Langmuir adsorption theory. Open Library - open library of educational information Fundamental adsorption equation
Adsorption can be considered as the interaction of adsorbate molecules with active sites on the surface of the adsorbent. Adsorption processes are classified according to the type of interaction between the adsorbate and the adsorbent. There are physical (molecular) adsorption, chemisorption (chemical addition of a molecule atom) and ion exchange. This section deals mainly with the physical adsorption of gases and vapors.
Physical adsorption is characterized by the interaction of adsorbent and adsorbate due to van der Waals forces and hydrogen bonds: these adsorption forces provide attraction. At close range, short-range repulsive forces appear. Van der Waals forces include three types of interactions:
Orientation forces act between polar molecules with a dipole moment greater than zero. The interaction of dipoles depends on their mutual orientation, which gives the name dipole-dipole interaction forces. These forces are maximum when the dipole moments of the molecules are located along one line due to the fact that in this case the distances between unlike charges are smaller than between like charges. As a result, the attraction of the dipoles exceeds their repulsion. Thermal motion continuously chaotically changes the orientation of polar molecules, but the average value of the force over all possible orientations has a value that is not equal to zero.
Inductive forces arise from the interaction of polar and nonpolar molecules. A polar molecule creates an electric field that polarizes a non-polar molecule. As a result, there is a displacement of electrical charges that are evenly distributed throughout the volume of the molecule before interaction. As a result, a dipole moment is induced in a nonpolar molecule.
Nature dispersion forces London-van der Waals (1930) was fully understood only after the advent of quantum mechanics. Their occurrence is due to the fact that even neutral atoms are systems of vibrating charges, as a result of which the instantaneous value of the dipole moment of an uncharged molecule is greater than zero. The fluctuationally generated dipole creates an electric field that polarizes neighboring molecules. The energy of interaction between nonpolar molecules is the average result of the interaction of all possible instantaneous dipoles with the dipole moments that they induce in neighboring molecules due to induction.
Dispersion forces act between all atoms and molecules, since the mechanism of their appearance does not depend on the value of the dipole moment of the molecule. An essential feature of dispersion interactions is their additivity: for two volumes of condensed phases located at a distance h, the attraction of individual molecules is summed up.
Dispersion effect(London forces) manifests itself in its pure form between non-polar molecules. The corresponding forces arise due to the fact that fluctuations in the electron density in one atom induce similar fluctuations in a neighboring atom. The resonance of such fluctuations leads to a decrease in the total energy of the system due to the attraction of atoms. Such forces are of a general nature and can arise between any atoms, which determines their universality.
Orientation effect(Kiesom forces) dispersion interaction is enhanced in the presence of permanent dipoles in molecules, characterized by the manifestation of dipole-dipole interaction. The greater the dipole moments of interacting molecules, the greater the component of the orientation effect.
Induction effect(Debye forces) manifests itself during the interaction between polar and non-polar molecules, reflecting an increase in attraction due to the fact that a polar molecule induces a dipole in a non-polar molecule; this effect is more significant, the greater the polarizability of the molecules.
The total potential energy of two interacting atoms (molecules) is satisfactorily described by the Lennard-Jones equation:
U x c 6 x b 12
Where x is the distance over which the forces of attraction act; c is a constant that takes into account the effect of each component of the van der Waals forces; b – empirical constant
During adsorption, interaction occurs between an atom (molecule) of the adsorbate with the surface of the adsorbent, i.e., with a large number of atoms (molecules) that make up the adsorbent. Therefore, the dependence of the attraction energy during adsorption on distance is different than that described by the Lennard-Jones equation. This is explained by the fact that dispersion forces, which make the main contribution to the interaction, have the property of additivity. Therefore, if one atom interacts with a system of atoms of 2, 3, 4, etc. atoms, then the energy of interaction is, respectively, 2, 3, 4, etc. times greater than the energy of two interacting atoms. Thus, in order to calculate the interaction energy during adsorption, it is necessary to sum the interaction energies of the adsorbed atom with each adsorbent atom.
U cn
6x3
This dependence indicates a slower decrease in the energy of attraction during adsorption and the long-range action of adsorption forces. The equation was used by London, and then by other scientists, to experimentally prove the dispersion nature of adsorption forces and the connection of adsorption energy with the properties of adsorbed molecules and the adsorbent. The total potential interaction energy during adsorption can be expressed by the equation
6x3 |
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m is the distance of atom A from individual atoms of the adsorbent. One of the important practical conclusions when considering the nature of adsorption
interaction is the conclusion about significantly better adsorption of substances in cracks and pores, when dispersion interaction is predominantly manifested, since there are a larger number of atoms of the solid body near the adsorbed molecule. If the electrostatic contribution to the adsorption interaction is significant, then in the cracks and pores the positive and negative charges compensate each other and the greatest potential appears on the protrusions, where adsorption will predominate, especially during the formation of hydrogen bonds (adsorption of water, methyl alcohol, etc.). In addition, the greater the number of atoms an adsorbate molecule has, the more energy it will be attracted to the adsorbent.
Henry's Law
Let us consider the distribution of substances between the bulk phase and the surface layer, and in particular during adsorption at the liquid-gas or liquid-liquid interface, when the activities of individual sections of the adsorption field are automatically equalized. The surface of solids, as a rule, is heterogeneous geometrically (porosity) and chemically, and in order to obtain the simplest laws of adsorption, it is necessary to assume that the surface of the adsorbent is homogeneous and the distribution of the adsorbate occurs in a monomolecular layer. If porosity is represented as a separate phase, then the process of substance redistribution can be considered as an equalization of the chemical potentials of the distributed substance in the adsorption layer and in the bulk phase
where μ0 and μ0 are the chemical potential of the distributed substance in the adsorption layer and in the bulk phase; a and a are the activities of the distributed substance in the adsorption layer and in the bulk phase; K is the Henry distribution constant, independent of concentration.
For non-electrolytes
where and are the activity constants of the distributed substance in the adsorption layer and in the bulk phase; D – distribution coefficient
Figure 11 – Dependence of the adsorption value on concentration (pressure)
Since in an infinitely dilute solution the activity coefficients are equal to unity, based on the equation the following pattern can be formulated: when the system is diluted, the distribution coefficient tends to a constant value equal to the Henry distribution constant. This is Henry's law. Regarding the magnitude of adsorption A, this law will be written as follows:
With , |
AK" |
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For an ideal gas KG = KG ’ RT
The equations represent adsorption isotherms of a substance at low concentrations. When adsorption on solid adsorbents, the range of action of this law is small due to the heterogeneity of the surface. But even on a homogeneous surface, with increasing concentration of the substance or vapor pressure, a deviation from the linear dependence is detected. This is due to the fact that, for example, with positive adsorption, the concentration of a substance in the surface layer increases faster than its increase in the bulk phase, and therefore the activity coefficients of the adsorbate on the surface of the adsorbent begin to deviate from unity earlier. At low concentrations of the distributed substance, deviations are caused mainly by the relationship between the interactions of molecules with each other and with the surface of the adsorbent. If the cohesive interaction of the adsorbate is greater, then the deviation from Henry’s law is negative - the activity coefficients are less than one (positive deviation from Raoult’s law), and the distribution coefficient increases (curve 1); if the adsorbate-adsorbent interaction is stronger, then the deviation from Henry’s law is positive (negative deviation from Raoult’s law) and the distribution coefficient decreases (curve 2). With a further increase in the concentration of the substance or vapor pressure, the free surface of the adsorbent decreases; which entails a decrease in its reactivity, expressed in an increase in the activity coefficients of the adsorbate on the surface of the adsorbent.
Monomolecular adsorption. Langmuir adsorption isotherm
Langmuir's theory was a fundamental contribution to the theory of adsorption. This theory makes it possible to take into account the strongest deviations from Henry’s law associated with the limited adsorption volume or surface of the adsorbent. The limitation of this parameter leads to adsorption saturation of the adsorbent surface as the concentration of the distributed substance increases. This position is fundamental in Langmuir’s theory and is clarified by the following assumptions:
1. Adsorption occurs at discrete adsorption centers, which can be of different nature.
2. During adsorption, a strictly stoichiometric condition is observed - one molecule is adsorbed on one center.
3. Adsorption centers are energetically equivalent and independent, that is, adsorption on one center does not affect adsorption on other centers.
4. The adsorption process is in dynamic equilibrium with the desorption process. The first position means that the adsorbed molecules are tightly bound to
adsorption centers; they are, as it were, localized at the centers (localized adsorption). From the second position it follows that only one adsorption layer can form on the surface, therefore Langmuir adsorption is called monomolecular. The third position means that the differential heat of adsorption is constant and that the interaction forces of adsorbed molecules can be neglected. And finally, according to the last position, adsorbed molecules, due to energy fluctuations, can be detached from the centers and return to the gas phase.
Based on these provisions, the adsorption isotherm equation can be obtained. The rate of adsorption from the gas phase Vadc (that is, the number of molecules adsorbed per unit time) is proportional to the gas pressure and the number of free centers on the surface of the solid. If the total number of centers is A, and during adsorption it turns out to be occupied by A centers, then the number of centers remaining free is equal to (A - A). Therefore V adc = k adc. p (A - A). Adsorption is dynamically balanced by the desorption process. The desorption rate is proportional to the number of adsorbed molecules V des = k des. A . At equilibrium, V adc = V des or k adc. p (A - A) = k des. A . Redesignating k ads / k des = K (where K is the adsorption equilibrium constant) and A/A = . (relative surface filling) we obtain
A A Kc 1 Kc
The equation is called equation Langmuir adsorption isotherms.
It should be noted that the Langmuir adsorption equilibrium constant characterizes the energy of interaction of the adsorbate with the adsorbent. The stronger this interaction, the greater the adsorption equilibrium constant. The Langmuir adsorption equation is often presented in relation to the degree of surface coverage - the ratio of the amount of adsorption to the capacity of the monolayer.
The expressions correspond to Henry's law: the adsorption value increases linearly with increasing concentration. Thus, the Langmuir equation is a more general relation, including the Henry equation. At high concentrations and pressures, when Kc> 1 and Kp > 1, the equations turn into the relations
A A and |
The ratios correspond to saturation, when the entire surface of the adsorbent is covered with a monomolecular layer of adsorbate.
According to the principle of independence of surface tension, which was introduced by Langmuir, the value of the limiting adsorption a ∞ is the same for all members of the homologous series, that is, it does not depend on the length of the hydrocarbon chain, but is determined only by the cross-sectional area of the molecules. This statement becomes clear if we consider the structure of the surface layer when it is completely filled. In this case, amphiphilic molecules can be located in the surface layer in the only possible way, when the hydrophilic parts of the molecules
are on the surface of the water and are tightly adjacent to each other, and the hydrophobic radicals are oriented towards the air (the so-called “Langmuir palisade”, which was already mentioned above).
Therefore, if the limiting adsorption is the number of moles of surfactant that completely occupies a unit of surface, then the reciprocal of the limiting adsorption
adsorption, will give the total cross-sectional area of one mole of molecules, then:
To find the length of a molecule, in addition to S of the molecule, it is necessary to know its volume:
Then |
V molecules |
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molecules |
S molecules |
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where M is the molar mass of the surfactant, ρ is the density of the surfactant, δ is the length of the surfactant molecule. Experimental results for determining the adsorption isotherm are usually
processed using the Langmuir equation written in linear form:
This linear dependence allows us to graphically determine both constant parameters of the adsorption isotherm.
When gases are adsorbed from their mixtures in accordance with the Langmuir isotherm equation, the adsorption values are summed up, and the concentration of free centers is common for the equilibrium multicomponent system.
K i p i |
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1 K i p i |
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An increase in the partial pressure of one component suppresses the adsorption of others, and the greater the adsorption equilibrium constant, the stronger it is.
Real surfaces of solids, as a rule, do not have energetically equivalent active centers. A significant approximation to real conditions is to consider the possible energy distributions of adsorption centers of the adsorbent surface. Having accepted the linear distribution of adsorption centers by energy (heat of adsorption), M. I. Temkin, using the Langmuir equation, obtained the following equation for the average degrees of filling of the adsorbent:
1 ln K 0 p
where is a constant characterizing the linear distribution; K0 is a constant in the Langmuir equation, corresponding to the maximum heat of adsorption.
The Langmuir equation can only be used in the absence of adsorption of a substance beyond the monomolecular layer. This condition is satisfied quite strictly during chemisorption, physical adsorption of gases at low pressures and temperatures above critical (in the absence of condensation on the surface of the adsorbent), and often during adsorption from solutions. The indicated restrictions for the application of the Langmuir equation are associated not so much with the formal
The founder of the monomolecular model of physical adsorption was Langmuir. In accordance with the concepts he developed, the adsorption layer consists of molecules localized on the exponential surface that do not interact with each other (there are no lateral - side - interactions)
At equilibrium pressure P and constant temperature T, the number of adsorbed molecules (per unit area of the adsorbent) is expressed by the number of moles ns or molecules Ns in the adsorption layer:
where N A is Avogadro's number.
This dependence is called the adsorption isotherm.
Adsorption is often expressed in terms of the volume equivalent to n s moles of adsorbate at normal temperature and pressure V s (cm 3). Therefore, the adsorption isotherm can be expressed as
Or . (1.3.5)
To describe an adsorption layer that is monomolecular at low degrees of filling, they resort to the model of a two-dimensional gas, and when the adsorption layer is saturated, it is considered to be a two-dimensional liquid. In this case, the equation of state of a two-dimensional gas is assumed to be valid for the unsaturated adsorbate layer
where is surface pressure; – area occupied by one mole of adsorbed substance; G – Gibbs adsorption (surface excess); s 0 , s – surface tension at the interface before and after adsorption, respectively.
Writing the Gibbs equation
and introducing the replacement , we get
Let us introduce the degree of filling of the adsorption layer
where is the limiting amount of adsorbate when the adsorption layer is saturated.
Multiplying the numerator and denominator of the right side of equation (1.3.8) by , we obtain
The chemical potentials of molecules in the adsorption layer and in the equilibrium gas phase must be equal, therefore
Integrating equation (1.3.12) provided that at P = 0, taking into account that P = s 0 - s, and denoting A m = A/n s m
we get
The equation of state of a two-dimensional ideal gas (1.3.6) can be rewritten in the form
Differentiating and comparing equations (1.3.14) and (1.3.15), we obtain
or where from
where B is a constant. Equation (1.3.17) is known as Henry's equation. Taking into account the intrinsic sizes of adsorbate molecules A 0 in the equation of state of a two-dimensional gas is carried out as
In general, the equation of state of a two-dimensional gas can be written as
Using the approach described above in deriving the Henry equation gives a generalized ideal model isotherm function, which is expressed by the Langmuir equation
Given equation (1.3.18), using the above approach leads to Volmer's equation
Introduction of a two-dimensional analogue of the van der Waals equation
where a and b are constants, gives the adsorption isotherm known as the Hill-de-Boer equation
Langmuir derived the above equation (1.3.21) based on consideration of the dynamic equilibrium between the adsorption layer and the volume of the gas phase.
From the kinetic theory of gases it follows that the number of gas molecules colliding per unit time with a unit area of the adsorbent (diffusion flow) will be
Considering that adsorption cannot occur if the molecule is in contact with an already occupied active site, only part of the j molecules from the diffusion flow will be captured by the adsorbent, the remaining molecules will be reflected. If the fraction of occupied active centers is Q, then the adsorption rate can be expressed as
The desorption rate will be
At equilibrium
where k a and k d are constant rates of adsorption and desorption processes.
Substituting the expression for i from (1.3.25) into equation (1.3.29), we obtain
where K is the equilibrium constant of the adsorption process, Q = Г/Г m.
A comparison of equations (1.3.30) and (1.3.21) shows that the integration constant is the inverse of the adsorption equilibrium constant, i.e. B = 1/K.
The Langmuir equation is valid only for an ideal localized monolayer, which excludes lateral interactions between molecules adsorbed on neighboring centers.
It is obvious that such interaction should manifest itself in real systems and statistical-thermodynamic consideration leads to the Fowler-Guggenheim equation
where e is the energy of lateral interaction.
Equation (1.3.33) gives isotherms almost identical to Hill-de-Boer equation (1.3.24). There are certain critical conditions under which lateral interaction begins to appear, i.e. two-dimensional condensation occurs and the Langmuir equation becomes inapplicable. In the case of the Fowler-Guggenheim isotherm, the critical conditions correspond to KP = 0.1353 and Q = 0.5; e/kT= 4.
All of the above adsorption isotherms based on the monomolecular model of the adsorption layer assume the exponentiality of the surface, which is considered homogeneous. In real systems, surfaces are heterogeneous, this is especially true for fibers of polymeric materials.
Adsorption. Lagmuir and Freundlich adsorption isotherms. BET equation and its analysis.
The surface energy tends to decrease spontaneously. This is reflected in a decrease in the interfacial surface or surface tension (s).
As a result of this tendency, adsorption occurs.
Adsorption– the process of spontaneous redistribution of system components between the surface layer and the bulk phase. That is, adsorption can occur in multicomponent systems; the component that most strongly reduces the surface tension passes into the layer. In the general case, adsorption can be the result of the chemical interaction of components with the surface - chemisorption.
Basic concepts.
Adsorbent– the phase that determines the shape of the surface; it is more dense and can be solid or liquid.
Adsorbate– a substance that is redistributed (gas or liquid).
Desorption -transition of a substance from the surface layer to the bulk phase.
To quantitatively describe adsorption, two quantities are used. One is measured by the mass of the adsorbent, that is, the number of moles or grams per unit surface area or per unit mass of the adsorbent. This value is designated “A” (the “finite thickness layer” method). Another characteristic of the adsorption value is determined by the excess of the substance in the surface layer compared to its amount in the same volume of the phase, also per unit area or unit mass of the adsorbent.
Physico-chemical classification.
1. physical (molecular),
2. chemisorption,
3. ion exchange.
The most widespread is physical adsorption.
During physical adsorption, the interaction between the adsorbate and the adsorbent occurs due to van der Waals forces and hydrogen bonds. Van der Waals forces include three types of interaction: dipole-dipole, induction and dispersion. For all types of interaction, one law of changing the energy of attraction depending on the distance between atoms is satisfied.
Surface phenomena.
Adsorption is a change in the concentration of a substance at the interface compared to the volume. This term also denotes the absorption process and the amount of absorbed substance G per unit surface area or mass of the adsorbent (mmol/m2 or mmol/g).
Adsorption occurs with the release of energy, therefore, this process is spontaneous.
Adsorbent is a substance on the surface of which adsorption occurs.
Adsorbate is an adsorbable substance.
Physical adsorption – adsorption caused by the forces of intermolecular interaction (usually reversible).
Chemisorption is the absorption of gases, vapors or dissolved substances by solid or liquid absorbers, accompanied by the formation of chemical compounds.
Heat of adsorption is the heat per mole of a substance that is released during its adsorption.
Adsorption is an exothermic process ( Q >0) . With constant adsorption(G, Q = const):
, 
.
Q value is an indirect criterion for determining the type of adsorption: if Q <30-40 кДж/моль – физическая адсорбция, если Q >40 kJ/mol – chemisorption.
Adsorption isotherm is a functional dependence of the amount of a substance adsorbed on the surface on the pressure or concentration of this substance in another phase Г = f (p) T = const, Г= f (с) T = const .For monolayer localized adsorption on a homogeneous surface Г= f(p ) is described by the Langmuir isotherm.
Langmuir equation
The absorption isotherm equation (1916–1918) was obtained based on the following assumptions:
1.) the surface of the adsorbent is energetically homogeneous, i.e. adsorption of molecules at any site occurs with the same thermal effect
2.) there is no interaction between adsorbed molecules, i.e. molecules cover the adsorbent only with a monomolecular layer. Maximum adsorption is observed when the entire surface is covered with a multimolecular layer
3.) adsorption is reversible, i.e. thermodynamic equilibrium is restored between the adsorption layer and the gas (liquid) phase.
At equilibrium, the adsorption rate V hell must be equal to the desorption rate V dec.
V hell = V des
In order for a molecule to be adsorbed, it must hit the surface and land on an unoccupied site. Since the number of impacts is proportional to the concentration C, and the probability of getting into an unoccupied place is proportional to the number of unoccupied places, then
V adc = k 1 C (1- q),
where k 1 – adsorption rate constant.q- share of occupied places, 1–q– share of unoccupied places.
A molecule is desorbed when its energy is sufficient to detach itself from the surface. The number of such molecules is proportional to the number of adsorbed molecules, therefore
V des = k 2 q,
where k 2 – desorption constant.
k 1 C (1- q) = k 2 q,k 1 C – k 1 q C= k 2 q,k 1 C = q(k 2 + k 1 C)
from here , divide the numerator and denominator by k2.
Bernoulli's equation
where b = k 1 / k 2.
If the number of places on the adsorbent is equal z, then adsorption Г= z qand the isotherm equation will be
(1) Langmuir equation.
Let's explore this equation:
1.) adsorption is small: either small k 1 , or C is small, then bC<<1. Г= zbC = , where is Henry’s constant, i.e. Langmuir's equation transforms into Henry's equation, therefore the adsorption isotherm must first be a straight line (Fig. 1).
Fig.1Fig. 2
2.) Adsorption is high: bc >> 1, then Г= z , i.e. limiting adsorption occurs ¥ . The ratio is called the degree of surface coverage. The Langmuir equation can be reduced to linear form (Fig. 2):
(2) or .
The segments and slopes of these straight lines cut off on the coordinate axis make it possible to determine the constants of the Langmuir equation z and b . However, the Langmuir equation does not interpret adsorption data satisfactorily. Deviation from Langmuir's theory is the result of a non-uniform surface, which is characterized by the presence of unequal adsorption centers with different affinities for the adsorbed substance. If the surface is energetically inhomogeneous, use the empirical Frendlich equation
where x is the amount of adsorbed substance,
m – mass of the adsorbent,
C is the equilibrium concentration after adsorption,
k, n – constants (approximation parameters).
Constant k – represents the amount of substance adsorbed by 1 g of adsorbent at C = 1 mol/liter. For each adsorbative k has its meaning for the same adsorbent, i.e. it characterizes the ability of a given adsorbate to be adsorbed by a specific adsorbent
(4)
where n is the slope of the straight line, and k – antilogarithm of a straight line segment. The Freundlich equation can be derived by assuming that the surface is energetically inhomogeneous and that adsorption on each type of adsorption center obeys the Langmuir equation. Then the constant k corresponds to the adsorption equilibrium constant, and n – degree of aggregation. According to the Freundlich equation, the amount of adsorbed substance increases indefinitely with increasing concentration and pressure, so this equation is not satisfactory for high surface coverages.
BET theory
For multilayer adsorption, the adsorption isotherm is described by the BET (Brunauer, Emmett, Teller) equation. They assumed that there are homogeneous localized adsorption centers on the surface of the adsorbent and that adsorption on one center has no effect on adsorption on neighboring centers, as in Langmuir's theory. They further suggested that molecules could be adsorbed in the second, third and n -th molecular layers, and the available area for molecules n th layer is equal to the area covered ( n -1) layer.
where ps – saturated vapor pressure of the adsorbate,
p – adsorbate pressure in another phase.
A distinctive feature of vapor adsorption is the transition to volumetric condensation at a limiting pressure equal to the saturated vapor pressure of the liquid, p = ps . The purpose of this equation is to find G ¥ with which you can calculate the accessible surface of the adsorbent.
To describe the adsorption process, in particular monomolecular adsorption, in addition to the fundamental Gibbs adsorption equation, a number of other analytical equations are used, which are named after their authors.
When the adsorbent is insignificantly filled with adsorbate, the ratio of the concentration of substances in the adsorption layer and in the volume tends to a constant value equal to G. This pattern can be expressed analytically as follows:
G(A) = to G s. (4.24)
Equation (4.24) characterizes the adsorption isotherm at low concentrations of the adsorbent (Fig. 4.5, section I) and is an analytical expression of Henry’s law. The kGhe coefficient depends on the concentration and is a distribution constant that characterizes the distribution of a substance in the adsorption layer in relation to its content in the bulk phase. Equation (4.24) obtained on the basis of Henry’s law and the corresponding linear dependence of adsorption on concentration in the initial section of the adsorption isotherm (section I) is observed only approximately, but this approximation is sufficient for practice.
In a more general form, the dependence of adsorption on the concentration of the adsorbent can be determined using the Freundlich equation
Г(А) = кс 1/n, (4.25)
where k, n are coefficients.
This equation was obtained based on the results of processing experimental data on surfactant adsorption. The coefficient k is numerically equal to the adsorption value when the concentration of the adsorbent, in this case the surfactant, is equal to unity (c = 1, k = G). The coefficient n characterizes the difference between the section of the adsorption isotherm (see Fig. 4.5, section II) and the straight line.
The coefficients of the Freundlich equation are easy to determine graphically. To do this, we take the logarithm of equation (4.25):
logГ(A) = logк + (1/n)logc. (4.26)
The relationship between lgГ and lgc (Fig. 4.6) is characterized by a straight line, the tangent of the angle of inclination of which is equal to 1/n, and the segment cut off on the ordinate axis is lgк.
Let us emphasize once again that Henry’s law and the Freundlich equation characterize absolute adsorption (A). However, given that the values of absolute and excess adsorption (G) are practically the same, the difference between them can be ignored.
The analytical expression of adsorption depending on the concentration of the adsorbent in the form of an adsorption isotherm is given in the Langmuir theory. The theory is based on kinetic concepts of the adsorption process, which determine the rates of adsorption and desorption under equilibrium conditions.
Let us schematically represent a unit area (for example, 1 m2) of the adsorption layer at the interface (Fig. 4.7). If an adsorbate molecule occupies area B0 in the surface layer, and the number of its molecules is n, then nB0 is the area that accounts for all molecules per unit area of the adsorption layer. The surface free from adsorbate molecules is equal to (1–nВ0); the free area determines the possibility of subsequent adsorption.
The dynamic nature of adsorption suggests the possibility of desorption of part of the substance from the adsorption layer with area nB 0. Desorption rate v d is proportional to this area and is determined by the equation
v d = k d nB 0 . (4.28)
In equations (4.27) and (4.28), k a and k D are the rate constants of adsorption and desorption.
Under equilibrium conditions, the rates of the forward and reverse processes are equal. On this basis, from equations (4.27) and (4.28) it follows
where b is the equilibrium constant of the adsorption process.
The equilibrium constant b is related to the standard value by the Gibbs energy as follows:
ΔG 0 = RT lnb; .
Let us carry out auxiliary transformations of equation (4.29) and express the number of adsorbate molecules:
In the case of limiting adsorption, the entire area of the interface is occupied by adsorbed molecules (see Fig. 4.7, b). In relation to the chosen unit of area, this can be expressed as follows:
n ∞ B 0 = 1, (4.31)
where n ∞ is the number of molecules in the saturated adsorption layer.
An unsaturated adsorption layer, unlike a saturated one, is not completely occupied by adsorbate molecules. The degree of saturation θ of the adsorption layer can be represented in the following form:
θ = n/n∞. (4.32)
During the adsorption process, the degree of saturation varies in the range 0< θ ≤ 1.
The number of molecules in the unsaturated n and saturated n ∞ adsorption layers can be expressed through adsorption Г(А):
n = Г(А)NA; n ∞ = Г ∞ (А ∞)N A , (4.33)
where N A is Avogadro’s number.
In equation (4.30) we substitute the values of n, n ∞ and B 0 according to formulas (4.31) and (4.33); Then
This is the Langmuir equation. The quantity b included in it, in accordance with condition (4.29), is the adsorption equilibrium constant.
Note that in accordance with equality (4.33), the absolute number of molecules in the adsorption layer, and therefore the absolute adsorption, was determined; but, as already noted, the large excess of molecules in the surface layer compared to their content in the volume allows us to use relation (4.3). Therefore, in formulas (4.33) and (4.34), adsorption is designated as Г(А).
Let us analyze the Langmuir equation (4.34) and compare it with the Henry equations (4.24) and Freundlich equations (4.25). At the beginning of the adsorption process, when c → 0 and 1>>bc, in accordance with equation (4.34) Г(А) = Г∞(А∞)bс. The product Г∞(А∞)b is a constant value, which corresponds to the coefficient to Г of Henry’s law, i.e. section I of the adsorption isotherm (see Fig. 4.5). The Freundlich equation is valid only for the middle part of the adsorption isotherm (section II). As с → ∞, bc >> 1, it follows from equation (4.34) that Г = Г∞; this corresponds to section III of the adsorption isotherm. Thus, the Langmuir equation determines all parts of the monomolecular adsorption isotherm, including the limiting adsorption.
In fact, the adsorption mechanism is more complex than it is shown in Fig. 4.3; This is confirmed by large deviations of experimental data from theoretical calculations. The surface of solid adsorbents, as a rule, is geometrically, energetically and chemically inhomogeneous; the adsorbent may have a complex composition, and the rate of adsorption at different points on the surface is not the same.
Adsorption is one of the most important and widespread surface phenomena. Based on adsorption, numerous methods are used to purify gases and liquids from harmful impurities, remove moisture, separate mixtures of substances and separate certain components from complex mixtures, as well as many other technological processes. The use of adsorption in industry will be discussed in Chapter. 6.
Exercises
1. What part of the absolute adsorption is excess adsorption if, as a result of adsorption, the adsorbate concentration increased 17 times?
According to the conditions of the problem, the concentration of the adsorbate in the adsorption layer is with B = 17c.
Based on equalities (4.1) and (4.2), excess adsorption can be determined:
G = A – ch = c in h – ch = (c in – c)h.
Ratio of excess and absolute adsorption
Excess adsorption is 0.941 parts, or 94.1% of absolute adsorption.
2. How does adsorption, expressed in mol/m 2 and mol/kg, relate to a powder with a particle diameter of 70 μm and a density of 1.25∙10 3 m 3 /kg?
Let us use formulas (1.1), (1.4) and (4.4):
3. The solid was placed in a gaseous environment. The chemical potential of a substance in the bulk phase of a gaseous medium μ i V is less than the chemical potential on the surface of a solid body μ i V. What process will occur - adsorption or desorption?
Since μ i V< μ i В, самопроизвольно будет протекать десорбция.
4. The binding energy between the adsorbate and the adsorbent is 215 kJ/mol. What type of adsorption takes place?
A high binding energy indicates that chemisorption is occurring.
If we consider the dynamic picture of adsorption, then its value will be greater, the greater the number of impacts of gas molecules on the surface (i.e., the greater the gas pressure) and the longer the time the molecule remains on the surface from the moment of impact until the moment of its transition back to the gas phase .
Therefore, according to de Beer, the adsorption value is:
a=n avg ∙τ (2.4)
where n cf is the average number of molecules hitting the surface per unit time, τ is the average time the molecules stay on the surface.
This formula assumes that each impact of a molecule is accompanied by its retention on the surface, regardless of whether there are other molecules already on it or not. In fact, a molecule hitting an already occupied site may be reflected back into the gas phase or be delayed. Taking these circumstances into account would require introducing a dependence on the surface occupancy, i.e. the proportion of its coverage with previously adsorbed molecules. That's why first simplifying position The model under consideration is that any molecule colliding with a surface is adsorbed on it, regardless of the presence of other molecules on the surface. Obviously, this assumption closely corresponds to the case of very low concentrations of adsorbed molecules, when, indeed, almost every molecule ends up in a free place and the probability of each falling into an occupied place is negligible.
Of course, the residence time of a molecule on the surface should depend on the adsorption energy. Molecules that find themselves in places where this energy is greater will remain on the surface longer, waiting longer for their “hour” when fluctuations in the surface energy push them back into the gas phase. Taking into account energy heterogeneity, however, would greatly complicate the description of adsorption. That's why second simplifying assumption consists in the assumption of surface homogeneity.
Using the theoretical principles of the kinetic theory of gases under the specified assumptions, the Henry adsorption isotherm equation was obtained:
a = K∙P, (2.5)
where K is Henry’s constant, depending on the Avogadro number, molecular weight, gas constant, absolute temperature and other quantities that are considered constant according to accepted assumptions; P - gas pressure.
The Henry equation constant K (the tangent of the straight line) depends on the temperature and energy of the adsorbate-adsorbent interaction. The lower the temperature and the greater the interaction of adsorbed molecules with the surface of the adsorbent, the greater the K, the steeper the adsorption isotherm.
The equation means that in this ideal model the amount of adsorption is directly proportional to the pressure of the vapor or gas. This dependence received this name by analogy with Henry’s law, known in physical chemistry, according to which the volume of gas dissolved in a solid or liquid is proportional to its pressure.
In accordance with this equation, Henry's law can be formulated: the amount of adsorption at low gas pressures (low concentrations of a substance in solution) is directly proportional to the pressure (concentration).
So, according to the accepted assumptions, the Henry isotherm should describe the experimental data obtained at small fillings on homogeneous surfaces.
The first assumption is justified when studying adsorption at very low pressures. As for the second, adsorption is almost always measured on heterogeneous surfaces. However, adsorption at very low pressures corresponds to very low degrees of coverage. This means that everything depends on how heterogeneous not the entire surface is, but only a small fraction of it, covered at low pressures. In real conditions, during adsorption on solids, the range of action of the law is small due to the heterogeneity of the surface, but even on a homogeneous surface, with increasing concentration, a deviation from the law is detected. At low concentrations of distributed matter, deviations are mainly due to the relationship between the interaction of molecules with each other and with the surface of the adsorbent.
