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 interesting experiments which show that trapping-time distributions in porous media obey a scaling law of the negative power-law type. Unfortunately, their theoretical interpretation of the experimental data has physical and mathematical inconsistencies and errors. Drazer and Zanette assume the existence of a distribution of local adsorption isotherms for which the random parameter is not a thermodynamic function, but a kinetic parameter, the trapping time. Moreover, they mistakenly identify the reciprocal value of a rate coefficient with the instantaneous (fluctuating) value of the trapping time. Their approach leads to mathematically inconsistent probability densities for the trapping times, which they find to be non-normalizable. We suggest a different theory, which is physically and mathematically consistent. We start with the classical patch approximation, which assumes the existence of a distribution of adsorption heats, and introduce two linear energy relationships between the activation energies of the adsorption and desorption processes and the adsorption heat. If the distribution of the adsorption heat obeys the exponential law of Zeldovich and Roghinsky, then both the adsorption isotherm and the probability density of trapping times can be evaluated analytically in terms of the incomplete beta and gamma functions, respectively. Our probability density of the trapping times is mathematically consistent; 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, which is consistent with the experimental data of Drazer and Zanette. By using their data we can evaluate the numerical values of the proportionality coefficients in the linear energy relations. The theory suggests that experimental study of the temperature dependence of the fractal exponents helps to elucidate the mechanism of the adsorption-desorption process.