Sorption isotherm

Adsorption isotherms of various substances at 20 ° C

A sorption isotherm describes the equilibrium state of sorption of a substance at constant temperature .

In addition, there are several prominent mathematically formulated adsorption isotherms in science. The parameters contained in these mathematically formulated adsorption isotherms can be obtained from measurement data by means of regression analyzes . Some mathematically formulated adsorption isotherms have arisen purely empirically and do not allow any statements about the underlying mechanisms and influencing variables. Others, however, are based on comprehensible kinetic or thermodynamic derivations; the parameters contained therein can provide state variables under suitable circumstances . Since sorption isotherms sometimes represent the sum of several, often not mathematically separable effects, their applicability to experimental data is not always given. Therefore, different adsorption isotherms can be found in the literature, each of which can be used under different boundary conditions.

The most commonly used isotherms are listed below. There are numerous other models, which are often modifications of the models mentioned. In the examples given, adsorption from solution is assumed and the amount adsorbed is expressed as a function of the concentration. If, on the other hand, the adsorbate is a gas, we replace the concentration of the adsorbate, C, with its partial pressure in the formulas. The dimension of the parameters used in the isotherms changes accordingly. The amount adsorbed is usually given in one of the following two sizes.

• the ratio of the adsorption amount to the amount of a hypothetical closed monolayer , ,${\ displaystyle \ Theta}$

or

• the ratio of the amount adsorbed to the amount of adsorbent, q

${\ displaystyle \ Theta}$is dimensionless, the dimension of q depends on which quantities we use to record the respective quantities. Ie q can be dimensionless or have the dimensions mass / volume , amount of substance / volume, or amount of substance / mass. One can convert to q by multiplying it by the product of the interfacial concentration of a closed monolayer of the adsorbate and the specific surface of the adsorbent ,,: ${\ displaystyle \ Theta}$${\ displaystyle \ Gamma _ {mono, max}}$${\ displaystyle S}$

${\ displaystyle q = \ Theta \ cdot \ Gamma _ {mono, max} \ cdot S}$

In the u. a. Formulas are used as a measure of the amount adsorbed , but in the literature, with the same justification, the left of the equal sign is used instead of and the terms to the right of the equal sign are left unchanged. Numerical values ​​and dimensions of the corresponding parameters change accordingly. ${\ displaystyle q}$${\ displaystyle \ Theta}$${\ displaystyle q}$

Linear isotherm

Henry's sorption isotherm with K H = 2.5
${\ displaystyle q = K_ {H} \ cdot C _ {\ text {eq}}}$
• q - sorbent load (mass of sorbate in relation to mass of sorbent)
• K H - Henry coefficient
• C eq - concentration of the sorbate in solution

Linear isotherms are very popular because they greatly simplify calculations. Therefore, they are often used when actually more complicated models should be used. They can mostly only be used for the range of low concentrations.

Linear isotherms are also called Henry's isotherms, especially for the sorption of gases in liquids . See also: Henry's Law .

Freundlich isotherm

Freundlich adsorption scheme. Active positions can be divided into more (red) or fewer (orange) active positions.
Freundlich isotherm with K F = 4, n = 3)
${\ displaystyle q = K_ {F} \ cdot C _ {\ text {eq}} ^ {1 / n}}$
• q - sorbent load (mass of sorbate in relation to mass of sorbent)
• K F - Freundlich coefficient
• C eq - concentration of the sorbate in solution
• n - Freundlich exponent

Freundlich isotherms take into account the fact that less sorbate can be absorbed when the sorption surfaces of the sorbent are heavily loaded. However, due to the increase in loading according to a power law , a complete loading of the surfaces cannot be mapped. This is practically the case for isotherms in which the saturation pressure of the adsorbent is comparatively high or cannot be reached ( superfluid media ). The Freundlich isotherm is a special form of the Zeldowitsch isotherm.

Langmuir isotherm

Langmuir adsorption scheme. The active sites (red) are equivalent and can only be occupied monomolecularly.
Langmuir isotherm with K L = 2
${\ displaystyle q = q_ {mono, max} \ cdot {\ frac {K \ cdot C _ {\ mathrm {eq}}} {1 + K \; C _ {\ mathrm {eq}}}}}$
• q - Relative coverage of the sorbent (number of adsorbed particles in relation to the number of adsorption sites)
• ${\ displaystyle q_ {mono, max}}$- the value that assumes when the adsorbent is covered with a closed monolayer of the adsorbate${\ displaystyle q}$
• K - Langmuir sorption coefficient
• C eq - concentration of the sorbate in solution

The Langmuir isotherm [ ˈlæŋmjʊə- ; according to the American physicist Irving Langmuir ] is the simplest adsorption model that describes complete adsorption on a surface. It is assumed that

• Adsorption takes place in a single molecular layer,
• all sorption sites are equivalent and the surface is uniform,
• there are no interactions between neighboring sorption sites and the adsorbed particles.

Frumkin isotherm

${\ displaystyle q = q_ {mono, max} \ cdot {\ frac {C _ {\ mathrm {eq}} \ cdot k \ cdot exp \ left (-b \ cdot q \ right)} {1 + C _ {\ mathrm {eq}} \ cdot k \ cdot exp \ left (-b \ cdot q \ right)}}}$

Brunauer-Emmett-Teller (BET) model

BET isotherms with K = 30, q max = 5 and c sat = 6.3
${\ displaystyle q = q _ {\ mathrm {mono, max}} \ cdot {\ frac {K {\ frac {C _ {\ mathrm {eq}}} {C_ {sat}}}} {(1 - {\ frac {C _ {\ mathrm {eq}}} {C _ {\ mathrm {sat}}}}) \ cdot \ left (1+ (K-1) \ cdot {\ frac {C _ {\ mathrm {eq}}} { C _ {\ mathrm {sat}}}} \ right)}}}$
• q - sorbent load (mass of sorbate in relation to mass of sorbent)
• K - sorption coefficient
• ${\ displaystyle q_ {mono, max}}$- the value that assumes when the adsorbent is covered with a closed monolayer${\ displaystyle q}$
• C eq - concentration of the sorbate in solution
• C sat - solubility of the sorbate

The BET model [according to Stephen Brunauer , Paul Hugh Emmett and Edward (Ede) Teller ] extends the Langmuir isotherm by the behavior at high sorbate concentrations close to solubility or saturation concentration . The model is based on the assumption that an adsorption site can bind several, up to an infinite number of molecules. The load can therefore increase to infinity. The model is used for BET measurement in surface chemistry .

Other models

Other well-known models for describing the equilibrium of adsorption and desorption are the model by Dubinin or Dubinin and Raduskevic based on the potential theory, the Toth isotherm, which in the simplest case can be converted into the Langmuir isotherm, and the model by Talu and Meunier, with which, for example, the adsorption of water on activated carbon can be described.

Derivation of the Langmuir isotherm

There is a certain number of binding sites on the surface. We assume that each of these binding sites can either not be occupied by a molecule or by one molecule.

Be it

• ${\ displaystyle A_ {total}}$ = the total surface area of ​​the adsorbent
• ${\ displaystyle A_ {1}}$ = the occupied surface of the adsorbent
• ${\ displaystyle A_ {0} = A_ {total} -A_ {1}}$ = the unoccupied surface of the adsorbent
• ${\ displaystyle \ Gamma _ {mono, max}}$ = the interfacial concentration of the adsorbed molecules when they form a closed monolayer
• ${\ displaystyle \ Gamma _ {1} = \ Gamma _ {mono, max} \ cdot {\ frac {A_ {1}} {A_ {total}}}}$ = the interface concentration of the adsorbed molecules,
• ${\ displaystyle \ Gamma _ {0} = \ Gamma _ {mono, max} - \ Gamma _ {1} = \ Gamma _ {mono, max} \ cdot {\ frac {A_ {0}} {A_ {total} }}}$
= the interface concentration of the free binding sites
• ${\ displaystyle p}$ = the partial pressure of the adsorbing gaseous substance
• ${\ displaystyle \ Theta _ {0} = {\ frac {A_ {0}} {A_ {total}}} = {\ frac {\ Gamma _ {0}} {\ Gamma _ {mono, max}}}}$
= the ratio of the unoccupied surface to the total surface
= the ratio of the number of free binding sites molecules to the total number of binding sites
• ${\ displaystyle \ Theta _ {1} = {\ frac {A_ {1}} {A_ {total}}} = {\ frac {\ Gamma _ {1}} {\ Gamma _ {mono, max}}}}$
= the ratio of the occupied surface to the total surface
= the ratio of the number of binding sites to which exactly one molecule is bound to the total number of binding sites.
• ${\ displaystyle \ Theta}$ = the ratio of the total number of bound molecules to the total number of binding sites.
= the ratio of the amount of adsorbed molecules to the amount of molecules in a closed monolayer.

The area balance results in:

${\ displaystyle \ Theta _ {0} + \ Theta _ {1} = 1}$

The mass balance results in:

${\ displaystyle \ Theta = \ Theta _ {1}}$

In the context of the Langmuir adsorption isotherm, values ​​between 0 and 1 can be assumed. ${\ displaystyle \ Theta}$

We describe the adsorption equilibrium using the law of mass action:

${\ displaystyle K = {\ frac {A_ {1}} {p \ cdot A_ {0}}} = {\ frac {A_ {1}} {p \ cdot (A_ {total} -A_ {1})} }}$
${\ displaystyle = {\ frac {\ Gamma _ {1}} {p \ cdot \ Gamma _ {0}}} = {\ frac {\ Gamma _ {1}} {p \ cdot (\ Gamma _ {mono, max} - \ Gamma _ {1})}}}$
${\ displaystyle = {\ frac {\ Theta _ {1}} {p \ cdot \ Theta _ {0}}} = {\ frac {\ Theta _ {1}} {p \ cdot (1- \ Theta _ { 1})}}}$

and solve for : ${\ displaystyle \ Theta _ {1}}$

${\ displaystyle \ Theta = \ Theta _ {1} = {\ frac {p \ cdot K} {1 + p \ cdot K}}}$

If you are interested in the ratio of the amount of adsorbate to the amount of adsorbent, we must do the above. Equation with the product of and the
specific surface area , multiply ${\ displaystyle q}$${\ displaystyle \ Gamma _ {mono, max}}$${\ displaystyle S}$

${\ displaystyle q = {\ frac {p \ cdot K \ cdot q_ {mono, max}} {1 + p \ cdot K}}}$

With: ${\ displaystyle q_ {mono, max} = S \ cdot \ Gamma _ {mono, max}}$

In practice, it is often determined from experimental adsorption isotherms by adjusting the parameters and K and the determination of and is dispensed with. ${\ displaystyle q_ {mono, max}}$${\ displaystyle S}$${\ displaystyle \ Gamma _ {mono, max}}$

Consider a adsorption from from a solution, we replaced the partial pressure in the above equation , by the concentration of adsorbing substance in the solution . The dimension of the equilibrium constant K changes accordingly${\ displaystyle p}$${\ displaystyle c}$

Derivation of the Frumkin isotherm

In contrast to the previous derivation, we assume that the binding constant K depends on and can be described by the following function: through appropriate substitution in the Langmuir isotherm we get: ${\ displaystyle \ Theta _ {1}}$${\ displaystyle K = k \ cdot exp \ left (-b \ cdot \ Theta _ {1} \ right)}$

${\ displaystyle \ Theta = {\ frac {p \ cdot k \ cdot exp \ left (-b \ cdot \ Theta _ {1} \ right)} {1 + p \ cdot k \ cdot exp \ left (-b \ cdot \ Theta _ {1} \ right)}}}$

Derivation of the Brunauer-Emmett-Teller (BET) isotherm

In contrast to the Langmuir isotherm, we assume that each binding site can bind any number of molecules. Let it be:

• ${\ displaystyle j}$ = the number of molecules bound to a binding site
• ${\ displaystyle \ Theta _ {j}}$ = the ratio of the number of binding sites to which exactly j molecules are bound to the total number of binding sites.

We describe the adsorption equilibrium with the law of mass action . We postulate a binding constant for the binding of the first molecule . The binding constant for further molecules is different from and not dependent on how many molecules are bound ${\ displaystyle K_ {1}}$${\ displaystyle K_ {2bis \ infty}}$${\ displaystyle K_ {1}}$

${\ displaystyle K_ {1} = {\ frac {\ Theta _ {1}} {p \ cdot \ Theta _ {0}}}}$
For  :${\ displaystyle j> 1}$
${\ displaystyle K_ {2} = {\ frac {\ Theta _ {j}} {p \ cdot \ Theta _ {j-1}}}}$

This results in:

${\ displaystyle \ Theta _ {1} = K_ {1} \ cdot \ Theta _ {0} \ cdot p}$
${\ displaystyle \ Theta _ {2} = K_ {2} \ cdot \ Theta _ {1} \ cdot p = K_ {1} \ cdot \ Theta _ {0} \ cdot K_ {2} \ cdot p ^ {2 }}$
${\ displaystyle \ Theta _ {3} = K_ {2} \ cdot \ Theta _ {2} \ cdot p = K_ {1} \ cdot \ Theta _ {0} \ cdot {K_ {2}} ^ {2} \ cdot p ^ {3}}$
${\ displaystyle \ Theta _ {j} = K_ {2} \ cdot \ Theta _ {j-1} \ cdot p = {\ frac {K_ {1}} {K_ {2}}} \ cdot \ Theta _ { 0} \ cdot {(K_ {2} \ cdot p)} ^ {j}}$

The area balance results in:

${\ displaystyle 1 = \ Theta _ {0} + \ sum _ {j = 1} ^ {\ infty} \ Theta _ {j}}$

and thus

${\ displaystyle 1 = \ Theta _ {0} + \ sum _ {j = 1} ^ {\ infty} \ Theta _ {j} = \ Theta _ {0} + \ sum _ {j = 1} ^ {\ infty} ({\ frac {K_ {1}} {K_ {2}}} \ cdot \ Theta _ {0} \ cdot {(K_ {2} \ cdot p)} ^ {j}) = \ Theta _ { 0} \ left (1 + {\ frac {K_ {1}} {K_ {2}}} \ cdot \ sum _ {j = 1} ^ {\ infty} {(K_ {2} \ cdot p)} ^ {j} \ right) = \ Theta _ {0} \ left (1 + K_ {1} \ cdot p \ cdot \ sum _ {j = 0} ^ {\ infty} {(K_ {2} \ cdot p) } ^ {j} \ right)}$

Knowing the solution formula for the geometric series

${\ displaystyle \ sum _ {j = 0} ^ {\ infty} x ^ {j} = {\ frac {1} {1-x}}}$

${\ displaystyle 1 = \ Theta _ {0} \ left (1 + {\ frac {K_ {1} \ cdot p} {1-K_ {2} \ cdot p}} \ right)}$

The mass balance results in:

${\ displaystyle \ Theta = \ sum _ {j = 1} ^ {\ infty} j \ cdot \ Theta _ {j}}$

and thus

${\ displaystyle \ Theta = {\ frac {K_ {1}} {K_ {2}}} \ cdot \ Theta _ {0} \ cdot \ sum _ {j = 1} ^ {\ infty} \ left (j \ cdot {(K_ {2} \ cdot p)} ^ {j} \ right)}$

We can map the series contained in the last term to the geometric series :

${\ displaystyle \ sum _ {j = 1} ^ {\ infty} \ left (jx ^ {j} \ right) = \ sum _ {j = 1} ^ {\ infty} \ left (x \ cdot {\ frac {d} {dx}} x ^ {j} \ right) = x \ cdot {\ frac {d} {dx}} \ sum _ {j = 1} ^ {\ infty} x ^ {j} = x \ cdot {\ frac {d} {dx}} \ left (x \ cdot \ sum _ {j = 0} ^ {\ infty} x ^ {j} \ right) = x \ cdot {\ frac {d} {dx }} \ left ({\ frac {x} {1-x}} \ right) = {\ frac {x} {{\ left (1-x \ right)} ^ {2}}}}$

${\ displaystyle \ Theta = \ Theta _ {0} \ cdot {\ frac {K_ {1} \ cdot p} {{\ left (1-K_ {2} \ cdot p \ right)} ^ {2}}} }$

Combining such as quantity and area balance, we get:

${\ displaystyle \ Theta = {\ frac {\ Theta _ {0} \ cdot {\ frac {K_ {1} \ cdot p} {{\ left (1-K_ {2} \ cdot p \ right)} ^ { 2}}}} {\ Theta _ {0} \ left (1 + {\ frac {K_ {1} \ cdot p} {1-K_ {2} \ cdot p}} \ right)}} = {\ frac {K_ {1} \ cdot p} {\ left (1-K_ {2} \ times p \ right) \ left (1 + K_ {1} \ times p-K_ {2} \ times p \ right)}} }$

This equation goes over into the Langmuir adsorption isotherm. The equation above has a pole at. This meant ${\ displaystyle K_ {2} = 0}$${\ displaystyle K_ {2} \ cdot p = 1}$

• If the pressure p approaches the value 1 / , the amount adsorbed becomes infinitely high. In other words, values ​​between 0 and 1 can be assumed within the Langmuir adsorption isotherm .${\ displaystyle K_ {2}}$${\ displaystyle \ Theta}$
• Values ​​of p> are not useful.${\ displaystyle K_ {2} = 0}$
• One can equate 1 / the equilibrium vapor pressure of a condensed 3-dimensional phase from the adsorbing substance ,,.${\ displaystyle K_ {2}}$${\ displaystyle p ^ {o}}$

Usually the BET isotherm is written in the following form:

${\ displaystyle \ Theta = {\ frac {K \ cdot {\ frac {p} {p ^ {o}}}} {\ left (1 - {\ frac {p} {p ^ {o}}} \ right ) \ cdot \ left (1 + {\ frac {p} {p ^ {o}}} \ cdot \ left (K-1 \ right) \ right)}}}$
with: and${\ displaystyle p ^ {o} = 1 / K_ {2}}$${\ displaystyle K = K_ {1} / K_ {2}}$

If we are interested in the ratio of the amount of adsorbate to the amount of adsorbent, we have to do the above. Equation with the product of and the
specific surface area , multiply ${\ displaystyle q}$${\ displaystyle \ Gamma _ {mono, max}}$${\ displaystyle S}$

${\ displaystyle q = q_ {mono, max} \ cdot {\ frac {K \ cdot {\ frac {p} {p ^ {o}}}} {\ left (1 - {\ frac {p} {p ^ {o}}} \ right) \ cdot \ left (1 + {\ frac {p} {p ^ {o}}} \ cdot \ left (K-1 \ right) \ right)}}}$

In practice, it is often determined from experimental adsorption isotherms by adjusting the parameters and K and using recognized values ​​for the gas used to determine the specific surface area of the adsorbent. At ${\ displaystyle q_ {mono, max}}$${\ displaystyle \ Gamma _ {mono, max}}$${\ displaystyle S}$

If we do not consider adsorption from the gas phase, but from a solution, we replace the partial pressure , by the concentration of the adsorbing substance in the solution, and the saturation vapor pressure, by the saturation concentration . ${\ displaystyle p}$${\ displaystyle c}$${\ displaystyle p ^ {o}}$${\ displaystyle c_ {sat}}$

Individual evidence

1. OECD Guidelines for the Testing of Chemicals / Section 1: Physical-Chemical properties, Test No. 106: Adsorption - Desorption Using a Batch Equilibrium Method, OECD Publishing, 2000, ISBN 978-92-64-06960-2 , doi: 10.1787 / 9789264069602-en .
2. a b P. W. Atkins : Physical Chemistry. Oxford University Press, 6th Edition, 1998, ISBN 0-19-850101-3 .
3. ^ SJ Gregg, KSW Sing: Adsorption, Surface Area and Porosity. Academic Press, London and New York, 1967
4. DM Young, AD Crowellm DM Young: Physical adsorption of gas. Butterworths, London 1962.
5. Herbert Freundlich : About the adsorption in solutions . In: Journal of Physical Chemistry . 57U, no. 1 , 1907, p. 385-470 , doi : 10.1515 / zpch-1907-5723 .
6. J. Zeldowitsch : adsorption site energy distribution. In: Acta Physikochim, USSR. No. 1, 1934, pp. 961-973.
7. ^ Irving Langmuir : Surface Chemistry. Nobel Lecture, December 14, 1932. In: Nobel Lectures, Chemistry 1922-1941. Elsevier Publishing Company, Amsterdam, 1966 ( PDF on nobelprize.org ).
8. ^ S. Brunauer, PH Emmett, E. Teller: Adsorption of Gases on Multimolecular Layers. In: J.Am.Chem.Soc. 60, No. 2, 1938, pp. 309–319 ( PDF file ( Memento of the original dated March 9, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. ).
9. Michail Michailowitsch Dubinin : The Potential Theory of Adsorption of Gases and Vapors for Adsorbents with Energetically Nonuniform Surfaces . In: Chemical Reviews . tape 60 , no. 2 , 1960, p. 235-241 , doi : 10.1021 / cr60204a006 .
10. Harald Jüntgen: Basics of Adsorption . Dust - cleanliness. Luft , , 36 (1976) No. 7, pp. 281-288
11. VDI 3674: 2013-04 Exhaust gas cleaning through adsorption; Process gas and waste gas cleaning (Waste gas cleaning by adsorption; Process gas and waste gas cleaning). Beuth Verlag, Berlin, p. 6