Adsorption- concentration of a substance from the volume of phases on the interface between them. Adsorption can be seen as absorption substance (adsorbate) surface of the adsorbent.
Adsorbent The substance on whose surface adsorption takes place.
Adsorbtiv - a gas or solute capable of being adsorbed on the surface of an adsorbent.
Adsorbate - adsorbed substance on the surface of the adsorbent. Often the concepts of "adsorbent" "adsorbate" are identified
Distinguish physical adsorption, occurring without chemical change of the adsorbate and chemical adsorption(chemisorption), accompanied by chemical interaction of the adsorbent with the adsorbent.
Adsorption happens at the phase boundaries: solid - liquid, solid - gas, liquid - gas, liquid - liquid.
When a substance is adsorbed in the form of molecules, it is called molecular adsorption, in the form of ions - ionic adsorption.
Adsorption is reversible, the reverse process is called desorption.
The rates of adsorption and desorption are equal to each other at adsorption equilibrium, which corresponds to equilibrium concentration adsorbate in solution or equilibrium pressure in the gas phase.
Adsorption value(A) is characterized by the equilibrium amount of absorbed substance (X) per unit mass of solid adsorbent (m): [mol/kg or kg/kg]
Adsorption isotherm- graphical representation of the dependence of the adsorption value on the equilibrium concentration or equilibrium pressure at a given constant temperature.
Distinguish adsorption monomolecular, at which the adsorbate covers the surface of the adsorbent with a layer one molecule thick and polymolecular, at which the adsorbate molecules can be located on the surface of the adsorbent in several layers.
Monomolecular adsorption isotherm has the form shown in Fig. 12 ( Langmuir isotherm)
A Site I - answers small equilibrium concentrations (pressures), when a small part of the adsorbent surface is occupied by adsorbate molecules, and the dependence A - c (p) is linear;
Section II - medium concentrations (pressures) at which a significant proportion of the adsorbent surface is occupied by adsorbate molecules;
c (p) Section III - observed at high equilibrium concentrations (pressures), when the entire adsorbent surface is occupied by adsorbate molecules and reached limit value of adsorption (A).
Isotherm of monomolecular adsorption well is described by the Langmuir equation:
where c, A constants individual for each individual substance during adsorption on a specific adsorbent;
s, r- equilibrium concentration or equilibrium pressure.
At low equilibrium concentrations, we can neglect the value With or R in the denominator. Then the Langmuir equation is transformed into the equation of a straight line passing through the origin:
A = A in c or A \u003d A in p
At high equilibrium concentrations can be neglected in the denominator v. Then the Langmuir equation is transformed into the equation of a straight line independent of With or R: A = A
For practical calculations it is necessary to know the constants of the Langmuir equation A and v. Transforming the equation into a linear form of a straight line that does not pass through the origin of coordinates: , allows you to build a graph of the dependence 1/A - 1/c (Fig. 13).
1/A The segment OB is equal to 1/A. Coefficient v can be found from the fact that v is equal to the concentration at which the amount of adsorption is half of the limit.
On the graph, interpolation determines the segment OD corresponding to 2/A and equal to 1/in. Then s = 1/OD.
The Langmuir equation was derived from the theory of monomolecular adsorption, which has the following basic provisions:
adsorption of molecules occurs only on adsorption centers (tops of irregularities and narrow pores);
each adsorption center can hold only one adsorbate molecule;
the adsorption process is reversible; adsorption equilibrium is dynamic. Adsorbed molecules are retained by adsorption centers only for a certain time, after which these molecules are desorbed and the same number of new molecules are adsorbed.
In addition to the Langmuir equation, in practice it is often used Freundlich equation:
A \u003d KS 1 / n or A \u003d KR 1 / n, where K and 1 / n are empirical constants.
The equation is more suitable to describe adsorption on the porous or powdered adsorbents in the area average concentrations (pressures).
The Freundlich adsorption isotherm does not have a horizontal straight line and adsorption increases with increasing concentration (pressure) (Fig. 14).
Rice. 14
For finding the constants of the Freundlich equation it is converted using the logarithm into the equation of a straight line that does not pass through the origin: lg A \u003d log K + 1 / n lg C.
In accordance with this, the graph of the dependence of lg A on lg C or (P), built according to experimental data, has the form shown in Fig. 15. By extrapolation to the ordinate axis, a segment OB is obtained equal to lg K. The tangent of the angle of inclination of the straight line BN to the abscissa axis is 1/n ( tg =)
Polymolecular adsorption- observed during adsorption on porous or powdered adsorbents (silica gel, activated carbon, powders and tablets of medicinal substances). In this case, adsorption continues until a dense monomolecular layer is formed, as shown in Fig. sixteen.
Rice. sixteen.
Such adsorption corresponds to another type of isotherm (Fig. 17), the so-called " S - isotherm".
capillary condensation- the phenomenon of vapor liquefaction in the pores or capillaries of a solid adsorbent, it is observed when easily liquefied gases or vapors (for example, water, benzene, etc.) are absorbed as a result of polymolecular adsorption. Wherein polymolecular layer represents thin film of liquid covering the inner surface of the pore. Layers of such a liquid merging with each other in narrow places form concave menisci, under which vapor pressure is created. Thereby pores draw in gas molecules (vapour) and filled with liquid formed during condensation.
When flowing adsorption complicated by capillary condensation, the isotherm corresponding to the filling of pores (1) does not coincide with the isotherm (2) corresponding to their emptying (Fig. 18). On the isotherm, condensation hysteresis loop. The processes of adsorption and desorption do not coincide.
Adsorption can be considered as the interaction of adsorbate molecules with active centers of the adsorbent surface. Adsorption processes are classified according to the type of interaction of the adsorbate with the adsorbent. There are physical (molecular) adsorption, chemisorption (chemical addition of an atom of a molecule) and ion exchange. This section deals mainly with the physical adsorption of gases and vapors.
Physical adsorption is characterized by the interaction of the 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 that have a dipole moment greater than zero. The interaction of dipoles depends on their mutual orientation, which gave the name to the forces of the dipole-dipole interaction. 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 less than between like charges. As a result, the attraction of dipoles exceeds their repulsion. Thermal motion continuously randomly 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.
Induction 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 electric charges uniformly distributed over the volume of the molecule before interaction. As a result, a dipole moment is induced in the nonpolar molecule.
Nature dispersion forces London-van der Waals (1930) was fully elucidated only after the advent of quantum mechanics. Their occurrence is due to the fact that even neutral atoms are systems of oscillating charges, as a result of which the instantaneous value of the dipole moment of an uncharged molecule is greater than zero. A fluctuationally formed dipole creates an electric field that polarizes neighboring molecules. The interaction energy between non-polar molecules is the average result of the interaction of all possible instantaneous dipoles with the dipole moments 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 magnitude 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 summation of the attraction of individual molecules takes place.
Dispersion effect(London force) appears in its pure form between non-polar molecules. The corresponding forces arise because fluctuations in the electron density in one atom induce similar fluctuations in the 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(Kiezoma forces), the dispersion interaction is enhanced if the molecules have permanent dipoles, which are characterized by the manifestation of dipole-dipole interaction. The greater the dipole moments of the interacting molecules, the greater the component of the orientation effect.
Induction effect(Debye force) manifests itself during the interaction between polar and non-polar molecules, reflecting an increase in attraction due to the fact that the polar molecule induces a dipole in a non-polar molecule, this effect is the 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 taking into account the effect of each component of the van der Waals forces; b is an empirical constant
During adsorption, an interaction occurs between an atom (molecule) of the adsorbate and 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 the distance is different than that described by the Lennard-Jones equation. This is explained by the fact that the 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 interaction energy 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 an adsorbed atom with each adsorbent atom.
Ucn
6x3
This dependence indicates a slower decrease in the attraction energy during adsorption and the long-range action of adsorption forces. The equation was used by London and later by other scientists to experimentally prove the dispersive nature of adsorption forces and the relationship between the adsorption energy and the properties of adsorbed molecules and the adsorbent. The total potential energy of interaction during adsorption can be expressed by the equation
6x3 |
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m is the distance of the atom A from the individual atoms of the adsorbent One of the important practical conclusions when considering the nature of the adsorption
interaction is the conclusion about a much better adsorption of substances in cracks and pores, when predominantly dispersion interaction is manifested, since a larger number of atoms of the solid body is located near the adsorbed molecule. If, however, the electrostatic contribution is significant in the adsorption interaction, then positive and negative charges compensate each other in the slots and pores, and the highest potential is on the protrusions, where adsorption will prevail, 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.
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 aligned. The surface of solids, as a rule, is geometrically (porosity) and chemically inhomogeneous, and in order to obtain the simplest patterns 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 substance being distributed 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 - The dependence of the adsorption value on the concentration (pressure)
Since the activity coefficients in an infinitely dilute solution are equal to unity, the following regularity can be formulated on the basis of the equation: when the system is diluted, the distribution coefficient tends to a constant value equal to the Henry distribution constant. This is Henry's law. With respect to the adsorption value A, this law can be written as follows:
With , |
A K" |
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For an ideal gas KG = KG ’ RT
The equations are adsorption isotherms of a substance at low concentrations. In the case of adsorption on solid adsorbents, the scope of this law is small due to the inhomogeneity of the surface. But even on a homogeneous surface, with an increase in the concentration of a substance or vapor pressure, a deviation from a 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 grows 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 mainly due to the ratio 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 unity (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 a substance or vapor pressure, the free surface of the adsorbent decreases; which entails a decrease in its reactivity, which is expressed in an increase in the activity coefficients of the adsorbate on the surface of the adsorbent.
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 adsorbent surface. The limitation of this parameter leads to adsorption saturation of the adsorbent surface as the concentration of the distributed substance increases. This provision is the main one in Langmuir's theory and is specified by the following assumptions:
1. Adsorption occurs on discrete adsorption sites, 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 strongly bound to
adsorption centers; they seem to be localized at the centers (localized adsorption). It follows from the second proposition that only one adsorption layer can form on the surface; therefore, Langmuir adsorption is called monomolecular adsorption. 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 provision, adsorbed molecules due to energy fluctuations can break away from the centers and return to the gas phase.
Based on these provisions, one can obtain the adsorption isotherm equation. The rate of adsorption from the gas phase Vads (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 A, and during adsorption turns out to be occupied by A centers, then the number of centers that remain free is (A - A). Therefore V ads = k ads. p(A - A). Adsorption is dynamically balanced by the desorption process. The rate of desorption is proportional to the number of adsorbed molecules V des = k des. A . At equilibrium, V ads \u003d V des or k ads. p(A - A) = k dec. A . Renaming k ads / k des = K (where K is the adsorption equilibrium constant) and A/A = . (relative surface filling) we get
A A Kc 1 Kc
The equation is called the equation Langmuir adsorption isotherms.
It should be noted that the Langmuir adsorption equilibrium constant characterizes the interaction energy of the adsorbate with the adsorbent. The stronger this interaction, the greater the adsorption equilibrium constant. The Langmuir adsorption equation is often presented with respect 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 relations
A A and |
The ratios correspond to saturation when the entire surface of the adsorbent is covered with a monomolecular layer of the 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, i.e., 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 at its maximum filling. 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 closely adjoin each other, and hydrophobic radicals are oriented towards the air (the so-called "Langmuir fence", which was already mentioned above).
Therefore, if the limit adsorption is the number of moles of surfactant that completely occupies a unit surface, then the reciprocal of the limit
adsorption, will give the total cross-sectional area of one mole of molecules, then:
To find the length of a molecule, in addition to the 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, and δ is the length of the surfactant molecule. Experimental results on the determination of the adsorption isotherm are usually
processed using the Langmuir equation, written in linear form:
Such a linear dependence makes it possible 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 an 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 stronger, the greater the adsorption equilibrium constant.
Real surfaces of solids, as a rule, do not have energetically equivalent active centers. An essential approximation to real conditions is the consideration of possible energy distributions of adsorption centers on the adsorbent surface. Having accepted the linear distribution of adsorption centers in terms of energies (heats 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 substance adsorption over the monomolecular layer. This condition is satisfied quite strictly during chemisorption, physical adsorption of gases at low pressures and temperatures above the critical one (in the absence of condensation on the surface of the adsorbent), and often during adsorption from solutions. These restrictions on the application of the Langmuir equation are not so much related to the formal
To describe the process of adsorption, in particular monomolecular, in addition to the fundamental Gibbs adsorption equation, a number of other analytical equations are used, which are named after their authors.
With an insignificant filling of the adsorbent with an 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 regularity 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 adsorbate (Fig. 4.5, section I) and is an analytical expression of Henry's law. The coefficient kHne depends on the concentration and is a distribution constant that characterizes the distribution of a substance in the adsorption layer with respect 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 adsorptive can be determined using the Freundlich equation
G(A) = ks 1/n , (4.25)
where k, n are coefficients.
This equation was obtained on the basis of the results of processing experimental data on surfactant adsorption. The coefficient k is numerically equal to the adsorption value when the concentration of the adsorbate, 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) from 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 lgG and lgc (Fig. 4.6) is characterized by a straight line, the tangent of the slope of which is 1 / p, and the segment cut off on the y-axis is lgk.
The analytical expression of adsorption depending on the concentration of the adsorptive 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 represent schematically a unit area (for example, 1 m 2) of an adsorption layer at the phase boundary (Fig. 4.7). If an adsorbate molecule occupies an area B 0 in the surface layer, and the number of its molecules is n, then nB 0 is the area that falls on 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.
v d = k d nB 0 . (4.28)
In equations (4.27) and (4.28) k a and k D are adsorption and desorption rate constants.
Under equilibrium conditions, the rates of the direct and reverse processes are equal. On this basis, equations (4.27) and (4.28) imply
where b is the equilibrium constant of the adsorption process.
The equilibrium constant b is related to the standard value of 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 a 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 as follows:
θ = n/n∞. (4.32)
During adsorption, the degree of saturation changes in the range 0< θ ≤ 1.
The number of molecules in unsaturated n and saturated n ∞ adsorption layers can be expressed in terms of adsorption Г(А):
n = G(A)NA; n ∞ = Г ∞ (А ∞)N A , (4.33)
where N A is the Avogadro 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 was determined, and, consequently, the absolute adsorption; but, as already noted, a large excess of molecules in the surface layer compared to their content in the volume allows one to use relation (4.3). Therefore, in formulas (4.33) and (4.34) adsorption is denoted as Г(А).
Let us analyze the Langmuir equation (4.34) and compare it with the Henry equations (4.24) and Freindlich equations (4.25). At the beginning of the adsorption process, when с → 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). When c → ∞, 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 segments 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 heterogeneous; 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. On the basis of adsorption, numerous methods are carried out for cleaning gases and liquids from harmful impurities, removing moisture, separating mixtures of substances and isolating certain components from complex mixtures, as well as many other technological processes. The use of adsorption in industry will be discussed in Chap. 6.
Exercises
1. What part of the absolute adsorption is excess adsorption, if as a result of adsorption the adsorbate concentration increased by 17 times?
According to the condition of the problem, the concentration of the adsorbate in the adsorption layer with В = 17с.
Based on equalities (4.1) and (4.2), it is possible to determine the excess adsorption:
G \u003d A - ch \u003d c in h - ch \u003d (c in - c) h.
The 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, compare on a powder with a particle diameter of 70 μm and a density of 1.25∙10 3 m 3 /kg?
We use formulas (1.1), (1.4) and (4.4):
3. The solid body 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 take place - 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 kind of adsorption takes place?
A high value of the binding energy indicates that chemisorption occurs.
Henry adsorption isotherm equation
If we consider the dynamic picture of adsorption, then its value will be the 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 stays on the surface from the moment of impact to the moment it passes back into the gas phase . Therefore, but de Boer, the adsorption value
where n is the average number of molecules hitting the surface per unit time, τ is the average residence time of the molecules on the surface. This formula assumes that each impact of a molecule is accompanied by its delay on the surface, regardless of whether there are already other molecules on it or not. In reality, a molecule hitting an already occupied place may be reflected back into the gas phase or delayed, but its retention time will be different.
Accounting for these circumstances led to the following formula:
This is the Henry adsorption isotherm equation. It means that in the ideal model the amount of adsorption is directly proportional to the vapor or gas pressure. This dependence received this name by analogy with Henry's law known in physical chemistry, according to which the volume of a gas dissolved in a solid or liquid is proportional to its pressure. So, according to the accepted assumptions, the Henry isotherm should describe the experimental data obtained at low fillings on homogeneous surfaces. The first assumption, as was said, is justified in the study of adsorption at very low pressures. As for the second, adsorption is almost always measured on inhomogeneous surfaces. However, adsorption at very low pressures corresponds to very low degrees of coverage. This means that everything depends on how inhomogeneous not the entire surface is, but only a small fraction of it, which is covered at low pressures. Therefore, in the literature one can find enough examples of both kinds. The constant K of the Henry equation (the tangent of the slope of the straight line) depends on the temperature and the interaction energy of the adsorbate - adsorbent, as can be seen from equation (4.4). The lower the temperature and the greater the interaction of adsorbed molecules with the surface of the adsorbent, the greater K, the steeper the adsorption isotherm.
Of course, the assumption that molecules are adsorbed with the same probability on any part of the surface, including those already occupied earlier, is too rough an assumption, suitable only for very small degrees of coverage. Another assumption can be made, which is that adsorption occurs only on free areas of the surface and that any hit of molecules on already occupied places does not lead to an adsorption event. This assumption is equivalent to the postulate of monolayer adsorption and, as we said earlier, it is indeed fulfilled in the case of chemical adsorption, but the situation is more complicated for physical adsorption.
Another assumption made in the derivation of the Henry isotherm equation is that the surface is homogeneous, i.e. equivalence of all its sections, we will keep unchanged. And, finally, the third assumption in the new model under consideration is the absence of interaction between adsorbed molecules, i.e. we will assume that the residence time of a molecule on the surface does not depend on where it hits - in the immediate vicinity of another molecule or at a great distance from it. All these assumptions were made by Langmuir in the derivation of the adsorption isotherm, which he made in 1918.
The Langmuir adsorption isotherm equation can be derived in various ways. Langmuir himself derived it by considering the dependence of the rates of adsorption and desorption on the degree of surface coverage and assuming that at equilibrium both rates become the same.
The thermodynamic derivation of this equation was given by Volmer, and the statistical derivation by Fowler.
In this form, the Langmuir equation is widely known. It contains two constants: a m briefly called monolayer capacity (maximum adsorption), and K is a constant depending on adsorption energy and temperature.
To describe the isotherm according to Fig. 2.10a use equations of the form:
where TO and To- constants.
The above equations are fundamental in Langmuir's theory of monomolecular adsorption. The first option is more often used, since in the case of surfactant adsorption, all equations containing the quantity A, since in this case the absolute and Gibbs adsorptions are practically the same ( A = G).
When deriving the Langmuir equation, the physical interaction on the surface can be represented as a quasi-chemical reaction:
where A- surface adsorption centers; V- distributed substance; AB is a complex formed on the surface.
As the concentration (pressure) of a substance increases V the equilibrium of the reaction shifts towards the formation of the complex and the number of free centers becomes smaller. The adsorption equilibrium constant according to the law of mass action has the form:
Let us introduce the notation: 
[B) = with; [lv]n=L and [a \u003d L 0, in which A- adsorption value; A^- the number of remaining free adsorption
centers per unit surface area or unit mass of the adsorbent. If A- the value of the limiting adsorption (capacity of the adsorption monolayer), then 
Substituting the accepted notation into the equation for the equilibrium constant, we obtain an expression for the constant 
, which, after transformations, gives the well-known Langmuir adsorption isotherm equation:
For gases, pressure is used instead of concentration d(since the concentration of gases and vapors during gas adsorption is practically proportional to partial pressures): 
To characterize adsorption, the degree of surface filling is used. Regarding the degree of filling, equation (2.9)
can be written in the form
Adsorption equilibrium constants in different types of Langmuir equations (TO, To and To") characterize the interaction energy of the adsorbent and adsorbate: the stronger this interaction, the greater the adsorption equilibrium constant.
Another version of the derivation of the Langmuir equation is known - the kinetic one, in which the main attention is paid to the speed of the onset of a dynamic equilibrium of the processes of adsorption and desorption. This derivation shows that the adsorption equilibrium constant is equal to the ratio of the adsorption and desorption rate constants:
To analyze the adsorption isotherm according to the Langmuir equation, we reproduce a typical adsorption isotherm according to the monomolecular mechanism (Fig. 2.26).
Rice. 2.26.
Analysis of the monomolecular adsorption isotherm:
At very low concentrations, when c^O, the product K s in the denominator can be neglected, so we get A \u003d A SS -Ks or A \u003d K G "S. The resulting ratios correspond to Henry's law and the proportionality coefficient K g is Henry's constant. According to Henry's law
the value of adsorption increases linearly with increasing concentration in section AB;
At high concentrations or pressures, when the product K-s" 1, adsorption tends to the limit value A = A#. This ratio in the SD section corresponds to the state of saturation of the adsorbent surface with adsorbate molecules, when the entire surface of the adsorbent is covered with a monomolecular adsorbate layer;
in the region of average concentrations in the BC section, the Langmuir equation is applicable in its full form.
The physical meaning of Henry's constant, sometimes also called the distribution constant, is explained by the following reasoning. If the surface layer is considered as a separate phase, then the redistribution of the substance between the surface layer and the volume of the phase will occur until the chemical potentials of both phases become equal:
where ps- chemical potential in the surface layer; p v is the chemical potential of the bulk phase.
Considering that , for the equilibrium state we have , whence
If in the region of low activity concentrations we assume equal concentrations, then the surface concentration is equal to adsorption
a s = with s = A and then 
From the presented ratios and semi
Henry's equation comes up: A \u003d Kr-s.
It is possible to obtain a similar expression in terms of pressure, given that in the region of low concentrations, the gas obeys the law of the state of an ideal gas pV=nRT, where 
. Substituting the afterbirth
its relation into the adsorption equation, we obtain: 
Henry's equations are simple in appearance, but sometimes they are quite enough for practical calculations. On solid surfaces, the scope of this law is small due to the inhomogeneity of the surface. But even on a homogeneous surface, a deviation from the linear dependence is found with increasing concentration (pressure). This is explained by a decrease in the fraction of the free surface, which slows down the growth of adsorption.
Deviations from Henry's law take into account the empirical equation established by Freundlich and Baedeker based on the study of gas adsorption on solid adsorbents. Later, this equation was theoretically substantiated by Zel'dovich and turned out to be applicable to solutions as well.
The theory of monomolecular adsorption was created by Langmuir while studying the adsorption of gases on solid surfaces. The main provisions of the theory are as follows:
Despite severe limitations, the theory is widely used and will give good agreement with practical results for a large number of adsorption types. It is currently being extended to adsorption at other interfaces.
The Langmuir theory explains the adsorption of surfactants at the water-air interface, when the polar group, having a high affinity for the polar phase, is drawn into water, while the non-polar radical is pushed into the non-polar phase (air) and, at low concentrations, hydrocarbon chains “float” on the surface water (this is possible due to their flexibility). As the concentration increases, the chains rise and occupy a vertical position in the saturated adsorption layer, while the water surface is completely covered with a “palisade” of vertically oriented surfactant molecules. The value of surface tension in this case approaches the value of pure liquid surfactant at the boundary with air. Maximum adsorption G a0 that is why it does not depend on the length of the hydrocarbon radical, but is determined only by the dimensions of the cross section of the molecules.
The existence of saturated adsorption layers makes it possible to determine the size of surfactant molecules. For the first time in the history of chemistry, the sizes of molecules were determined precisely by the colloid-chemical method and later confirmed by other methods. Since the molecules in the saturated layer are densely packed and have a vertical orientation, it is possible to calculate the important characteristics of the monomolecular layer:
The size of the cross section of molecules, that is, the area occupied by one surfactant molecule in the surface layer (“landing pad”):
The length of the surfactant molecule, equal to the thickness of the adsorption layer:
where N A- Avogadro's number, R and M- density and molecular weight of the surfactant.
To determine the constant parameters, the Langmuir equation is transformed to the equation of a straight line
Presenting the experimental data in reverse axes or in the axes , in the first case, the value is determined by the segment cut off on the y-axis at . The tangent of the slope of the straight line allows you to determine the ratio and calculate the value of pre
specific adsorption, which can be used to calculate the adsorption constant TO. In the second case, on the contrary, the segment on the ordinate is related to the value of the inverse limiting adsorption , and according to the tangent of the slope angle
Consider a variant of determining the constants of the Langmuir equation on the example of adsorption in the system water - isoamyl alcohol. The table presents experimental data on the values of surface tension o solutions of various concentrations With:
The temperature of the experiment is 296 K, at which the surface tension of water is 72.28 mJ/m
We will apply the method of graphical differentiation, for this we construct the surface tension isotherm
and calculate the adsorption values using the Gibbs equation: 
To simplify calculations, we denote by Z the value , then hell
sorption is determined by the expression
Rice. 2.27.
size Z corresponds to a segment cut off on the ordinate by a tangent and a horizontal drawn to a point corresponding to the desired concentration. For example, finding the value of Z for a point corresponding to a concentration of 0.125 kmol/m 3 is shown. In the example, the Z value is 3.9 mJ/m 2 . The remaining results are presented in table. 2.3. After that, we calculate the reciprocal values of concentrations and adsorptions necessary to work with the Langmuir equation in a linear form: 
Table 2.3
Processing of experimental data_
Continuation of the table. 2. 3
On fig. 2.28 a graph is built in the "reverse" axes, find the constants of the Langmuir equation from it TO and G"simple, but even easier to do
this is with Excel.
In this case, we write the dependence equation 
, from which 
(according to the schedule, this is from
sharp, cut off on the ordinate at ). Then the value of the limit is G oo \u003d 2.098 10 "6 mol / m 2. This is one of the constants of the Langmuir equation.
The second constant is found from the coefficient before the reciprocal concentration, equal to 15500, i.e. 
. For a known value
The dimension of the adsorption constant = m 3 / kmol.
Rice. 2.28.
Let us finally write down the adsorption equation with the found constants:
Let us emphasize the legitimacy of equating the values of excess Gibbs and absolute adsorption, since the Langmuir theory extends to all surfaces (liquid and solid) filled by a monomolecular mechanism.
Based on the results obtained, an adsorption isotherm can be constructed in the example under consideration in two ways, substituting the concentration into the resulting equation or directly constructing a graph according to the data of the first and third columns of Table. 2.3 (Fig. 2.29).
Rice. 2.29.
This is a visual verification of the correctness of the calculations. The equation obtained using Excel has an approximation confidence value of 0.99. When plotting points for which adsorption is calculated according to the equation, small deviations are found in comparison with the location of points for which adsorption is determined by graphical differentiation (from tangents). This is due to the closeness of the values \u200b\u200bof the limiting adsorption (2.098-10 6 mol / m 2) and adsorption at a concentration of 0.5 kmol / m 3 (2.073-10 6 mol / m 2), as well as (to a lesser extent) rounding during calculations.
When plotting manually, you need to pay attention to such practical features as data averaging. The isotherm line must be drawn smoothly, located between the points, and not separate straight lines between adjacent points (Fig. 2.30).
Rice. 2.30.
On fig. 2.30 shows a family of tangents when manually processing the sodium oleate adsorption isotherm (on the y-axis, surface tension with the dimension mJ / m 2).
