Download simulation of kinetics for selected types of chemical reactions using

SILESIAN UNIVERSITY OF TECHNOLOGY
FACULTY OF CHEMISTRY
DEPARTMENT OF PHYSICAL CHEMISTRY
AND TECHNOLOGY OF POLYMERS
SIMULATION OF KINETICS FOR SELECTED
TYPES OF CHEMICAL REACTIONS USING
COMPUTER SOFTWARE
Opiekun ćwiczenia: Tomasz Jarosz
Miejsce ćwiczenia: Zakład Chemii Fizycznej
ul. M. Strzody 9, p. II, sala nr 209/210
LABORATORIUM Z CHEMII FIZYCZNEJ
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I.
Aim of the exercise
The purpose of this exercise is to familiarise students with the possibility of
conducting a computer simulation of chemical processes on the example of selected types of
simple and complex reactions.
In the course of the exercise practitioner should establish a basic familiarity with
chemical kinetics, in particular to become able to specify the type of chemical reaction and to
determine the reaction rate constants, based on a set of data showing the change in the
concentration of the reactants as a function of reaction time.
II.
Introduction
Programs simulating the progress of a chemical reaction are based on the knowledge
of the integral form of the kinetic equation, which describes the course of a given type of
process. On the basis of this equation, a series of complex calculations is carried out, often
designed to reflect the actual concentrations of various reactants as a function of process time.
In order for the mathematical model of the process to be complete, it is necessary to
know and understand the kinetics, the mechanism and the order of occurring reactions, as well
as to know the process parameters, such as initial concentration, reaction rate constants. Some
of the parameters included in the mathematical model are determined experimentally, while
others may be derived via computer simulation. In such a case, the procedure is to assume a
value of one of the unknown parameters, carry out a series of simulations and compare their
results with those obtained experimentally. The application of this procedure reduces the
number of experiments necessary to provide a detailed description of the investigated reaction
system.
Computer assistance is particularly useful in the case of real-world processes, in which
a series of co-interacting phenomena occur alongside the reaction that is being considered,
such as mass transfer and heat transfer. Taking into account that in industrial conditions,
starting materials used in the chosen manufacturing process can come from different sources
or be products of other processes conducted in the company and, therefore, their performance
is characterised by a certain variability, we can conclude that the ability to comprehensively
simulate the occurring phenomena, supported by significant computing power, is a valuable
tool for control and diagnostics.
III.
Elementary reactions
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a. Zero order reactions
The rate of a zero order reaction is independent of the concentration of chemical
species taking part in it. Consequently, increasing the concentration of the reactants will not
accelerate the process. Typical zero order reactions include photochemical reactions, in which
the rate depends on the intensity of irradiation or catalytic reactions, in which the rate depends
on the concentration of the catalyst.
The zero order reaction rate equation for a substrate A can be expressed as the
following differential:
r=−
d[A ]
=k
dt
(1)
The parameter k, called the rate constant of the reaction, is an experimental value.
Integrating the above equation yields the integral form, representing the concentration
changes of the substrate A in time:
[A]t = [A]0 − k ⋅ t
(2)
In the above notation [A]t is the concentration of the reactant A at a given time t,
elapsed from the start of the reaction, while the [A]0 is the initial concentration of the reagent
A.
The zero order reaction can be easily identified if the plot of reactant concentration
versus time is a straight line.
b. First order reaction
For a first order reaction, the reaction rate is proportional to the concentration of one
of the reactants. Other reagents may take part in the reaction, however, they do not affect the
reaction rate (the reaction is zero order in respect to them).
The differential equation of the first order reaction kinetics, relative to reagent A is:
r=−
d[A ]
= k[A ]
dt
(3)
Upon integration, the following logarithmic relation is obtained:
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ln([A ]t ) = ln([A ]0 ) − k ⋅ t
(4)
For a first order reaction, the plot of the natural logarithm of the concentration of the
reagent A versus time is a straight line with a slope equal to -k.
c. Second order reactions
In the second order reaction, the reaction rate is proportional to the square of the
concentration of one of the reactants (Equation 5a) or linearly proportional to the
concentrations of each of two separate reactants (Equation 5b):
r=−
d[A ]
2
= k[A ]
dt
(5a)
r=−
d[A ]
= k[A ] ⋅ [B]
dt
(5b)
Integrating the above equations, we obtain equations 6a and 6b.
1
1
=
+k⋅t
[A]t [A]0
(6a)
[A]t = [A]0 ⋅ e ([A ] −[B] )kt
[B]t [B]0
0
0
(6b)
d. Higher order reactions
Dla reakcji trzeciego rzędu charakterystyczną jest liniowa zależność odwrotności
kwadratu stężenia substratu od czasu.
Elementary reactions of the third order are rare cases, not to mention the reactions of
higher order than the third. In cases where the reaction involves two or more substrates, the
total order of reaction can reach values higher than three.
For a third order reaction the linear dependence of the inverse square of the
concentration of the substrate versus time is a characteristic feature.
IV.
Complex reactions
a. Parallel reactions
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When a single substrate (or a set of substrates) undergoes reactions, leading to the
simultaneous formation of various products, it is said that, parallel reactions occur in the
reaction system, according to the scheme:
Based on the knowledge of simple reactions, it can be concluded that the rate equation
for the above case will depend on the order of each of the occurring reactions.
Consider the simplest case, wherein both reactions are first order. Then, the substrate
A will be consumed in the reaction, according to the equation:
[A] = [A]0 ⋅ e−(k + k
1
2
)t
(7)
Changes in the concentration of the products are given by:
d[B]
= k1 ⋅ [A ]t = k1 ⋅ [A ]0 ⋅ e − (k1 + k 2 )t
dt
(8a)
d[C]
= k 2 ⋅ [A ]t = k 2 ⋅ [A ]0 ⋅ e − (k 1 + k 2 )t
dt
(8b)
Upon integration and calculation of the integration constant, using the boundary
conditions of [B] = 0 and [C] = 0 for t = 0, we obtain equations 9a and 9b, describing the
relation between the concentration of the reaction products and time:
[B]t
=
k 1 ⋅ [A ]0
⋅ 1 − e −(k1 + k 2 )t + [B]0
k1 + k 2
(9a)
[C]t
=
k 2 ⋅ [A ]0
⋅ 1 − e −(k1 + k 2 )t + [C]0
k1 + k 2
(9b)
(
(
)
)
From the above equations, we can derive the conclusion that at any moment of the
reaction increases the ratio of the increases in B and C concentrations, in respect to their
initial values, corresponds to the ratio of the values of the k1 and k2 constants (assume
[B]0=[C]0=0 and divide [B]t by [C]t).
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b. Consecutive reactions
Consecutive reactions are simultaneously occurring reactions, in which the product of
one reaction is the substrate in another reaction occurring in this system. A special case of the
secondary reactions are chain reactions.
The simplest case of a consecutive reaction is the following system of two reactions:
The rate equations for each of the reagents are given by:
d[A ]
= − k1 ⋅ [A ]t
dt
(10a)
d[B]
= k1 ⋅ [A ]t − k 2 ⋅ [B]t
dt
(10b)
d[C]
= k 2 ⋅ [B]t
dt
(10c)
Based on these equations, after integration and inclusion of the boundary conditions
the following system of equations is obtained:
[A] = [A]0 ⋅ e− k t
1
[B] = [A]0
[C] = [A]0 ⋅ 1 + k1 ⋅ e

(11a)
k1
⋅ e −k1t − e −k 2t + [B]0 ⋅ e −k2t
k 2 − k1
(
)
(11b)
− k 2 ⋅ e − k1t  −k1t −k 2t
 ⋅ e − e
+ [B]0 ⋅ 1 − e −k2t + [C]0
k 2 − k1

− k 2t
(
)
(
)
(11c)
c. Equilibrium reactions
An equilibrium process can be defined as a pair of reactions proceeding at the same
time in opposite "directions", meaning that one reaction leads to the conversion of reagent A
into reagent B, while the other reaction involves the transformation of reagent B into reagent
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A. Each of the reactions has a separate rate constant, whose ratio is the equilibrium constant
of the reaction.
For first order equilibrium reaction kinetics, the mathematical model has the form:
[A] = [A]0 ⋅
1
k2
⋅ k 2 + k1 ⋅ e − (k1 + k 2 )t + [B]0 ⋅
⋅ 1 − e − ( k 1 + k 2 )t
k1 + k 2
k1 + k 2
)
(12a)
[B] = [A]0 ⋅
k1
1
⋅ 1 − e − (k1 + k 2 )t + [B]0 ⋅
⋅ k1 + k 2 ⋅ e − ( k 1 + k 2 ) t
k1 + k 2
k1 + k 2
)
(12b)
(
)
(
)
(
(
d. Autocatalytic reactions
Autocatalytic reactions are a group of reactions, in which the increase in the
concentration of the reaction product leads to the acceleration of the reaction progress. For
autocatalytic conversion of the substrate A into product P, the kinetic equation in the simplest
case takes the form of:
r=−
d[A ]
= k[A ] ⋅ [P ]
dt
(13)
Integrating and transforming equation 13, expressions for the concentrations of
substrate A and product P can be obtained:
[A]t = [A]0 − [P]0 ⋅
e([A ]0 + [P ]0 )kt − 1
[P]
1 + 0 ⋅ e([A ]0 + [P ]0 )kt
[A]0
e ([A ]0 + [P ]0 )kt − 1
[P]t = [P]0 + [P]0 ⋅ [P]
0
1+
⋅ e ([A ]0 +[P ]0 )kt
[A]0
V.
(14a)
(14b)
Effect of temperature on the reaction
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Every chemical reaction features a change in the Gibbs free energy during its progress,
however, even in the case of a reaction occurring spontaneously under given conditions, the
process needs to be initiated by a given amount of energy, required to overcome the chemical
potential barrier. This energy is called the activation energy of the reaction.
The molecules may have a wide range of different energy levels - translational,
rotational, vibrational, therefore, at a given temperature different molecules can exist at
different energy levels. A quantitative description of this phenomenon utilises the Boltzmann
distribution. At a given temperature the average energy of molecules in a given system is
considered, therefore, even when the average energy of the particles does not allow the
potential barrier, allowing the occurrence of a chemical reaction to be overcome, a population
of molecules can be at a sufficiently high energy level and, consequently, undergo reaction.
If, in such a situation, the temperature of the system is raised, the share of the particles
having the energy required for conversion will also increase, giving rise to an increase in the
rate of reaction taking place. The relationship between the reaction rate constant and the
temperature of the reaction system is given by the Arrhenius equation:
k = k0 ⋅ e
−
EA
RT
(15)
where k is the value of the rate constant at temperature T expressed in Kelvin, k0 is the
pre-exponential factor, independent of the temperature resulting from the mechanism of the
reaction, the EA is the activation energy of the reaction expressed in Joules, and R is the gas
constant.
VI.
Experimental
The exercise is based on a computer simulation of three cases: an elementary reaction,
a complex reaction and a reaction occurring at different temperatures. Spreadsheets,
corresponding to each part of the exercise allow calculations to be performed, leading to the
generation of a table of concentrations of the different reactants as a function of time for the
inputted control codes, given by the overseer of the exercise.
When you turn on your computer and load the operating system, open the accordingly
labelled spreadsheets to carry out different parts of the exercise.
1. Running the simulation
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Open the spreadsheet "Symulacja.xls" and in the "Reakcja prosta - Dane" tab enter
Code 1 given by the overseer. After entering the code, the program will generate an array of
data, affected by random error, which will also be shown in the tab "Wykres danych".
Note! Make sure that the code has been entered correctly - otherwise it will be impossible to
complete the exercise.
2. Selection of the investigated time range
The time range of the obtained data is modified by means of manipulation of the
numerical value in the "krok czasowy" ("Reakcja prosta – dane wyjściowe" tab). The range
should be chosen in such a way that the generated chart comprised 60-80% of the total
reaction. Examples of time range selections are shown in Fig.1.
Figure 1 Examples of time range selection for the same reaction. a) Too short timespan;
b) Too wide time range; c) Correctly selected timespan.
3. Saving the data and continuation of the exercise
Having selected the time range, copy the generated data to a new Excel workbook in
order to analyse the results. These instructions must be repeated for complex reactions
("Reakcja złożona – dane wyjściowe") and for the analysis of the effect of temperature on the
reaction ("Reakcja temperatura – dane wyjściowe").
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VII. Safety rules
During this exercise maintain caution typical for work with equipment powered from
the 230V mains.
VIII. Data processing
1. Determine the type of reaction (for complex reactions) and its order (all types of reactions,
see Appendix 1).
2. Define the roles of each of the species present in the reaction environment (indicate
substrates, products, etc.).
3. Using the selected kinetic model, determine: the reaction rate constants, initial
concentrations of the reactants, the reaction stoichiometry and the Arrhenius equation
coefficients (for analysis of the effect of temperature on the reaction only).
IX.
Error analysis
1. Based on the selected model of kinetics and determined parameters of the rate equation,
calculate the concentrations of all reagents as a function of time.
2. Compare the calculated concentrations with the data generated by the simulation and for
each individual data point, calculate the deviation of the modelled value from the
simulated value, using equation 16:
si2 =
(CSIMULATED − CMODELLED )2
(16)
3. Sum up the deviations for all the data points and divide the obtained value by the number
of data points, using equation 17:
Si
2
∑s
=
2
i
n
(17)
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4. Minimise the obtained mean deviation for each of the determined parameters of the rate
equation. Present the relations between the mean deviation and each of the reduced
parameters graphically (see Appendix 2).
X.
Report
The report is to contain each of the following (the lack of any of these components
results in failing the exercise):
•
Short theoretical introduction, summarising your knowledge of chemical kinetics and
modelling of chemical processes,
•
Printout of the datasets for each of the reaction types obtained by the „Sym
dopasowanie.xls” spreadsheet,
•
Description of conduct during fitting the model to experimental values,
•
Brief analysis of the effect of each of the parameters on the progress of the reaction,
•
Error analysis,
•
Conclusions
XI.
Sample questions for study
a. What is the difference between elementary and complex reactions ?
b. Define the rate constant of a chemical reaction.
c. What are the units of the rate constant ?
d. Define the order of the chemical reaction. Define the total and partial orders of
the reaction.
e. Define the half-time of a chemical reaction. Derive the equations describing t1/2
for a zero order, first order and second order reactions.
f. What determines the total rate of a consecutive reaction ?
g. Explain the phenomenon of autocatalysis and give examples of autocatalytic
reactions.
h. What parameters effect the value of the rate constant ? What is the Arrhenius
equation used for ?
i. Define the activation energy of the reaction.
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XII. Literature
a. Group work, „Chemia fizyczna”, PWN W-wa, 1965
b. R. Brdicka, „Podstawy chemii fizycznej”, PWN W-wa, 1969
c. K. Gumiński, „Wykłady z chemii fizycznej”, PWN W-wa, 1973
d. P. Atkins, „Chemia fizyczna”, PWN W-Wa, 2007
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