values obtained compared favorably with the acidimetric analysis. Percentage compositions for hydrazine were calculated by
yohydrazine
=
Analogous formulas were used for the other components. Calculated responses were based upon prepared mixtures of known composition and were dependent upon the acidimetric analysis of hydrazine and MMH, which is accurate to *0.2%. A chromatogram of a typical mixture of hydrazines and water is shown in Figure 1. The analysis was calculated by using the reciprocal response ratios obtained from the analysis of a previously prepared mixture of similar composition. The sample shown in Figure 1 had a calculated weight of 25.0% water, 40.4% MMH, and 34.6% hydrazine. The chromatographic analysis of this mixture, using the previ-
ously obtained res onse ratios, showed 24.2% water, 4 0 . 9 g MMH, and 34.9% hydrazine. Analyses of other mixtures by this procedure showed similar results. Because of the high reactivity of the hydrazine, the prepared column should not be used for the analysis of other materials. RESULTS
The statistical analysis of the data obtained in the analyses of hydrazine, MMH, and water mixtures indicated that 99% of all of the values collected should fall within f 0.71% of the average value of three data points a t the 95% confidence level. If the mean of three data points is used, the mean will be within 0.41% of the true value of each of the components as calculated by weight. The column described has been used for over 200 analyses with no apparent fatigue. The method employed in the study has the advantages of short
analysis time, minimal tailing, excellent resolution, and column stability. One of the disadvantages is the inability to separate UDMH and water. ACKNOWLEDGMENT
The author thanks W. R. Carpenter and E. M. Bens for their assistance and advice in the development and evaluation of this gas chromatographic procedure. LITERATURE CITED
(1) Cain, E. F. C;,, Stevens, M., “Gas Chromatography, p. 343, Academic Press, New York, 1961. (2) Kuwada, D. M., J . Gus Chromatog. 1, (3), 11 (1963). (3) O’Donnell, J. F., Mann, C. K., ANAL. CHEM.36,2097 (1964). (4) Penneman, R. A., Audrieth, L. F., Ibid., 20, 1058 (1961).
RICHARD M. JONES U. S. Naval Ordnance Test Station China Lake, Calif. 93557
Method of Proportional Equations for Analysis of Closely Related Mixtures by Differential Reaction Rates Where Concentration Reactants of Reagent
[Reactants], overcame most of the above mentioned limitations. The chief advantages associated with this kinetic method are (1, 2): Prior knowledge of total initial concentration of the species of interest need not be known; more than two components can be analyzed; and only a fraction of the total reaction time is required for analysis. However, because of the mathematical framework of this method, small ratio of rate constants-i.e., 4 to 1 or less-could not be tolerated (2). A Method of Proportional Equations (or double point method) which is applicable for reactions pseudo-firstorder with respect to the reagent is reported in this paper. This method has the above advantages of Garmon and Reilley’s method plus the additional advantage that small ratio of rate constants-i.e., 2 to 1 or less-can be used. The method was tested by the analysis of several different carbonyl mixtures using the reaction mentioned above. The method could be utilized, however,
for any closely related mixtures where the concentration of reagent can be followed experimentally as a function of time. PRINCIPLE
Roberts and Regan (6) presented a kinetic method for simultaneously determining two components, A and B, reacting with a common reagent, R. If the concentrations of A and B are much greater-Le. 50 to 1 or largerthan the reagent, R, the following pseudo-firsborder rate expression holds:
where K* is the overall pseudo-firstorder rate constant for both species which is given by where k A and kg are the second-order rate constants of A and B respectively reacting with R, and [AIoand [ B ] ,are the initial concentrations of reacting species A and B (the amount of A and B that reacts during the reaction is negligible as the concentrations of these species are in such great excess). The analysis is accomplished by measuring K* (as described by Papa,
limited by the accuracy of determining the second order rate constants of the individual pure species present and K* for each reaction condition or media. PROCEDURE
I
*A MsOH in Reoction Medium IHZO-MeOH Mixture)
Figure 1 . Second-order rate constants for 3-pentanone, A, and anisaldehyde, 6, reacting with hydroxylamine hydrochloride as a function of reaction medium
et al. (4)) k A and kg, and by determining the total concentration of [AIoplus [ B ] ,(found by an independent method). The method employed in this paper is based on the principle that two nonparallel simultaneous equations of the form of Equation 2 can be obtained by simply altering the reaction conditions. The equations are thus :
where the numerical subscripts signify different reaction mediums or conditions. The initial concentrations cai: be found directly from the solution of the above equations provided that k ~.k ,B , # k ~ *k B., . As the reaction is pseudo-firstorder with respect to the reagent, only a medium or condition change which results in a variation of the ratio of rate constants can give the proportional nonredundant equations. Changing t or [ E ] , only leads to two redundant equations. This method, in theory, can also be employed for the analysis of mixtures containing more than two components. If N components A , B , + N , are present, reacting with a common reagent, N equations of the form
can be written for N different reaction media or conditions. Of course, in actual practice, the maximum number of components that can be determined with a satisfactory degree of accuracy is
The rate of the reaction is automatically followed by a direct recording conductance apparatus in the same manner as described previously (6). A water-jacketed cell, 150 ml. in capacity, is connected to a circulating water bath maintained a t 25.0' f 0.1' C. Stock solutions of hydroxylamine hydrochloride (2 X 10-3M) are prepared in varying alcohol-water mixtures and placed into the water bath for a t least 1 hour prior to the analysis. Seventy-five ml. of this reagent is placed into the cell, equipped with a magnetic stirring motor and a stirring bar made of Teflon, and the conductance electrodes are placed into the cell. A recorder is started (employing a 3 to 12 inches per minute time base d e pending on the reaction rate) , and when a steady base line is maintained, a sample of pure aldehyde or ketone, previously placed in the water bath, is injected into the cell using a 1-ml. hypodermic syringe. The needle is immersed into the solution to avoid bubbling. The second-order rate constant, for the pure aldehyde or ketone, is calculated in the usual manner (4) from the conductance curve. This same procedure is performed for other water-alcohol reaction mediums with both pure carbonyls found in the unknown mixture, until a reaction medium rate spectrum (with respect to solvent medium) is obtained for each component. Typical spectra for 3pentanone and anisaldehyde are shown in Figure 1. The variation of the ratio of rate constants as a function of solvent composition for these two compounds is shown in Figure 2. The analysis of any unknown mixture is made with the two mediums exhibiting the largest change in the ratio of rate constants. The most suitable reaction mediums and the second-order rate constants for the pure compounds in the mediums are given in Table I. Pseudo-firstorder rate constants, K*, for both reaction mediums are calculated in the usual manner from the conductance curve and the initial concentration of unknowns is calculated by means of Equations 3 and 4. RESULTS AND DISCUSSION
Table I lists the results obtained for the analysis of three different carbonyl mixtures. Several different ratios of initial concentration of reactants, [ A ] , / [ B ] , , were analyzed. The average relative error, based on total initial concentration, for all the mixtures analyzed was 2.4%. The uniqueness of this kinetic method is clearly demonstrated with the 3pentanone cyclopentanone mixture. Small ratio of rate constants (less than 1.4 to 1for this mixture) did not hinder
-
-
0 10 20 30 40 50 60 70 80 90 0 */e MeOH in Reaction Medium lHZO-MeOH M i K l U d
Figure 2. Ratio rate constants for 3pentanone, A, and anisaldehyde, 6, as a function of the reaction medium
the analysis and all concentration ratios of the reacting species could be analyzed. The second-order logarithmic extrapolation method has also been employed for the analysis of this particular ketone combination ( 7 ) . It was reported that only mixtures which contained less than 10% of cyclopentanone could be successfully determined as would be expected ( 2 ) . Also, it would not be expected that Garmon and Reilley's Method of Proportional Equations could tolerate this low ratio of rate constants ( 2 ) . The advantage over the single point mathematical modification of Roberts and Regan method, as explained above, is that no independent measurement of total carbonyl concentrations is required. While performing the calculations from the conductance curve, it is necessary to ascertain the optimum fractional life a t which to take the rate data chosen. This fraction should be that which produces the minimum error in the determination of the two components. This optimum fractional life, as explained by Papa, et al., (4) occurs when [R]z/[R]1 equals l / e (to ensure homogeneity of the solution on mixing [ R ] Iis taken as the reagent concentration measured 3 seconds after the reaction is initiated and not that a t time, t = 0). However, although nearly correct, this optimum fractional life takes into consideration only the error resulting from deviations in the measured parameter [R]2. As explained previously ( 2 ) ,other experimentally measured parameters, namely [RI1,k , and ka and t, have an effect on the optimum fractional life that produces the minimum error in the analysis. Taking all parameters into account, they obtained a value of optimum fractional life VOL. 38, NO. 2, FEBRUARY 1966
341
~~~~~
Table I.
Mixture A. 3-Pentanone B. Anisaldehyde
A. 3-Pentanone B. Cyclopentanone
A. CyClOentanone B. *ðyl-2pentanone
~~
~
-
Determination of Carbonyls by Reaction with Hydroxylamine Hydrochloride
Molarity Taken Found 6.44 2.64 6.44 6.53 2.64 2.80 8.97 9.17 0.44 .26 8.97 8.47 0.44 0.83 3.94 3.82 4.92 5.21 3.82 3.75 5.20 4.92 2.15 2.40 6.31 6.16 2.30 3.02
i:::
8.53 4.88 5.46 7.16 2.72
8.00 4.97 5.26 6.93 2.91
2.70
2.68
6.08 2.70 6.08 6.38 3.48 6.38 3.48
6.11 2.57 6.27 6.69 3.01 6.35 3.82
Reaction Medium 1 Medium 2 78% MeOH) (42% MeOH) (227,H~O 58% HzO
70%MeOH (30% HIO
ANALYTICAL CHEMISTRY
)
80% MeOH (20%H20
slightly smaller than l / e , approximately equal to 0.35. This value was used for all measurements. The time intervals corresponding to the optimum fractional life for each pure component are given in Table I. There is no general differential kinetic method suited for all possible reaction properties and conditions. Each method has its own particular place and use in the field of analysis by differential reaction kinetics. The Method of Proportional Equations proposed here has its advantages, but it also has its limitations and disadvantages. As the ratio of rate constants increases, the accuracy in determining the slower reacting component decreases and only mixtures which contain a high amount of slower reacting component (or small amount of faster reacting component) can be analyzed. As a rule of thumb, the contribution of the slower reacting component to K* in Equations 3 and 4 should be a t least 7 to 10%. For example, if the ratio of rate constants is 50:1,and the ratio of k,[A~,]/ka[Bo] is no larger than 10:1 ( B is considered the slower reacting component) only mixtures which contain 15% or less of A can be analyzed. An attempt was made ta resolve mixtures of 3-pentanone and acetophenone, where the ratio of rate constants is approximately 50: 1. Ac342
~
)
lO%MeOH (90% HzO
)
30% MeOH (70%&0
)
Rate constant, liter/mole sec. Medium 1 Medium 2 A. 0.00257 A. 0.00700 B. 0.00497 B. 0.00644
Optimum time, sec. Medium 1 Medium 2 A. 43.2 A. 15.8 B. 25.5 B. 19.7
A. 0.00345
A. 0.00790
A. 32.1
A. 14.0
B. 0.00466
B. 0.00699
B. 20.0
B. 13.3
A. 0.00355
A. 0.00905
A. 26.2
A. 10.2
B. 0.000957
B. 0.00403
B. 136.0
B. 32.5
curate results were obtained only when the faster reacting component, 3pentanone, was in small concentration. Large errors occurred for acetophenone in mixtures that contained more than 15% 3-pentanone. It should be acknowledged a t this point that Siggia and Hanna also observed that better results were obtained for the analyses of mixtures containing small amounts of the faster reacting component employing the second-order extrapolation method when the concentration of the reactants in the reaction mixture was greater than that of the reagent (7). The principle is, of course, the same as that for the method of Roberts and Regan (2, 6). The ideal situation for the method presented in this paper occurs when the order of the ratio of rate constants reverses-i.e., larger than one in the first reaction medium while smaller than one in the other medium- as found in the 3-pentanone-cyclopentanone system and in the anisaldehyde-3-pentanone system. As each component contributes significantly to K* in both reaction mediums, either component can be analyzed in concentrations as small as 3% (5% of anisaldehyde was successfully analyzed in the 3-pentanoneanisaldehyde system). It was found that a successful analysis
by this method was not possible if the ratio of rate constants does not change a t least by a factor of 1.5. Although Siggia and Hanna (8) had reported that the rate constants of the individual components in a mixture may differ from the rate constants for the compounds in pure form, this synergistic effect was not observed for the carbonyls in the reaction mediums stated in Table I. However, if this effect does occur, a calibration curve can be used to overcome this difficulty. Numerous other parameters, such as temperature, pH, catalysts, two different reagents, and viscosity, can alter the activation energy of reactions. Although these parameters were not utilized here, they certainly would enhance the probability of producing a change in the ratio of rate constants. The fact that one can often alter the ratio of rate constants of a system enhances the flexibility and use of all differential kinetic methods. One can easily adjust the reaction medium, etc., in most systems in order to obtain the ratio of rate constants that would give the best results for a particular method. For example, the use of the second order logarithmic extrapolation method under the reported reaction conditions was restricted to the determination of only small amounts (7) (less than 10%) of
cyclopentanone, the faster reacting component, in a cyclopentanone-3pentanoiie mixture. However, by judicious selection of reaction medium, 3-pentanone could have been actually made the faster reacting component. This would, therefore, also allow 3pentanone-cyclopentanone mixtures with small amount of 3-pentanone to be successfully analyzed by the secondorder extrapolation method. LITERATURE CITED
(1) Garmon, R. G., Reilley, C. N., ANAL. CHEM.34, 600 (1962).
(2) Mark, H. B. Jr., Greinke, R. A., Papa, L. J., “Proceedings of Society of Analytical Chemists Conference,” Nottingham, England, 1965 (in press),. (3) Mark, H. B. Jr., Papa, L. J., Reilley, C. N., “Advances in Analytical Chemistry and Instrumentation,” C. N. Reillev. Vol. .2, p. 255, Inter~. . ~ ed.. ~ ~ science, New York, 1963. (4) Papa, L. J., Patterson J. H., Mark, H. B., Jr., Reilley b. N., ANAL. CHEW 35.1889 ( 1 ~ f i ~ I (5) Reilley,. C. N;, J. Chem. Educ. 39, A853 (1962). (6) Roberts, ‘J. D., Regan, C., ANAL. CHEM.24, 360 (1952). (7) Siggia, S., Hanna, J. G., Ibid., 33, 896 (1961). I,
(8) Siggia, S., Hanna, J. G., Ibid., 36,
228 (1964).
RONALD A. GREINKF, HARRY B. MARK,JR. Department of Chemistry The University of Michigan Anp Arbor, Mich. RESEARCH supported in art by a grant from the U.S. Army gesearch Office, Durham, Contract No. DA-31-124-AROD-284. One of us (RAG) is indebted to the National Aeronautics and Space Administration for a Graduate Traineeship in 1965. which made ossible. this work. Division ofAnalytica? Chemistry, 150th ACS Meeting, Phoenix, Arizona, 1966.
Application of Adsorption Electroanalysis to Trace Determination of Tetrabutylammonium Ion SIR: Investigations of the effects of adsorbable substances on the kinetics of electrochemical reactions have shown that the faradaic current falls rapidly toward zero for many depolarizer-surfactant combinations following adsorption of the surfactant ( 6 , 7 , 10, 11). The polarographic method has been most commonly used in these studies, and theoretical interpretations have been based on solution of Fick’s second law equation for an expanding plane electrode with the restriction that the electrode reaction is quasi-reversible or totally irreversible (6, 8, 11). In addition to these kinetic studies, the inhibiting effect of adsorbable substances on electrode reactions should be applicable to the electroanalytical determination of the particular surfactant under consideration. Because the adsorbing species need not be electrochemically reactive, a method based on adsorption electroanalysis would be especially useful in determining such substances as tetrabutylammonium ion and Triton X-100, surfactants which are normally irreducible or difficult to reduce. Such an analytical application is implied in the polarographic investigations of Schmid and Reilley (10) where, for several substances, a linear relation was obtained between the time a t which the instantaneous current fell to zero and the reciprocal of the square of the surfactant concentration. However, under conditions of diffusioncontrolled adsorption, considerable time is required before attainment of adsorption equilibrium, SO that analytical applications using a dropping mercury electrode are limited to about l O - 5 M because of the short drop-times normally used ( 3 ) . On the other hand, a stationary electrode should be more sensitive than a dropping electrode because the adsorbing substance is permitted to ac-
cumulate on the electrode surface. That is, for a stationary electrode, the fractional coverage a t a particular applied potential is limited by the bulk concentration of surfactant and the adsorption-desorption rates. In addition to these factors, the fractional coverage is also limited by the finite lifetime of the drop a t a dropping mercury electrode @). For this reason, a stationary electrode was used in the experimental procedure for the analytical applications investigated here. Because adsorption equilibrium is more rapidly attained under conditions of forced convection (S), a stirred solution was used. CONCENTRATION-TIME RELATION
Because a stationary mercury drop is used as the working electrode, Fick’s second law equation for spherically symmetrical mass transfer must be solved, Under steady-state conditions (stirred solut,ions), this equation may be written in the form
In this equatiop, r is the radial distance measured from the center of the electrode toward the interior of the solution, and CA is the time-independent concentration of the adsorbable substance. Equation 1 applies to diffusion of this substance through a thin layer oi solution immediately adjacent to the electrode surface : the Nernst diffusion layer (3). If the boundaries of this layer are defined as a and b [the inner and outer radii, respectively, of a spherioally symmetrical shell (1)1, then the appropriate boundary conditions required to solve Equation 1 are: r = a : CA=O r
=
b: CA = CA”
(2)
(3)
Equation 2 specifies that the surfactant concentration at the electrodesolution interface, r = a , is zero throughout the experiment. I n Equation 3, it is assumed that the concentration of the surfactant over the entire solution volume immediately adjacent to the electrode surface a t T = b is constant and equal to its value in the bulk of the solution, CA”. The flux of the adsorbable substance at the electrode surface is obtained from solution of Equations 1 to 3 :
Here, DA is the diffusion coefficient of the surfactant. Equating the material flux in Equation 4 to the rate of surface coverage, d r / d t , and integrating gives the desired relation between the bulk surfactant concentration, CA”, and the time required, to, for coating the electrode with the equilibrium concentration of surfactant, re: to =
a(b - a) I’. bDA
_1_ ’
CA”
(5)
I n deriving Equation 5, it has been tacitly assumed that there is no adsorption before the start of the experiment. Because adsorption processes are charge dependent (2, 9 ) , this assumption may often be realized experimentally by applying a sufficiently large anodic or cathodic initial potential. According to Equation 5, to varies as l/cAo, while for diffusion-controlled adsorption, t o l l 2 varies as l/Ca” (3). A comparison of these two relations shows that smaller values of CA’ are required to attain a particular value of to for a stirred solution, so that complete coverage under this condition is more rapidly attained than for diffusioncontrolled adsorption. Delahay and Trachtenberg predict a similarly more rapid coverage a t a planar electrode VOL. 38, NO. 2, FEBRUARY 1966
343
