[adsorption_help]

If you fit your kinetic (effect of contact time to adsorption capacity) data onto weber morris intraparticle diffusion model (WMID), wouldn't it always gurantee that there would be at least 2 linear sections?

The first one from t = 0, qt = 0 to the nearest data point (contact time). The second one is all the data after that.

Another thing im confused about, is do you include data where the adsorption is at equilibrium? because wouldn't this create potentially another linear section (although the R2 is quite bad since i think its just random error at this point since the system is already at equilibrium).

I have this kinetic data and I don't know how to determine how many linear segements (rate limiting steps) it should have (2 or 3)

[Corribus]

I'm afraid I am unfamiliar with this model. Could you provide more info?

[adsorption_help]

In the weber moris intraparticle diffusion model, you plot adsorption capacity (I use mg adsorbate/g adsorbent) at a certain time (qt) to the square root of time.

qt = k.t^0.5+c

and you can see that the data can show multiple linear segements. Each linear segements indicates a change in the rate limiting step of the adsorption process.

So for example if there are 2 linear segments one from 0-30 minutes, the other one from 30-60 minutes. The first linear segment is from 0-30 minutes the rate limiting step of the adsorption is bulk and/or film diffusion. After 30 minutes there is a change in the rate limiting step from bulk/film diffusion to intraparticle diffusion.

Here is an example
https://www.researchgate.net/figure/Weber-Morris-intraparticle-diffusion-plot-for-the-removal-of-hydroquinone-by-granulated_fig8_228325722
(keep in mind that the x axis is still in square root of time, so you need to square it to get the time in which the rate limiting step changes).

Another thing to note is that these adsorption process (bulk/film diffusion, intraparticle diffusion, adsorption onto active site) is all happening simultaneously, its just that at different stages of the adsorption the rate limiting steps differ since it is driven by diffusion (so its driven by the difference in concentration?)

[Corribus]

The equation has no linear segments. It is exponential.

Digging in a little further, it seems the standard practice is to plot your data on a sq. rt time (t^1/2) axis. In this case, if there is a single dominating interparticle diffusion step, the data should be linear. If there are multiple rate-limiting steps, you will have more than one linear region in your plot.

See,  for an example, Hwang et al. Phys. Chem. Chem. Phys., 2015,17, 21974-21981   https://pubs.rsc.org/en/content/articlelanding/2015/cp/c5cp03416g

I suppose if your rate constants are very close together, it may be hard to deconvolute these even on a (t^1/2) axis. Nevertheless, it is common to plot diffusion curves on a t^1/2 axis due to the frequency with which this shows up in solutions to the diffusion equation.