138 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS Third Order. The rate is proportional to the concentration of three reacting substances. For these purposes, third order reactions are considered to be too rare and will not have to be considered. Other complex reactions are noteworthy, but similarly, they will not affect our use of chemical kinetics. By complex reactions are meant consecutive reactions, competing reactions, and reverse reactions in which equilibria are set up which mass-wise are neither near to the be- ginning nor near to the point of completion. The reason confusion does not run rampant, considering the many realistic possibilities, is that, in general, some limiting step predominates such that experimentally we are able to observe the behavior of some single entity or parameter. Before we try to organize and use the just-discussed physical-chemical information, one more concept, the role of temperature, should be noted. That reaction rates increase with temperature is a well-known observa- tion amply illustrated by every-day experience. This fact has been quantitated and incorporated in the scheme of chemical kinetics by the well-known Arrhenius equation: log (k2/kl)= (AHa/2.303I•)Cl•- ]•l/T2T•) where: /XHa = heat of activation R = gas constant T = absolute temperature ( øC 4- 273 ø) Graphically, a straight line of negative slope results when log k is plotted rs. I/T the slope = (--Atta/2.303R). Now that the foundation has been laid, we must choose which type of edifice to build and then justify the nature of the architecture. First, it is obvious that the role of temperature can be incorporated into the scheme of things by employing the well-known "accelerated studies" in which formulations are stressed by storage at various elevated temperatures. Secondly, a variation of the half-life concept will be used in which a too is employed, i.e., the time it takes for 10% degrada- tion or for a drop to 90% of the original concentration of a material. This will be illustrated later. Thirdly, first order kinetics will be utilized because of the ease of application. Amplification of these points will be effected in reverse order. First, the validity of using first order kinetics should be verified. One should be sure that use of this order will not be misleading if the sys- tem under observation is, in fact, degrading according to second order kinetics. The existence of confusion or the lack of it, which would enter

PRODUCT STABILITY--PART I 139 Table I Calculated Results of Model Second Order Kinetic Rate Equation Illustrating Degradation of J and B: Initial Concentration o[ A = 10, B = 1 x t x 102 (A) (B) log (zi) log (B) O. 1 O. 46 0.2 0.98 O.3 1.58 0.4 2.27 O. 5 3.10 0.6 4.12 0.7 5.46 0.8 7.36 O. 9 10.66 99 98 97 96 95 94 93 92 91 0.9 0.996 --0.046 0.8 0.991 --0.097 0.7 0.987 --0.155 0.6 0.982 --0.222 0.5 0.978 --0.301 0.4 0.973 --0.398 0.3 0.968 --0.523 0.2 0.964 --0.699 0.1 0.959 --1.000 Table II Calculated Results of Model Second Order Kinetic Rate Equation Illustrating Degradation of C or D when Initial Concentration of C = D = 1 x t (C) log (C) 0.1 0.11 0.9 0.2 0.25 0.8 0.3 O.43 0.7 0.4 0.67 0.6 0.5 1.00 0.5 0.6 1.50 0.4 0.7 2.33 0.3 0.8 4.00 0.2 0.9 9.00 0.1 -- O. 046 - O. O97 -0. 155 -0 222 --0 301 -0 398 -0 523 -0 699 - 1 000 into an attempt at analysis of a system which was degrading in second order fashion but which was treated as first order, could be determined by setting up model systems, as will be illustrated next. The second order model data shown in Table I may best be examined by plotting them as if the system were following first order kinetics. These data were obtained for the case in which a =• b letting: a = original concentration of A = 10, b -- original concentration of B = 1, x = amount of degradation = 0.1, 0.2, ..., 0.8, 0.9, and k = 2.303. Then: t = •/.• log (0.1) (a-- x) / (b - x) in arbitrary units. Solving for t as A decomposes to about 90% and B to about 10% of original concentrations produces the figures of Table I.