platinum microelectrode (12). If i t is assumed that a bright platinum microelectrode is platinized or partially platinized by electrolysis of liquid NH3 solutions of ammonium salts, a-ave A of Figure 1 represents the reduction of ammonium ion aggregates at a bright platinum electrode, and wave B, C, or D represents the reduction at a platinized electrode. Since this is the reverse of the order observed by Laitinen and Kolthoff the reduction process in liquid XH3 apparently is different from that in aqueous solution. The reduction of ammonium ion pairs, or ammonium ions, to free ammonium (Equation 2) represents an alternate possibility. The division of the original m-ave into two more or less distinct waves (see Figure 1) can bc attributrd to the reduction of ion aggregates a t two different rlectrode surfaces. Although the data indicate that the polarographic lvave which occurs a t a rotating platinum electrode in liquid ammonia solutions of ammonium salts ~
arises from the reduction of ammonium ions and ion aggregates containing the ammonium ion and that the diffusion current is proportional to the concentration of reducible species in solution, the gradual disruption of the electrode surface makes i t difficult to use this technique for the quantitative investigation of acid-base behavior in liquid ammonia solutions. LITERATURE CITED
(1) Bjerrum, N., Kgl. Danske Selskab. 7, KO.9 (1926). (2) Booman, G. L., Ph.D. dissertation, University of Washington, Seattle, Wash., 1954. (3) Brunner, E , 2. physzk. Chem. 47, 56 (1904) (4) Coustal, R., Spindler, H., Compt. rend. 195, 1263 (1932). (5) -Hammer, R. K., Ph.D. Dissertation, Lniversity of Illinois, Urbana, Ill., 1954. (6) Hnizda, V. F., Kraus, C. A,, J . A m . Chem. Soc. 71, 1565 (1949). (7) Hogge, E. rl., Kraichman, &I. B., Ibid., 7 6 , 1431 (1954). (8) Hogge, E. A4.,Kraichman, M. B., J . Phys. Chem. 59, 986 (1955).
(9) Jablczynski, K., 2. physik. Chem. 64, 748 (1908). (10) Kolthoff, I. M., Lingane, J. J., “Polarography,” Vol. I, p. 122, Interscience, New York, 1952. (11) Ibid., pp. 52, 90. (12) Laitinen, H. -4.,Kolthoff, I. hl., J. Phys. Chem. 45, 1061 (1941). (13) Laitinen, H. A , Nyman, C. J., J . Am. Chem. Sac. 70, 3002 (1948). 114) Laitinen. H. A.. Shoemaker. C. E.. Ibid.. 72. 4975 (1950). (15) Levicfi, B ’ A’cla Physicochim. U.S.S.R. 17, 2Zi (1942). (16) Lord, S . S., Rogers, L. B., ANAL. CHEM.26, 284 (1954). (17) Nernst, W.,Merriman, E. S., 2 vhusik. Chem. 53. 235 (1905). (18) vStoll, W.,Berbalk, H:, Xonatsh. Chem. 84, 1179 (1953). (19) Taukamota, T., Kambara, T., Tachi, I., Sbornik mezindrod. polarog. sjezdu. praze. Inst. Cong. Pt. I, 524 (1951). (20) Van Name, R. G., Edgar, G., Am. J . Sci. 29, 237 (1910). (21) Wiesner, K., Collection Czechoslov. Chem. Commztns. 12, 64 (1947). ~
RECEITED for review September 28, 1961. Accepted March 5, 1962. Taken in part from the Ph.D. dissertation of J. J. Lagoaski, Michigan State University, 1957.
Kinetic Analysis of Mixtures by the Method of Proportional Equations RONALD G. GARMON‘ and CHARLES N. REILLEY Department o f Chemistry, University of North Carolina, Chapel Hill, N. C. The initial concentration of a species undergoing a simple or complex firstorder reaction is directly proportional to the amount o f product formed a t any given time or to any parameter directly proportional to the product. This proportional relationship i s utilized in two ways. First, measurements o f the extent o f reaction for several species undergoing competing reactions a t a number of times equivalent to the number of species present permits a simultaneous determination of the mixture. Second, the change in the extent o f a reaction between any two given times i s directly proportional to the initial concentration of the reacting species. Judicious selection o f a time interval in which to observe the reaction greatly reduces the effect of interfering species. An illustrative example of each technique is given. The use of variables other than time for kinetic analysis is discussed.
the utility and scope of these methods, they are operative n ith certain limitations whose naturevaries from method to method. First, these methods are usually restricted to two-component mixtures. Second, a knowledge of the sum of the concentrations of the species being determined is usually required. Hence, the reactions would either have to be carried to completion--a time consuming process-or an independent analysis of the sum is necessary. Third, most of the methods require simple first-order or second-order kinetics and are not, for example, applicable to consecutive reactions; the work of Blaedel and Petitjean ( I ) and the empirical calibration curves described by Kolthoff, Lee, and hlairs (8) proride notable exelltions. The method of proportional equations proposed in this paper circumvents the first two limitations and is applicable to all first- and pseudo-first-order reactions, both simple and complex.
A
The method of proportional equations, while new, is based on the existence of proportional relationships 11-hich are well known.
PROPORTIONAL RELATIONSHIPS
of general methods have been developed for t h e determination of mixtures based on differences in reaction rates (6,9-11,14). Despite
600
NuhIBER
ANALYTICAL CHEMISTRY
Simple First-Order Kinetics. Consider a compound, A , a-hich reacts by a first-order or pseudo-first-order process t o produce product C according to A
-+
nC
The concentration of A and C a t any time is given by the well known kinetic espressions : [A]t = [ 9 ] o e - k a t (1) [C]t = n[4]0 (1 - e c k a f ) = k’a,t[Alo (2)
where k , is the first-order or pseudo first-order rate constant. Hence, the concentration of C at any given time is directly proportional to the initial concentration of A, the proportionality constant, KA t , being a function only of the stoichiometry, the reaction time, and the rate constant (hence, upon those parameters which affect the rate constant such as temperature and ionic strength). The course of a reaction is usually followed by observing some parameter 1 Present address, Chemstrand Research Center, Inc., Durham, N. C.
directly proportional to the concentration of product, C (such as the absorbance of light, electrical conductivity of the solution, polarographic diffusion current, or volume of reagent required for a titration). I n such cases, the experimentally determined parameter, P I , can itself be directly related to the initial concentration of A by: Pt
e
m(1 - e - k ~ t ) [ A = ] OK ’ A , t [ . ! i ] o (3)
n here y is the proportionality constant. For example, in a case where C follows
Deer’s lam, P t corresponds to the ahsorbaiice measured a t any set time, t , and K‘A t = sbn(1
=:
17
K ” A , [Ajo ~ (4)
n here CY is the proportionality constant for A . Hence, a simple proportional relationship is again obtained. I n cases where the experimental parameter is related in a nonlinear manner to concentration, it is only necessary to convert the measured parameter-Le., by a n empirical calibration curve-into a corrected parameter proportional to concentration and then employ Equations 3 or 4. Complex First-Order Kinetics. illthough t h e existence of a proportiona1 relationship is well known for simple first-order reactions, less nidely recognized is t h e fact t h a t the most complex reaction mechanism, if made u p of only first-order reactions, exhibits this property. For example, consider the reaction scheme of t n o consecutive reversible first-order reactions:
The detailed kinetic expression for this reaction mechanism has been given by Lonry and John (12) and is one of the more complex first-order mechanisms that has been considered rigorously. Severthclcss, it can be shon-n t h a t : = K1[.1110
(5)
provided that [A2]0and [&lo arc zcro a t = 0. d proportional relationship, of the form of Equation 5 , can also he rigorously derived for as a function (If [-41]o.
t
kl_ u k2 +
+
Pt = [Ala [m(l - e%,)
[’431t
A
- e-”,)
nhere E and 6 are the molar absorbtivity for C and the length of the light path, respectively. Where the magnitude of experimental parameter P is determined by both the concentration of reactant A and of product C-Le., where the course of a reaction is followed by observing the absorbance a t a wavelength n-here both 9 and C have finit
