operation oaxaca: freundlich isotherm.

pre-lab jargon, extra (011)

Carlos Manuel Jarquín Sánchez

6 min readFeb 9, 2024

this is carlos.

this is part two of isotherms & models.

then, i’ll speak about pseudo-models,

and explain the problem of water in united states (this time).

as always, imma explain what the things i’m researching are & why it’s relevant for my thang.

AND!

because this information is a continuation of the previous one,

i recommend read the first one before this one.

operation oaxaca: langmuir models.

pre-lab jargon, extra (010)

cjarquin.medium.com

enjoy, imma just go dive into it.

research the freundlich isotherm (via scihub papers), and condense it.

then, i will speak about the magnesium/calcium water in usa. this will be a goody.

then the pseudo-models.

freundlich isotherm.

the old isotherm was known as langmuir model.

that one did this:

it assumes that adsorption occurs at specific homogeneous sites at the adsorbent surface.

adsorption → sticks to the surface only. does not get sucked in.

homogenous → “evenly distributed” in every direction. (aka, ur cup of coffee, no sugar, cream, milk)

but this one, the freundlich isotherm, says this:

freundlich isotherm is a model based on the adsorption of the adsorbate (metal ions) onto heterogeneous surfaces* (or heterogeneous spots on a surface).

*in this case, the heterogenous surface is the adsorbent surface… my mango peel.

heterogeneous → a mixture of components in a solution… and these components are not distributed evenly over the solution.

**solution is the milk. components are cereal flakes. and the flakes are not evenly distributed across the bowl.**

in langmuir models, the assumption was this:

langmuir assumes that the adsorbate molecules (our metal ions) adsorb onto the surface until a full, single layer (of these ions) is formed on the surface of the peel.

for freundlich models, the assumption is this:

on heterogeneous sites on the peel, the spots where stronger bonding can occur will get filled with ions first.
but as those sites get full, the tendency for the ions to bond to the peel decrease.

but how is it expressed, graphically & mathematically?

‘tis the plot.

on a graph, the x-axis will have the variable: ln/C(e)

and on the y-axis will look have the variable: ln/Q(e)

ln → natural log function, ln(x)

but why this?

and what does C,Q, & e mean?

we’ll find out by the equation used to define the freundlich model.

so what do these symbols mean?

note: the majority of defined terms have the word “equilibrium” in it. i’ll define that too (in terms of this context).

equilibrium → where the rate of adsorption (of metal ions) equals the rate of desorption.

q(e) → the amount of adsorbate that’s adsorbed when equilibrium is acheived, measured in milligrams per gram (mg/g)

K(f) → a freundlich constant, measured in (mg/g)

n → a freundlich constant.

C(e) → the concentration of the adsorbate under equilibrium conditions, measured in (mg/L)

but what do those freundlich constants give us in terms of our isotherm model?

the constant K(f) tells us how much of our adsorbate (ions) can be adsorbed onto the surface of the adsorbent (peel).

higher values of K(f) mean that there’s more adsorption occuring.
but with more adsorption, it implicitly means that the bonding between the adsorbate (ions) and the adsorbent (peel) are strong…
perhaps hydroxyl, carboxyl, or carboxylate groups are in play?

the constant n tells us the intensity* of the adsorption of the adsorbate (ions) onto the adsorbent (peel).

*this typically means “how fast is adsorption occuring”.

the value of n typically ranges between 0 < n ≤ 1.
but n > 1 is possible, and it means something too.
as n goes to one, we’re approaching an idealistic adsorption rate using freundlich’s model.
when n = 1, this is a linear adsorption rate. this is the ideal.
anything less than one creates a monolayer adsorption of the adsorbate (ions) onto the adsorbent (peel).

remember, a maximum monolayer is when the adsorbate molecules (our metal ions) adsorb onto the surface…

until a full, single layer (of these ions) is formed above the surface of the peel.

but what happens when n > 1?
we get a “cooperative adsorption”.
what is means is this:
the adsorption of ions on the peel allow the adsorption of additional ions/adsorbate.
this could be done via a multilayer adsorption or via chemical bonding.
BUT!
for peels, this won’t happen, i highly doubt it.
why?
how can a metal ion, that’s positively-charged, create another layer of positively-charged ions on top of each other?
like charges repel.

now, how did we get the nautral log’s on the graph?

the old formula is to just calculate it.

but it won’t suffice to put it on a graph.

we must convert this formula to into a linear equation “y = mx+ b”,

…so that we can plot it on the xy-plane.

step 1: take the natural log (ln) on both sides of this equation.

we would get:

since “q” is the variable we want to plot (from looking at our definitions of each symbol)…

rearrange the variables so it looks like this (in y = mx + b format):

final answer.

in terms of linear equation format:

ln(q) = y

1/n = m (slope)

ln(C(e)) = x (variable)

ln(K(f)) = b (constant, shifts up/down our output for adsorption)

and it makes sense.

n & K(f) were our freundlich constants.

and m & b (slope & y-intercept) are constants in a line-equation.

ex: y = 3x — 7 … m = 3 & b = -7
y-intercept is the value of the line when x = 0, btw.
try plugging x = 0.. answer will be y = -7

final stuff on isotherms.

cool.

these two: langmuir & freundlich, are the models we want to use when describing adsorption on an adsorbate and an adsorbent.

but which one should i use?

and which one makes more sense?

for my cases, the langmuir makes more sense to my peel & ions, by a mile.

why?

let’s see the general cases.

it’s better to use langmuir IF:

adsorbate molecules (ions) can form only one layer on the surface of the adsorbent (peel).
the adsorbent has homogeneous surfaces with evenly distributed sites.
langmuir assumes that once an ion has occupied an adsorption site, another ion can’t steal its place and/or stack on top of the ion.

this is not an exhastive list. imma just list the top three.

it’s better to use freundlich IF:

the adsorbent is heterogeneous and lacks even distribution of bonding sites.
allows multi-layer adsorption of the adsorbate onto the surface (of the peel).
if you want to find out how fast the adsorbate is being adsorbed and/or how much it can adsorb.
higher-complexity adsorption mechanisms (depending on the adsorbent).

without a doubt, langmuir > freundlich… with one caveat.

my peels cannot stack positively-charged ions on top of each other.
like charges repel.
so this rules out multi-layer.
mango peels tend to be homogeneous from one end to another.
there’s no extra peel on one side or the other.
and the easiest way for an ion to give up it’s bonding site is by increasing/decreasing the water’s pH level.
so langmuir fits all this criteria.

the caveat?

i still need to know the adsorption’s maximum,

and how fast it can adsorb a certain percentage of the ions in the water.

this can tell me if i am progressing on quick PLUS effective filtration.

so yes, i also need freundlich isotherm model.