# Weisz-Prater Criterion

The Weisz-Prater Criterion is a method used to estimate the influence of pore diffusion on reaction rates in heterogeneous catalytic reactions.[1] If the criterion is satisfied, pore diffusion limitations are negligible. The criterion is
$N_{W-P} = \dfrac{\mathfrak{R} R^2_p}{C_s D_{eff}} \le 3\beta$
Where $\mathfrak{R}$ is the reaction rate per volume of catalyst, $R_p$ is the catalyst particle radius, $C_s$ is the reactant concentration at the particle surface, and $D_{eff}$ is the effective diffusivity. Diffusion is usually in the Knudsen regime when average pore radius is less than 100 nm.
For a given effectiveness factor,$\eta$, and reaction order, n, the quantity $\beta$ is defined by the equation:
$\eta = \dfrac{3}{R^3_p} \int_{0}^{R_p} [1-\beta (1-r/R_p)^n] r^2\ dr$
for small values of beta this can be approximated using the binomial theorem:
$\eta = 1-\dfrac{n \beta}{4}$
Assuming $\eta \ge 0.95$ with a 1st or zero order reaction gives values of $\beta$, 0.6 and 6 respectively. Therefore for many conditions, if $N_{W-P} \le 0.3$ then pore diffusion limitations can be excluded.[2]

## References

1. ^ Weisz, P. B.; Prater, C. D. (1954). "Interpretation of Measurements in Experimental Catalysis". Advances in Catalysis. Advances in Catalysis 6: 143. doi:10.1016/S0360-0564(08)60390-9. ISBN 978-0-12-007806-6.
2. ^ Vannice, M. Albert (2005). Kinetics of Catalytic Reactions. New York: Springer Science+Business Media. pp. 63–65.