Critical Experimental Evaluation of Limitations of Graohical Extrapolation and the Roberts and Regan Differential Reaction Rate Methods Effects of Ratio of Rate Constants and Composition of Mixture on Accuracy Ronald A. Greinke and Harry B. Mark, Jr. Department of Chemistry, Unioersity of Michigan, Ann Arbor, Mich. Two differential reaction rate methods, the graphical extrapolation method and the single point modification of the method of Roberts and Regan, are evaluated for their application limitations by actual experimental analysis of binary mixtures of amines, The maximum and minimum tolerable mole per cent of the faster reacting component in the mixture are experimentally Obtained for each method as a function Of the ratio of rate constants of the two species in the mixture. The results agree with the theoretical error analysis which estimated the magnitude of the expected error, resulting from probable errors in the measured experimental parameters, as a function of ratio of rate and per cent Of the
THEFIRST
ORDER GRAPHICAL extrapolation method (GEM) ( I ) and the single point modification of the method of Roberts and Regan (MRR) ( 2 , 3) are two differential reaction rate methods which have been frequently employed for the analysis of mixtures of closely related compounds. Recently, the limitations and applicability of these two methods were theoretically evaluated by a general error analysis which took into account all sources of error ( 4 ) . Mathematical expressions were derived that related the expected error in the determination of the initial concentration of the species to be analyzed to given probable errors in the experimental parameters (such as rate constants, analysis time, total concentration of reactants, etc.). The magnitude of expected error in the initial concentrations of the analyzed species which resulted from errors in the measurement of these experimental parameters was studied as a function of time of analysis, ratio and magnitude of the rate constants, and the per cent composition of the mixture. From the results of this theoretical error analysis, a prediction was made for the GEM regarding the maximum tolerable mole of the faster reacting component for five different ratio of rate constants. Because these limitations are based on a theoretical error analysis, its applicability to real systems was questionable. An estimation was also made for the MRR that a ratio of rate constants closer than 2 to 1 could be used. However, no estimation of the maximum or minimum tolerable mole % of the faster reacting component as a function of the ratio of rate constants was made for the MRR based on the above error analysis. Another paper has predicted these limitations for the MRR by stating that the minimum and maximum tolerable mole % of the faster reacting component, A , are 10 and 90 %, respectively (5). However, several papers which applied the MRR to mixtures of closely related compounds have reported the
(1) B. E. Saltzman, ANAL.CHEM., 31, 1914 (1959). (2) J. D. Roberts and C. Regan, Ibid.,24, 360 (1952). (3) L. J. Papa, J. H. Patterson, H. €3. Mark, Jr., and c. N. ReilleY, Zbid., 35, 1889 (1963). (4) H. B. Mark, Jr., R. A. Greinke, and L. J. Papa, Proc. SOC. Anal. Chemists Con$ Nottingham, England, Heffer, Cambridge, England, 1965, p. 490. (5) F. Willeboordse and R. L. Meeker, ANAL.CHEM.,38, 854 (1966).
48104
accurate analysis of amounts of A greater than 90% and less than 10% (3, 6). Also, frequently reported in the literature is the fact that the GEM fails to give accurate results because the mixture contains a large amount of the faster reacting component, However, these statements are made without mentioning the ratio of rate constants of the two components in the mixture (5,7). at this Point Of Because existing literature regarding the GEM’S and MRR’s limitations was vague, incomplete, or contradictory, a critical evaluation of these methods’ limitations by a systematic experimental analysis was necessary. The limitations obtainedfrom actual analysis of closely related mixtures coupled with the theoretical considerations brought out in the error analysis provide a more reliable and practical guide for the analyst. This paper is designed to help the analyst decide which of these two methods will best yield an accurate analysis for a particular system. In order to obtain the limitations, binary mixtures of amines reacting with the reagent, methyl iodide, were analyzed as previously described employing the above two differential reaction methods (6, 8). One should note that several variations of the GEM exist; namely, the first or pseudo first order logarithmic extrapolation method ( I ) , the pseudo fractional order extrapolation method (a),the second order logarithmic extrapolation method [(Reagent], # [ A ] , [B],)(7) and the second order linear extrapolation method [(Reagent], = [ A ] , [B],)(9). Although the subsequent theoretical error analysis is formally based on the first order logarithmic extrapolation method, the conclusions drawn from this error analysis can be applied to all graphical extrapolation methods. This generality applies because all the above extrapolation methods are based on the principle that the faster reacting component must completely react before the data points become useful. Thus, the mathematical expressions employed by these methods for analysis are quite similar. Therefore, results of the theoretical error analysis for all the methods would be the same, The experimental results are obtained by employing both the pseudo first order and the pseudo fractional order graphical extrapolation methods for mixtures of amines. For the above reason we feel that the experimental concentration limitations obtained by the pseudo fractional order method can also apply to the other graphical extrapolation methods. This assumption is experimentally verified at a particular ratio of rate constants for the second order and the fractional order GEM and is presented in the results and discussion section.
+
+
EXPERIMENTAL
The experimental methods employed for the analysis of mixtures by these two methods is identical to that previously reported (6, 8). (6) (7) (8) (9)
R. A.Greinkeand H. B. Mark, Jr., ANAL.CHEM., 38,1001 (1966). S. Siggia and J. G. Hanna, Ibid.,33, 896 (1961). R. A. Greinke and H. B. Mark, Jr., Ibid.,39, 1572 (1967). C. N. Reilley and L. J. Papa, Ibid.,34, 801 (1962). VOL. 39, NO. 13, NOVEMBER 1967
* 1577
GRAPHICAL EXTRAPOLATION METHOD IA+B] 0 ,25 (4).
proaches zero. At long times, the contribution to A[B], by y is negligible compared to the coefficients a and 0. At short times, however, y makes a n appreciable contribution to the small probable error A[Bl0because the coefficients a and @ are small. The ratio of rate constants has no effect on the shape of the curves in Figure 1 since the coefficients a , 0,and y, are independent of the ratio of rate constants. However, the probable error, A[B],, is dependent on the ratio of rate constants. The differential form, Equation 9, of the rate expression, (Equation l), is helpful in explaining the dependence of the probable error, A[B],, on the ratio of rate constants.
d3!?!! = dt
k.4[AIz
+ ks[B],
(9)
In the differential form the time, t, when the term kA[AIt becomes negligible with respect to the term kB[BII,corresponds to the time, f, when the term [AIoe-k~l >[Reagent]o
+
14
0
a
Error Coefficients and Theoretical Considerations. The equation employed in the single point modification of this method for calculating the initial concentration of [AIo in the mixture is: (3, IO) :
-I2
a d 0
10
W W
$ 8
m
+
where [MI0 = [ A ] , [B],and must be measured independently. The value of [BI0 must be obtained by subtracting [AIofrom [MI,. The parameters ka” and ks” are the true second order rate constants for the reaction of A and B with the reagent, R, respectively. During the analysis, significant errors may arise in any or all of the variables of Equation 10 and the general expression for the error analysis can be written in the form: A[Alo2
=
+
(~u‘)’(A[Rlt)~ 4- @’>’ (A1R3d2 ( Y ’ ) ~(at2) ( W 2(A[W,)~ (E’)’ ( A k A ’ r ) 2 ({’)’(Akg”)’
+
+
+
0 K
a
6
4
2
10
20
40
30
50
60
0
kA‘
(11)
Figure 2. Variation of probable error, a[A],,in [ A ] ,as a function of reaction time k.i/ks = 7.5 ( [ A ] , [B],) = O.lM, [RI0= 1 X 10-3M, kA” = 0.1 liter-mole-’ minute-’, A[R]t and A[R], = 1 X 10-6M, A [ M ] , = 0.00015i34, At = 1 minute, Ak.4” = 0.003, AkB“ = 0.0004 Curve A . [A],/[B],= 119 Curve B. [A],/[B],= 1 Curve C . [A],/[B],= 9
The error coefficients of Equation 11 are:
+
The variations of these individual error coefficients as a function of time, ratio of rate constants, and concentration have already been reported (4). Figure 2 represents a plot of probable error, A[&, GS. time (employing all the error coefficients) for three systems; [A],:[B], = 119, curve A [A],/[B],= I , curve B; and [A]&?], = 911 curve C; [A], [B], = 0.1M, [R], = 1.0 x 10-3M, ka” = 0.1 liter-mole-I minute-’, k B = 0.0133 liter-mole-’ minute-l, A[RI1 and A[R], = 1.5 X A[&’], = 0.00015, At = 1 minute, Aka” = 0.003 and AksN = 0.0004. Note that minima in probable error are obtained in the three curves. The tal,,,, reported previously as the optimum analysis time (3) cor1 responds to the time when the ratio [R],/[R],= - = 0.368 e but is really only the time for minimum probable error arising from error in the measurement of [RII (4). The overall optimum analysis time, io,,, occurs at longer times than re(,,,, because of the contributions of error in [ R ] , and t (the coefficients /3’ and y’ both decrease as time increases in Equations 13 and 14. The value of the ratio of reagent
+
concentrations, [RIt/[R],,at t,,t is approximately 0.333. As indicated in Figure 2, for a given ratio of rate constants, the probable error ALA],, increases as the mole of [ A ] , increases. This is because the contributions to ALA], from an experimental error in the rate constants increases as [A], increases (Equation 16 and 17). Although not indicated in this figure, the probable error would decrease for the three curves if a higher ratio of rate constants were used, By taking the values of probable error, A[& at tOptfrom the three curves in Figure 2, the error in the determination of [A],for the three mixtures can be estimated. As an example, take A[A],to be negative in all three cases. The results of the analysis would be as follows: for curve A , 0.01M or 10 [A],taken ([AIo [B],= O.lM),0.01M - A[AJ0=0.01M - 1.2 X 10-3iM = 8.8 X 10-3M or 8.8% found; curve B, 0.05M or 50% [A],taken, 0.05M - A[AI, = 0.05 - 3.3 X = 4.67 X 10-2M or 4 6 . 7 z found; curve C, o.o91?d or 90% [A],taken; 0.09M - 5.6 X 10-3M = 0.0844 or 84.4% found.
+
RESULTS AND DISCUSSION As previously seen: the magnitude of the expected error (A [B],or A[A],)which results from errors in measurement of the experimental parameters is a function of the ratio of rate constants, the per cent composition of the mixture, and the time that measurements are made. The optimum time for an analysis can be picked mathematically by the analyst for any system. However, the optimum per cent composition and the optimum ratio of rate constants cannot be controlled by the analyst (although in some cases medium changes can vary VOL. 39, NO. 13, NOVEMBER 1967
1579
Table 1. Analysis of Amines by Pseudo First and Fractional Order Graphical Extrapolation Methods
% [B!o remaimng at [AIt s 0 11.0
Reaction meduim Acetonitrile
2.1
10.0
Nitrobenzene*
3.1
10.0
Acetone
5.8
7.0
Acetonitrile
7.2
7.7
Acetonitrile
12.2
3.5
Acetonitrile
66
4.0
Acetonitrile
[AI, 72
Mixture
a. tz-Butylamine
B. Triethylamine A. Triethylamine
Taken 14.8 21.8 36.2
Found 16.3 6.7 36.8 40.6 58.9 56.7 72.9 67.2 82.3
49.4 B. Tributylamine 56.0 A. Triethylamine 62.9 B. Tributylamine 73.2 A. Triethylamine 76.6 B. I-Nethyl-4-piperidone 82.3 A. +Butylamine 90.3 85.7 B. 1-Methyl-4-piperidone 94.6 92.7 A. Diethylamine (94.6 93.20) B. I-Methyl-4-piperidone 95.4 94.5 A. Trirnet hylamine B. Tributylamine a By visual extrapolation from the experimental rate curve. * By pseudo first order method.
the ratio of rate constants as demonstrated below (12). Mixtures analyzed for this paper are specifically designed to define the composition limits at a given ratio of rate constants. Optimum times were employed for all measurements. Maximum Mole % [A], Tolerable for GEM. Table I gives the results for the analysis of mixtures of amines employing the GEM. Two different mixture compositions are reported for each binary system. The first composition represents the maximum tolerable mole [A],(as found experimentally) that could be analyzed accurately at the given system kA/kg value. (Mixtures having smaller amounts of the faster reacting component, A , at each ratio of rate constants were accurately analyzed but are not reported here.) The second mixture analyzed for each binary system illustrates a typical example of the larger error obtained by increasing the mole % [A],. Low results occur for A because the faster reacting component is still reacting significantly even when only small amounts of B remain. The maximum mole [A],tolerable obtained from Table I is plotted as a function of the ratio of rate constants in Figure 3, curve A . Also indicated in Table I is the % B that remains at the point when the faster reacting component has “completely” reacted for the maximum tolerable mole % [AI0case. As the ratio of rate constants increases, accurate experimental data can be obtained at lower amounts of B remaining (Table I). This experimental fact can qualitatively be explained by examining the theoretical error coefficients for the GEM. As kAJkBincreases for a constant k g , the time interval becomes smaller (even for a higher mole % [A],)when the faster reacting component has “completely” reacted. Thus, the coefficients a: and /3 are smaller, and hence, the probable error in [B],, A[B],, resulting from errors in measurement of [C10 and [ C ] ,are smaller. For example, experimentally, 1000 seconds were required for triethylamine to react “completely” in the triethylamine-1-methyl-4-piperidone mixture (73.2 [A],, kai k B = 5.8) while only 300 seconds were required for diethylamine to react “completely” in dietkylamine-l-methyl-4piperidone mixture (94.6% [AIo,kA/kB = 12.2). A shorter extrapolation distance occurs in the latter case. The error in the data points is not magnified by a long extrapolation distance as in the former case. (12) R. A. Greinke and H. B. Mark, Jr., AKAL.CHEM., 38, 340
(1966). 1586
e
ANALYTICAL CHEMISTRY
kalks
1.25
When the ratio of rate constants is above 12, accurate results (within 2 7 3 can also be obtained for mixtures having 90 or more A present by a visual extrapolation from the rate curve. This visual extrapolation from the conductance rate curve is illustrated in Figure 4 for the diethylamine-l-methyl4-piperidone mixture. The actual results are given in Table I. In order to obtain an accurate visual extrapolation from the rate curve, the amount of the slower reacting component, B, that reacts during the time that all of A has “completely” reacted should be small. Thus, the rate of reaction of B, d[B]/dt,has not changed appreciably during this time period-
With kA/ks < 12 this approximation is not valid. However, with ka/kB > 12, especially for the trimethylamine-tributylamine mixture, kA/kB = 66, where a sharp bend occurred in the rate curve, at small times, this approximation is valid. The use of visual extrapolation from the rate curve has previously been reported (13,14). Koltoff and Lee (14) stated that visual extrapolation is not accurate and must be rejected because “the slope of the slower reacting component changes continuously.” Several examples were given where tangents were drawn on the rate curve at various times after [A]2 0. The results obtained became increasingly poorer as time increased because the approximation became less valid. Thus, for visual extrapolation, the tangent is placed on the rate curve immediately after the sharp bend in the curve (Figure 4). This sharp bend, which indicates that A has completely reacted, is easily observable at a ratio of rate constants above 12. Because a 2 z uncertainty was found when [A], employing visual extrapolation, the maximum mole tolerable is approximately 98 with a ratio of rate constants above 12 (Figure 3, curve A). It is interesting to note that the probable error, A[B],in Figure 1 (50 % [A],)for the case where the ratio of rate constants is 211 is extremely high; however, the probable error for (13) E. 0. Schmalz and G. Geiseler, Z . Anal. Chem., 188, 241 (1962). (14) I. M. Kolthoff and T. S. Lee, J . Polymer Sei., 2, 206 (1947).
4I 30-
k ~ / b
z
TIME 3 0 0 s e c / i n .
Figure 3. Maximum tolerable mole [AIoin a mixture as a function Of ka/kB Curve A. Graphical extrapolation method Curve €3. Method of Roberts and Regan
kA/kB = 4 is much lower, These theoretical results compare well with the experimental results obtained in Figure 3, curve A for a 50% [ A ] ,mixture. Only one case could be found in the literature where k.4/kB was given for a mixture that failed. Willeboordse and Critchfield ( 1 5 ) , employing the second order graphical extrapolation method (7),reported that a mixture of 1-propanol-2-propanol (kA,’kB= 2.2) could not be analyzed if 50% [ A ] , was present while a 39 % [A],mixture could accurately be analyzed. This also agrees with the experimental results obtained here. When employing the second order graphical extrapolation method, the reaction used for analysis may become reversible or behave poorly near the end of the reaction. Hence, the maximum mole [ A ] ,tolerable at a given kA/kswill be lower than indicated in Figure 3 curve A for the pseudo first order irreversible amine-methyl iodide reaction. However, curve 1 Figure 3 can also be considered the limiting curve for the second order graphical extrapolation method when the reaction is irreversible. In this same direction, Siggia and Hanna (7) suggested that, if the mole of [A],in a mixture is too high for a successful analysis, the addition of a known amount of B to the mixture could lower the mole % [ A ] , to a point where a successful analysis can be obtained. In some cases, employing a different reaction medium may increase kA/ka,which will allow a higher tolerable mole percentage [ A ] , . For example employing acetone as a reaction medium in place of nitrobenzene for the triethylamine-tributylamine system resulted in a larger kA:kB ratio and higher tolerable mole percentage of [ A ] ,(Table I). Minimum Mole Percentage [A], ToIerabIe for the GEM. Because of uncertainties always present when obtaining the data points used for the extrapolation of the slower reacting component, a 1 % error is common for the determination of [B],and, hence, also of [ A ] , . Analysis of amine mixtures by the pseudo-fractional and first-order GEM resulted in an average deviation from the true value of 0.9% for A (8). The minimum sensitivity or detectable amount of A is approxi-
Figure 4. Analysis of diethylamine-l-methyI-4-piperidone mixture (94.6 [ A ] , )by visual extrapolation from conductance rate curve. [ A ] , [B], 2 is maximum 98 (one can usually expect a 2 % average deviation from the true value at high concentrations of A ) . Figure 3, curve B, plots the maximum mole % of [A],tolerated as a function of the ratio of rate constants. The shape of this curve was actually obtained from the error coefficients by employing the same probable errors in the experimentally measured parameters as previously given above and by assuming that at a given kA/kBvalue, the maximum tolerable probable error, 4 [A],, is 1 x 10-ZM. The interesting fact that this method can tolerate a ratio of rate constants of 2, even at high concentrations of A , can be explained qualitatively by looking at the mathematical framework of the error coefficients for this method, All the coefficients (with the exception of Equations 13 and 14 where the rate constants also appear in the numerator) are proportional to the reciprocal of the difference in the rate constants. Hence, for a constant value of kA, the probable error, A [A],, would only be approximateIy doubled at a kA/kB = 2 when compared to a kA/kB of infinity, and would only be approximately four times higher at k A / k s of 1.5 when compared to a kJkB ofinfinity. However, as ka/ks approaches 1, the error coefficients (Equations 12, 13, 14, 15, 16, and 17) approach infinity and the probable error, 4[A],, also approaches infinity. Although the MRR cannot be employed for kA/kB < 1.4, of the GEM is applicable at kA/kB< 1.4 when the mole [A],5 15 % (Figure 3). Accurate results were obtained when the GEM was used for the first two systems in Table II (4.6% triethylamine found in 90% DMSO-10 MeOH by the GEM and 4.1 triethylamine found in 85 DMSO-15 % MeBH). ole % [A],Tolerable for kfRR. In contrast to the GEM, the minimum mole of [A], that can be detected for the MRR is a function of the ratio of rate constants. The kinetic equation employed for a determination by this method is (6, aO> IC” = kA”[A], kB”[B], (18)
z
DMSO-15
75.6 48.0 82.2 5.7
51.2 83.Q
A. Tertiarybutylamine B. Isobutylamine A. n-Butylamine B. Tributylamine
4.0
165
DMSO-20 % n-propyl alcohol
DMSO
where K* is the pseudo first order overall rate constant for both species, and i s equal to (In [R],/[RI,)/tin Equation 10. Figure 5 is obtained by assuming that the minimum contribution to K* from the faster reacting component, [A],, must be at least 5%-i.e., ka”[A],/M*X PO0 = 5 % , Hence, as the ratio of rate constants increases, the method becomes quite sensitive to very small amounts of the faster reacting component. This is verified in Table I1 as 2 parts per thousand of diethylamine was accurately detected in methylaniline. However, for the terliarybutylamine mixture, kA/kB = 2.2, a 2 to 3 % deviation is obtained at a small mole % [A],. This is also predicted in Figure 5. SUMMARY
The experimental limitations obtained in Figures 3 and 5 are meant to be guidelines for the analyst. The values of the limitations may shift somewhat from analyst to analyst depending upon his own personal tolerances and also upon the accuracy in measuring his experimental rate curves. The limitations for the GEM can also be guidelines for other types of competitive decay reactions. For example, the use of a semilogarithmic plot of disintegration rate us. time after termination of activation for a mixture of isotopes differing in half lives is a well known simple means of measuring the concentrations of a radio active mixture (16). Also, recently, the GEM was used to measure two component mixtures of phosphorescent compounds (17).
RECEIVED for review March 27, 1967. Accepted August 14, 1967. Division of Analytical Chemistry, 153rd Meeting, ACS, Miami Beach, Fla., April i967. Research supported in part by a grant from the U. S. Army Research Office, Durham, (Contract No. DA-31124-ARD-D-284). One of us ( R A G . ) is indebted to the National Aeronautics and Space Administration for a Graduate Traineeship in 1966 and 1967 which made possible this work.
(16) G. E. Boyd, ANAL.CHEM.,21, 335 (1949). (17) P. A. St. John and J. D. Winefordner Ibid., 39, 500 (1967).
