4 LABORATORY PRACTICUM: KINETICS OF CHEMICAL
REACTIONS
4.1. THEORETICAL INTRODUCTION
Chemical kinetics is a section of physical chemistry devoted to studying of the chemical reaction rates depending on various factors (nature and concentration of reactants, temperature, reaction time, presence of catalysts, the reactant surface value for heterogeneous reactions, the pressure for the gaseous reactants, etc.).
Chemical kinetics determines the kinetic parameters of the most important chemical reactions, the reaction molecularity, the order of reaction, the rate constants, the activation energy, the half-life, etc.
A rate of chemical reaction (υ) is determined as the change in concentration of one of the starting materials or reaction products in the time unit
υ = ± dcdt . |
(4.1) |
If the reaction rate is calculated from the concentration change of the starting substances a negative sign in the equation should be chosen, while a plus should be chosen in the case of the reaction products.
Dependence of the chemical reaction rate from the reactant concentrations is determined by following law. The reaction rate is proportional to a multiplication of the reactant concentrations (Ci) taken in some exponents (ni), which are determined experimentally. For example, in the case of reaction
aA + bB → products
the dependence of the reaction rate from the reactant concentrations is described by the equation
υ = k CAn1 CBn2 , |
(4.2) |
which is called the kinetic equation of a chemical reaction; where k is the rate constant of a chemical reaction, CA and CB are concentrations of substances A and B, respectively, n1 is the reaction order at substance A and n2 is the reaction order at substance B.
90
Reaction rate constant is numerically equal to the reaction rate at reactants concentrations equal to unity. The rate constant as well as the reaction rate is dependent from the nature of the reactants, temperature, presence of a catalyst, but does not depend on the concentration and reaction time.
The partial reaction order (ni) is determined by the exponent at the concentration in the kinetic equation of reaction. The total reaction order (n) is equal to a sum of the exponents at the reactant concentrations (sum of the partial orders), e.g. for the above shown reaction
n = n1 + n2. |
(4.3) |
In the general case n = 0, 1, 2, 3, but may be a fractional number also. Only in the case of simple elementary reactions the partial order value coincides with the stoichiometric coefficient in the reaction equation, i.e.
n1 = a and n2 = b.
To determine the order of reaction it is necessary to exam the concentration change of reactants (or reactant) with time.
If the chemical reaction rate depends upon the concentration of only one component, for experimental data processing often use the chemical
reaction rate equation for a first order reaction (A → products)
− dc |
= k CA , |
(4.4) |
dt |
|
|
where k is the rate constant of a first order reaction, СA is concentration of substance A at the time t.
Integration of the equality (4.4) leads to the equation
k = |
1ln |
C0 |
, |
(4.5) |
|
C |
|||||
|
t |
|
|
where С0 is initial concentration of the starting substance A. Equation (4.5) may be also represented in the form of
1 |
|
C |
|
, |
(4.6) |
|
k = t ln |
|
|
0 |
|
||
C |
|
− |
x |
|||
0 |
|
|
|
|
||
where х is concentration of unreacted substance at time t.
91
The rate constant of a first order reaction has dimension of (time)-1. Constancy of the k values calculated at different times is a proof for a
first order of the investigated reaction. This method of the reaction order determination is called the substitution method.
The order of reaction can be also defined graphically.
The equation (4.5) converted into a linear form looks as follows
ln C = ln C0 – kt. |
(4.7) |
On the basis of experimental data and using equation (4.7), the dependence of lnC as function from t was plotted (Fig. 4.1). A straight line indicates that the reaction is a first order reaction.
Fig. 4.1. The logarithm of the concentration versus time for first order reactions
The reaction rate constant may be determined by the slope of α,
k = - tgα = |
a |
. |
(4.8) |
|
|||
|
b |
|
|
Besides the reaction rate constant a half-life (t1/2) is a useful characteristic of the reaction rate. A half-life is a time interval during which a half of the initial substance quantity was reacted.
The half-life of first-order reactions may be determined using the equation
t1 2 |
= |
ln 2 |
. |
(4.9) |
|
|
k |
|
|
92
As seen, the half-life for first order reactions is not dependent from the initial concentration of the starting substance.
In order to determine the half-life by the graphical method it is necessary to construct the C=f(t) dependence. The time value corresponding to a half of the whole concentration is the half-life (Fig. 4.2).
Fig.4.2. Graphical determination of the half-life
The half-life can be found also from the first graph (Fig. 4.3). The time which corresponds to the ln(C0/2) is the half-life.
Effect of temperature on the rate of chemical reactions
As the temperature increases the rate of most reactions increase. Van't Hoff rule indicates that the rate constant of the chemical reaction increases in 2-4 times when temperature increases on 10 degrees,
γ = kT +10 ≈ 2 ÷ 4 kT
where γ is a temperature coefficient of the chemical reaction rate. In the common case of temperature increasing from T1
constants ratio is equivalent to
k2 |
T2 −T1 |
|
= γ 10 |
||
k1 |
(4.10)
to T2 the
(4.11)
Van't Hoff rule is approximation and applicable only in a limited temperature range since the temperature coefficient varies with temperature. For most reactions, a temperature dependence of the rate constant is described by the Arrhenius equation:
93
− Ea
k = Ae RT . (4.12) This equation contains two parameters that do not depend on the temperature; A is a pre-exponential factor, which is determined only by the type of reaction, Ea is the activation energy of a chemical reaction that
characterizes the height of the energy barrier for the reaction. Pre-exponential factor A has the same dimension as the rate constant. Arrhenius equation (4.12) in differential form can be written as
d ln k |
= |
Ea |
. |
(4.13) |
|
dT |
RT 2 |
||||
|
|
|
The activation energy of a chemical reaction can be calculated from the rate constants at two different temperatures. We write the equation (13) for temperatures T1 and T2, and subtract the first equation from the second:
|
k2 |
|
Ea |
|
1 |
|
1 |
|
|
|
|
|
|
|
|||||
ln k |
= |
|
|
−T |
(4.14) |
||||
R T |
|
||||||||
1 |
|
|
|
1 |
2 |
|
|
||
The equation for the activation energy of the chemical reaction is as follows:
Ea = |
R(T2T1 ) |
ln |
k2 |
(4.15) |
|
T2 −T1 |
k1 |
||||
|
|
|
If the data about the rate constant values are not known, then a rate constant in the Arrhenius equation may be substituted by other associated parameters. For example, the initial reaction rate at different temperatures, or the time at which the reaction yield will have a predetermined value at different temperatures.
The activation energy (Ea) in the Arrhenius equation is the minimum energy that must have reacting molecules to their clash led to a reaction. Thus, during transition from the initial to the final state system must overcome kind of an energy barrier (Fig. 3).
The activation energies in the Arrhenius equations for the forward and reverse reactions are connected with each other through the change in internal energy for the overall reaction.
94
Fig. 4.3. Energy diagram of a chemical reaction, where Ein is the mean particle energy of the initial reactants, Eprod is the mean particle energy of the reaction products, U is the internal energy change of the system.
Heterogeneous Chemical Reactions
Reactions occurring at the interface between adjacent phases are called heterogeneous reactions. These include reactions between substances located in different phases, such as fuel combustion, the oxidation of metals by atmospheric oxygen, processes on the catalysts surface, etc.
For chemical reactions between substances located in different phases of a heterogeneous system the basic postulate of chemical kinetics becomes inapplicable. In the heterogeneous reactions molecules linked chemically to the interface usually play role of intermediates.
There are the following stages of heterogeneous chemical reactions:
1.Diffusion of reactants to the reaction zone located at the interface;
2.Adsorption of reactants on the interface;
3.Chemical conversion of the adsorbed particles;
4.Desorption of the reaction products formed;
5.Diffusion of the reaction products from the reaction zone.
Stages 1 and 5 are called diffusion stages, stages 2, 3, and 4 are kinetic stages.
Due to the fact that these stages occur sequentially one after the other, the rate of the overall process is determined by the slowest stage. If the stage defining the process is occurring at the interface, it is customary to say that the heterogeneous process occurs in the kinetic region. If the slowest stage is the supply and the removal of the substances by diffusion, the heterogeneous process occurs in the diffusion region.
95
Diffusion is important in heterogeneous processes since it occurs due to the change in concentration in the interface layer, which affects the kinetics of the process. Diffusion is described by Fick's laws.
The First Fick's law states that the mass of the substance dm, transported by diffusion in the x direction through the area perpendicular to this direction, is proportional to the value of this area S, time dt and the
concentration gradient dcdx along the chosen direction:
dm = −DS dc dt |
or υdiff = |
dm |
= – DS dc |
, |
(4.16) |
dx |
|
dt |
dx |
|
|
where D is a diffusion coefficient, υdiff = dm/dt is a diffusion rate. The minus sign in equation (16) indicates that the diffusion process is directed toward reducing of the concentration.
The diffusion coefficient depends from temperature. However, the activation energy of the diffusion process is significantly less than the activation energy of most chemical reactions. Therefore, the temperature has less influence on the rate of diffusion processes than on the chemical process rate.
The Second Fick's law states the time dependence of the concentration in the volume of one of the contacted phases:
dс |
= D |
d 2c |
(4.17) |
|
dt |
dx2 |
|||
|
|
At steady-state diffusion the concentration varies only with the distance x, and is independent from time. In this case, the dc/dt = 0 and equation (4.17) yields dc/dx = a = const, after the integration
c = с0 + ах, (4.18) where с0 corresponds to the coordinate х = 0.
Thus, at steady-state diffusion the concentration changes linearly in the direction of diffusion and a concentration gradient may be written using finite values,
a = dc |
= |
c −c0 |
, |
(4.19) |
|
δ |
|||||
dx |
|
|
|
where δ is a finite value of the x.
Substituting this equation in (16), we obtain steady-state diffusion equation,
υdiff = |
dm |
= DS |
c0 −c |
or υdiff = β(с0 – с), |
(4.20) |
|
dt |
δ |
|||||
|
|
|
|
where β is so called a mass transfer coefficient.
96
Consider that the heterogeneous chemical reaction is a first order reaction and proceeds steadily. Suppose that only two successive stages (a chemical reaction and a diffusion process) can be identified in the total process. Since the process is stationary there is no accumulation of the raw materials or reaction products at the interface, and the rates of both stages are the same, kc = β(с0 – с). Hence, we find the concentration of the
substance at the interface between phases с = kβ+с0β , and insert into an equation of the first order reaction rate:
υ = |
kβс0 |
or υ = |
|
|
с0 |
|
, |
(4.21) |
|
k + β |
|
1 |
1 |
|
|||||
|
|
|
|
|
|
||||
|
|
|
|
|
+ |
|
|
|
|
|
|
|
|
β |
k |
|
|
|
|
where 1/β is the diffusion resistance, 1/k is the chemical resistance.
When k >> β the heterogeneous reaction rate is equivalent to βc0 and it is determined only by the β value characterizing diffusion. The process occurs in the diffusion region. Otherwise β >> k, υ = kc0 and the overall process is determined by the chemical stage and proceeds in the kinetic region. In other cases the rate of the overall process depends from the rates of the both considered stages.
Dissolution of solids in liquids
A. N. Shchukarev experimentally established the following equation for dissolution rate of a solid in a liquid
υ = kS(csat – c), |
(4.22) |
where S is the contact surface area between the solid body and a liquid, c is concentration of solute in the bulk of liquid, csat is concentration of the saturated solution, k is a coefficient that depends on the temperature, the nature of the bodies and the dissolution conditions.
In accordance with this equation, the greater the contact surface area between the phases and the concentration difference between reached in the given time and maximum possible values of the concentrations, the greater the rate of dissolution.
Nernst suggested that liquid near the surface of the solid body forms a so-called custom diffusion layer, outside of which the concentration with good stirring is maintained throughout the same. In addition, he believed
97
that the dissolution rate is much more than the diffusion rate, therefore directly at the surface of a solid body solution is close to saturation. Since the diffusion is considered to be the slowest stage, the speed of the overall process can be taken for the rate of diffusion.
Further studies have shown that the Nernst`s theory is not entirely true. Diffusion layer thickness calculated on the basis of this theory is so large (about one million molecular layers) that it cannot be considered as a not attracting in the process of mixing. In fact, the mass transfer between the interface and internal regions is ensured not only by molecular diffusion, but also by convection associated with the movement of layers of substances (convective diffusion). Despite the fact that the equation (4.22) is often performed a certain value of the δ = D / k obtained with this equation does not correspond to the actual near-surface layer and may be called a diffusion layer effective thickness.
Catalytic reactions
In the presence of the catalyst the reaction rate increases significantly. Catalysts are substances involved in the reaction, altering the reaction rate, but not expendable in the reaction and remaining chemically unchanged thereafter reaction. Catalysts interact with reactants, form with them certain intermediates, are included in the active complex composition, and are reallocated after the reaction. A positive catalysis accelerates the reaction, while a negative catalysis slows down the reaction (i.e., inhibition occurs).
Catalysis divided into homogenous (all reactants and the catalyst itself are in the same phase) and heterogeneous (reactants and the catalyst are in separate phases, and the catalytic reaction occurs at the interface).
General features of catalytic reactions.
1.Catalyst reduces the activation energy of the reaction, which leads to a significant increase in reaction rate.
2.Catalyst has no effect on the thermodynamic equilibrium. It changes only the rate to reach equilibrium, i.e. rates forward and reverse reactions.
3.Catalyst`s action is specific. Catalyst selectively enhances only a few of all the possible reactions of the reactants.
4.Very small amount of catalyst is usually sufficient for appreciable catalytic action.
5.Catalyst`s action is strongly affected by the presence of foreign matter. Certain substances, known as promoters, enhance the action of the
98
catalyst. Substances that reduce the activity of the catalyst are called catalytic poisons.
6. For heterogeneous catalytic reactions the catalyst`s action depends from the surface area.
Mechanisms of Catalytic Reactions
Chemical reactions proceed with breaking certain chemical bonds in the initial compounds and the formation of new bonds in the reaction products. The activation energy is usually much less than the energy used to break chemical bonds. Progress of the reaction is due to the fact that when driving along the reaction way (via transition state) a path of the energy required to break old bonds is compensated by the energy liberated during the formation of new bonds. The degree of compensation determines the reactivity of substances. Interacting with the reactants entering into the activated complex catalyst increases the degree of compensation, reduces the activation energy, and thereby increases the rate of chemical conversion.
Consider these reasons for the increased rates of catalytic reactions for the two mechanisms of catalytic reactions (stepwise and confluent).
Stepwise (separate) mechanism of the catalytic reactions is substitution of one catalytic reaction by several consecutive stages of interaction between initial reactants and the catalyst with the possible formation of an activated complex at each stage.
In the catalysis reaction of the A + B → C + D type can occur in the following two stages
1)A + K → (AK)≠ → AK,
2)AK + B → (ABK)≠ → C + D + K,
where AK is intermediate stable compound with catalyst, (AK)≠ and (ABK)≠ are intermediate activated complexes.
According to confluent (synchronous) mechanism the simultaneous interaction of all the initial reactants with the catalyst and the formation of single activated complex occur during the reaction:
A + B + K → (ABK) ≠ → С + D.
Thus, at least not less than two activated complexes are formed during the stepwise catalytic reactions, while there cannot be more than one activated complex occurrence at the confluent catalytic reactions (Fig. 4.4.).
99
Fig. 4.4. Effect of catalyst on the activation energy of the reaction described by the stepwise mechanism (a) and by the confluent mechanism (b)
Heterogeneous Catalysis
Typically used heterogeneous catalysts are solids. The catalytic reaction in this case takes place at the interface between the solid phase and gas or between solid and liquid phases and its rate is proportional to the interface area. In this regard, an important characteristic of the heterogeneous catalysts is the interface area per unit mass of catalyst (the specific interface area)
Ssp = |
S |
, |
(4.23) |
mc
where S (m2) is the total interface area, mc (g) is the mass of catalyst, Ssp (m2/g) is the specific interface area of catalyst.
High value of the specific interface area is provided by developed porous structure of solid catalysts. The porous walls elongated from the outer surface of the grain into the interior of the solid body form an inner surface, which has the specific interface area of 5-500 m2/g for industrial catalysts. The outer surface area of the catalyst grains typically is less than 0.01-1 m2/g. In additional, the effectiveness of heterogeneous catalysts is determined by the structure and the chemical composition of the surface layer. The catalytic effect of solids associated with the presence on the surface of certain structural elements of the crystal lattice of atoms and functional groups where chemical reactions occur directly, which are called active centers.
Any heterogeneous catalytic reactions occurring at the presence of a solid catalyst comprising a number of stages:
100
1.mass transfer of the initial reactants by diffusion from the bulk of liquid phase or gas to the external surface of the catalyst granules;
2.mass transfer by diffusion through the pores of the granules to the inner surface of the catalyst;
3.adsorption of reactants on the surface of the catalyst;
4.chemical act on the surface with one or more chemisorbed species;
5.desorption of the reaction products and the reverse mass transfer of them into the bulk of the liquid or gaseous phase.
Adsorption
Adsorption is a process of spontaneous concentrating of substances from the bulk of the phases at the interface. Adsorption is the second (after diffusion) stage of many heterogeneous chemical reactions.
Differences between physical adsorption and chemisorption.
At the physical adsorption absorbing molecules (adsorbate) are held by absorber surface (adsorbent) due to weak molecular forces of attraction.
In the case of chemisorption absorbing molecules form a surface chemical compound, usually with considerable overcome the energy barrier and it is called activated adsorption.
Consider the relatively simple case of adsorption, which is explained by the theory of Langmuir, whose main provisions are as follows.
1.The surface of the adsorbent (absorbing material) has a limited number of the adsorption centers, i.e. sites where adsorption takes place.
2.Only one molecule of adsorbate (a substance that is absorbed) can be adsorbed on the each adsorption center.
3.All adsorption centers are identical.
4.The adsorbed molecules do not affect each other.
5.Adsorbed molecules are in dynamic equilibrium with the adsorbate molecules located in the environment from which absorption occurs.
Adsorption is characterized by the degree of surface filling Θ, which is the ratio of the number of occupied adsorption centers to the total number of adsorption centers in a given surface.
Adsorption rate is proportional to the concentration of the substance and to the number of free centers on the surface of the adsorbent, i.e.
υаds = k1c(1 – Θ) , |
(4.24) |
while desorption rate (an inverse process of adsorption) is proportional to the degree of surface filling Θ, i.e.
υdes = k2Θ , |
(4.25) |
101
where k 1 and k 2 are the rate constants.
At equilibrium, the rate of adsorption and desorption are equal to each
other |
|
|
|
|
k1c(1 – Θ) = k2Θ , |
(4.26) |
|||
wherefrom |
k1c |
|
||
Θ = |
(4.27) |
|||
k2 + k1c |
|
|||
Dividing the numerator and denominator of this equation for k2 and entering the equilibrium constant for adsorption (adsorption coefficient)
b = |
k1 |
(4.28) |
|
k2 |
|||
|
|
we obtain the equation of Langmuir isotherm
Θ = |
bc |
(4.29) |
|
1 +bc |
|||
|
|
Since υdes=k2Θ, i.e. the rate is proportional to the degree of filling, the order (true order) of chemical transformation is unity. Using (4.29), we obtain
υ = 1kbc+bc ,
comprising a rate dependent from the concentration of the initial substance. At low concentrations (bc << 1) and υ ≈ kbc. In these conditions, the reaction has an apparent order of unity. When bc >> 1 υ ≈ k = const, hence the apparent order is zero. Obviously, at average concentrations the apparent order must be fractional and lies between zero and one.
Theory of Heterogeneous Catalysis
There are several general theories of heterogeneous catalysis. Most common in the past years have three theories. The multiplet theory, the active ensembles theory, and the electron theory.
According to the multiplet theory of heterogeneous catalysis (A.A. Balandin) it is assumed that the formation of the surface compound (complex multiplet) involved a group of active surface atoms (multiplets) with certain geometric and energetic properties. The multiplet theory covers the principles of geometrical and energy compliance.
The theory of active ensembles (N.I. Kobozev) assumes that the catalytically active center is a collection (ensemble) of free atoms of the
102
catalyst located on a separate block of the surface area of the solid. These atoms are not included in the crystal lattice and can migrate freely within each block.
The electron theory of catalysis is based on a quantum-mechanical band theory of solid (semiconductor). When temperature is different from absolute zero in the conduction band of the crystal there are electrons providing free valences on its surface. The initial molecules interacting with them on the surface of the catalyst are adsorbing with the formation of free radicals or atoms. Interaction of the particles adsorbed and weakly coupled to the surface can lead to formation of reaction products.
LABORATORY EXERCISE 9.
DISSOLUTION RATE OF HARD-SOLUBLE SALT
Purpose. In this lab you will determine the dissolution rate constant at given temperature using the concentration dependence of the resistivity (or conductivity) of a hard-soluble salt (gypsum) solution.
Background. To calculate the rate of the dissolution process following equation is used
w = |
dc |
= k(с0 – с), |
(4.30) |
|
dt |
|
|
where k is the dissolution rate constant, с0 is the solute concentration at the surface, с is the solute concentration in the bulk. Equation (30) was obtained by Shchukarev under the assumption that the diffusion rate of dissolution products in the bulk of solvent is small in comparing with rates of the other stages.
The dissolution rate constant corresponding to equation (4.30) is calculated as follows
1 |
|
с0 |
|
||
k = t ln |
|
|
|
. |
(4.31) |
с |
0 |
−с |
|||
|
|
|
|
|
|
For poorly soluble substances, such as gypsum, the rate constant can be determined by measuring of the aqueous solution resistance at various times. Equipment for studying of the dissolution kinetics includes a temperature control system, a reaction vessel, and apparatus for measurement of the solution resistance (conductometer). The reaction vessel is a glass with a capacity of 300-400 mL, which contains the stirrer.
103
Procedure.
1.Turn on the thermostat system and set the temperature (20-25 °C).
2.Prepare reaction vessel. Rinse the vessel several times with distilled
water.
3.Fill the vessel by 250 mL of distilled water heated to a temperature of the thermostat and turn on the stirrer setting a uniform mixing mode. Insert the solute plate that has been previously soaked with water into the vessel so that it is above the water surface.
4.Put pre-prepared cylindrical vessel with a saturated solution of gypsum into the thermostat.
5.Put electrode to the reaction vessel with distilled water and turn on the conductometer.
6.After equilibration of the reaction vessel at constant temperature during 15 to 20 minutes start the measurements. For this lower the gypsum plate in the glass of water. Moment of the dive is considered as the start time of the process. Three minutes later from the beginning of the process make the first measurement of the resistance of the solution. Since at the beginning of process the solution resistance is rapidly changing, several first measurements should be carried out every 3-5 minutes, while subsequent measurements should be carried out every 10-15 minutes. Measurements should continue for an hour (9-10 measurements). Solution resistance will decrease with time.
7.Determine the resistance of a saturated solution of gypsum. The electrode previously rinsed by separate portions of the gypsum saturated solution should be lowered into the cylindrical vessel with the saturated solution.
8.At the end of the experiment drain solution from the reaction vessel and rinse it with water.
9.Store the measurement results in the Table 1 using following example:
The solute _____________________
Temperature _______ºС
Resistance of the saturated solution, Rs =__________Ω The volume of solution, __________ mL
104