Henry adsorption isotherm. Surface phenomena and adsorption
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 AND, since in this case the absolute and Gibbs adsorptions are practically the same ( AND = G).
When deriving the Langmuir equation, the physical interaction on the surface can be represented as a quasi-chemical reaction:
where AND- surface adsorption centers; AT- distributed substance; AB is a complex formed on the surface.
As the concentration (pressure) of a substance increases AT 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 AND- adsorption value; A^- the number of remaining free adsorption
centers per unit surface area or unit mass of the adsorbent. If AND- 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 AND = AND#. 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 = AND 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:
- - there are active centers on the surface of a solid adsorbent, all of them are energetically homogeneous (the surface is equipotential) and their number per unit area is constant for a given adsorbent;
- - each active center retains only one molecule of the adsorbate, which is fixed to it by the forces of physical nature (adsorption is reversible). The adsorbed molecule forms a strong complex with the center and is unable to move over the surface;
- - only the forces of interaction between the molecule and the adsorption center are taken into account (without taking into account the interaction between the molecules of the adsorbate).
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 according to the 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 when conducting 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 ordinate axis, surface tension with the dimension mJ / m 2).
The equation quantitatively describing the process of adsorption is called by the names of their authors.
When deriving this equation, the surface layer will be considered as a separate phase. The redistribution of the substance between the surface layer and the volume of the phase will occur until the chemical potentials in the surface layer and the volume of the phase are equalized, i.e. μ=μ s (1)
Where μ s is the chemical potential of the substance in the surface layer;
μ is the chemical potential for the bulk phase.
If μ μ s , then the adsorption is positive,
if μ μ s, then the adsorption is negative. (The substance leaves the surface.)
Considering that μ= μ 0 + RT∙ln a, (2)
μ s = + RT∙ln a s , (3)
where a is the activity of the adsorbate in the bulk phase;
a s is the activity of the adsorbate on the surface.
Substituting (2) and (3) into (1), we get:
μ 0 + RT∙ln a = + RT∙ln a s , (4)
Transforming, we have:
= const= Kg (5)
The constant Kg is called Henry's distribution constant. It does not depend on concentration, but depends only on temperature - Kg \u003d f (T).
If in the region of low concentrations the activity can be considered equal to the concentration (а = с, а s = с s), then the surface concentration С s =А.
From equation (5) we will have:
Kg or A \u003d Kg ∙C (6)
Considering that P=C∙R∙T; C= ,
You can get the expression for adsorption in terms of pressure:
A= or A= Kg ’ ∙P (7)
Equation (6) and (7) express Henry's Law for adsorption:
the value of adsorption at low gas pressures (solution concentration) is proportional to the pressure (concentration).
This equation is simple, but sometimes it is quite enough for practical calculations. On solid surfaces, the scope of the 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 an increase in P or C. This is explained by a decrease in the fraction of the free surface, which leads to a slowdown in the growth of adsorption.
Deviation from Henry's law takes into account the empirical adsorption equation established by freindlich.
The equation looks like:
for adsorption of gases: A= = K∙P 1/ n (1)
for adsorption from solutions: A = = K ’ ∙ c 1/ n (2)
where x is the amount of adsorbed substance;
m is the mass of the adsorbent;
P, C - equilibrium pressure or concentration;
K, K',1/n are constants, and n 1, i.e. 1/n 1 .
For gases 1/n= 0.2-0.9, for solutions: 1/n=0.2-0.5.
Value n characterizes the degree of deviation of the isotherm from linearity.
Let us consider which sections of the adsorption isotherm are described by the Freundlich equation.
In the OM section, adsorption is directly proportional to the concentration; for it, the constant 1/n must be equal to unity.
In the DE- section, adsorption is independent of concentration.
In order for equation (1) to describe this section, it is necessary that 1/n=0, but in the Freundlich equation 1/n is a fractional value. Therefore, this equation is valid only for the transitional part of the isotherm in the HP section, i.e. for the region of average concentrations, where 0 1/n 1.
Thus, the Freundlich equation describes only the transitional part of the isotherm, and does not determine the limiting adsorption A.
The initial section of the OF - obeys the Henry equation. The Freundlich equation is widely used in practice, but only for approximate calculations.
Schematically, the adsorption isotherm has the form:
Section 1 - rising steeply, almost rectilinear, shows that at low pressures (or C) adsorption grows linearly or proportionally to these values:
A = K ∙ C or A = K ∙ R
Section 3 is horizontal, corresponding to high pressures (or C), the surface of the adsorbent is completely saturated with the adsorbent:
Section 2 - the middle section of the curve corresponds to intermediate degrees of surface filling:
A \u003d K ∙ C 1 / n, where 0< <1 - эмперическое уравнение Фрейндлиха,
used to analytically express the adsorption isotherm
adsorption isotherm. Freundlich equation.
Adsorption value (absolute AND or excess G) in each case is determined by the temperature T and pressure R(with a gaseous adsorptive) or temperature T and concentration With(when adsorbed from solutions). As a rule, in the theory of adsorption, when considering the adsorption equilibrium, one of these parameters is kept constant. So, an equation of the form A \u003d f (p) T or G \u003d f (c) T, relating the amount of adsorption with pressure or concentration at a constant temperature, is called the adsorption isotherm. Adsorption (if it is expressed not as an excess, but as a total content) always increases with increasing equilibrium pressure or concentration. Since adsorption is an exothermic process, when the temperature rises, the value adsorption decreases. On fig. 26.9 shows the main types of adsorption equilibrium curves. Adsorption isotherms at three temperatures (T 1 > T 2 > T 3) corresponds to Fig. 26.9a.
Figure 26.9. Curves of adsorption equilibrium: isotherms (a), isobars (b) and isosteres (c) of adsorption
Equation relating adsorption value to temperature at constant equilibrium pressure A \u003d f (T) p or constant equilibrium concentration G \u003d f (T) s, is called, respectively, adsorption isobars or isopycnes (Fig. 26.9-b); here p 1 > p 2 > p 3. Type equation R= f (T) A, the adsorption isostere (Fig. 26.9-c), relates the equilibrium pressure to temperature at a constant adsorbed amount; in this case A 1 > A 2 > A 3.
The task of any adsorption theory is to compile its mathematical description on the basis of a certain model of the adsorption process. Ideally, the equation should describe the dependence of the equilibrium value of adsorption on the concentration of the adsorbate in the bulk phase at different temperatures, as well as predict the change in the heat of adsorption depending on the filling of the adsorbent. Most often, the adsorption isotherm equation is found in this case. The shape of the adsorption isotherm on solids depends on many parameters: the properties of the adsorbent and adsorbate, the interaction of the adsorbent adsorbate, the interaction of adsorbate molecules with each other in the gas phase and in the adsorbed state. In the range of low pressures (or concentrations) and the corresponding small surface coverages, the interaction between adsorbate molecules is insignificant and the dependence A = f(p) T reduced to its simplest form, called Henry's law:
A \u003d kp or A \u003d k "c(26.20)
where k and to"- adsorption coefficient (or Henry coefficient), with is the adsorbent concentration in the bulk phase, R is the vapor pressure of the adsorbate. Henry coefficient k is a measure of the intensity of adsorption. It can be shown that any theoretical isotherm should, in the limit (for small fillings), go over into the Henry equation.
In area medium concentrations, the concentration dependence of the adsorption of solutes is well described by the empirical Freundlich equation:
(26.21)
where X- amount of adsorbed substance, m- adsorbent mass, βi P - constants characteristic of each adsorption system, with 0< 1/n< 1 . According to Freindlich, n does not depend on padding, although this statement is not entirely accurate. This empirical equation is often used for approximate calculations of adsorption. Most often it is used in logarithmic form:
which allows constructing a linear dependence ln AND - ln c and graphically determine both constant parameters β and n.
Surface phenomena and adsorption. Types of adsorption interactions. Gas adsorption isotherms. Henry and Langmuir equation. Polymolecular adsorption, BET theory.
SURFACE PHENOMENA AND ADSORPTION
surface energy. Adsorption
Until now, the properties of heterogeneous systems have been described using parameters and state functions that characterize each of the phases as a whole. However, the properties of the phase area adjacent to its surface differ from the properties of the phase in the volume: in fact, the particles located on the surface of each phase form a special surface phase, the properties of which differ significantly from the properties of the internal regions of the phase. The particles located on the surface are in a different environment compared to the particles located in the volume of the phase, i.e. interact both with homogeneous particles and with particles of another kind. The consequence of this is that the average energy g s of a particle located on the phase interface differs from the average energy of the same particle in the volume of the phase g v (moreover, the energy of a particle on the surface can be either greater or less than the energy of a particle in the volume). Therefore, the most important characteristic of the surface phase is surface energy G s is the difference between the average energy of a particle located on the surface and a particle located in the volume of the phase, multiplied by the number of particles on the surface N:
(26.1)
It is obvious that the total value of the surface energy of a phase will be determined by the value of its surface S. Therefore, to characterize the interface separating a given phase from another, the concept is introduced surface tensionσ is the ratio of the surface energy to the area of the interface; the magnitude of the surface tension depends only on the nature of both phases. Like the surface energy of a phase, surface tension can be either positive or negative. The surface tension is positive if the particles on the surface interact with particles of the same phase more strongly than with particles of another phase (and, therefore, g s > g v). According to the principle of minimum free energy, any phase will tend to spontaneously reduce its surface energy; therefore, in the case of positive surface tension (σ > 0), the phase tends to reduce its surface. If σ< 0, поверхностная энергия фазы будет уменьшаться при увеличении площади поверхности.
The influence of the surface layer of a phase on its general properties is determined by the proportion of particles located on the surface of the total number of particles that make up this phase, i.e. the value of the specific surface of the phase S/V (surface per unit volume). The free energy of the phase G can be represented as the sum of the surface G s and volume G v energies proportional to the surface area and volume of the phase, respectively:
Dividing this expression by the volume of the phase, we get:
It follows from equation (IV.4) that with the same amount of phase (i.e., constant volume), the contribution of the surface energy to the total energy of the phase increases with an increase in the specific surface area, or, in other words, degree of dispersion(fragmentation) phase. In the case when the degree of dispersion of the phase is small (the specific surface is insignificant), the contribution of the surface energy to the total energy of the phase is usually neglected. The contribution of the surface layer to the properties of the phase and the system as a whole is taken into account when studying dispersed systems– heterogeneous systems, one of the phases of which is continuous ( dispersion medium), and the other is fragmented ( dispersed phase).
At the boundary of a condensed (i.e., solid or liquid) phase with a gas, the surface tension is always positive, since the particles of the condensed phase interact with each other more strongly than with gas molecules. According to the principle of minimum free energy, the condensed phase will tend to spontaneously reduce its surface energy. This can be the result of either a decrease in the surface area of the phase (that is why a drop of liquid in weightlessness takes the form of a sphere), or a decrease in surface tension when new particles appear on the phase interface - gas molecules or a dissolved substance. The process of spontaneous change in the concentration of a substance at the interface between two phases is called adsorption. Adsorbent a substance is called, on the surface of which there is a change in the concentration of another substance - adsorbate.
Adsorption at the solution-vapor interface
In liquid solutions, surface tension σ is a function of the solute concentration. On fig. 4.1 shows three possible dependences of surface tension on the concentration of the solution (the so-called surface tension isotherms). Substances whose addition to a solvent reduces surface tension are called surface-active(surfactants), substances, the addition of which increases or does not change the surface tension - surface-inactive(PIAV).
Rice. 26.1 Surface isotherms 26.2 Adsorption isotherm
tension of surfactant solutions (1, 2) and PIAV. Surfactant at the solution-vapor interface
PIAV (3)
A decrease in surface tension and, consequently, surface energy occurs as a result of surfactant adsorption on the liquid-vapor interface, i.e. the fact that the concentration of surfactant in the surface layer of the solution is greater than in the depth of the solution.
The quantitative measure of adsorption at the solution-vapor interface is surface excess G (gamma), equal to the number of moles of solute in the surface layer. The quantitative relationship between the adsorption (surface excess) of a solute and the change in the surface tension of the solution with increasing solution concentration determines Gibbs adsorption isotherm:
The plot of the surfactant adsorption isotherm is shown in fig. 26.2. It follows from equation (26.5) that the direction of the process - the concentration of a substance in the surface layer or, conversely, its presence in the bulk of the liquid phase - is determined by the sign of the derivative dσ / dС. The negative value of this derivative corresponds to the accumulation of the substance in the surface layer (G > 0), the positive value corresponds to a lower concentration of the substance in the surface layer compared to its concentration in the bulk of the solution.
The value g = –dσ/dС is also called the surface activity of the solute. The surface activity of surfactants at a certain concentration of C 1 is determined graphically by drawing a tangent to the surface tension isotherm at the point C = C 1 ; in this case, the surface activity is numerically equal to the tangent of the slope of the tangent to the concentration axis:
It is easy to see that with increasing concentration, the surface activity of surfactants decreases. Therefore, the surface activity of a substance is usually determined at an infinitesimal concentration of the solution; in this case, its value, denoted g o, depends only on the nature of the surfactant and solvent. Investigating the surface tension of aqueous solutions of organic substances, Traube and Duclos established the following rule of thumb for the homologous series of surfactants:
In any homologous series at low concentrations, the elongation of the carbon chain by one CH2 group increases the surface activity by a factor of 3–3.5.
For aqueous solutions of fatty acids, the dependence of surface tension on concentration is described by the empirical Shishkovsky equation:
Here b and K are empirical constants, and the value of b is the same for the entire homological series, and the value of K increases for each subsequent member of the series by 3–3.5 times.
Rice. 26.3 Limit orientation of surfactant molecules in the surface layer
Molecules of most surfactants have a amphiphilic structure, i.e. contain both a polar group and a non-polar hydrocarbon radical. The location of such molecules in the surface layer is energetically most favorable under the condition that the molecules are oriented by the polar group to the polar phase (polar liquid), and the nonpolar group to the nonpolar phase (gas or nonpolar liquid). At a low concentration of the solution, thermal motion disrupts the orientation of surfactant molecules; with an increase in concentration, the adsorption layer is saturated and a layer of "vertically" oriented surfactant molecules is formed on the interface (Fig. 26.3). The formation of such a monomolecular layer corresponds to the minimum value of the surface tension of the surfactant solution and the maximum value of adsorption G (Fig. 26.1-26.2); with a further increase in the surfactant concentration in the solution, the surface tension and adsorption do not change.
Adsorption at the solid-gas interface
When gases are adsorbed on solids, the description of the interaction between adsorbate and adsorbent molecules is a very complex problem, since the nature of their interaction, which determines the nature of adsorption, can be different. Therefore, the problem is usually simplified by considering two extreme cases, when adsorption is caused by physical or chemical forces - respectively, physical and chemical adsorption.
physical adsorption arises due to van der Waals interactions. It is characterized by reversibility and a decrease in adsorption with increasing temperature, i.e. exothermicity, and the heat effect of physical adsorption is usually close to the heat of liquefaction of the adsorbate (10 – 80 kJ/mol). Such is, for example, the adsorption of inert gases on coal.
Chemical adsorption(chemisorption) is carried out by chemical interaction of adsorbent and adsorbate molecules. Chemisorption is usually irreversible; chemical adsorption, in contrast to physical adsorption, is localized; adsorbate molecules cannot move over the surface of the adsorbent. Since chemisorption is a chemical process that requires an activation energy of about 40-120 kJ/mol, an increase in temperature contributes to its occurrence. An example of chemical adsorption is the adsorption of oxygen on tungsten or silver at high temperatures.
It should be emphasized that the phenomena of physical and chemical adsorption are clearly distinguished in very rare cases. Intermediate options are usually carried out, when the bulk of the adsorbed substance binds relatively weakly and only a small part is firmly bound. For example, oxygen on metals or hydrogen on nickel at low temperatures are adsorbed according to the laws of physical adsorption, but as the temperature rises, chemical adsorption begins to occur. As the temperature rises, the increase in chemical adsorption from a certain temperature begins to overlap the drop in physical adsorption, so the temperature dependence of adsorption in this case has a clearly defined minimum (Fig. 26.4).
Rice. 26.4 Dependence of the volume of hydrogen adsorbed by nickel on temperature
At a constant temperature, the amount of adsorbed substance depends only on the equilibrium pressure or concentration of the adsorbate; the equation relating these quantities is called the adsorption isotherm.
Theories of adsorption
There is no unified theory that would adequately describe all types of adsorption on different phase interfaces; Therefore, let us consider some of the most common adsorption theories that describe individual types of adsorption on the solid-gas or solid-solution interface.
Langmuir's theory of monomolecular adsorption
The theory of monomolecular adsorption, which was developed by the American chemist I. Langmuir, is based on the following provisions.
1) Adsorption is localized and is caused by forces close to chemical ones.
2) Adsorption occurs not on the entire surface of the adsorbent, but on active centers, which are protrusions or depressions on the surface of the adsorbent, characterized by the presence of the so-called. free valencies. Active centers are considered independent (i.e. one active center does not affect the adsorption capacity of others), and identical.
3) Each active center is able to interact only with one adsorbate molecule; as a result, only one layer of adsorbed molecules can form on the surface.
4) The adsorption process is reversible and balanced– the adsorbed molecule is retained by the active center for some time, after which it is desorbed; thus, after some time, a dynamic equilibrium is established between the processes of adsorption and desorption.
Rice. 26.5 Monomolecular adsorption isotherm
In the state of equilibrium, the rate of adsorption is equal to the rate of desorption. The desorption rate is directly proportional to the proportion of occupied active centers (x), and the adsorption rate is directly proportional to the product of the adsorbate concentration and the fraction of free active centers (1 – x):
(26.9)
From here we find x:
Dividing the numerator and denominator of the right side of equation (26.10) by k A , we get:
(26.11)
The maximum possible value of adsorption T o is achieved under the condition that all active centers are occupied by adsorbate molecules, i.e. x = 1. Hence it follows that x = r / r o. Substituting this into equation (26.11), we get:
Equation (26.13) is monomolecular adsorption isotherm, which relates the value of adsorption G to the concentration of adsorbate C. Here b is some constant value for a given adsorbent-adsorbate pair (the ratio of desorption and adsorption rate constants), numerically equal to the adsorbate concentration, at which half of the active centers are occupied. Schedule Langmuir adsorption isotherms shown in fig. 26.5. The constant b can be determined graphically by drawing a tangent to the adsorption isotherm at the point C = 0.
When describing the process of adsorption of gases in equation (26.13), the concentration can be replaced by a proportional value of the partial pressure of the gas:
Langmuir's theory of monomolecular adsorption is applicable to describe some processes of adsorption of gases and dissolved substances at low pressures (concentrations) of the adsorbate.
Polanyi's theory of polymolecular adsorption
In practice, often (especially in the adsorption of vapors) there are so-called. S-shaped adsorption isotherms (Fig. 4.6), the shape of which indicates the possible, starting from a certain pressure value, the interaction of adsorbed molecules with the adsorbate.
Rice. 26.6 Polymolecular adsorption isotherm
To describe such adsorption isotherms, M. Polyani proposed theory of polymolecular adsorption based on the following main principles:
1. Adsorption caused purely physical forces.
2. Adsorbent surface homogeneous, i.e. there are no active centers on it; adsorption forces form a continuous force field near the surface of the adsorbent.
3. Adsorption forces act at a distance greater than the size of the adsorbate molecule. In other words, at the surface of the adsorbent there is some adsorption volume, which is filled with adsorbate molecules during adsorption.
4. The attraction of an adsorbate molecule by the adsorbent surface does not depend on the presence of other molecules in the adsorption volume, as a result of which it is possible polymolecular adsorption.
5. Adsorption forces do not depend on temperature and, consequently, with a change in temperature, the adsorption volume does not change.
Freundlich equation
The theoretical concepts developed by Langmuir and Polanyi largely idealize and simplify the true picture of adsorption. In fact, the surface of the adsorbent is inhomogeneous, there is an interaction between the adsorbed particles, active centers are not completely independent of each other, etc. All this complicates the form of the isotherm equation. G. Freindlich showed that at a constant temperature, the number of moles of adsorbed gas or solute per unit mass of the adsorbent (the so-called specific adsorption x / m) is proportional to the equilibrium pressure (for gas) or equilibrium concentration (for substances adsorbed from solution) of the adsorbent raised to a certain power, which is always less than one:
Adsorption at the solid-solution interface
Molecular adsorption from solutions
Adsorption isotherms of dissolved substances from a solution are similar in appearance to adsorption isotherms for gases; for dilute solutions, these isotherms are well described by the Freundlich or Langmuir equations, if the equilibrium concentration of the solute in the solution is substituted into them. However, adsorption from solutions is a much more complex phenomenon compared to gaseous one, since the adsorption of a solvent often occurs simultaneously with the adsorption of a solute.
Rice. 26.8 Orientation of surfactant molecules on the adsorbent surface
The dependence of adsorption on the structure of adsorbate molecules is very complex, and it is rather difficult to derive any regularities. Molecules of many organic substances consist of polar (hydrophilic) and non-polar (hydrophobic) groups, i.e. are surfactants. When adsorbed on a solid adsorbent, surfactant molecules are oriented on its surface in such a way that the polar part of the molecule faces the polar phase, and the nonpolar part faces the nonpolar phase. So, during the adsorption of aliphatic carboxylic acids from aqueous solutions on a nonpolar adsorbent - activated carbon - the molecules are oriented by hydrocarbon radicals towards the adsorbent; when adsorbed from benzene (non-polar solvent) on a polar adsorbent - silica gel - the orientation of the acid molecules will be reversed (Fig. 4.8).
Adsorption from electrolyte solutions
Adsorption from aqueous solutions of electrolytes occurs, as a rule, in such a way that ions of the same type are adsorbed from the solution on the solid adsorbent. Preferential adsorption from a solution or an anion or a cation is determined by the nature of the adsorbent and ions. The mechanism of adsorption of ions from electrolyte solutions can be different; allocate exchange and specific adsorption of ions.
Exchange adsorption is a process of ion exchange between a solution and a solid phase, in which the solid phase absorbs ions of a certain sign (cations or anions) from the solution and instead releases an equivalent number of other ions of the same sign into the solution. Exchange adsorption is always specific, i.e. for a given adsorbent, only certain ions are capable of exchange; exchange adsorption is usually irreversible.
At specific adsorption adsorption on the surface of the solid phase of ions of any kind is not accompanied by the release into the solution of an equivalent number of other ions of the same sign; the solid phase acquires an electric charge. This leads to the fact that near the surface, under the action of electrostatic attraction forces, an equivalent number of ions with opposite charges are grouped, i.e. an electrical double layer is formed. The interaction of charges concentrating on the surface leads to a decrease in the surface energy of the system. For the case of specific electrolyte adsorption, Peskov and Fayans formulated the following empirical rule ( Peskov-Faience rule):
On the surface of a crystalline solid, an ion is specifically adsorbed from an electrolyte solution, which is able to complete its crystal lattice or can form a poorly soluble compound with one of the ions that make up the crystal.
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, according to de Baer, the value of adsorption:
a=n cf ∙τ (2.4)
where n cf is the average number of molecules hitting the surface per unit time, τ is the average residence time of 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 fact, a molecule hitting an already occupied spot may be reflected back into the gas phase or be delayed. Accounting for these circumstances would require the introduction of a dependence on the occupancy of the surface, i.e., the fraction of its coverage by previously adsorbed molecules. That's why first simplifying provision of the model under consideration is that any molecule colliding with the 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, in fact, almost every molecule falls into a free place and the probability of its falling into an occupied place is negligible.
Of course, the residence time of a molecule on the surface must depend on the adsorption energy. Molecules that have fallen into places where this energy is greater will stay longer on the surface, waiting longer for their “hour”, when surface energy fluctuations will push it back into the gas phase. Accounting for the energy inhomogeneity, however, would greatly complicate the description of adsorption. That's why second simplifying assumption consists in assuming a uniform surface.
Using the theoretical provisions of the kinetic theory of gases under the indicated 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 the accepted assumptions; P - gas pressure.
The constant K of the Henry equation (the slope 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 K, the steeper the adsorption isotherm.
The equation means that in this 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.
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 pressure (concentration).
So, according to the accepted assumptions, the Henry isotherm should describe the experimental data obtained at low 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 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. Under real conditions, during adsorption on solids, the scope of the law is small due to the inhomogeneity of the surface, but even on a homogeneous surface, with increasing concentration, a deviation from the law is detected. At low concentrations of the distributed substance, deviations are mainly due to the relationship between the interaction of molecules with each other and with the surface of the adsorbent.
Henry's law can be formulated as follows: when the system is diluted (pressure decreases), the distribution coefficient tends to a constant value equal to the Henry distribution constant. With respect to the adsorption value A, this law can be written as follows:
E 
These equations are adsorption isotherms of a substance at low concentrations. In accordance with them, Henry's law can be formulated as follows: the amount of adsorption at low gas pressures (substance concentrations in solution) is directly proportional to pressure (concentration).
Deviations from Henry's law, expressed by changes in the activity coefficients in the phases, usually do not allow one to describe and predict the course of isotherms with increasing concentration.
(pressure) of the adsorbate. To obtain a theoretical adsorption isotherm describing a wider range of concentrations, it is necessary to use concepts of the adsorption mechanism and specific models.
A large fraction of deviations of the adsorbate activity coefficient in the surface layer from unity can be taken into account using the idea of adsorption as a quasi-chemical reaction between the adsorbate and the adsorption centers of the adsorbent surface. This is the main idea of Langmuir's adsorption theory. This provision is clarified by the following assumptions:
1) adsorption is localized (molecules do not move over the surface) on separate adsorption centers, each of which interacts with only one adsorbate molecule; as a result, a monomolecular layer is formed;
2) adsorption centers are energetically equivalent - the surface of the adsorbent is equipotential;
3) adsorbed molecules do not interact with each other.
Lyophilic disperse systems. Classification and general characteristics of pav. Thermodynamics and mechanism of micellization. The structure of surfactant micelles in aqueous and hydrocarbon media. Solubilization.
All dispersed systems, depending on the mechanism of their formation, according to the classification of P. A. Rebinder, are divided into lyophilic, which are obtained by spontaneous dispersion of one of the phases (spontaneous formation of a heterogeneous free-dispersed system), and lyophobic, resulting from dispersion and condensation with supersaturation (forced formation of a heterogeneous free-range system).
If, with increasing concentration of a substance, the surface tension at the interface decreases, then such a substance is called surface-active. For such substances, the surface activity
The presence of hydrophilic and oleophilic parts in surfactant molecules is a characteristic distinguishing feature of their structure. According to the ability to dissociate in aqueous solutions, surfactants are divided into ionic and nonionic. In turn, ionic surfactants are divided into anionic, cationic and ampholytic (amphoteric).
1) Anionic surfactants dissociate in water to form a surface-active anion.
2) Cationic surfactants dissociate in water to form a surface-active cation.
3) Ampholytic surfactants contain two functional groups, one of which is acidic and the other basic, such as carboxyl and amine groups. Depending on the pH of the medium, ampholytic surfactants exhibit anionic or cationic properties.
All surfactants with respect to their behavior in water are divided into truly soluble and colloidal.
Truly soluble surfactants in solution are in a molecularly dispersed state up to concentrations corresponding to their saturated solutions and the separation of the system into two continuous phases.
The main distinguishing feature of colloidal surfactants is the ability to form thermodynamically stable (lyophilic) heterogeneous disperse systems (associative, or micellar, colloids). The main properties of colloidal surfactants, which determine their valuable qualities and wide application, include high surface activity; the ability to spontaneous micelle formation - the formation of lyophilic colloidal solutions at a surfactant concentration above a certain specific value, called the critical micelle concentration (KKM); the ability to solubilize - a sharp increase in the solubility of substances in solutions of colloidal surfactants due to their "introduction" into the micelles; high ability to stabilize various disperse systems.
At concentrations above KKM, surfactant molecules are collected into micelles (associate) and the solution transforms into a micellar (associative) colloidal system.
A surfactant micelle is understood as an associate of amphiphilic molecules, the lyophilic groups of which are facing the corresponding solvent, and the lyophobic groups are connected to each other, forming the core of the micelle. The number of molecules that make up a micelle is called the association number, and the total sum of the molecular weights of the molecules in the micelle, or the product of the mass of the micelle and the Avogadro number, is called the micellar mass. A certain orientation of amphiphilic surfactant molecules in a micelle provides a minimum interfacial tension at the micelle-environment boundary.
P 
At concentrations of surfactants in an aqueous solution that are somewhat higher than KKM, according to Hartley's concepts, spherical micelles (Hartley micelles) are formed. The inner part of Gartley micelles consists of intertwining hydrocarbon radicals, the polar groups of surfactant molecules are turned into the aqueous phase. The diameter of such micelles is equal to twice the length of surfactant molecules. The number of molecules in a micelle grows rapidly within a narrow concentration range, and with a further increase in concentration, it practically does not change, but the number of micelles increases. Spherical micelles can contain from 20 to 100 molecules or more.
As the surfactant concentration increases, the micellar system passes through a series of equilibrium states that differ in association numbers, sizes, and shapes of micelles. When a certain concentration is reached, spherical micelles begin to interact with each other, which contributes to their deformation. Micelles tend to take a cylindrical, disc-shaped, rod-shaped, lamellar shape.
Micellization in non-aqueous media, as a rule, is the result of the action of attractive forces between the polar groups of surfactants and the interaction of hydrocarbon radicals with solvent molecules. The inverted micelles formed contain non-hydrated or hydrated polar groups inside, surrounded by a layer of hydrocarbon radicals. The association number (from 3 to 40) is much less than for aqueous solutions of surfactants. As a rule, it grows with an increase in the hydrocarbon radical up to a certain limit.
The phenomenon of dissolution of substances in surfactant micelles is called solubilization. The way in which solubilizate molecules are included in micelles in aqueous solutions depends on the nature of the substance. Non-polar hydrocarbons, penetrating into micelles, are located in the hydrocarbon cores of micelles. Polar organic substances (alcohols, amines, acids) are incorporated into a micelle between surfactant molecules so that their polar groups face water, and the lipophilic parts of the molecules are oriented parallel to the surfactant hydrocarbon radicals. A third way of incorporating the solubilizate into micelles is also possible, which is especially characteristic of nonionic surfactants. Molecules of a solubilizate, such as phenol, do not penetrate into micelles, but are fixed on their surface, located between randomly bent polyoxyethylene chains.
Solubilization is a spontaneous and reversible process; a given surfactant concentration and temperature corresponds to a well-defined saturation of the solution with solubilizate. As a result of solubilization, stable disperse systems are obtained similar to spontaneously formed ultramicroheterogeneous emulsions.
Determine the surface and total (internal) energy of 4 g of water mist, which has particles with a dispersion of 5 10 7 m -1 , t= 20ºC, σ = 72 mJ/m 2 ; dσ/ dT= - 0.16 mJ/(m 2 ·TO); ρ = 1000 kg/m 3 .
Examination ticket number 10
Theory of polymolecular BET adsorption: initial positions, derivation of the isotherm equation and its analysis. Linear form of the BET equation. Determination of the specific surface of adsorbents, catalysts and other porous bodies.
The Langmuir equation can be used only under the condition that the adsorption of a substance is accompanied by the formation of a monomolecular layer.
In most cases, a monomolecular adsorption layer does not fully compensate for the excess surface energy, and the influence of surface forces can extend to the second, third, and subsequent adsorption layers, resulting in polymolecular adsorption.
With 
The modern form of the polymolecular adsorption equation - the basic equation of the generalized Langmuir theory - was proposed by Brunauer, Emmett and Teller.
In this theory, an additional assumption to those that were used as the basis for the derivation of the Langmuir isotherm equation is the idea of the formation of "successive complexes" of adsorption centers with one, two, three, etc. adsorbate molecules on the surface of the adsorbent. Then the adsorption process can be represented as successive quasi-chemical reactions:
The equilibrium constants of these reactions are respectively equal to
Denote:
The total number of active sites on the adsorbent, or the capacity of the monolayer, will be equal to
After a series of calculations using the theory of series, we finally get:
D 
This relation is the basic equation of the generalized Langmuir theory and is called the BET polymolecular adsorption equation.
When processing experimental results, the BET equation is usually used in a linear form:
It allows you to graphically determine both constant parameters A ∞ and С:
The experimental determination of A ∞ makes it possible to calculate the specific surface area of the adsorbent (surface area per unit mass of the adsorbent): 
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