A quantitative characteristic that determines the dynamics of the chemical transformation process is the reaction rate. The rate of a chemical reaction is the change in the number of moles of one of the components per unit volume:
where V is volume; N is the number of moles formed or reacted; t is time.
Introducing the concentration of the substance C as the number of moles per unit volume, i.e. C=N/V, we obtain
, (2)
. (3)
For reactions occurring at constant volume in a closed system, the second term of equation (4.3) is equal to zero. This equation is reduced to the form
or 
(4)
Here the sign (+) indicates that the substance accumulates as a result of the reaction, the sign (–) indicates that the concentration of the substance decreases.
This definition of speed chemical reactions is valid for homogeneous reactions, where all the reacting components are in one phase with a volume V. If we assume that the mechanism causing the reaction to occur is the collision or interaction of one molecule A with a molecule B, which leads to the formation of one molecule of the reaction product C, then the number collisions of molecules A and B is proportional to the rate of reaction A + B. Equation 
is called the stoichiometric reaction equation. Since the number of collisions at a given temperature is proportional to the concentration of the reactants in the mixture, the rate of disappearance of substance A can be expressed by the equation
(5)
where k is the rate constant of the chemical reaction (actually equal to the reaction rate at unit concentrations). The dependence of the reaction rate constant on temperature is usually expressed by the Arrhenius law:
,
where k 0 is a pre-exponential factor depending on the number of collisions of reacting molecules; E – activation energy, kJ/kmol; R – gas constant, kJ/(kmol*deg); T – absolute temperature.
For a given type of molecules at a given temperature k=const. The rate of reaction, expressed in terms of the powers of their stoichiometric coefficients, is called the law of mass action. In the general case, for a reaction involving jcomponents with concentration C j and stoichiometric coefficient j, the reaction rate can be described by the relation
Where 
- starting materials.
All reactions are divided into simple and complex, elementary and non-elementary. If one stoichiometric equation is sufficient to describe the course of a given reaction, then it is classified as a simple reaction; if not, then it is classified as a complex reaction. Complex reactions are divided into the following types:
Consecutive
Parallel
Mixed
Reactions that are considered to occur in one step are called elementary; their speed is uniquely obtained from the stoichiometric equation. In the absence of a direct connection between the stoichiometric equation and the rate expression, this reaction is non-elementary.
The molecularity of a reaction is the number of molecules participating in the elementary act of a reaction, which determines the rate of the process. Mono-, bi- and trimolecular reactions are known. Molecularity refers only to an elementary reaction and is expressed as an integer.
The rate of reaction in which substances A, B, C, …, S take part can be expressed by the law of mass action
The value of n, found from the relation n = a + b + c +…+s, is considered the general order of the reaction. The experimentally determined values of a, b, c,...,s do not necessarily coincide with the stoichiometric coefficients of the reaction equation. The power to which the concentrations (a,b,c, ...,s) are raised is called the order of reaction for this substance.
When studying the kinetics of a chemical reaction, two problems are considered: direct and inverse. Direct task consists in finding kinetic curves using known reaction mechanisms and kinetic equations for given values of reaction rate constants. Curves that describe changes in the concentrations of substances over time are called kinetic curves. Inverse problem consists in determining the reaction mechanism and unknown rate constants from known experimental data (kinetic curves).
Methodology for solving the direct problem of kinetics for non-elementary reactions.
Let us denote by N j 0 the number of moles of component j before the reaction, and by N j the number of moles of the same component after the reaction. If j is the stoichiometric coefficient for the j component, then the value of X, determined from the relation
(6)
called the extensive degree of completeness. It expresses the total amount of change and is constant for a given reaction.
For a reaction occurring at a constant volume V=const, the molar concentration of the substance j is
. (7)
Then formula (6) is written as follows:
(8)
. (9)
Value x=X/V – intensive degree of completeness:
;
Where 
,m – number of reactions.
For V=const, dividing all terms of equation (4.13) by volume, we obtain
, (10)
Where 
,m – number of reactions.
IN physical chemistry the rate of a chemical reaction is determined according to the equation:
Where dq– change in the mass of the reactant, mol.
dt– time increment, s.
V– measure of reaction space.
There are homogeneous chemical reactions in which all participating substances are within one phase (gas or liquid). For such reactions, the measure of the reaction space is volume, and the dimension of speed will be: .
Heterogeneous chemical reactions occur between substances in different phases (gas-solid, gas-liquid, liquid-liquid, solid-liquid). The chemical reaction itself is realized at the phase interface, which is a measure of the reaction space.
For heterogeneous reactions, the speed dimension is different: .
The change in the mass of reacting substances has its own sign. For starting substances, the mass decreases during the reaction, the change in mass has a negative sign, and the rate takes negative meaning. For the products of a chemical reaction, the mass increases, the change in mass is positive, and the sign of the speed is also taken to be positive.
Consider a simple chemical reaction
Simple reactions include those that are carried out in one stage and go to the end, i.e. are irreversible.
Let us determine the rate of such a chemical reaction. To do this, first of all, it is necessary to decide which of the substances will determine the reaction rate: after all, A and B are the starting substances, and the change in their masses is negative, and C is the final product, and its mass increases with time. In addition, not all stoichiometric coefficients in the reaction are equal to unity, which means that if the consumption of A for some time is equal to 1 mole, the consumption of B during the same time will be 2 moles, and accordingly the rate values calculated from the change in masses of A and B will differ twice.
For a simple chemical reaction, a single measure of rate can be proposed, which is defined as follows:
Where r i– speed according to the i-th reaction participant
S i– stoichiometric coefficient of the i-th reaction participant.
Stoichiometric coefficients for starting substances are assumed to be positive; for reaction products they are negative.
If reactions take place in an isolated system that does not exchange substances with the external environment, then only a chemical reaction leads to a change in the masses of substances in the system, and, consequently, their concentrations. In such a system, the only reason for changes in concentrations is WITH is a chemical reaction. For this special case
The rate of a chemical reaction depends on the concentrations of the substances involved and on the temperature.
Where k– rate constant of the chemical reaction, C A, C B– concentrations of substances, n 1, n 2– orders for the relevant substances. This expression is known in physical chemistry as the law of mass action.
The higher the concentration values, the higher the rate of the chemical reaction.
Order ( n) is determined experimentally and is associated with the mechanism of a chemical reaction. The order can be an integer or fractional number; there are also zero-order reactions for some substances. If the order is i th substance is zero, then the rate of the chemical reaction does not depend on the concentration of this substance.
The rate of a chemical reaction depends on temperature. According to Arrhenius's law, the rate constant changes with temperature:
Where A– pre-exponential factor;
E– activation energy;
R– universal gas constant, constant;
T- temperature.
Like the reaction order, the activation energy and pre-exponential factor are determined experimentally for a specific reaction.
If a chemical reaction is carried out in a heterogeneous process, then its speed is also influenced by the process of supplying starting substances and removing products from the chemical reaction zone. Thus, a complex process takes place, in which there are diffusion stages (supply, removal) and a kinetic stage - the chemical reaction itself. The speed of the entire process observed in the experiment is determined by the speed of the slowest stage.
Thus, by influencing the speed of the diffusion stage of the process (mixing), we influence the speed of the entire process as a whole. This influence affects the value of the pre-exponential factor A.
Most chemical reactions are not simple (i.e. they do not occur in one stage and not to completion) - complex chemical reactions:
a) AB – reversible;
b) A→B; B→C – sequential;
c) A→B; A→C – parallel.
For a complex chemical reaction there is no single measure of speed. Unlike simple, here we can talk about the rate of formation and destruction of each chemical substance. Thus, if chemical reactions occur in a system and involve n substances for each n substances have their own speed value.
For any substance, the rate of formation and destruction is the algebraic sum of the rates of all stages involving this substance.
The mathematical description of the physical and chemical processes listed above has great importance when creating dynamic models that reproduce the behavior of processes over time. Such models make it possible to predict the future state of the process, determine the optimal trajectories of its flow, and, consequently, ways to increase productivity or efficiency. This also opens up the possibility of automating control using a computer.
Features of the kinetics of homogeneous and heterogeneous reactions
The rates of chemical reactions depend on a number of factors: the concentration of reactants, temperature, pressure (if gaseous substances are involved in the reaction), the presence of catalysts, and in the case of heterogeneous transformations, in addition, on the state of the surface, conditions of heat and mass transfer. Let us consider, in this regard, the features of the kinetics of homogeneous and heterogeneous reactions. In homogeneous reactions, the starting substances and reaction products are in the same phase (gas or liquid), while molecules, atoms or ions can interact throughout the occupied volume. An example is the combustion reactions of and, which are part of coke oven (natural) gas:
In heterogeneous reactions, the interacting substances are in different phases, and the process of chemical transformation occurs at the interface between these phases.
The oxidation reaction of carbon in the slag-metal system, as an example of a heterogeneous reaction
Reaction
An example is the reaction of carbon oxidation in the slag-metal system in relation to the bath of an open-hearth or electric furnace.
Three stages of reaction
At least three stages can be distinguished here:
- diffusion oxygen from the slag into the metal to the place of reaction (interface: metal - gas bubble, unfilled pores on the hearth or surface of pieces of ore and lime);
- a chemical reaction between oxygen and carbon of the metal at the interface between the mentioned phases;
- release of a gaseous reaction product from the metal.
It should be noted that with more detailed analysis each of the listed stages can be divided into several more stages, reflecting, in particular, adsorption-chemical acts at the phase boundaries (see Fig. 1.3 - 1.5). The rate of such a complex heterogeneous reaction is limited by the slowest stage of the process. For the conditions of open-hearth and electric furnace melting processes, this stage is diffusion oxygen from slag to metal. In the converter process, due to the high intensity of oxygen blowing and the high degree of dispersion of the interacting phases, adsorption-chemical acts at the phase interface, the magnitude of which increases by several orders of magnitude compared to hearth steelmaking processes, may be limiting.
Description of diffusion and mass transfer
Diffusion
Before continuing to describe the kinetics, let us dwell on the laws of diffusion, which is of great importance in heterogeneous processes, since their rates can be determined by the supply of reactants and the removal of reaction products.
Diffusion is the process of spontaneous movement of a substance aimed at equalizing concentrations in the volume. Moving force diffusion is the concentration gradient determined by the change in the concentration of the substance per segment of the path in the direction of diffusion. The increment in the amount of a substance transferred by diffusion is proportional to the diffusion coefficient, the concentration gradient, the cross-sectional area of the medium through which the substance is transferred, and time.
and moving on to infinitesimal increments and the rate of diffusion (mass flow through a unit area)
we get the equation
(3.57) describing stationary diffusion and called Fick’s first law.
Diffusion of a system with distributed parameters according to Fick's law
For the case of a system with distributed parameters, when the concentration changes along all three coordinates, in accordance with Fick’s second law, the diffusion equation takes the following form:
(3.58) where is the density of substance sources, for example, the amount of substance formed as a result of chemical reactions per unit volume per unit time.
Conditions for the applicability of molecular diffusion
It must be emphasized that equations (3.57) and (3.58) relate to molecular transport in a stationary medium and are valid for an isothermal process and the case when diffusion of this component does not depend on the diffusion of other components.
Stokes-Einstein formula
Under these conditions, the dependence of the diffusion coefficient on temperature, viscosity of the medium and the radius of diffusing molecules is determined by the Stokes–Einstein formula:
(3.59) where
And – gas constant and Avogadro’s number.
Turbulent diffusion
In most metallurgical units, especially steelmaking, the predominant role is played not by molecular, but by turbulent diffusion, caused by thermal convection and the work of mixing rising bubbles and jets of purge gas penetrating into the bath.
For example, the value of the atomic diffusion coefficient in stationary molten iron at 1500 – 1600°C is – . The value of the coefficient of turbulent diffusion in an open-hearth bath, depending on the decarbonization rate, is 0.0025 -0.0082, and in the converter process 2.0 -2.5, i.e. three orders of magnitude higher.
Diffusion taking into account the influence of convection
Taking into account the influence of convection, the diffusion equation takes the following form:
(3.60) where is the speed of matter transfer, m/s.
More often, in cases of the predominant influence of turbulent diffusion, an empirical equation of the form is used
– diffusion flow;
– concentration difference;
– coefficient of mass transfer (turbulent diffusion).
Empirical equation for turbulent diffusion
When assessing the conditions of mass transfer and possible areas of use of the above equations, it is advisable to use the methods of theory similarities, which, as shown in the analysis of the second theorem similarities, opens up the possibility of generalizations.
First of all, it should be noted that diffusion, viscosity and thermal conductivity are similar processes that characterize similar types of transfer: diffusion– mass transfer, viscosity – momentum transfer, thermal conductivity – heat transfer. Molecular transfer coefficients (viscosity, diffusion and thermal diffusivity) have the same dimension ().
Reynolds number
In accordance with the second theorem similarities you can significantly reduce the dimension of the problem and increase its generality if you move from the primary physical parameters to their dimensionless complexes, called criteria or numbers similarities. One of these well-known criteria is Reynolds number, which allows one to evaluate the nature of fluid movement depending on its average speed, pipeline (flow) diameter and kinematic viscosity:
(3.62) This criterion is a measure of the ratio of inertial forces, characterized by speed, to internal friction forces, characterized by viscosity. Reynolds number reflects the degree of flow stability in relation to external and internal disturbances. The value of the number at which the stability of fluid motion is disrupted is called critical and is designated . When any disturbances occurring in the flow fade over time and do not change the general laminar nature of the flow. When disturbances can spontaneously increase, which leads to turbulization of the flow. In reality, there is no sharp boundary in the transition from laminar to turbulent movement; there is a transition regime in which the turbulent regime predominates in the main part of the flow, and laminar movement is possible in the layer adjacent to the walls.
When value<2300 поток является ламинарным. В этой области для описания диффузии могут использоваться уравнения (3.57) или (3.60). Область значений 2300<<10000 является переходной. Здесь, в зависимости от степени развития турбулентности и наличия ламинарного слоя, целесообразно использовать уравнения (3.60) или (3.61).
For values >10000, as a result of the predominant influence of inertial forces, the flow becomes turbulent. Under these conditions, it is unlawful to use equations in which molecular diffusion coefficients appear. With this type of flow, equations of the form (3.61) are used to describe mass transfer, in which the mass transfer coefficient is determined either through the work of mixing or experimentally-statistical methods based on measured process speed and concentration differences.
Equations of the kinetics of homogeneous reactions
Speed reaction
The reaction rate is the derivative of concentration with respect to time
Molecularity of the reaction
Chemical reactions vary in molecularity and reaction order. Molecularity is determined by the number of molecules participating in an elementary act of chemical interaction. Based on this criterion, reactions are divided into mono-, bi- and trimolecular. Each type of chemical reaction corresponds to certain kinetic equations that express the dependence of the reaction rate on the concentration of the reactants. In accordance with the laws of formal kinetics, including the law of mass action, the rate of any reaction of the form
In the forward direction it is proportional to the concentrations of the reactants and is represented by the equation
(3.63) where
is a rate constant that makes sense at .
Reaction order
Definition
The order of a reaction is the sum of the exponents in which concentrations enter the kinetic equations. The above reaction is therefore third order. In reality, third-order reactions are rarely observed. Equations similar to expression (3.63) are based on simplified ideas that reactions occur during the simultaneous collision of such a number of molecules that correspond to the sum of the stoichiometric coefficients. Most real reactions proceed according to more complex laws with the formation of intermediate products. Therefore, equations like (3.63) are correct only for elementary reactions occurring in one stage, i.e., the order of reactions cannot be determined by the form of the stoichiometric equation; most often it is determined experimentally. For this purpose, the reaction rate is found at a constant temperature depending on the concentration of the reagents; by the type of dependence obtained (the exponents at concentrations), one can judge the order of the reaction. For this purpose, you can use one of the parametric identification methods discussed in Chapter. 5.
Let us dwell on the form of kinetic equations depending on the order of the reaction.
Zero order reaction
In zero-order reactions, the rate is constant over time
(3.64) After integration we obtain
– integration constant, which has the meaning of the initial concentration at =0.
Thus, in the case under consideration, the concentration of the reagent decreases linearly with time.
First order reaction
The first order reaction is represented schematically as follows:
The kinetic equation has the form:
(3.65) and its solution
shows that the concentration of the original component decreases exponentially with time (Fig. 3.2).
Rice. 3.2 Change in concentration and its logarithm over time during first-order reactions
The solution to this equation can be presented in another form, more convenient for determining the reaction rate constant. As a result of separation of variables and choice of integration limits
At a temperature
we get a solution
from which we can distinguish that it linearly depends on time. If the experimental data fits on a straight line (see Fig. 3.2), then this indicates the first order of the reaction. The value is determined from the angle of inclination of the straight line.
Second order reaction
The second order reaction scheme has the form
Or, for example,
And the reaction rate is described by the equation
(3.66) which, at the same concentrations, takes the form
After separating the variables and integrating the relation
we get the relation
(3.67) which can be used to determine . If the initial concentrations of the reactants are unequal and equal to and , respectively, and the concentration of the product at the moment is , then we obtain the equation
Taking the logarithm of which gives
(3.68)
Backlash
All of the above kinetic equations relate to reactions occurring only in the forward direction, i.e., under conditions far from equilibrium, which can, for example, be ensured by the continuous removal of reaction products. In the general case, a reverse reaction can also occur, then the overall rate for a reaction of the form
(3.69) As the reagents are consumed and the product is formed, the rate of the direct reaction decreases and increases. When the total speed is zero, equilibrium occurs. Then
or
(3.70) i.e. the equilibrium constant is equal to the ratio of the rate constants of the forward and reverse reactions. At the same time, relation (3.70) is nothing more than the expression law of mass action, obtained in this case through the kinetics equation.
The effect of temperature on the rate of a chemical reaction
Let us now dwell on the issue of the influence of temperature on the rate of chemical reactions. The dependence of the reaction rate constant on temperature was first obtained empirically by Arrhenius, and somewhat later found theoretical confirmation based on the mechanism of active collisions. In differential form it has the following form:
– activation energy.
After integration, provided that , we obtain
– constant, meaning the logarithm of the rate constant at infinite temperature ().
This relationship can also be represented in the form
(3.73)
Activation energy
The value can be determined from the tangent of the angle of inclination of the straight line (3.72), constructed in coordinates, for which it is necessary to measure the rate constants at different temperatures.
The physical meaning of activation energy and the mechanism of chemical reactions can be explained on the basis of the theory of active collisions.
The probability of an elementary chemical reaction occurring depends on the nature of the reacting substances (bond energy) and on the temperature, which increases the overall energy level of the chaotic movement of molecules. In Fig. 3.3, where and are the activation energies of the forward and reverse reactions, it is clear that as a result of an exothermic reaction, the internal energy of the system decreases by an amount equal to the thermal effect of the reaction.
Rice. 3.3 On the issue of activation energy
However, on the way from the initial state to the final state system must cross a certain energy barrier, and the lower the barrier (lower the activation energy), the larger the proportion of molecules at any given moment is capable of reacting and the higher the reaction rate will be.
A more detailed presentation of molecular kinetics, which was further developed in the theory of the transition state, is beyond the scope of this manual.
Relationship between mass transfer and kinetics in heterogeneous reactions
A visual representation of the relationships between mass transfer and kinetics in heterogeneous processes is given by the diagram of the generalized model shown in Fig. 3.4.
Interfaces
In the first case, the processes are not accompanied by changes in the chemical composition in the boundary layer. The interaction at the interface of multicomponent systems is characterized, most often, by a change in the composition of the boundary layer, while the overall rate of the process is determined by the rate of concentration equalization in the boundary layer, i.e., the rate of diffusion. The diffusion boundary layer is a thin layer adjacent to each phase of a two- or multicomponent system (Fig. 3.6).
Rice. 3.6 Diffusion boundary layer
- – solid
- – diffusion boundary layer
- – liquid
With increasing mixing intensity, the thickness of this layer decreases and, consequently, the influence of diffusion on the rate of the entire process decreases. Similar phenomena are observed when pieces of coke and agglomerate are dissolved in blast furnaces or pieces of lime in steel-smelting units.
IN systems, which are characterized by sequential occurrence of chemical and physical processes, the speed of the entire process is determined by the slower stage. In this regard, the reaction can be in the kinetic or diffusion region. If the rate of chemical reaction and diffusion are comparable, the process is a complex function of kinetic and diffusion phenomena and is considered to occur in the transition region.
Stages of heterogeneous reactions
In most cases heterogeneous reactions proceed through a number of stages, the most characteristic of which are the following:
- diffusion particles of starting substances to the phase interface (reaction zone);
- adsorption of reagents on the surface;
- chemical reaction on the surface;
- desorption reaction products at the interface;
- diffusion of these products from the reaction zone deep into one of the phases.
Stages 1 and 5 are diffusion, and stages 2 – 4 are kinetic.
Kinetic resistance of a heterogeneous reaction
The observed kinetic resistance of a heterogeneous reaction proceeding through a number of successive stages is equal to the sum of the kinetic resistances of its stages
(3.74) where
– rate constant of the total (observed) process;
– rate constant of the kinetic stage;
– rate constant (diffusion coefficient) of the diffusion stage.
The stage that has the greatest resistance is limiting.
Features of processes in the kinetic region
Let us consider the main features of processes in the kinetic region:
The first three features can also be observed if the process is in the transition region. The fourth feature is the main experimental confirmation that the process is in the kinetic region.
Features of processes in the diffusion region
Main features of processes in the diffusion region:
- first order process;
- weak dependence of the process speed on temperature and on the size of the phase interface;
- 3) a sharp influence on the speed of the process of hydrodynamic and aerodynamic conditions of the process.
The most important sign that a process is in the diffusion region is the first and third features.
Lime dissolution as an example of a heterogeneous process
Let us consider, as an example, the process of dissolving lime in the main steel-smelting slag, which takes place in open-hearth, electric furnaces and converters. This process, which is typically heterogeneous, depends, first of all, on the convective flows developing in the bath, i.e., on the mixing power, and consists of the following stages: supply of slag components (, etc.) to the surface of the lime pieces; penetration of solvents into the pores of pieces of lime, which facilitates the transition of calcium oxide into the liquid phase due to the formation of fusible compounds; removal of these products, saturated, from the surface of pieces of lime in the volume of slag. The supply of solvents to the surface of pieces of lime and the removal of dissolved ones is determined by the laws of convective diffusion within the diffusion boundary layer at the surface of pieces of lime. The diffusion equation has the form.
Theoretical part
Goal of the work
Familiarize yourself with methods for constructing kinetic models of homogeneous chemical reactions.
Construct a kinetic model of a given chemical reaction.
Investigate the effect of temperature on the yield of products and the degree of conversion.
kinetic chemical reaction homogeneous
Kinetics of homogeneous chemical reactions
The rate of a chemical reaction is the change in the number of moles of reactants as a result of chemical interaction per unit time per unit volume
One of the basic laws of chemical kinetics, which determines the quantitative laws of the rates of elementary reactions, is the law of mass action.
The rate constant of a chemical reaction is a function of temperature, and its dependence on temperature is expressed by the Arrhenius law:
Work order
Create a kinetic model in accordance with the given reaction scheme based on the law of mass action.
Develop an algorithm for calculating the compiled kinetic model using the Euler numerical method.
Develop a program for calculating kinetics taking into account the temperature dependence of the rate constants of a homogeneous chemical reaction.
Discuss the results. Draw conclusions from the work.
Practical part
Task No. 11
A → B ↔ C
Ca0=0.8 mol/l; Cb0=Cc0=0mol/l;
h1=20; K10=0.3; K20=0.25; K30=0.18; E1=10; E2=8; E3=9; T1=393; T2=453; R=8.31;lb2;mas=array of real;tempk=10.0; h=1.0; Ca0=0.8; Cb0=0; CC0=0; K10=0.3; K20=0.25; K30=0.18; E1=10; E2=8; E3=9; T1=393; T2=453; R=8.31; h1=20;C, f: mas; i:integer; T, k1, k2, k3, temp:real;:=T1;T<=T2 do begin:=k10*exp(-E1/(R*T));:=k20*exp(-E2/(R*T));:=k30*exp(-E3/(R*T));("T=", T:5:2, " ",
"k1=", k1:5:5, " ",
"k2=", k2:5:5, " ",
"k3=", k3:5:5, " ");:=0; C:=Ca0; C:=Cb0; C:=Cc0;temp<=tempk do begin:=-k1*C;:=K1*C-K2*C+K3*c;:=K2*C-K3*C;i:=1 to 3 do C[i]:=C[i]+h*f[i];("C=", C:5:2, " ",
"C=", C:5:2, " ",
"C=", C:5:2, " ",
"temp=", temp:5:2);:=temp+h; end;
T:=T+h1; end;.
Temperature 393 K
k1=0.29908k2=0.24939k3=0.17950Concentration C1TimeConcentration C2TimeConcentration C3Time0.5600.240000.3910.3510.0610.2820.3920.1420.1930.430.2130.1440.3940.2 740.0950.3950.3250 ,0760,3760,3660,0570,3770,3970,0380,3680,4180,0290,3590,4390,02100,35100,4410
Temperature 413 K
k1=0.29913 k2=0.24942 k3=0.17953Concentration C1TimeConcentration C2TimeConcentration C3Time0.56 00.24 00.00 00.39 10.35 10.06 10.28 20.39 20.14 20.19 30.40 30.2 1 30.14 40.39 40.27 40.09 50.39 50.32 50.07 60.37 60.36 60.05 70.37 70.39 70.03 80.36 80.41 80.02 90.35 90.43 90.02 100.35 1 00.44 10
k1=0.29917 k2=0.24944 k3=0.17955 Concentration C1TimeConcentration C2TimeConcentration C3Time0.56 00.24 00.00 00.39 10.35 10.06 10.28 20.39 20.14 20.19 30.40 30.2 1 30.14 40.39 40.27 40.09 50.39 50.32 50.07 60.37 60.36 60.05 70.37 70.39 70.03 80.36 80.44 80.02 90.35 90.43 90.02 100.35 1 00.41 10
Temperature 453 K
k1=0.29920 k2=0.24947 k3=0.17957 Concentration C1TimeConcentration C2TimeConcentration C3Time0.56 00.24 00.00 00.39 10.35 10.06 10.28 20.39 20.14 20.19 30.40 30.2 1 30.14 40.39 40.27 40.09 50.39 50.32 50.07 60.37 60.36 60.05 70.37 70.39 70.02 80.35 80.41 80.02 90.35 90.44 90.03 100.36 1 00.43 10
Conclusion
During laboratory work, kinetic models of a homogeneous chemical reaction were compiled; built a kinetic model of a chemical homogeneous reaction; calculated the concentrations of the reactant, intermediate substances and reaction product at various temperatures and reaction times. We plotted the dependence of the change in concentration on time; as the reaction time increases, the concentration of the starting substance decreases, the intermediate substances increase, and the concentration of the reaction product increases. Recommendations on the conditions for carrying out a chemical reaction in order to obtain the maximum yield of the target product C: temperature 393 K and time 10 s.
^ Matrix method
In addition to the directed graph method, there are other methods for solving stoichiometric problems for complex systems of chemical reactions. The matrix method allows you to reduce the problem to a form that is most suitable for its further solution using computer technology.
Let's consider solving the previous problem using the matrix method. A system of 4 chemical reactions involves 7 substances. The equations of chemical reactions involving these substances can be written as if all substances were involved in them simultaneously. If a substance does not take part in some chemical reaction, formally this means that the stoichiometric coefficient for this substance is equal to zero. Let us also agree that the stoichiometric coefficients for the starting substances will be taken as positive, and for the products as negative. Then the first of the chemical equations of the system of chemical reactions considered in the previous example can be written as follows:
A + 2B - 2C + 0D + 0E + 0F + 0H = 0.
Reasoning similarly for all substances and all reactions, we will compose a system of linear equations that describe the ratio of the masses of all substances participating in the reactions. The dimension of the system is 4x7, where 4 is the number of equations, 7 is the number of substances participating in chemical reactions. The coefficient matrix of these equations is given below, and the column vector is zero.
To the resulting system of equations it is necessary to add several more equations that have a non-zero right-hand side. These equations are written based on the initial conditions of the problem.
A B C D E F H
1 2 -2 0 0 0 0 0
1 0 0 -2 0 0 0 0
0 0 1 -1 0 -1 0 0
0 0 0 1 0 -2 -1 0
Under certain conditions, when the values of the initial and current masses of some components of the system are known, it is possible to obtain a unique solution using linear algebra methods.
Description of systems by calculating the stoichiometry of chemical reactions from a practical point of view makes it possible to calculate the masses of all participating substances. Thus, it is possible to predict the behavior of the system, the composition of products, and the amount of substances consumed.
Stoichiometric calculations assume that all chemical reactions in a given technological process proceed all the way to the right.
^ Modeling equilibrium in chemical reaction systems
A significant part of the chemical reactions that make up the main content of technological processes in non-ferrous metallurgy are reversible. Consider an example of a reversible chemical reaction:
Equilibrium in such a chemical reaction is achieved at certain values of the activities of the participating substances. If these substances are in solution and their concentrations are small (diluted solutions), then with some approximation, instead of activity values, concentration values can be used. Equilibrium in a chemical reaction is characterized by the value of the equilibrium constant:
.
The value of the equilibrium constant is related to the change in the Gibbs energy and can be calculated from the thermodynamic data of the substances involved:
Where Δ G T- change in Gibbs energy for a given chemical reaction, T- temperature, R- universal gas constant.
By calculating the value of the equilibrium constant for a chemical reaction occurring at a given temperature, it is possible to determine the ratio of the concentrations of starting substances and products that will be established when equilibrium is reached.
It is somewhat more difficult to determine the equilibrium composition of a system in which several reversible chemical reactions occur simultaneously. Consider the following example. Let there be a system of reversible chemical reactions involving substances A, B, C and D. In this system, substance A is sequentially and reversibly converted into substance C, previously forming B. A parallel path is also possible: substance A, in parallel with the formation of B, decomposes with the formation of D. Under given conditions (temperature, pressure), equilibrium will be established in the system and equilibrium concentrations of substances will be achieved.
To calculate equilibrium concentrations, we write expressions for the equilibrium constants of all reactions in terms of equilibrium concentrations:
A B 
; 
B C 
;
A D 
; 
.
Let at the initial moment there are no intermediate substances B and C, as well as the final product D:
; C B0 =0; С С0 =0; C D 0 =0.
We calculate the values of the equilibrium constants for each of the reactions using thermodynamic data: 
. Thus, we will consider the values of the equilibrium constants to be known quantities.
Per unit volume of a given system, C A0 – C A represents the number of moles of component A consumed. In accordance with the stoichiometry of chemical reactions and the law of conservation of matter, the loss of mass A is equal to the sum of the masses of the resulting substances B, C and D, which can be expressed by the equation:
C A0 – C A = C B + C C + C D.
Let's transform the equation to the following form:
C A0 = C A + C B + C C + C D,
And let’s substitute the expressions for the corresponding concentrations of substances on the right side:
C A0 = C A + k 1 C A + k 1 k 2 C A + k 3 C A .
Let us group homogeneous terms of the equation
C A 0 = C A (1 + k 1 + k 1 k 2 + k 3)
and obtain an expression for the equilibrium concentration CA
.
The equilibrium concentrations of other substances are easy to determine, since the values of all equilibrium constants are known to us from the previous calculation, and the expressions contain C A .
When calculating equilibria in systems of chemical reactions, it is necessary to know k p of each reaction and the initial composition of the system - this makes it possible to calculate the equilibrium composition of the system.
Real problems of calculating the equilibrium composition of systems are much more complex: the equations in these problems are nonlinear; it is necessary to take into account that the components involved in the reaction are in different phases; Instead of concentrations, it is correct to use the activity values of the components. The practical meaning of calculating equilibria in such complex systems comes down to the fact that the calculated equilibrium composition of the system is the physical and chemical limit to which a real process can reach if unlimited time is allocated for its implementation.
^ Simulation of chemical reaction kinetics
In physical chemistry, the rate of a chemical reaction is determined according to the equation:
,
Where dq– change in the mass of the reactant, mol.
dt– time increment, s.
V– measure of reaction space.
There are homogeneous chemical reactions in which all participating substances are within one phase (gas or liquid). For such reactions, the measure of the reaction space is the volume, and the dimension of the rate will be: 
.
Heterogeneous chemical reactions occur between substances in different phases (gas-solid, gas-liquid, liquid-liquid, solid-liquid). The chemical reaction itself is realized at the phase interface, which is a measure of the reaction space.
For heterogeneous reactions, the rate dimension is different: 
.
The change in the mass of reacting substances has its own sign. For starting substances, the mass decreases as the reaction progresses, the change in mass has a negative sign, and the rate takes a negative value. For the products of a chemical reaction, the mass increases, the change in mass is positive, and the sign of the speed is also taken to be positive.
Consider a simple chemical reaction
A + 2B = 2C.
Simple reactions include those that are carried out in one stage and go to the end, i.e. are irreversible.
Let us determine the rate of such a chemical reaction. To do this, first of all, it is necessary to decide which of the substances will determine the reaction rate: after all, A and B are the starting substances, and the change in their masses is negative, and C is the final product, and its mass increases with time. In addition, not all stoichiometric coefficients in the reaction are equal to unity, which means that if the consumption of A for some time is equal to 1 mole, the consumption of B during the same time will be 2 moles, and accordingly the rate values calculated from the change in masses of A and B will differ twice.
For a simple chemical reaction, a single measure of rate can be proposed, which is defined as follows:
,
Where r i– speed according to the i-th reaction participant
S i– stoichiometric coefficient of the i-th reaction participant.
Stoichiometric coefficients for starting substances are assumed to be positive; for reaction products they are negative.
If reactions take place in an isolated system that does not exchange substances with the external environment, then only a chemical reaction leads to a change in the masses of substances in the system, and, consequently, their concentrations. In such a system, the only reason for changes in concentrations is WITH is a chemical reaction. For this special case
,
The rate of a chemical reaction depends on the concentrations of the substances involved and on the temperature.
Where k – rate constant of the chemical reaction, WITH A ,WITH IN– concentrations of substances, n 1 , n 2 – orders for the relevant substances. This expression is known in physical chemistry as the law of mass action.
The higher the concentration values, the higher the rate of the chemical reaction.
Order ( n) is determined experimentally and is associated with the mechanism of a chemical reaction. The order can be an integer or fractional number; there are also zero-order reactions for some substances. If the order is i th substance is zero, then the rate of the chemical reaction does not depend on the concentration of this substance.
The rate of a chemical reaction depends on temperature. According to Arrhenius's law, the rate constant changes with temperature:
Where ^A– pre-exponential factor;
E– activation energy;
R– universal gas constant, constant;
T- temperature.
Like the reaction order, the activation energy and pre-exponential factor are determined experimentally for a specific reaction.
If a chemical reaction is carried out in a heterogeneous process, then its speed is also influenced by the process of supplying starting substances and removing products from the chemical reaction zone. Thus, a complex process takes place, in which there are diffusion stages (supply, removal) and a kinetic stage - the chemical reaction itself. The speed of the entire process observed in the experiment is determined by the speed of the slowest stage.
Thus, by influencing the speed of the diffusion stage of the process (mixing), we influence the speed of the entire process as a whole. This influence affects the value of the pre-exponential factor A.
Most chemical reactions are not simple (i.e. they do not occur in one stage and not to completion) - complex chemical reactions:
A) AB – reversible;
B) A→B; B→C – sequential;
B) A→B; A→C – parallel.
For a complex chemical reaction there is no single measure of speed. Unlike simple, here we can talk about the rate of formation and destruction of each chemical substance. Thus, if chemical reactions occur in a system and involve n substances for each n substances have their own speed value.
For any substance, the rate of formation and destruction is the algebraic sum of the rates of all stages involving this substance.
Speed of a complex chemical reaction
Let us consider modeling the kinetics of a system of complex chemical reactions using the following example. Let there be a technological process, the essence of which is reflected by the following chemical reactions:
K 1 ; 1 to B
K2; 0.7 C
K 3; 1 to A; 0.35 by N
K4; 1 to C; 1 to D
K5; 2 on E;
R A = –k 1 C B + k 2 C C 0.7 – k 3 C A C H 0.35
R B= –2k 1 C B + 2k 2 C C 0.7
R C = k 1 C B – k 2 C C 0.7 – k 4 C C C D + k 5 C E 2
R D = k 3 C A C H 0.35 – k 4 C C C D + k 5 C E 2
R E = k 4 C C C D – 3k 5 C E 2
Kinetic constants (orders for substances and values of rate constants for stages) are determined experimentally. The process diagram above the arrows corresponding to the stages shows the order values for the substances. Orders not specified are zero.
6 substances take part in the process: A and B are the initial substances, C and D are intermediate, E is the final product, H is a catalyst for one of the stages. Three chemical reactions have five stages, three of which are direct, two are reverse.
All reactions are carried out homogeneously and take place in a system closed in substance, which gives grounds to use the following expressions to characterize the rate:
.
Based on the above, we will write down expressions for the velocities for each participating substance. In total we get 6 expressions for the number of substances. For each substance, the rate of consumption or formation is the algebraic sum of the rates of all stages involving this substance. Thus, substance A participates in three stages, in the first as a starting substance, in the second as a product, in the third again as a starting substance. The speed components for the first and third stages will be negative; for the second stage the speed has a positive sign. According to the law of mass action, the velocity values for each stage are the product of the rate constant of the corresponding stage and the concentrations of substances in powers equal to the orders of the substances. Taking this into account, the expressions for the velocities of substances will be as follows:
= –k 1 C B + k 2 C C 0.7 – k 3 C A C H 0.35
= –2k 1 C B + 2k 2 C C 0.7
= k 1 C B – k 2 C C 0.7 – k 4 C C C D + k 5 C E 2
= k 3 C A C H 0.35 – k 4 C C C D + k 5 C E 2
= 3k 4 C C C D – 3k 5 C E 2
= 0.
The final rate for substance H, the catalyst of the third stage, is zero. The mass of the catalyst does not change during the reaction.
On the left side of all equations there is a derivative of the concentration of a substance with respect to time, therefore, the kinetics equations are differential. The concentrations on the right side of the equations at an arbitrary moment of time must simultaneously satisfy all the equations, which means that the set of kinetics equations in the mathematical sense is a system of equations.
A chemical kinetics model is a system of differential equations whose solution is a set of functions C i = f i (t) :
C A =f 1 (t)
In order to establish a specific type of functions, it is necessary to solve a system of differential equations, i.e. integrate the system of kinetics equations. We will consider the integration of kinetic equations below using a simpler example, and after that we will return to the problem discussed above.
^ Integrating Kinetics Equations
Let there be a chemical reaction of decomposition of substance A, as a result of which substance B is formed. It has been experimentally established that it is of the first order in concentration A, and the value of the rate constant for the conditions for its implementation is equal to k. This is shown in the reaction diagram below.
k; 1 to A
The reaction rate is r a = –kC A, or
.
Let us determine the initial conditions for solving the differential kinetics equation. We will assume that at the initial moment of the reaction we know the concentration of substance A, let us denote it as C Ao. Let us write the initial conditions in the form 
. Let's integrate the resulting equation using integrals with substitution of limits. The limits of integration are determined from the initial conditions: when time is zero, the concentration A is equal to the initial one, at an arbitrary moment t the concentration is WITH A :
.
As a result of integration we have:
,
Replacing the difference of logarithms with the logarithm of the quotient we have further:
,
Carrying out potentiation we get:
.
After all transformations, the solution to the differential equation is an exponential decreasing function:
.
Let's check whether the resulting solution contradicts the conditions of our problem. At t= 0, i.e. at the start of a chemical reaction WITH A = C A 0 , since the exponent goes to unity. Indeed, at the initial moment the concentration of substance A is equal to the initial one. At t→∞ an exponential with a negative exponent tends to zero in magnitude. Over an infinitely long time, due to a chemical reaction, the entire substance decomposes and forms B.
^ Numerical integration methods
Let us now return to the previous problem. Obviously, integrating a system of differential equations is a more complex problem compared to the one considered earlier. The use of analytical methods of integration is hardly possible, since the right-hand sides of differential equations contain concentrations of several substances at once, and it will not be possible to separate the variables.
Let's use the numerical integration method. To do this, we divide the time axis into small segments (steps). Considering that the derivatives of concentrations of substances with respect to time is the mathematical limit of the ratio of concentration increments to time increments, with Δ t , tending to zero:
,
Let's transform the system of differential equations into a system of algebraic ones. On the left side, we know the time increment, since we choose the time step ourselves. The only important thing is that this step should be small.
On the right side, the values of all rate constants are also known to us from experiment, and the same should be said about order values. Let us also substitute the concentration values of all substances into the right side, using the initial conditions. Each of the equations of the system contains in this case only one unknown quantity - the change in concentration Δ C i. Essentially, this is the change in concentration during the first step of the solution, when time changes from zero (the beginning of a chemical reaction) to Δ t. We sum the change in concentration with its own sign with the initial concentration and determine the concentration of each of the substances at the end of the first step of the solution.
At the next solution step, we substitute the concentration values from the previous solution step into the right-hand side and again obtain Δ C i, but now for the next step of the solution as shown in the figure.
At each step of the solution we obtain ordinates corresponding to the change in the concentration of all substances participating in the reactions. The geometric location of the points that are ordinates will give for each of the substances a graph of the function of changes in concentration over time. Note that as a result of numerical integration we do not obtain an analytical expression specifying the change in concentration over time; the ordinates on the graph are obtained by calculation. However, constructing graphs of functions of changes in concentrations over time is possible, and the appearance of the curves allows us to draw a number of conclusions that have practical meaning.
It is obvious that the concentrations of the starting substances decrease over time as they are consumed in the reaction. It is equally obvious that the concentrations of the final products are increasing.
The behavior of intermediate substances deserves separate consideration. The concentration graphs of intermediate substances have maxima corresponding to a certain reaction duration. If the intermediate substance is the target product of chemical reactions, then the maximum concentration corresponds to the optimal duration for obtaining this target substance.
This happens because at the initial moment of a chemical reaction, the concentrations of the starting substances are high, and the rate of the chemical reaction involving the starting substances is proportional to their concentrations. Reactions involving starting substances initially occur at high rates. This means that intermediate substances are also formed at a high rate.
On the other hand, the rate of decomposition of intermediate substances is also proportional to their concentration, and is small at first. The rate of formation of intermediate substances is greater than the rate of their decomposition, which contributes to the accumulation of intermediate substances, their concentration increases.
As a chemical reaction develops, the rate of formation of intermediate substances decreases and the rate of their destruction increases. When the rates become equal, the increase in concentration stops, and a maximum concentration of the intermediate substance is observed in the system.
Further, the rate of formation of the intermediate substance decreases as the concentrations of the starting substances continue to decrease. The rate of destruction of the intermediate substance also decreases, remaining in magnitude greater than the rate of formation, and this leads to the consumption of the intermediate substance in the system and to a drop in its concentration.
;
T C is the optimal time to obtain substance C.
Let us consider the behavior of substance C: at the initial moment of time C C = 0.
K 1 C B >k 2 C C 0.7
Thus, kinetics modeling makes it possible to determine the formation and consumption of all substances in chemical reaction systems, to establish the type of concentration function depending on time, and in some cases to determine the optimal conditions for conducting a chemical reaction.
^ Chemical reactions in a flow of matter
Many technological devices operate in continuous mode. Let us consider, as an example, a smelting furnace for processing a charge of copper concentrates and fluxes. The diagram of such a device is shown in the figure below.
N 
A continuous flow apparatus is a flow reactor in which a certain set of chemical reactions is carried out.
The presence of substance flows affects the conditions for chemical reactions.
Real flows of matter have quite complex properties:
hydrodynamic regime – laminar, turbulent, transitional;
number of phases – multi- and single-phase.
An example is a flow moving through a pipe. The speed of the flow within one section is not the same: the highest value of the speed is on the flow axis, and near the walls, due to the braking of the flow by viscous forces, this speed differs little from zero. However, if the volumetric flow rate of the flow medium is Q and the cross-sectional area is F, it is not difficult to determine the average flow rate of the flow equal to Q/F.
Q m 3 /s
Even more difficulties arise when describing multiphase flows, and real flows most often are just that.
In this regard, it is quite difficult to take into account the properties of real flows when creating a mathematical model. Therefore, to create a model of flow-type devices, there are several idealized flow models.
^ 1. Ideal displacement model – this idealized flow model is based on the following assumptions (an apparatus of this type could be a tubular kiln):
the flow is stationary, the volumetric flow rate of the medium does not change over time;
in such a flow, the speeds at all points of the flow are the same;
the volume element dV in such a flow is a system closed in matter (does not exchange with neighboring elements);
in an ideal displacement flow there is no longitudinal mixing;
There is also no transverse mixing in the flow.
To model the kinetics in the case of an ideal displacement flow, an approach applicable to systems isolated by substance is quite suitable.
Let us consider a first-order reaction that takes place in an ideal displacement apparatus.
K 1 ; 1 to A
WITH 
Let's create a model that allows us to calculate the output concentration A. The constant is known, first order.
– residence time of the substance in the apparatus
The greater the rate constant k, the faster the concentration tends to the concentration at the exit point.
Within the ideal displacement apparatus, the concentration of a substance does not remain constant - it drops from the concentration at the entry point to the concentration at the exit point.
^ 2. Ideal mixing model (an apparatus of this type is, for example, a KS furnace, a hydrometallurgical leaching reactor, etc.).
Assumptions:
the flow is stationary, the volumetric flow rate of the substance (Q) through the apparatus must be constant;
the concentration at all points of the ideal mixing apparatus is the same.
The consequence of the second assumption is that the concentration of the substance at the exit point is equal to the concentration inside the apparatus. 
Average residence time of the substance in the apparatus – 
.
The residence time of different portions of the flow in the ideal mixing apparatus is not the same.
The volume element in such a device is an open system; a closed system approach is not suitable for such a device. To describe the kinetics in this case, we use the law of matter and consider the apparatus as a single whole, the concentration at all points is the same. Based on the law of conservation of matter, we write down the material balance equation for the entire apparatus as a whole (per unit time):
Income – Expense = 0
Let a first-order decomposition reaction occur under the conditions of an ideal mixing apparatus:
K 1 ; 1 to A
The material balance for substance A will be the sum of the components:
1 term – the number of moles of substance A introduced by the flow per unit time;
2nd term – removal of substance from the apparatus per unit of time;
3rd term – the mass of the substance consumed in the chemical reaction. Let's divide both sides of the equation by the volume flow rate Q≠0:
.
Let us create the same conditions for chemical reactions in both apparatus (same temperature, 
k 1 =k 2). Let's assume that at a certain temperature k 1 =k 2 =1. Let's set C A0 = 1 mol/m3. Va = 1m 3, Q 1 = Q 2 = 1 m 3 /s. Then:
.
The surprising thing is that the result of the same chemical reaction turns out to be different in different devices. More efficient is the ideal displacement apparatus, in which the output concentration is lower.
The reason for this is not the speed of the chemical reaction (it is the same in both devices), but the presence or absence of mixing of the flow elements. In an ideal mixing apparatus, a concentration will be established at the outlet, which is the result of mixing portions of the substance that were inside the apparatus for different times. Some portions of the substance pass through the apparatus quickly, and the duration of the reaction in such portions is short, while the concentration of substance A, on the contrary, is high. Other portions of the substance are inside the apparatus for quite a long time, the duration of the chemical reaction is long, and the residual concentration of A is small.
^ Cellular flow model . According to this model, the real technological apparatus is replaced by an idealized circuit - a sequence of ideal mixing cells.
1 ; 1 to A P 
let n=2, then at the output of the 1st cell: 
If n cells then 
Considering that 
– let’s move on to the solution for the ideal displacement apparatus. When n=1 we have an obvious solution for an ideal mixing apparatus.
Let us show on the graphs how increasing the number of cells can allow us to move, using the cell model, from an ideal mixing apparatus to an ideal displacement apparatus.
H 
To eliminate longitudinal mixing in the flow, the working volume of the apparatus is sectioned.
Cascading devices are also used - a series connection of technological devices to equalize the results of chemical reactions.
Modeling the kinetics in chemical reaction flows allows, taking into account the characteristics of the flow, to calculate the operating characteristics of the equipment (output composition).
