Kinetic Systems
How a Reaction Evolves in Time
Consecutive Irreversible Reactions
How a Reaction Evolves in Time
Chemical reactions are processes in which a substance or substances (known as reactants) are transformed into other products. When this change occurs directly, providing a complete description of the reaction mechanism presents few difficulties. However, complex processes in which the substances undergo a series of stepwise changes (each constituting a reaction in its own right) are much more common.
A simple case of the above is a single-step irreversible reaction, in which the products cannot be converted back to the original reactants. In this case, the rate of forward reaction decreases until all the reagents have been consumed, and the reaction terminates. On the contrary, in a single-step reversible reaction, the apparent rate of forward reaction will decrease in line with the accumulation of reaction products until a state of dynamic equilibrium is finally established. At equilibrium, the forward and backward reactions proceed at equal rates.
Generally speaking, model reactions are set up and the appropriate differential equations recorded. The equations are then integrated so as to arrive at an expression relating well established thermodynamic parameters of the reactions (like activation energy, equilibrium constants) to concentrations that vary with time. The approach (excluding elementary simple reactions) is cumbersome and requires a good command of differential-calculus knowledge. Here, much simpler iterative procedures are used, without the need for integration.
Approximate solutions can often be found by using some simplifying assumptions. In this way, the model can be used as a basis to describe the process. Quite often, with a tiny increase in programming efforts and further refinement, and the elimination of approximation, considerable modification and improvements can be achieved.
In kinetics, the parameters of interest are the quantities of reactants and products, and their rate of change. Starting with a single irreversible reaction, expressions for reactants are given a negative sign as the reactants are used up during a reaction. Product amounts increase and their rate of change is therefore positive. As they are not constants, the rates are written as differentials. Thus, in terms of a general reaction
aA + bB + ....... → cC + dD + ....
The reaction rates for the individual components are:
r1 = -1/a • d[A]/dt r2 = -1/b • d[B]/dt r3 = +1/c • d[C]/dt r4 = +1/d • d[D]/dt
a,b,c,d are the so-called stoichiometric coefficients, necessary to balance the reactants and products. Their inverses divide the prime derivative of concentrations, so that r1 = r2 = r3 = r4 = R (overall rate of the reaction)
The square brackets denote concentrations in moles per liter, symbolised M. On the contrary, in seawater calculations, concentrations are given in moles per kg, thereby avoiding variances due to increases in water pressure and a resulting decrease in volume (in the depth of the oceans).
In addition, unless otherwise indicated only closed, homogeneous systems are considered, in which there is no gain or loss of material during the reaction. Reactions are considered to proceed isothermally, so that temperature can be treated as an independent variable.
The rate (R) of a reaction at a fixed temperature is proportional to the concentration of reactants as can be seen in the following form
R = r1 = r2 = r3 = r4 = k([A]^{α} • [B]^{β}) (2.1)
The proportionality constant k in 2.1 is called the rate constant. The k unit can be deduced on examination of the rate expression; it has dimensions of (concentration)^{(1-n)}•(time)^{-1}.
The sum of all the exponents of the concentrations, n = α + β + .... is the overall order of reaction, while α is the order of reaction with respect to A, and β is the order of reaction with respect to B, and so on.
In the case of an irreversible reaction, the reaction order for each reacting compound should be determined experimentally since it cannot be predicted from the equations describing the reaction. The exponents may be positive integers (as is usual for simple reactions) or fractions (when a reaction occurs through different intermediate steps). Apart from simple reactions, they do not have to be equal to the stoichiometric coefficients of the reactant in the net reaction. We shall see that this does not apply to reversible reactions; when they reach equilibrium the stoichiometric coefficients are equal to the reaction order for each component.
Up to now we have considered irreversible reactions. In the following chapter we shall see that reversible reactions occur as well. In such cases, one must also consider the transformation from reactants to products.
A→B (rate constant = k1)
The rate of a first-order reaction is proportional to the first power of the concentration of only one reactant. This means that the amount d[A], which undergoes chemical change in the short time interval dt , depends only on the amount of A present at that instant, assuming that there is no change in volume, temperature, or any other factors that could affect the reaction.
The rate expression which describes a first-order reaction is
- d[A]/dt = k1•[A] (2.2)
As can be seen in chemistry textbooks, the equation above can be rearranged and integrated, introducing [A]_{0} as the initial quantity of the reacting substance A in a given volume and x as the amount which reacts in time t. It follows that ([A]_{0} - x) is the amount of A remaining after time t. The exponential form of the integrated equation is
[A] =[A]_{0 }• (1 – exp(-k1•t)) (2.3)
This analytical expression is shown here merely for comparison, but all the subsequent reactions will be solved in this handbook using iterative computer procedures. The code listed below is all that is needed to solve and display the variations of concentrations during a first-order irreversible reaction.
The iterative procedure consists
of a for..next cycle in the program. This is repeated until the
reaction reaches its end point. In the example, 100 cycles are
sufficient. Code lines, or parts thereof beginning with the
symbol (') are comments.
dim A(100),B(100) A(0) = 60 ' initial concentration of A B(0) = 0 ' initial concentration of B k1 = 0.1 ' reaction constant t1 = 100 ' final time for t = 1 to t1 ' reaction takes place A(t) = A(t-1) - A(t-1)*k1 B(t) = B(t-1) + A(t-1)*k1 next t call DrawScreen |
The 'DrawScreen' routine is used for graphical presentation of the results and is listed in the appendix. Its graphical output is shown in Fig.2.1.The initial condition of this simulation is [A]_{0} = 60 mMol/L , k1 = 0.1 sec^{-1} , time in seconds from 0 to 100 and a time-step of 1 second. The concentration of A exponentially decreases to zero, whilst that of B rises to 60. All the values are stored in two matrices to be used for the plot (or other purposes).
Fig.2.1
First-order
irreversible reaction and plot of the A and B concentration versus
time.
When two reactants, A and B, react in such a way that the reaction rate is proportional to the first power of the product of their respective concentrations, the compounds are said to undergo a second-order reaction.
If [A]_{0 }and [B]_{0} designate the initial quantities of the two reacting chemicals A and B, and x is the number of moles of A or B which react in a given time interval t, then the rate of formation of product C can be described by the following mechanism
time |
A |
B |
C |
0 |
[A]_{0} |
[B]_{0} |
0 |
t |
[A]_{0} - x |
[B]_{0} - x |
x |
and by the following differential equation:
dx/dt = k1•([A]_{0} - x)•([B]_{0} – x) (2.4)
This differential equation can be integrated, but in this handbook it will be solved by a much simpler iterative procedure. The results in Fig.2.2 show that after a certain time the concentrations reach a stable value (or more precisely asymptotically). One of the two reactants (in our simulation B) reaches zero concentration. Consequently, the reaction can no longer evolve and comes to a halt.
dim A(100),B(100),C(100) A(0) = 60 ' initial concentration of A B(0) = 40 ' initial concentration of B C(0) = 0 ' initial concentration of C k1 = 0.003 ' reaction constant t1 = 100 ' final time for t = 1 to t1 ' reactions take place A(t) = A(t-1) - A(t-1)*B(t-1)*k1 B(t) = B(t-1) - A(t-1)*B(t-1)*k1 C(t) = C(t-1) + A(t-1)*B(t-1)*k1 next t call DrawScreen |
Fig.2.2
Second-order irreversible reaction.
Consecutive Irreversible Reactions
A → B → C → D (rate constants = k1,k2,k3)
The mechanism implies a series of three, four or even more consecutive reactions, the products of the first being simultaneously the reagents of the second, and so on.
The mathematical treatment is
really cumbersome, but as usual a simple iterative procedure can help
us. Let’s
see the table first:
time |
A |
B |
C |
D |
0 |
[A]_{0} |
[B]_{0} |
[C]_{0} |
[D]_{0} |
t |
[A]_{0} - x |
[B]_{0} + x - y |
[C]_{0} + y - z |
[D]_{0} + z |
where x is the amount of A transformed in B after a certain time t, y the amount of B transformed in C and z of C to D.
The corresponding differential equations are
- d[A]/dt = k1•[A]
+ d[B]/dt = k1•[A]- k2•[B]
+ d[C]/dt = k2•[B]- k3•[C]
+ d[D]/dt = k3•[C]
The
code is here below
dim A(100),B(100),C(100),D(100) A(0) = 50 ' initial concentration of A B(0) = 10 ' initial concentration of B C(0) = 0 ' initial concentration of C D(0) = 0 ' initial concentration of C k1 = 0.08 ' reaction constant of reaction A-->B k2 = 0.04 ' reaction constant of reaction B-->C k3 = 0.08 ' reaction constant of reaction C-->D t1 = 100 ' final time for t = 1 to t1 ' reactions take place A(t) = A(t-1) - A(t-1)*k1 B(t) = B(t-1) + A(t-1)*k1 - B(t-1)*k2 C(t) = C(t-1) + B(t-1)*k2 - C(t-1)*k3 D(t) = D(t-1) + C(t-1)*k3 next t call DrawScreen |
and the corresponding plot is in Fig.2.3
Fig.2.3
Consecutive irreversible reactions.