Effect of intraparticle diffusion on experimental parameters

When intraparticle diffusion occurs, the kinetic behaviour of the system is different from that which prevails when chemical reaction is rate determining. For conditions of diffusion control 0 will be large, and then the effectiveness factor tj( 1/ tanh 0, from equation 3.15) becomes. From equation 3.19, it is seen therefore that rj is proportional to k Ul. The chemical reaction rate on the other hand is directly proportional to k so that, from equation 3.8 at the beginning of this section, the overall reaction rate is proportional to k,n. Since the specific rate constant is directly proportional to e E/RT, where E is the activation energy for the chemical reaction in the absence of diffusion effects, we are led to the important result that for a diffusion limited reaction the rate is proportional to e E/2RT. Hence the apparent activation energy ED, measured when reaction occurs in the diffusion controlled region, is only half the true value  

A further important result which arises because of the functional form of is that the apparent order of reaction in the diffusion controlled region differs from that which is observed when chemical reaction is rate determining. Recalling that the reaction order is defined as the exponent n to which the concentration CAm is raised in the equation for the chemical reaction rate, we replace f(CA) in equation 3.8 by CJ . Hence the overall reaction rate per unit volume is (1 - e)rjkCA. When diffusion is rate determining, tj is (as already mentioned) equal to f1 from equation 

20 it is therefore proportional to C i l)/2. Thus the overall reaction rate depends on = CHL /2. The apparent order of reaction nD as measured when 

A zero-order reaction thus becomes a half-order reaction, a first-order reaction remains first order, whereas a second-order reaction has an apparent order of 3/2 when strongly influenced by diffusional effects. Because k and n are modified in the diffusion controlled region then, if the rate of the overall process is estimated by multiplying the chemical reaction rate by the effectiveness factor (as in equation 3.8), it is imperative to know the true rate of chemical reaction uninfluenced by diffusion effects. 

The functional dependence of other parameters on the reaction rate also becomes modified when diffusion determines the overall rate. If we write the rate of reaction for an nth order reaction in terms of equation 3.8 and substitute the general expression obtained for the effectiveness factor at high values of, where rj is proportional to 1/ and is defined by equation 3.20, we obtain  

When intraparticle diffusion is rate limiting, the kinetic behaviour of a chemically reacting system is generally different from that which would prevail if chemical reaction were rate limiting. It is therefore extremely important to develop criteria to assess whether intraparticle diffusion effects may be neglected and thus define the conditions of experiment which would reveal true chemical kinetics rather than overall kinetics disguised by intraparticle diffusion effects. 

If intraparticle diffusion controls the overall reaction rate, the Thiele modulus will be large (0 2) and then the effectiveness factor 77 is approximately 0. From eqn. (10) defining the Thiele modulus, it follows that, for a given reaction, the effectiveness factor will be 

The observed activation energy under such conditions of diffusional control is thus half the true reaction order. 

From eqn. (11) and the functional dependence of the Thiele modulus on concentration [eqn. (10)], it is also obvious that the overall rate of reaction in the diffusion-limited regime is proportional to the product of Cg and j2. Hence the apparent order of reaction which would be 

A zero-order reaction thus becomes a half-order reaction, a first-order reaction remains first-order, whereas a second-order reaction would have an apparent order 3/2 for diffusion-limited conditions. 

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