolation of this two concentrations, A , latter plot to t = 0 should yield an intercept equal to A,. The suggested extrapolations are represented by dashed lines on Figure 1. Values obtained from these extrapolaBo)’ on the figure. In tions are labeled A,’, A,”, and ( A , principle, for the ideal situation when the rate constants are sufficiently different, there should be agreement among the Bo and values A,, A,’, and A,” and between the values A, (A, Bo)’. However, for the more practical situation presented here, which involves contributions from both reactants throughout much of the reaction time, there are differences among these values. The manner in which these differences depend upon the relative concentrations of the reacting species is discussed here. It is quite apparent from this plot that the value, A,‘, differs significantly from both A , and A,”. On the other hand, the values A, and A,” are virtually identical. Similarly, there is good agreement between the values of A , Bo and ( A , Bo)’. Clearly, in this situation, in which AO = Bo and k,Jkb 10, component B has a large effect on the plot during the early part of the reaction, but component

+

+

+

+

+

+

+

A has little effect on the plot after one half life of component B has been exceeded. Thus, it appears that for equimolar mixtures of species with rate constants having a ratio of 10: 1, best results can be obtained using an extrapolation from the slower reacting component to both the t = 0 and t + axes. For ratios of Bo/Aogreater than unity, the situation becomes even less favorable for extrapolation from the plot for component A and more favorable for extrapolation from the plot for component B. Thus, the observations made above apply with at least equal validity for situations in which Bo/Ao> 1. The question arises as to the concentration ratio at which the extrapolation from the plot for component A becomes valid. This question is partially answered using Figure 2. This plot represents the situation in which component A is present at a level ten times that of component B. Careful examination of this plot demonstrates that the data through about the first quarter-life of component A yield a valid extrapolation to the t + m axis while data in excess of one halflife are showing significant deviations from the ideal situation. Also, it will be observed that there is little contribution from the A component to the plot for the B component after about one half-life of this latter reaction has been exceeded. This latter observation can be a little misleading, since a small absolute error in the extrapolated intercept at t = 0 can result in a large relative error for component B. In this instance, it is best to extrapolate both plots to the t -+ m axis. It is desirable to consider concentration ratios (Bo/&) between 0.1 and l. Based upon the observations made already, the important question is how well do the intercept values at t + 03 for component A represent the concentration of this component. The inset in the upper right-hand corner of Figure 2 shows the intercept values for several concentration ratios. The horizontal lines mark the points of intersection on the ordinate at t + and the numerical values associated with each point represent the concentration ratio, B,/A,. The vertical scale is the same as that of the ordinate in the remainder of the figure and the intercept values span an error range from 0 (at Bo/Ao= 0.1) to about 10% (at B,/A, = 1.0). These results indicate that for values of Bo/Ao greater than about 0.4, systematic positive errors greater than about 5Z ANALYTICAL CHEMISTRY, VOL. 44, NO. 4, APRIL 1972

769

Figure 3. Circuit for generating exponential function


OAl OA2 OA3 OA4 B1

= Philbrick SP2A = = = =

Philbrick P25A Philbrick P65 Philbrick P65 Philbrick P66

P,,Pn = 100 K 10-turn Helipot potentiometer P3 = 10 K 10-turn Helipot potentiometer P4 = 5 K trim pot Q1 = Philbrick PLlP transistor C, = 2 pf Polystyrene Capacitor (Southern Electronics)

can be expected in the determination of the faster component from an extrapolation of its response curve to the t + axis. It is reemphasized that the observations made here apply to the situation when k,/kb 10.

Taking account of the integrator response rate (T = 0.09(l/RlC,)V/sec) and assuming switches SI and S? are in the positions shown, it follows that the output of OA3 is given by

INSTRUMENTATION

Substituting Equation 11 into Equation 10 and rearranging yields

N

The procedures outlined above would be quite tedious to implement if data were to be collected and plotted manually. An instrumental system designed to generate the required plots is described below. It consists of the time base and exponential circuitry to generate continuously varying values of and e-kat. These functions drive the X axis of an X-Y recorder and an absorbance circuit which monitors the product concentration drives the Y axis. The data represented by solid circles and squares in Figure 1 can be used to illustrate the operation of the system. The system is started at t = 0 (reagent mixing point) with the e-katfunction driving the X axis. When the plot begins to deviate from linearity, then the circuitry is switched so that the e - k a t function now drives the X axis. The system is permitted to continue to operate in this fashion until sufficient data are recorded to permit a reliable extrapolation of the slower reacting component response. The switching operation is implemented without disturbing the time axis. Details of the circuitry are presented below. Exponential Circuit. The circuit used to generate the exponential function is represented in Figure 3. Amplifier OAl is an integrator generating the time base, t. Amplifier OA2 scales the output of OAl by appropriate factors (determined by the settings of SZ,PI, Pz, and P3) to generate the products kat and kbt. Amplifier OA3 allows proper biasing of the transistor, QI, which in conjunction with OA4 generates the desired exponential. The output of OA4 is given by eo4 = P410 exp(eos/Eo)

(10)

where lo is a characteristic of the transistor and Eo has the value of 0.059 V at 25 OC (12). This expression holds with good accuracy for values of e03 between 0.43 V and 0.60 V (5). The output of OA3 is the algebraic sum of the output from OA2 and the output from the voltage divider at the input of OA3. This voltage divider at the input of OA3 biases the output of this amplifier near the upper operating limit (0.60 V) of QI when the output of OA2 is zero volts (SIclosed). (12) “Application Manual for Computing Amplifiers,” Philbrick Researches Inc., Nimrod Press, Boston, Mass., 1966. 770


e03 = 0.58 V - l’(Pz/Pl)t

eo4 = P J , exp[(0.58/E0) - (T/Eo)(Pz/P~)t1

(11)

(12)

For a given set of operating conditions, the parameters I,, E,, and T are constant. Under these conditions, Equation 12 reduces to e04

=

k’P4 exp[ -(T/E,)(P~/P~)~I

(1 3)

where K = lo exp (0.58/E0). The exponential term in Equation 13 represents e-”. Potentiometers PI and PZ are used to establish the desired value of the rate constant and P4 is used to adjust the output of OA4 to match the full scale range of the recorder used. In principle, it should be possible to adjust T to be some decimal multiple of E, so that the ratio Pz/P1could be an integral multiple of the first order rate constant. However, since E, varies with temperature, and since the temperature of QI may not be known accurately, it is more practical to adjust this ratio empirically as described later. The ratio PzjPl is adjusted to represent one rate constant ( k , for example) and the ratio P,/Pl is adjusted to represent the other rate constant (ko). The generation of the exponential corresponding to e - k o t is initiated by connecting PZ in the feedback loop of OA2 and opening SI. Switching SZfrom Pz to Ps to generate the exponential corresponding to e-kat does not disturb the time base. Amplifiers OAl and OA2 must be low current drift amplifiers and Clmust be a low leakage capacitor to ensure a reproducible time base. In addition, the transistor Ql has a large temperature coefficient and must be thermostated for reliable performance. The system described here exhibited a drift at the output of OA4 of less than 0.2% of full scale. The output followed an exponential function to within 1 % over 95 % of its operating range and the exponential function generated is reproducible to within =t0.5%. Spectrophotometer. Because of the relatively low value of the molar absorptivity of CuCyDTA (1.60 X lo3l./m-cm), the fact that the kinetic measurements are to be made over a period of several minutes, and the fact that high sensitivity is desired, it is imperative that the spectrophotometer used have a good long term stability and low electrical noise. Recent work has demonstrated that ultra-high photometric sta-


bility can be achieved using the optical feedback principle for control of source intensity (13-15). A stabilized spectrophotometer utilizing the optical feedback principle and operating into the near ultraviolet region was utilized in this work. The optical system and circuitry utilized are similar to those reported earlier and only differences will be mentioned here. The energy source is a quartz tungsten-iodine lamp (General Electric No. 1958) rated at 150 watts. A high intensity UV grating monochromator (Bausch & Lomb No. 33-8625-01)is used as a dispersing element. The energy beam is split into signal and control portions using a quartz plate as a beam splitter and detected with RCA 935 phototubes (S-5 response). A dc programmable power supply (Harrison Laboratories No. 6267A) rated at 40 V and 10 A is used to power the lamp. With the components mentioned above, the spectrophotometer gives a useful output in the range of 280 nm to 400 nm with a band pass of about 10 nm, long term drift of about *0.05% T per hour, and a signal noise level equivalent to 0.005% T . The instrument can be extended to the visible region by using a visible grating. A logarithmic amplifier is used to convert the spectrophotometer output to a signal linear in absorbance. A scaling amplifier is used to calibrate the output in absorbance or concentration units as desired. The conversion is accurate to 0.1Z over the range of 1 absorbance unit. The spectrophotometer, including the logarithmic conversion circuit, is stable to +0.00027 absorbance unit per hour. The reaction cell is thermostated at 25 "C. Mixing is accomplished by a stirring rod rotating at 3200 rpm and located so that the tip is just above the light path. EXPERIMENTAL Reagents. Solutions were prepared using procedures outlined earlier (8, 9). All solutions were prepared to be 1 .OMin sodium acetate and 0.1M in sodium perchlorate with a final pH of 3.5. Stock solutions (1.25 X 10-3M) of LaCyDTA- and NdCyDTA- were prepared by adding required amounts of buffer and ionic strength components (sodium acetate and sodium perchlorate) and a 10% excess of CyDTA to solutions of the metal oxides, diluting just short of the final volume with deionized water, adjusting the pH to 3.5 using 1M HC104, and diluting to the final volume. Desired working concentrations of the metals are prepared by diluting the stock solutions with an acetate-perchlorate solution buffered at pH 3.5. A Cu(I1) solution (2.5 X 10-2M)is prepared by dissolving copper metal in nitric acid, adding appropriate amounts of sodium acetate and sodium perchlorate, and adjusting the pH to 3.5 at the desired final volume. Procedure. INSTRUMENT ADJUSTMENTS. The spectrophotometer circuitry is permitted to warm up for at least 1 hour prior to use. The absorbance scaling amplifier is adjusted to give the desired display of absorbance data. In this work the instrument was calibrated to read in absorbance units, and chemical concentrations were computed from an experimental value of the molar absorptivity of CuCyDTA (1.60X lo3 @ 300 nm). Calibration plots using MCyDTA as prepared above had a positive intercept on the concentration axis. This difficulty can be removed by maintaining the MCyDTA solutions at a higher pH than is used to carry out the determination. It was mentioned above that a knowledge of the exact values of all parameters in Equation 12 coupled with a knowl(13) P. A.Loach and R. J. Loyd, ANAL.CHEM.,38,1709 (1966). (14) H.L. Pardue and P. A . Rodriguez, ibid., 39,901 (1967). (15) H.L. Pardue and S. N. Deming, ibid., 41,986(1969).

Time

e-kt

Figure 4. Absorbance cs. exponential functions for CyDTA complexes of La and Nd reacting with Cu(I1)

kLa = 4.11 X lo-* sec-l La, 'u 5 'u lO-SM ~ C ~ C ~ D=T 1.6 A = 103 @ 300 nm

k

~ = d

5.06 X 10-3 sec-l = 10-5M

Nd, 'u 5

edge of rate constants k , and ka would permit the ratios P2/Pl and P3/P1to be set directly. However, since certain of the parameters are not known with high reliability, an empirical procedure is used to establish these ratios. Simply stated, the procedure is to set the ratio of P2/P1 (or P3/P1) so that the output of OA4 changes over an amount equivalent to a given percentage ( L e . , 99%) of the recorder during the period required for the reaction to proceed t o the same percentage completion. This is accomplished as follows. The first order rate constant for the reaction is used to compute the time required for the reaction to proceed to the desired percentage completion. Then the integrator is permitted to operate for this time period at which point S3 is closed. Then the ratio P2/Pl (or P3/P1) is adjusted until the recorder monitoring the output of OA4 reads the same percentage of full scale deflection as the percentage completion for which the reaction time was computed. This procedure is repeated for each reaction component to be determined. It should be noted that once P1is set for one component, it must not be changed in making adjustments for another component. MEASUREMENT STEP. All solutions are adjusted to the analysis temperature by immersion in a water bath at this temperature for several minutes prior to performing the analysis. Switch SZis set for the generation of the exponential function for the faster reacting component. Then 2.0 ml of the MCyDTA solution is added to the cell and the stirrer is started. Then 0.5 ml of Cu(I1) solution is added and S I is opened immediately. Switch Sz is left in its oiiginal position as long as the plot remains linear. When the plot begins to show noticeable nonlinearity, S2 is changed to the position to generate the exponential function for the slower reacting component. COMPUTATIONS. The concentration of metal ion is computed from data obtained by extrapolation of the linear portions of the recorded plots to the absorbance axes. The major component (that with the longer linear region) is determined by extrapolating the plot to both axes (1 = 0 and t m ) and reading the difference in absorbance. The minor component plot is extrapolated to the nearest axis ANALYTICAL CHEMISTRY, VOL. 44, NO. 4, APRIL 1972

771

Table I. Results for Automatic Differential Kinetic Analysis for Lanthanum and Neodymium Concentration (moles/liter X los) Total metal La Nd Error, Taken Found Taken Found Taken Found La Nd

21.0 2.00 21.6 20.0 12.1 10.0 12.4 2.00 20.P 19.8 10.0 10.3 5.00 10.0 a Re1 std dev, La l.O%, 22.0 22.0 12.0 12.0

2.33 20.0 18.9 +16.5 - 5 . 5 2.00 1.76 -1.0 -12.0 19.8 -4.0 2.00 1.92 +2.0 10.2 2.17 10.0 10.2 +8.5 +2.0 9.97 10.0 9.78 -0.3 -2.2 5.08 5.00 5.21 4-1.6 $4.2 Nd 0.3%.

-


(t = 0 for the faster component and t m for the slower component) and the concentration of this component is determined by difference. RESULTS AND DISCUSSION

The instrument system was tested initially for simulated reaction rate curves generated by the first order decay of series resistance-capacitance circuits. The ratio of “effective rate constants” (1/RC) was 8.9 and equal “concentrations” were used (identical voltages on the capacitors). Results obtained agreed with predicted values to within 1.6%, and relative standard deviations for replicate runs were within 1 %. The exchange reaction between the lanthanium and neodymium complexes of CyDTA using Cu(I1) as a scavanger as reported by Margerum et ai. (9) was selected as the chemical system to evaluate the instrument system. Using data reported by these authors, the rate constants for the exchange of H+with La and Nd in the complexes (Reaction 1) at pH 3.5 are computed to be kL, = 4.11 X sec-I and k N d = 5.06 X sec-’. Values measured in this work agree with the computed values to within 2 %. Figure 4 represents a tracing of the recorder plot obtained for a typical analysis. The neodymium concentration is in slight excess over the lanthanum concentration. The similarity between these data and the computed data in Figure 1 is readily apparent. As expected, the intercepts marked Y and Y’ do not correspond. In this case, it is probable that the Y value is the better of the two. This is the one used in the computations. The quantities Y - X and 2 - Y would be used to compute the La and Nd concentrations, respectively. Results for the quantitative determination of mixtures of La and Nd at different ratios are presented in Table I. Relative errors are generally in the range of a few per cent with occasional errors as large as 10-207& As expected, the larger errors occur for the larger ratios of concentration. Relative standard deviations are reported for equimolar mixtures of the metals (1 X 10-4M). These data (1.02% for La and 0.3%for Nd) are typical of many observed.

772


The reliability of these data is comparable to that reported using digital computers by both batch (9) and on-line (11) processing methods for the particular example examined in this work. The relatively low cost and simplicity of this instrumental system could make it attractive for such favorable cases. On the other hand, the more sophisticated digital systems will certainly be superior in handling less favorable situations, such as those involving multiple component samples or rate constants which differ only slightly from one another. The maximum and minimum rate constants that can be used are dependent upon the switching time and integrator time constant. For very fast reactions, it would be necessary to replace the manual switches with electronic switches. For very slow reactions, the integrator time constant should be increased so that amplifier OAl does not go into saturation before the run is completed. Several advantages for the approach described here can be cited. The data are presented in a form which is easily interpreted to yield concentration as the reaction proceeds. Thus the tedious and time consuming steps of plotting the data are eliminated. If either component of the expected mixture is not present, then the portion of the plot corresponding to that component will show no linear region. Thus the method quickly indicates the presence or absence of a component. This method does not require a separate determination of the total concentration of reactants as this information is available at the end of each run. The extrapolation utilizes many points and does not depend upon a relative few points as is the case with many methods. Finally, the circuitry is straightforward and inexpensive and is simple to use. There are, of course, limitations to the method. The reactions must follow first order kinetics, there must be no interaction, and they must go to completion. The rate constants of all reactions must be known accurately. Also, there are limitations on the minimum ratios of rate constants which can be utilized, and errors rise rapidly as concentration ratios become large. Finally, the method is applicable only in those situations in which a reactant or product can be monitored continuously by a suitable detector. In principle it should be possible to extend this method to three or more components by adding other potentiometers and switch positions in the feedback loop of OA2. However, it is probable that except for very favorable situations, insufficient linear regions would be observed in the plots for the intermediate components to permit the required extrapolations. ACKNOWLEDGMENT

The authors thank G. L. Chen for technical assistance. RECEIVED for review March 9, 1970. Resubmitted October 14, 1971. Accepted November 4, 1971. Study supported in part by Grant 1212-67, Air Force Office of Scientific Research, by an ACS Analytical Division Summer fellowship and by a Procter and Gamble fellowship.

Simple analog system for simultaneous kinetic analysis

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