UNIVERSITY OF UKRAINE

«IGOR SIKORSKY KYIV POLYTECHNIC INSTITUTE»

**I. M. Ivanenko, T. A. Dontsova, Yu. M. Fedenko**

**ADSORPTION, ADSORBENTS ** **AND CATALYSTS BASED ON **

**THEM **

*Approved by the Academic Council of National Technical University of Ukraine *

*«Igor Sikorsky Kyiv Polytechnic Institute» *

*as Manual for Master students of the specialty 161 «Chemical technologies and *
*engineering» *

*specialization «Chemical technologies of inorganic substances and water purification» *

Kyiv

Igor Sikorsky Kyiv Polytechnic Institute 2019

Reviewers: Chygyrynets O.E., Dr. Sci., Prof.

* Sabliy L.A., Dr. Sci., Prof. *

Responsible editor Kontsevoy A.L., PhD, Assoc. Prof.

*The Grief is provided by the Academic Council of National Technical University of Ukraine "Igor *
*Sikorsky Kyiv Polytechnic Institute" (protocol № 11 from 09.12.2019) *

Electronic educational edition

*Ivanenko Iryna Mykolaivna, PhD, Assoc. Prof. *

*Dontsova Tetiana Anatoliivna, PhD, Assoc. Prof. *

*Fedenko Yurii Mykolayovich, PhD *

### ADSORPTION, ADSORBENTS AND CATALYSTS BASED ON THEM

Adsorption, Adsorbents and Catalysts Based on Them [Electronic resource]: Manual for Master students of the specialty 161 Chemical technologies and engineering specialization

«Chemical technologies of inorganic substances and water purification» / Igor Sikorsky Kyiv Polytechnic Institute; redactors: І.M. Ivanenko, Т.А. Dontsova, Yu.M. Fedenko. – Electronic text data (1 file: 3,43 Mbyte). –Кyiv : Igor Sikorsky Kyiv Polytechnic Institute, Faculty of Chemical Technology, 2019. – 256 p.

The material of this manual is set out in accordance with The Program of the discipline «Adsorption, Adsorbents and Catalysts Based on Them», covers and reveals most of the lecture material. It can have used by students at preparation to practical and laboratory classes, at performance individual and independent kinds of works, and also at preparation to control works and examination.

The tutorial contains three sections. The first section extended the technology of carbon sorbents. The second section contains the technologies of main mineral pigments, the classification and appointment of mineral pigments. The third section contains laboratory work, the purpose of which is the practical mastering of the material described in the two preceding sections.

© I. M. Ivanenko, T. A. Dontsova, Yu. M. Fedenko, 2019

© Igor Sikorsky Kyiv Polytechnic Institute, 2019

**CONTENT **

**INTRODUCTION ... 5 **

**CHAPTER 1 TECHNOLOGY OF CARBON SORBENTS ... 7 **

1.1 PHYSICOCHEMICAL BASES ... 7

1.1.1 Sorption of gases... 7

1.1.2 Sorption from water solutions ... 29

1.2 Structural chemistry of activated carbon ... 38

1.3 TECHNOLOGIES OF OBTAINING CARBON ADSORBENTS ... 48

1.3.1 Method of vapor-gas activation ... 49

1.3.2 Production of granular activated carbon by steam activation ... 56

1.3.3 Production of crushed activated carbon by steam activation ... 66

1.3.4 The technology of carbon molecular sieves ... 72

3.5 Production of activated carbon from fossil raw materials ... 77

1.3.6 Getting active carbon by chemical activation ... 80

1.3.7 Production technology of peat adsorbent by sulfur-potassium activation 88 1.3.8 The technology of carbon adsorbents with lignite, modified potassium hydroxide ... 94

1.3.9 The technology of carbon adsorbents from waste wood, modified phosphoric acid ... 100

QUESTIONS FOR SELF-CONTROL ... 102

**CHAPTER 2 TECHNOLOGY OF MINERAL PIGMENTS ... 104 **

2.1 MINERAL PIGMENTS AND THEIR BASIC PROPERTIES ... 104

2.1.1 Classification and designation of mineral pigments ... 104

2.1.2 Application of mineral pigments ... 105

2.1.3 Main pigment properties ... 107

2.2 SOURCE OF RAW MATERIALS ... 113

2.2.1 White pigments based on titanium oxide compounds ... 113

2.2.2 Colored pigments containing iron ... 121

2.3 GENERAL CHARACTERISTICS OF MINERAL PIGMENTS ... 122

2.3.1 Characterization and chemical properties of titanium (IV) oxide ... 122

2.3.2 Characteristics, structure and chemical properties of iron– containing pigments ... 127

**Akageneite ... 130 **

2.4 TECHNOLOGY FOR PRODUCING OF MINERAL PIGMENTS ... 134

2.4.1 Production technology of pigmentary titanium (IV) oxide by sulfate method ... 136

2.4.2 Technology for pigmentary titanium (IV) oxide chloride process ... 159

2.4.3 Production of iron– containing pigments ... 163

2.4.4 Production of natural iron-containing pigments ... 181

QUESTIONS AND TASKS FOR SELF-CONTROL ... 184

**CHAPTER 3 LABORATORY WORKSHOPS ... 187 **

3.1 DETERMINATION OF SORPTIVE PROPERTIES OF TITANIUM (IV) OXIDE ... 187

3.2 SYNTHESIS OF ACTIVATED CARBON ... 192

3.3 DETERMINATION OF TOTAL PORE VOLUME AND BOUNDARY VOLUME OF ADSORPTION SPACE OF ACTIVATED CARBON ... 199

3.4 RESEARCH PORE VOLUME AND SPECIFIC SURFACE AREA OF ACTIVATED CARBON ... 207

3.5 STUDYING SURFACE PROPERTIES OF ACTIVATED CARBON ... 216

3.6 TEST METHOD ACTIVATED CARBON ... 225

3.7 DESIGNING THE SORPTION ISOTHERM OF CARBOXYLIC ACID .. 232

3.8 SYNTHESIS OF FINE SORBENT – NICKEL (II) FERROCYANIDES ... 237

3.9 BIOSORPTION OF HEAVY METALS FROM WATER SOLUTIONS .... 243

3.10 PREPARATION OF SILICA GEL BY SOL-GEL METHOD ... 248

**REFERENCES ... 254**

**INTRODUCTION **

The discipline «Adsorption, Adsorbents and Catalysts Based on Them»

taught in the first year of master’s studies of the specialty 161 «Chemical technologies and engineering», attends to selective academic disciplines, i.e.

disciplines of a free choice of students, and are profiling in the relevant curriculum.

The discipline «Adsorption, Adsorbents and Catalysts Based on Them» takes important role in formation outlook of modern specialist in specialization

"Chemical technologies of inorganic substances and water purification" and is final in preparation of specialists of inorganic substances chemical technologies.

Educational material of discipline «Adsorption, Adsorbents and Catalysts Based on Them» based on knowledge of normative disciplines «Applied chemistry»,

«Physics», «General and inorganic chemistry», «Physical chemistry», «General chemical technology», «Processes and apparatuses of chemical manufactures»,

«Chemical technology of inorganic substances».

The material of this manual is set out in accordance with The Program of the discipline «Adsorption, Adsorbents and Catalysts Based on Them», covers and reveals most of the lecture material. It can have used by students at preparation to practical and laboratory classes, at performance individual and independent kinds of works, and also at preparation to control works and examination.

The tutorial contains two sections. The first section extended the technology of carbon sorbents, including physical and chemical bases of adsorption, structural chemistry of activated carbon and main technologies of synthesis of carbon adsorbents. The second section contains the technologies of main mineral pigments, the classification and appointment of mineral pigments, branches of its applications, the main properties of pigments and review of raw base has been presented.

The material contributes to assimilation of material of such branches in chemical technology, as adsorbents and its producing, and also sorption processes

and its application in different branches: pigments and technologies of its obtaining (can have used for getting of metal oxides with other purpose).

**CHAPTER 1 TECHNOLOGY OF CARBON SORBENTS **
**1.1 PHYSICOCHEMICAL BASES **

**1.1.1 Sorption of gases **

Any technological adsorption process, no matter at what sequence it is carried – periodically or continuously, includes a row of mandatory stages, the first of all, these are adsorption and desorption. Only a comprehensive analysis of equilibrium and kinetic correlations of adsorption-desorption cycle and auxiliary stages (cooling, drying, etc.) reveals the best conditions for the whole process and recommend reasonable method of engineering calculation.

Most industrial adsorption processes are based on selective absorption of specific components of the gas-vapor mixture flow. During the absorption of gas or vapor, adsorption capacity depends on the type of sorbent, its porous structure, the nature of the substance absorbed is its partial pressure and temperature [1].

At equilibrium for the selected adsorbent-adsorbate system, amount of

absorbed gas or vapor is a function of the gases/velour’s partial pressure and temperature:

*a = f (p, T). * (1.1)

Equation (1.1) is fair for all ranges of temperature, but for the characterization of adsorption processes in porous matrix, usually the dependence of adsorption capacity of the pressure at a constant temperature – the so-called «isotherm of adsorption» is used:

*a = f (r) for T = const. * (1.2)
In real process of the purification and separation of gases, influence of
adsorption of bulk gas and other impurities, and kinetic factors may cause the need

to make adjustments in calculation of the adsorption capacity, which was initially determined by isotherms of pure components. However, in all real adsorption process, curve of thermodynamic equilibrium is the main comparative characteristics of different types of adsorbents and it determines the choice of optimal operating conditions of the process.

Simultaneously, the adsorption isotherm is a source of information about the structure of the adsorbent, adsorption heat and several other physic-chemical and technological characteristics.

C. Brunauer [1] highlighted five main types of adsorption isotherms, which are presented in Fig. 1.1. In the case of technical adsorbents, type I can be characterized as microporous adsorbents that contain virtually no transient pores. Initial bulging area of type II and IV isotherms indicate presence of macro pores with more or less substantial amount of micro pores in conjunction. Less steep initial ascent isotherm curves can be explained by mono- and multimolecular adsorption only for adsorbents with transitional porous type. Initial curved section isotherms types III and V, which are rarely found, are common for adsorbent- adsorbate systems, when interaction of molecules of adsorbate with an adsorbent much less than intermolecular interaction between adsorbate molecules, for example, caused by a presence of hydrogen bonds.

**Fig. 1.1. The main types (IV) of adsorption isotherms. **

*a* I

*P*

II III IV V

*P*_{S}*P*_{S}*P*_{S}*P*_{S}*P*_{S}

The main difference between II to IV and III of V types is that the volume transition pore (IV and V types) in a result of capillary condensation are filled up with adsorbate earlier than relative pressure become equal 1.1. As a result, on the isotherms appears top, almost horizontal, section.

The basis of engineering calculation of almost any technological process of adsorption is the proceeding of adsorption isotherms, despite the fact, that most of them run dynamically (not statically).

*Theory of monomolecular adsorption *

The first fundamental equation of adsorption isotherm was Langmuir’s equation. It is based on the assumption that the adsorption localized and occurs at active centers with the equal energy. They situated relatively rarely on the surface of the adsorbent. Consequently, the interaction between adsorbed molecules is absent. Each active center can adsorb only one molecule. According to this theory, with increasing of pressure, the part of solid surface, which is covered with molecules of adsorbate, increase. After reaching saturation pressure throughout the surface a monolayer of adsorbate is formed [2,3].

The ratio of filling of the surface is expressed as ratio of adsorption capacity
at equilibrium pressure p to the adsorption capacity at monomolecular filling of
surface a*m**. The parameter a**m* being called monolayer capacity. Accordingly, the
equation of Langmuir’s adsorption isotherm written as follows:

𝑎 = ^{𝑎}^{𝑚}^{𝑏𝑝}

(1+𝑏𝑝), * * (1.3)

where b – a factor that takes into account the ratio of the speed of adsorption and desorption.

The equation of Langmuir covers a wide range of pressures. At the starting point of isotherm bp<<1, the equation takes the following form:

𝑎 ≈ 𝑎_{𝑚}𝑏𝑝. (1.4)
On this site adsorption capacity increases linearly with increasing of
equilibrium pressure (Henry ‘s equation). With high pressure (bp>>1) the surface
starts to be covered by monolayer of molecules, and isotherm is parallel to the X
axis:

𝑎 ≈ 𝑎_{𝑚}. (1.5)

To simplify the use of equation (1.3), it can be transformed as follows:

𝑝
𝑎 = ^{1}

𝑎_{𝑚}𝑏+ ^{1}

𝑎_{𝑚}𝑝, (1.6)

or

1
𝑎 = ^{1}

𝑎_{𝑚}+ ^{1}

𝑎_{𝑚}𝑏
1

𝑝. (1.7)

The slope of the line and the segment formed by interception by the curve of
the Y axis, gives information how calculate constants a*m* and b. Using monolayer
capacity (a*m**, mole/g) the surface area of the adsorbent (S**sp*, m^{2}/g) can be determined
as:

*S**sp** = а**m** N**A**m* * * (1.8)
where N*A* – Avogadro's number; m – the area occupied by a molecule of adsorbate
in a dense layer on the surface of the adsorbent.

The equation of Langmuir, and hence the method for determining the surface area can be applied to systems in which the process is not complicated by multimolecular adsorption, adsorption in micro pores and capillary condensation.

To such systems can be attributed the case of adsorption of gases at temperatures above the critical on non-porous or adsorbents with large pores. Despite this limitation, the equation of Langmuir commonly used in technical adsorption.

*Theory of multimolecular adsorption *

A large number of adsorption systems described by isotherms type II. For
these isotherms a sharp rise after reaching the relative pressure (p/p*S*)>0,2 very
common, which is associated with the formation of the second and subsequent
layers of molecules that cover the molecules of the first layer.

Brunauer, Emett and Teller [4,5] in justifying theory multimolecular adsorption accepted that, despite the change in the total process model, the behavior of each adsorbed layer separately consistent with the concept Langmuir, adsorption localized and occur in the absence of interaction between the molecules of adsorbate. Each adsorbed layer generally obeys the Langmuir’s equation. To create the equation of multimolecular adsorption, the authors started from the point that the rate of condensation of molecules on a clean surface is equal to the evaporation rate on the first layer. Similar assumptions made when comparing the rate of condensation in each of the previous and the evaporation rate in each subsequent layer.

Detailed analysis of equation multimolecular adsorption, called by the initial letters of the names of the authors (BET), is presented in a large number of articles and monographs. The final form of the equation is as follows:

𝑎 = ^{𝑎}^{𝑚}^{𝐶}

𝑝
𝑝𝑠
(1−^{𝑝}

𝑝𝑠)[1+(𝐶−1)^{𝑝}

𝑝𝑠]. (1.9)

BET equation is fair in the range of relative pressures from 0,05 to 0,35. It is widely used to determine the specific surface of different porous bodies.

Determination of the specific surface area is usually carried out using an experimental isotherm of adsorption of standard vapor on the sample. The linear form of BET equation is used:

𝑝
𝑝𝑠
𝑎(1−^{𝑝}

𝑝𝑠) = ^{1}

𝑎_{𝑚} + ^{𝐶−1}

𝑎_{𝑚}𝐶
𝑝

𝑝_{𝑠}. (1.10)

Representing isotherm adsorption in the coordinates (

𝑝
𝑝𝑠
𝑎(1−^{𝑝}

𝑝𝑠) , ^{𝑝}

𝑝_{𝑠}), from the
segment that is formed by the interception of the Y-axis, the value of a*m* can be
found and the angle of inclination of the line to the X-axis gives the value of C (Fig.

1.2). Specific surface area (S*spec**) is determined from the capacity monolayer, as *
described above. Constant C is directly linked to true molar heat of adsorption
[19,20], which corresponds to a subtraction from heat absorption in the first
layer Q*1* the molar heat of steam condensation adsorbate λ:

𝐶 = ^{exp(𝑄}^{1}^{−𝜆)}

𝑅𝑇 . (1.11)

Last equation is used to calculate the value of the true heat of adsorption from experimental data. The value of the constant C determines the type isotherms. In the case of low value of the true heat of adsorption (p <2) isotherm has a concave shape (type III). If C>2, the isotherm gains S-shape (type II).

**Fig. 1.2. Graphic representation of the adsorption isotherm in BET ****coordinates. **

*x*
*0,15*

*0,1*

*0,05*

*0,1* *0,2* *0,3* *0,4* *0,5*
*tg а*

1

*S*

*S*

*p*
*p*
*a* *p*

*p*

−

*S*

*p*
*p*

If isotherm of the investigated substance can be classified as type II (Fig. 1.1),
the calculation of specific surface area can also be done using point B as the start
point of inflection on the isotherm, which indicates the complete filling of
monolayer when a steep climb of isotherm moves in gentle part. The value of a*m*

can be found by projection on the vertical axis the extension of gentle parts of the
isotherm in coordinates *. *

Calculations using two methods consistent in case of isotherms with a large slope which is typical for substances with a high heat of adsorption. Determination of B point in the case of flat isotherms (typical for substances with low heat absorption) can lead to significant errors in the evaluation of surface area.

Usually, for measurement of the specific surface area the nitrogen gas is used
as adsorbate. Experiment carried out at a temperature of minus 196 °C. The size of
covered area by nitrogen molecule on almost all solids is 0,162 nm^{2}, and packing
density of molecules in the adsorbed layer equal to their packaging in a normal
liquid [6,7].

*The theory of volume filling of micropores *

The principal difference between adsorption phenomena occurring in the micro pores or on the surface of transient pores in non-porous adsorbents requires different theoretical approaches to their description and interpretation. All theories of physical adsorption, despite their apparent physical differences, come from the same physical pattern. This physical pattern is concerned about the concept of geometric surface of phase interaction, at which the adsorption occurs with formation of one or more successive adsorption layers [8,9].

The idea of micro pores as a region of space in solids, which size is comparable with the size of adsorbed molecules, suggests that in any kind of

adsorption interactions (independently whether it is dispersion, electrostatic or other forces) that cause physical adsorption, in the whole space of micro pores the adsorption field appears. Limited adsorption space of micro pores leads to the fact that adsorbed in micro pores molecules do not form adsorption layers. Adsorption in micro pores is characterized volume filling of adsorption space. Therefore, the main geometric parameter that characterize microporous adsorbent is the volume of micro pores, not their «surface».

The concept of volume filling of micro pores results in pattern of threshold
adsorption magnitude a*0* corresponding to the filling of the entire adsorption space
micro pores by adsorbed molecules. Dependence a*0* from temperature is determined
by the thermal coefficient of threshold absorption:

𝛼 = − ^{1}

𝑎_{0}
𝑑𝑎_{0}

𝑑𝑇 = ^{𝑑𝑙𝑛𝑎}^{0}

𝑑𝑇 . (1.12)

The coefficient is almost constant over a wide temperature range. If the
threshold adsorption magnitude experimentally determined for a certain temperature
T0, then according to (1.12) thresholds adsorption a*0* for other temperature T
expressed as follows:

𝑎_{0} = 𝑎_{0}^{0}𝑒𝑥𝑝[−𝛼(𝑇 − 𝑇_{0}]. (1.13)
To calculate a*0* in equation (1.13) it is necessary to know the thermal
coefficient of threshold adsorption. K. Nikolaev and N. Dubinin [5,7] proposed a
method for calculating the density of substance in the adsorbed state (adsorbate) for
the temperature range from normal boiling temperature Tbp to critical Tcr using the
physical constants of adsorbed material. This method can be used to calculate α:

𝛼 = ^{𝑙𝑔}

𝑎00 𝑎0∗

0,434(𝑇_{𝑐𝑟}−𝑇_{𝑏𝑝}) = ^{𝑙𝑔}

𝜌00 𝜌0∗

0,434(𝑇_{𝑐𝑟}−𝑇_{𝑏𝑝}). (1.14)

Quite reliably calculated thresholds adsorption magnitude a*0*, allows instead
of adsorption magnitude a use dimensionless parameter, which expresses the degree
of filling of micro pores [10,11]

*Ѳ = a/a**0**.* (1.15)

The theory of volume filling of micro pores has thermodynamic origin, so to
describe the adsorption equilibrium, the following thermodynamic functions like
enthalpy, entropy and free energy are used. To calculate changes in these functions
as a standard condition at a given temperature, we assume that the liquid phase
which is in equilibrium with its saturated steam has a pressure PS, or volatility of f*s.*

The main thermodynamic function is the differential maximum performance of adsorption A, which is the free Gibbs energy of adsorption G with the sign

«minus»:

*А = – G = RT ln (p**s**/p), * (1.16)
or

*А = RT ln (f**s**/f), * (1.17)

where p – the equilibrium pressure or volatility f of the vapor at temperature T.

Introduction volatility instead of pressure takes into account the imperfection of gas phase.

If adsorption is expressed in dimensionless units, then the differential work of adsorption advisable to express also in the form of dimensionless ratio, where E – the characteristic free energy of adsorption, the physical meaning of which will be mentioned below. Then the thermodynamic equation of adsorption can be represented in general form:

𝜃 = 𝑓 (^{𝐴}

𝐸, 𝑛). (1.18)

Equation (1.18) is the distribution function of filling the micro pores on the differential work of adsorption, and E is one of the parameters of this function.

Since most distribution functions in normalized form characterized by two parameters, the second of them, which is conventionally denoted by n, is in a permanent setting in the analytical expression for the function (1.18).

According to equation (1.18) we obtained equation for the so-called characteristic curve:

*А = Е ( , n). * (1.19)

If for different vapor function * and parameter n remain unchanged, then fair *
the equation:

𝐴
𝐴_{0} = ^{𝐸}

𝐸_{0} = 𝛽, * * (1.20)

i.e. characteristic curves in coordinates А- is affine. In other words, taken under
the same values of * is *ratio of ordinates is constant and equal to coefficient of
affinity in the range of variation in which assumptions about the immutability
of function and sustainability of the parameter n are fair.

In the formula (1.20) A0 and E0 is a value for a standard pair. Good adherence of affinity conditions of characteristic curves was grounded in research of academician N. Dubinin on microporous carbon adsorbents. Also, it was shown that the development of microporous structure as a result of activation of coal within the errors of experiment do not affect the coefficient of affinity for various vapors, although the absolute values of the characteristic free energy change significantly [12,13].

Equation (1.19) shows that E=A for some filling *0 *or characteristic point,
which is defined in general case from:

*(**0**, n) = 1, * (1.21)

and at constant functions filling *0* will be the same for different vapors. The role
of n will be discussed below. The above is the basis for experimental determination
of characteristic free absorption energy by one point of adsorption isotherm, which
corresponds to filling *0** and expressed by equation (1.21). Obviously, the absolute *
value of *0* depends on the type of function *. *

The distribution of the filling for differential molar work of adsorption is expressed as:

𝜃 = 𝑒𝑥𝑝 [− (^{𝐴}

𝐸)^{𝑛}]. (1.22)

Analysis of many adsorption systems showed that equation (1.22) is consistent with the results of experiments with parameters n, which is represented by small integers.

If we put in the equation (1.22) the degree of filling through adsorption magnitude from (1.15) the following equation adsorption is obtained:

𝑎 = 𝑎_{0}𝑒𝑥𝑝 [− (^{𝐴}

𝐸)^{𝑛}]. (1.23)

This equation can be represented in linear form:

𝑙𝑔𝑎 = 𝑙𝑔𝑎_{0}−^{0.434}

𝐸^{𝑛} 𝐴^{𝑛}. (1.24)

In the coordinate axes (lg a, A^{n}) the equation (1.24) is presented by a straight
line, and intercept with the Y axis in point which is equal lga0, and the angular
coefficient of the line is equal to ^{0,434}

𝐸^{𝑛} (Fig. 1.3). If the exponent n is known, then
the using graph showing linear equation (1.24), it can be easily defined threshold
adsorption *a**0* and the characteristic energy of absorption E based on one
experimental adsorption isotherm. In this case, for each experimental point isotherm
*(a, p) the formula (1.16) can be used to calculate the corresponding value of the *

differential molar work of adsorption in J/mol. With R = 8,32 J/mole·deg formula (1.16) takes the form:

*A = 2,3 · 8,32 Т lg (p**s**/p) =19,14 Т lg(p**s**/p). * (1.25)
As already explained above, the exponent n in the equation of adsorption
(1.23) is represented by a small integer. Almost all adsorption systems, which
occurs in practice, the parameter n known, and its determination is not necessary.

However, during the research this problem may occur. Therefore, it is useful to
consider reasonable way to estimate the exponent n and determine the parameters a*0*

and E of (1.24).

Usually the initial experimental adsorption isotherms determined for the
temperature not to exceed the normal boiling point of adsorbate and includes a
range of relative equilibrium pressures of up to decimal, in which almost completed
filling of micro pores occur. Therefore, the previous value of the threshold
adsorption can be obtained from the graph of isotherm by interpolation of
adsorption values a0 in the range of the high equilibrium relative pressures ^{𝑝}

𝑝_{𝑠} > 0,3,
where adsorption is almost constant or slightly increases with pressure. This vapor
adsorption isotherm is changed for adsorption that occurs on the surface of transient
pores.

Next the adsorption to the characteristic point is calculated:

*а**x**=0,368 а*0. (1.26)

From the graph of original adsorption isotherm the equilibrium relative
pressure ^{𝑝}^{𝑥}

𝑝_{𝑠} is determined by for the characteristic points and the preliminary value
of the characteristic energy of adsorption E is calculated:

𝐸 = 4,574𝑇𝑙𝑔^{𝑝}^{𝑥}

𝑝_{𝑠}. (1.27)

**Fig. 1.3. Adsorption isotherms in linear coordinates theories surround ****filling the micropores. **

Having approximate values a*0* and E the parameter n in equation (1.23) can
be evaluated. After double logarithm of its left and right parts we get:

𝑛 = 𝑙𝑔[2,3031𝑙𝑔^{𝑎0}
𝑎]
𝑙𝑔^{𝐴}

𝐸

. (1.28)

Equation (1.28) allows for each point of the adsorption isotherm estimate the
value of n. However, at values a close to a0 the 𝑙𝑔^{𝑎}^{0}

𝑎 becomes close to zero, and
determining n is unreliable. The same applies to the points isotherms that are close
to the characteristic point when the 𝑙𝑔^{𝑎}^{0}

𝑎 in denominator approaches zero. For
fillings significantly smaller than for the characteristic points *0*, can be observed
deviation from the conditions of invariance, making assessment n formula (1.28) is
less reliable. Therefore, to determine n using this formula is possible even using
only one point of isotherms for fillings, which are about 2 times higher than *0* and
corresponds to the absorption of about 0,7-0,8 on a0. The obtained value n is usually
close to an integer, which is taken as a parameter n [14,15].

A specific feature of (1.24) is the deviation in the preliminary evaluation of
parameter *n for one to two decimals of an integer virtually has no effect on the *

lg a

*0,5*

*0*

*0,5*

*1* *2* *3* *4* *5* *6* *8* *9* *A*^{n}*·10*^{–6}

accuracy of a linear relationship. Relatively small changes in the parameters *a**0* and
*E compensate such deviations. This allows for the specified determination of these *
parameters, representing all points of the experimental adsorption isotherms on the
graph in the linear form of the equation (1.24) for evaluated integer values of the
parameter n. Usually the experimental points fairly good fit to a straight line. Using
the segment that is cut off on the Y-axis and the angular coefficient, the corrected
value of threshold adsorption a00 for the temperature T=T0 (at which the initial
adsorption isotherm was defined) can be calculated.

The obtained value a0 and E for accepted integer values n are the parameters
of equation (1.23) at constant temperature T0, i.e. equation of initial adsorption
isotherm. Among them, only a00 depends on temperature. This dependence is
expressed by equation (1.13). As mentioned above, under the conditions of
temperature invariance, E and n are independent of temperature. It should be noted
that because T0 is designated normal boiling temperature of adsorbate, and a*0**0* is
respective value of threshold adsorption only for calculating the thermal coefficient
of thermal expansion of adsorbate α using the formula (1.14). It is easy to show that
equation (1.13) *a*00 and T0 may represent values for Tbp*<T*0*<T*cr at a value of α,
which is defined by the formula (1.14) [16,17].

Equation (1.23) allows to calculate adsorption equilibrium of vapor at different temperatures. Suppose you want to calculate the adsorption of vapor a for a given equilibrium values of pressure p and temperature T. First, using the formula (1.25) we find the corresponding value of p and T differential molar work of adsorption:

*А = 4,574 Т ln (p**s**/p), * (1.29)

where the vapor pressure for the temperature T is taken from tables or
calculated by known formulas. Threshold adsorption a*0* for the temperature T is

calculated from the equation (1.13) (see. Above). Substituting the calculated values a0 in equation (1.23), written for ease of computation in logarithmic form

𝑙𝑔𝑎 = 𝑙𝑔𝑎_{0}− 0,434 (^{𝐴}

𝐸)^{𝑛}, (1.30)

you can find specific adsorption a. In this way can be calculated adsorption isotherms for temperatures T that lie in the mentioned above temperature range, which is of practical interest. To increase the accuracy of calculations original experimental isotherm should be determined at temperature T0, which is close to the average for the interval [18,19].

According to equation (1.23) and (1.13), thermal equation of adsorption can be written in the form:

𝑎 = 𝑎_{0}^{0}𝑒𝑥𝑝 [− {(^{𝐴}

𝐸)^{𝑛} + 𝛼(𝑇 − 𝑇_{0})}]. (1.31)
In this case, the parameters of equation (1.31) is constant values of a00, E and
the known *n. Differential molar work of adsorption A is calculated as before by *
formula (1.25) for given values of equilibrium pressure p and temperature *T. *

Including the accepted values of deviation not more than 10% the magnitude of adsorption is calculated, because of (1.31) can be used in the range of fillings from 0,15-0,20 to 1,0. Equation (1.31) describes the adsorption only in micro pores. With significant development in microporous adsorbent volume and surface transient pores parameters a00 and E get effective values if the initial experimental adsorption isotherms are not adjusted for adsorption in transient pores [19,20].

If equation (1.31) consider as thermal [19,20] equation of adsorption of
standard vapor of parameters *a**0 **0*, E0 and n, then the equation of adsorption for
another vapor following defined earlier assumptions, expressed as:

𝑎 = 𝑎_{0}^{0 𝑝}^{∗}

𝑝_{0}𝑒𝑥𝑝 [− {( ^{𝐴}

𝛽𝐸_{0})^{𝑛}+ 𝛼(𝑇 − 𝑇_{0})}], (1.32)

where affinity coefficient; – appropriate density of adsorbate; A – differential molar work adsorption and α – thermal factor limiting absorption for this pair.

The coefficient before the exponent in equation (1.32) expresses the limit
adsorption at temperature T0, which does not necessarily coincide with the same
temperature for standard vapor. For some microporous adsorbents such as carbon,
the coefficient of affinity can be calculated, but using the physical constants of
adsorbate and standard substance. In this case, the transition from equation (1.31) to
the total equation (1.32) is not accompanied by an increase in the number of
constants that can be determined experimentally. They still remain two: *a*00 and
*E*0, assuming exponent n known.

*Defining parameters of transition pore *

The main parameters of transient pores of activated carbon is the value of the
volume of pores, specific surface and function of the distribution of equivalent
radius. Volume of transient pores in conventional samples of activated carbon is
within the range 0,02-0,10 cm^{3}/g. In this case, the specific surface of transient pores
is in the range of 20 to 70 m^{2}/g. The effective range of transient pores for up
distribution curves are usually stacked in the range of 4 to 20 nm. Pores of highly
porous silica gel, alum gel and alum-silicate catalysts also belong to the transition
type.

Volume of transient pores usually calculated from the equation:

*V*п = W*S* – Vmic, (1.33)

where W*S* – threshold volume of sorption space, which, in turn, is equal to:

*W**S** =а**S** · V**m*, (1.34)

where V*m* – molar volume of adsorbed material (adsorbate); *a**S* – threshold
absorption, which is corresponds to ^{𝑝}

𝑝_{𝑠} ≈ 1. (Defined by desiccator method in
conditions of full saturation of adsorbent sample by adsorbate vapour at ^{𝑝}

𝑝_{𝑠} ≈ 1.)
For samples of activated carbon with significant volumes of transient pores
formula (1.33) is not sufficiently satisfactory. A more precise calculation is
necessary to introduce an amendment to the adsorbed amount of vapor on the
surface of transient pores.

Then the corrected amount of transient pores 𝑉_{𝑝}^{∗} is:

𝑉_{𝑝}^{∗} = 𝑊_{𝑠} − (𝑎_{0} − 𝑎_{п})𝑉_{𝑚}, (1.35)
where V*m* – molar volume of adsorbate; a0 – value of adsorption of adsorbate vapor,
such as benzene or methanol at ^{𝑝}

𝑝_{𝑠} = 0,175 and ^{𝑝}

𝑝_{𝑠} = 0,533 respectively; a*п *– value
of adsorption of adsorbate vapor on the surface of transient pores prior to capillary
condensation.

Calculating the value of specific surface adsorption film formed in transient pores prior to capillary condensation, is performed by A. Kiselyov equation [15].

The general thermodynamic equation of capillary condensation formulated be A.V. Kiselyov (assuming that condensed film that is formed in the primary adsorption process can be considered as liquid phase) is:

– σ' · ds' = А*a* · da, (1.36)

where σ' – the surface tension of adsorption film; s' – its surface; A*a* – differential
molar work adsorption pair; a – the amount of adsorbed substance.

Based on the study of the properties of adsorption layers is assumed that the film adsorbed substance at the start of capillary condensation has practically normal value of the surface tension of the liquid σ.

Then the equation for determining the specific surface film s', which starts capillary condensation, will look like:

,

(1.37)

where σ – surface tension of liquid, vapor of which is absorbed (for benzene σ=28,9
ergs/cm^{2}, for methanol σ=22,6 ergs/cm^{2}); a0 – the amount of adsorbed substance at
the beginning of capillary condensation; a*S* – the amount of adsorbed substance at
full saturation, i.e., ^{𝑝}

𝑝_{𝑠} = 1; 𝐴_{𝑎} = 𝑅𝑇𝑙𝑛^{𝑝}^{𝑠}

𝑝 – differential molar work of adsorption.

Graphically, this calculation can be represented as in Fig. 1.4.

**Fig. 1.4. Graphic representation of dependence of differential molar of ****adsorption on the amount of adsorbed substance. **

After determination of the value of specific surface of transition pores s’, a
value of the adsorption of adsorbate vapors in transient pores prior to capillary
condensation is calculated a*n* using formula:

*a*n* = γ s’, * (1.38)

where γ – adsorption at the beginning of capillary condensation for surface unit non-
porous carbon adsorbent, which has temperature of pretreatment that is close to the
temperature of activated carbon. Adsorption isotherms of benzene at 20°C (γ=f(h))
for non-porous carbon adsorbent (e.g. soot) in a wide range of relative pressure
(^{𝑝}

𝑝_{𝑠} = 10^{−5÷0,3})) is expressed by the empirical Freundlich equation:

𝑙𝑔𝛾 = − (1,908 + 0,384𝑙𝑔 ^{𝑝}

𝑝_{𝑠}). (1.39)

After determination of a*n*, you can find corrected volume 𝑉_{п}^{∗} from the
equation (1.35) and V*mic**as:

*V**mic** = WS – Vp*. (1.40)

Calculation of the equivalent radius of transition pore and creation of integral and differential distribution curves of pore volume by equivalent radius (rn , nm) are performed using the equation of Thomson-Kelvin:

𝑟_{п=}2𝜎𝑉𝑚100
2,3𝑅𝑇𝑙𝑔𝑝𝑠
𝑝

, (1.41)

where σ – the surface tension of a liquid, a vapor of which is adsorbed; V*m* – its
molar volume.

Volume of transient pores Vp that corresponds to the radius obtained by the equation:

*V*p = α´ · V*m*. (1.42)

To calculate rp and Vp it is necessary to have values of ^{𝑝}

𝑝_{𝑠} and a' which are
taken from the desorption curve of adsorption isotherm hysteresis in capillary
condensation (^{𝑝}

𝑝_{𝑠} = 0,175 − 0,90 for benzene). Using obtained information the
integrated structural curve (V*п**, r) is built (Fig. 1.5), which describes the increase in *
the volume of pores Vп with a corresponding increase of r. Then, if necessary, a
differential curve of volume of transition pore distribution is built dependently from
their equivalent radius (^{∆𝑉}^{п}

∆𝑟 ,−

𝑟 ) (Fig. 1.6).

**Fig. 1.5. Graphic representation of integrated structural curve. **

**Fig. 1.6. Graphic representation of the distribution curve number of transient ****pores in the equivalent radius. **

The maximum on the distribution curve indicates the radius of the pores that
prevails in this adsorbent. This value r*max* should be increased by the value of the
thickness of the pre-adsorbed layer of molecules t.

The value of (t, nm) for the sorption of benzene vapor, which is accepted as the standard adsorbate on the carbon surface, can be estimated with satisfactory accuracy using different equations, such as equations adsorption (Harkins-Jura) [16]

𝑡 = ^{0,4715}

(𝑙𝑔^{𝑝𝑠}

𝑝−0,035)^{1/2}

. (1.43)

**Table 1.1. Parameters of porous structure of activated carbon [2] **

**Brand **

**Volume of common pores types, **

**cm**^{3}**/g ** **W****01****, **

**cm**^{3}**/g **

**W****02****, **

**cm**^{3}**/g ** **X****1****, nm ** **X****2****, nm **
**V****mic** **V****mez** **V****mac**

SCT 0,40-0,48 0,18-0,19 0,26-0,28 0,40-0,48 — 0,54-0,57 — SCT-1А 0,45-0,55 0,14-0,15 0,18-0,24 0,46-0,57 — 0,60-0,62 — SCT-1B 0,42-0,50 0,15-0,17 0,18-0,22 0,43-0,59 — 0,61-0,64 — SCT-2А 0,37-0,42 0,18-0,22 0,20-0,22 0,38-0,45 — 0,50-0,52 — SCT-2B 0,35-0,40 0,17-0,20 0,12-0,18 0,37-0,42 — 0,54-0,55 — SCT-3 0,37-0,46 0,06-0,09 0,25-0,32 0,37-0,46 — 0,47-0,55 —

**Continuation of Table 1.1 **

SCT-3С 0,35-0,45 0,06-0,08 0,15-0,20 0,43-0,55 — 0,50-0,55 — SCT-3U 0,37-0,42 0,24-0,28 0,21-0,24 0,39-0,43 — 0,70-0,75 — SCT-4 0,40-0,42 0,15-0,20 0,12-0,20 0,42-0,46 — 0,59-0,60 — SCT-6А 0,57-0,60 0,15-0,25 0,15-0,25 0,59-0,62 — 0,70-0,73 — SCT-6B 0,55-0,61 0,17-0,28 0,15-0,30 0,57-0,60 — 0,66-0,69 — SCT-7А 0,47-0,50 0,20-0,22 0,15-0,20 0,48-0,53 — 0,64-0,70 — SCT-7B 0,48-0,52 0,21-0,23 0,16-0,17 0,49-0,55 — 0,66-0,72 — SCT-7С 0,44-0,49 0,15-0,17 0,11-0,25 0,45-0,53 — 0,62-0,66 — SCT-10 0,40-0,42 0,20-0,21 0,21-0,27 0,43-0,44 — 0,59-0,65 — АРТ-1 0,43-0,45 0,15-0,20 0,12-0,30 0,44-0,47 — 0,60-0,67 — АРТ-2 0,45-0,48 0,10-0,20 0,19-0,32 0,45-0,50 — 0,54-0,56 — АG-PR 0,30-0,35 0,10-0,12 0,40-0,49 0,20-0,32 0,10-0,12 0,70-0,80 1,00-1,20 AG-ОС 0,45-0,47 0,05-0,15 0,10-0,20 0,47-0,52 — 0,70-0,72 —

For nitrogen, which is more often used abroad as a standard vapor, most recommended a formula of J. de Boer:

𝑡 = ^{0,4584}

(𝑙𝑔^{𝑝𝑠}
𝑝)^{1/3}

, (1.44)

then

*r*p* = r**max **+ t. * (1.45)

Parameters of porous structure of major industrial brands of carbon adsorbents for adsorption of gases and vapors are given in Table 1.1 and 1.2.

**Table 1.2. The specific geometric surface of micro pores ****Brand of carbon ** **S, m**^{2}**/g **

SCT 1200-1500

АCB 1000-1200

AG-ОС 900-1000

PAU-1 2000-2500

There are the following main brands of activated carbon of this type: SCT, SCT-1, SCT-2, SCT-3, SCT-3С, SCT-3U, SCT-4, SCT-6, SCT-7, SCT-10, SCPТС,

АРТ, АG-PR, AG-ОС, which are used for separation from air vapors of organic compounds and for the removal of gas emissions.

Activated carbon of this type is characterized by: high adsorption and holding capacity; sufficiently high mechanical strength; high activity (coal for recovery of organic solvents vapor, gasoline, ethanol, ethyl acetate, dichloroethane, etc.).

Depending on the design of the absorber (stationary, «boiling» or non- stationary layer) different requirements for fractional composition of coal are imposed.

Comparative evaluation of adsorption and strength of active carbon gas type (Table 1.3) shows almost complete identity of its adsorption characteristics by the bulk density, the volume of micro pores and the values of the time of protective action for substances that are badly and well absorbed. Domestic gas coal inferior to foreign models by strength.

**Table 1.3. Parameters of industrial gas type activated carbon [5] **

**Brand **

**Apparent **
**density, **

**kg/dm**^{3}

**Mechanical **
**strength, % **

**The volume **
**of micro- **
**pores, cm**^{3}**/g **

**Duration of protective **
**action, min **
**chloroethyl ** **benzene **

AG-2 0,58 75 0,32 45 50

SCT-2 0,49 74 0,45 70 60

SCT-6 0,42 73 0,58 65 75

PK 0,52 - 0,30 57 55

PB 0,50 93 0,46 55 76

PC 0,56 93 0,26 52 45

**1.1.2 Sorption from water solutions **

Sorption of liquid solutions is more complicated than the gas-vapor mixture, so that involves the interaction of the sorbent with the substance that is absorbed and the solvent (water). It also should take into account the interaction of the solvent with adsorbate. Therefore, despite the fact that the sorption from aqueous

solutions is studied and used nearly 200 years, it has been studied much less adsorption than sorption of gas-vapor phase. In general, mechanism of adsorption from solutions in one form or another is explained by concepts, derived from the gas phase, supplementing by limiting conditions specific to the liquid phase. Differences in the approach to such a transition influence on the form and accuracy of models and calculations of sorption of water purification.

**Fig. 1.7. The classification of adsorption isotherms according to BET [19]. **

Basics sorption properties of the material and the nature of sorption on it of certain substances can be obtained, as in the case of sorption process in gaseous phase, from sorption isotherms, describing the dependence of sorption capacity A Concentration C component that is absorbed and at a constant temperature: A=f(C) for the liquid phase. Brunauer, Emmet and Teller (BET) divided sorption isotherms into 5 main groups (Fig. 1.7). Convex plots of isotherms I, II and IV types indicate the presence of micro pores in sorbents, but also, sorbents II and IV are also macro pores. Isotherms III and V types are less common and described strong intermolecular interactions in solution. The steepness isotherms of type I characterizes the size micro pores adsorbents: a – ultra microporous, – a

microporous. Isotherm IVb correspond to transitional porous sorbent; IVc – uniformly macro porous and IVa – with a mixed structure.

The most common and complete classification of sorption isotherms of the liquid phase was given Smith (Fig. 1.8). Concave isotherm S-type is rare. Isotherms Langmuir (L-type) correspond to I and III types classification BET (Fig. 1.7).

Type H is typical for substances with high affinity (i.e., high ratio of molar
volumes 𝑉_{𝑚}/𝑉_{𝑚}^{𝑐𝑚}*, where V*m and 𝑉_{𝑚}^{𝑐𝑚}– molar volumes of investigated and standard
materials), in which a large sorption capacity is achieved at very low concentrations.

In case when the law of Henry is valid (type C), sorption capacity proportional to a
final concentration of the solution. This is a common case in water purification,
examples of which are given in Fig. 1.9. Type sorption isotherms often depends on
the concentration of substances in solution. For example, sorption on soot of
surfactant with a concentration of 0,1; 0,2 and 0,3 g-eq/dm^{3} isotherms are shaped
L2, L3 and H3 (Fig. 1.8), due to the volume association of ions in surfactant.

**Fig. 1.8. The classification of solutions adsorption isotherms according by ****Smith. **

a) b)

**Fig. 1.9. Sorption isotherms of biochemical treatment of industrial wastewater ****(a) and pattern for definition of sorption capacity by linear isotherms in standard **

**conditions (B), (Г - excessive amount of substances absorbed in the adsorbed state): **

charcoal AG-3 - ● and ■; AB from lignin - ○, □ and Δ; wastewater production of synthetic rubber - ●; production of brake fluid - ■; hydrolysis plants - ○, □ and Δ; experimental

points - ▲; estimated value Г_{р}^{ст}* at С*_{р}^{і} = С_{р}^{с𝑡}* — ▼). *

In practice, engineering studies and calculations often use a simple empirical Freundlich equation:

*a = KС** ^{n}*,
where K and n – constants.

Using Freundlich equation in range of average concentrations well coincides
with the experimental data; isotherm is linear in coordinates (lg C, lg *a). The *
coefficients *K and n for Freundlich equation for certain substances listed in *
Table 1.4.

The equation of Langmuir sorption isotherms derived from molecular-kinetic theory and ideas about the monomolecular nature of the sorption. For solutions, it is:

𝑎 = ^{𝑎}^{𝑚}^{𝑏𝐶}

1+𝑏𝐶, (1.46)

where a *m* – capacity of monolayer; b – constant; C – the concentration of the
substance.

At low concentrations equation Langmuir transforms into the Henry equation:

*a = KС (K = a**m**b), * (1.47)

i.e. sorption capacity is directly proportional to the concentration of substances in
solution. Langmuir sorption isotherms is linear in coordinates (a^{–1}, *с*^{–1}), which
allows the graphically determine the coefficients a*m* and b.

Fundamentals of adsorption thermodynamics in solutions were first formulated by Gibbs more than 100 years ago [13]. He introduced the concept of excess adsorption of Г, i.e. excessive content of adsorbed substance in the adsorbed phase compared with its content in the solution. The magnitude of the excess adsorption can be easily determined by the formula:

Г = (С_{0}− С_{𝑘})^{𝑉}

𝑚, (1.48)

where *C**0* and *C**k* – concentration of substances in solution before and after
adsorption; V – volume of the solution; m – mass of adsorbent.

**Table 1.4. Coefficients K and n in Freundlich equation **

**Substances ** **K ****n **

Amyl acetate 4,8 0,49

Aniline 25 0,32

Benzenesulfonic acid 7 0,17

Butanol 4,1 0,44

Vinyl chloride 0,37 1,09

The thermodynamic approach to the problem of sorption is the most
common and allows you to evaluate sorbability of molecules using the value of the
maximum of mass transfer from the solution to the surface of the sorbent. During
the sorption material from the water, the free energy of the system ∆G*ads* decreases,
A. Koganovskiy [10] proposed to use this value to predict effectiveness of
removal of solutes from water. The equilibrium constant of sorption from
diluted solutions K*ads* associated with ∆Gads by ratio 𝑙𝑔𝐾_{𝑎𝑑𝑠} = ^{∆𝐺}^{адс}

𝑅𝑇 , which implies that the bigger ∆Gads the better sorbability of substance. The calculated value of ∆Gads for a number of compounds and functional groups during the sorption on KAD-Iodine active carbon, WAU and OC-A presented in Table 1.5.

**Table 1.5. The values of adsorption ∆G****ads**** to compounds and functional groups **
**on activated carbon [17] **

**Compound ** **∆G****ads****, kJ/mol ** **Functional groups ** **∆G****ads****, kJ/mol **

*n-Nitroaniline * 24,8 -C6H5 21,16

Naphthalene 24,6 -NO2 2,60

Naphthol 23,4 -OH (primary) 2,30

Aniline 22,3 =CH2 (in the

alcohols and acids) 2,18

Phenol 21,3 -COOH 1,63

Chlorobenzene 19,4 -Cl l, 38

Chloroform 18,6 -NH2 1,05

Dichloroethane 18,3 =C=C= 0,88

Methylamine 18,3 -CH3 0,46-0,59

Triethanolamine 17,8 OH (secondary or

tertiary) 0,25

Acetic acid 17,8 -OH 0,042-0,084

Nylon acid 17,7 -OH (if NHx is

presented) -0,25

**Continuation of Table 1.5 **

Formic acid 17,7 =CH2 (if -NH2 is

presented) -0,42

Ethylamine 17,7 -SO3H -1,09

Ethylenhlorhidryn 14,3

Butyric acid 13,7

Oxalic acid 13,5

Some surfactant ∆Gads=21-29 kJ/mole. With ∆Gads lower than 16-17 kJ/mole (or less than 42 kJ/g – to surfactants that are presented in the water as micelles) sorption is relatively low.

The obtained value ∆Gads for functional groups allows to calculate with sufficient accuracy ∆Gads for a broad class of different organic compounds (as the sum of ∆Gads components), thus predicting sorption capacity of these substances.

However, data for calculation ∆Gads is reliable only for aromatic compounds.

The idea of the possibility of applying the theory of volume filling of micro pores to describe the sorption of liquid phase was designed by Eltekov and Stadnik [15]. This theory uses the idea of the absence of impact of physical properties of sorbate in the bulk phase on sorption capacity in micro pores of coal and absence of associative, ionic and hydrogen bonds between the molecules of a substance that is absorbed and water, as well as within sorbate. This theory can be used to calculate the sorption of very dilute solutions of substances with limited solubility. The equation of sorption isotherms on microporous AC in this case takes the form:

𝑙𝑔Г_{р} = 𝑙𝑔^{𝑊}^{0}

𝑉_{𝑚}− 2,3^{𝐵𝑇}^{2}

𝛽^{2} (𝑙𝑔^{𝐶}^{𝑠}

𝐶_{𝑝})

2

, (1.49)

where W*0* – the threshold volume of micro pores of the adsorbent; V*m* – molar
volume adsorbate is calculated by Fig. 1.10; 𝐵 = (^{4,574}

𝐸 )^{2}– structural and energy
constant; C*S* – solubility of compounds in water at a given temperature; C*p* –
equilibrium concentration in water.