KINETICS OF HAIR REDUCTION 283 that the sigmoidal, acceleratory, or deceleratory nature of the reaction mechanism is not an indication of reaction-controlled or diffusion-controlled behavior.) The most popular method of identifying the appropriate kinetic model for a reaction is the reduced time method of Sharp et al. (8). Consider the integral form of the first order equation -In(1 - or) = kt (6) When ot = 0.5, this expression reduces to: 0.693 = kto. 5 (9) Using this expression as the normalizing function, i.e., dividing (6) by (9), then -In(1 - or) kt 0.693 - kt0.5 (10) or -In(1 - o0 = 0.693 t (11) t0.5 Hence master curves of or versus the reduced time (t/to.5) can now be constructed for the various theoretical mathematical models, and comparison of these master plots with the experimental data allows the best-fitting mechanism to be selected. Tables of reduced time data for the more common kinetic models have been given by Jones et al. (9). A slight modification of this method has also been proposed by Jones and coworkers in which either O/.experimental is plotted against O[theoretical, or (t/to.5)experimental is plotted against (t/to.5)theoretica 1 for a given model. Therefore, this method of analysis has the advantage that it gives rise to a more preferred linear relationship when the correct theoretical model has been identified. In this work we have used the more traditional method of Sharp et al. Using the integral form of the basic kinetic equation, Wickett has represented the pseudo first order reaction in terms of the tensile properties as rF(t)l -,n[F--•J = kC0t (12) Comparison of this equation with equation (6) shows that F(r)/F(0) can be equated to (1 - o0, and if the initial concentration (C o) is incorporated into the specific reaction rate constant, then both equations are the same. Similarly, the moving boundary model i IF(r)] 2 Cor3/2 n[F•-6•] - 3 k•- (13) can be expressed as [' ] - • In(1 - or) 2/3 = kt (14) if it is assumed that the temperature can also be incorporated into the constant. Therefore, the normalization can now be carried out to produce the theoretical reduced- time curves for these two models, as shown in Figure 1. To gain a better understanding of the physical significance of these two expressions, Figure 2 shows the derivative of these two plots expressed as a function of the fraction of reaction.

284 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Moving Boundary _ First order -- 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 t/to.s Figure 1. Theoretical reduced-time plots for the first order model and the moving boundary expression. I I I I 0.7 • 0.6 % ..-.. % o ..o 0..5 - • 0.4- • 0..• 0.2 - 0.( 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 .0 Fraction of Reaction Figure 2. First derivatives of the reduce-time plots for the first order and moving bounda• models, expressed as a function of the fraction of reaction. One might expect that both reaction- and diffusion-controlled mechanisms would give rise to a deceleratory process. For diffusion-controlled processes, it is imagined that it will become more difficult for the reducing agent to advance further into the hair fibers. This results from a combination of the distance the molecules have to travel and the decreasing surface area of the reaction interface due to the cylindrical geometry. There- fore, the maximum rate would probably be expected to be in the initial stages of the