Substrate Concentration Limit Unsaturated Kinetics

We now examine the situation when there is concentration polarization of substrate in the layer. First we look at the situation of low substrate concentration, the unsaturated situation where kj K - 

Hence we see that the problem reduces to a diffusion/reaction with first-order kinetics. The solution to this equation is given 

This concentration profile is illustrated in Fig. 2.31 as a fimction of the Thiele modulus 3 . From this diagram we note that when 1 is small, concentration profiles are fairly shallow, since the entire layer is being used. However when 4 is large, the concentration of substrate falls rapidly with distance into the polymer film. The reaction is essentially complete within a thin reaction zone (thickness 2fjt) near the polymer/solution interface. 

This discussion can be amplified by examining Eqn. 116 in more detail. When 3 0.3 we note that tanh I I). In this case assuming that I 1 (i.e., L X ), the expression for the flux in Eqn. 116 reduces to 

Hence provided s°° and we use thin films, the steady-state current response depends linearly on the substrate concentration, the catalyst concentration in the layer, and on layer thickness. The reaction zone encompasses the entire layer, and the rate of reaction at the catalyst surface is much lower than the rate of substrate transport. In this sense when 4 1, the rate is reaction-controlled. Note also that for 4 1, cosh4 1, and so cosh ( l j) = 1. From Eqn. 117 we obtain u(x) 1 indicating a uniform concentration distribution of substrate through the layer. 

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Let us examine this expression in some detail. First we note that terms involving kc and KmIK are important when the Michaelis-Menten kinetics are rate-limiting. The first of these terms involving kc corresponds to the saturated catalyst the kinetics are zero-order with respect to substrate concentration. The second term involving Km Ike corresponds to the unsaturated catalyst, where the kinetics are first-order with respect to substrate concentration. If the flux j approaches the limit imposed by the substrate transport through the diffusion layer, then concentration polarization results. As a consequence the catalyst is less saturated than would be expected from the value of the bulk concentration j". This effect is described by the... 

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