Rate of Diffusion-controlled Adsorption Processes - Stifel, George (My University, 1966)
Rate of adsorption data for gases on molecular sieve·s and coals have been interpreted using equations for unsteady state diffusion derived from Fick's law for spheres usually, ignoring the amount adsorbed and the shape of the adsorption isotherm. These inappropriate equations result in calculated diffusivities that are too low and activation energies that are too large.
Numerical. solutions of Fick's law were made for diffusion and adsorption in a porous sphere of radius R by finite difference methods for the following conditions: a. Diffusion is the rate-controlling step, and the diffusivity, D, is constant. b. Within an increment of the particle the total amount of adsorbate per unit volume, T is related to the "effective" concentration, c, by a Langmuir-like isotherm T = abC/(1 + bC). c. At zero tine the particle containing no adsomate is surrounded by adsozbate of concentration, Co, which remains constant throughout the rate process, and d. Equilibrium is established immediately at the periphery of the sphere.
The solutions are obtained in terms of Z = Q/Q and T=(DCo/QR^2) t = kt, where t is time, k is a constant equal to the term within the brackets, and Q and Q are the amounts adsorbed per unit volume at time t and at equilibrium. The quantity within brackets is also a valid expression for linear and Freundlich-like adsorption isotherms and probably holds for other isotherms. Plots of z as a function of T shift systematically as the parameter B = bCo increased from 0, corresponding to a linear adsorption isotherm, to large values; the value of Z at a given T increasing with increasing values of B. For B = 0 the numerical solution is identical with analytical solution for the linear adsorption isotherm which for values of z <0.87 is given by kt = (2/π) { (-1 - πZ/6) - (1 ~πZ/3) ^1/2 } where k = DCo/R^2Q. For large values of B the numerical solutions approach as a limit the parabolic law kt = (1/2) {(1- 2Z/3) - (1- Z) } The value of (1/k~) o.zidt~ at short times increases fran 3.385 for B = 0 to 4 .. 243 for very large values of B.. From experimental data the value of k derived using the equation for B = 0 is 1.56 larger than for the parabolic equation. Hence the values of D obtained from the initial linear portions of t.he rate curve change by only a factor of 1.56 when the type of isotherm is changed from linear to rectangular.
Rates of adsorption and the adsorption isotherm were determined for N2 , CH4, co2 , and C2H6 on samples of Linde 4A molecular sieve at several temperatures from -78° to +50°C in a manostatic volumetric aborption apparatus. The Langmuir equation satisfactorily approximatedthe isotherms and the values of B were moderately large at the lower temperatures of each series of experiments, eg., for N2 at -78°C, 10.6; for CH4 at -78°C, 7.3; for co2 at 0°C, 64; for Ci!6 at 0° and 30°C, 37 and 10.3.
The rate data plotted as Z against t^1/2 were not linear at short times but curved upward initially before becoming linear. The initial (, nonlinear portion persisted significantly longer than the brief uncertain period at the beginning of the experiment. This phenomena could result from the equilibration at the periphecy of the particles requiring a finite time rather than being instantaneous.
An equation based on the parabolic law model and a first order equilibration process was derived, which fits. the experimental data for 0.05 < Z < 0.95. This equation is appropriate only to data with a large value of B, but is probably a reasonable approximation for other rate data.
The rates of adsorption for different molecules were co2 > N2 > CH4 > c2H6 · The activation energies for the diffusivity were found to be 4.1 and 6.0 kcal./nole for methane and ethane. The _heats of adsorption were found to be 7.2 and 8.3 kcal/mole for methane and ethane.
Keywords: Diffusion-controlled Adsorption; molecular sieves; state diffusion; adsorption isotherm