Statistical fractal adsorption isotherms, linear energy relations, and power-law trapping-time distributions in porous media

Drazer and Zanette [Phys. Rev. E 60, 5858 (1999)] have reported on interest ing experiments which show that trapping-time distributions in porous media obey a scaling law of the negative power-law type. Unfortunately, their th eoretical interpretation of the experimental data has physical and mathemat ical inconsistencies and errors. Drazer and Zanette assume the existence of a distribution of local adsorption isotherms for which the random paramete r is not a thermodynamic function, but a kinetic parameter, the trapping ti me. Moreover, they mistakenly identify the reciprocal value of a rate coeff icient with the instantaneous (fluctuating) value of the trapping time. The ir approach leads to mathematically inconsistent probability densities for the trapping times, which they find to be non-normalizable. We suggest a di fferent theory, which is physically and mathematically consistent. We start with the classical patch approximation, which assumes the existence of a d istribution of adsorption heats, and introduce two linear energy relationsh ips between the activation energies of the adsorption and desorption proces ses and the adsorption heat. If the distribution of the adsorption heat obe ys the exponential law of Zeldovich and Roghinsky, then both the adsorption isotherm and the probability density of trapping times can be evaluated an alytically in terms of the incomplete beta and gamma functions, respectivel y. Our probability density of the trapping times is mathematically consiste nt; that is, it is nonnegative and normalized to unity. For large times it has a long tail which obeys a scaling law of the negative power-law type, w hich is consistent with the experimental data of Drazer and Zanette. By usi ng their data we can evaluate the numerical values of the proportionality c oefficients in the linear energy relations. The theory suggests that experi mental study of the temperature dependence of the fractal exponents helps t o elucidate the mechanism of the adsorption-desorption process.