The kinetics of the transformation processes

The kinetics of the reactions study, the speed of development of a transformation reaction, which, starting from certain reagents, gives rise to certain products. In fact, all reactions take place at a certain characteristic speed k = 1 / Δt_r, where Δt _r is a characteristic time of the reaction considered. A typical characteristic time is that of half-life, that is, the time necessary for 50% of the reactants to be transformed into the products, also called t₅₀. This time must be compared with the time scales typical of other processes, transport processes such as advection and diffusion, which take place, together with the transformation reactions, within the surface water body. Self Δt_p is the time scale of the transport processes, 3 cases can occur:

- if the reaction takes place very quickly, on a time scale lower than that of the transport processes, i.e., it is Δtr << Δtp, you can make the hypothesis that the products are formed as soon as the reagents are available and, therefore, the reaction kinetics can be neglected. The reactions of this type are called snapshots and are controlled by the availability of reagents. In other words, the speed of formation of the products depends on that of formation of the reactants and not on the intrinsic speed of the transformation reaction:

 - if the reaction takes place slowly, on a time scale to that of the transport processes, i.e., it is Δtr >> Δtp, the presence of the reaction can be neglected, and the substance can be considered conservative, i.e., non-reactive.

- If the reaction time scale is similar to that of the transport processes, i.e., Δt r = Δtp, the reaction must be considered, and its kinetics also plays an essential role. Assuming that the reactants are available, the reaction is controlled precisely by its kinetics, and the rate of formation of the products depends precisely on the kinetics of the transformation reaction.

Consider again the generic reaction, which takes place under conditions of complete mixing:

aA + b B↔ c C + d D  (1)

Where a, b, c, and’s’ are the stoichiometric coefficients of the reaction, and A, B, C, and ‘D’ are the substances involved in the reaction. The reaction law or kinetic law of generic substance A is:

D[A]/dt= R_A       (2)

Where R_A is a function that describes the reaction law for species A

At this point, it is necessary to define the expression of the reaction law Ri of the generic species i. The general form of the kinetic law for the generic product formed from a number of reagents A equal to j is:

R_i=k_r-i[A1]ⁿ¹ A2]ⁿ² A3]^n3……. A_j]^nj     (3)

Where k r-i is the speed constant of the reaction [T -1], and n j is the order of the reaction with respect to the generic constituent j. The sum of the orders of reactions of all constituents, namely:

It is the total order of the reaction, which therefore serves to identify the influence that the concentration of some or all of the reactants has on the speed of the reaction. The law of reaction is often determined on an experimental basis.

The kinetics of transformation processes

In the simple case of the reaction:

A+B -àP  (4)

It is reaction law for P holds:

D[p]/dt=k_r[A]^nA [B]^ns    (4)

The rate constant of a reaction k r depends on the temperature and, in particular, generally increases with it. To define the relationship between k r and T, we can use the Arrhenius-Van't Hoff equation:

K_r= A exp( -E_a/R_gas T)    (5)

Where A is the frequency factor, E a,  are the activation energy characteristic of the reaction, R gas the universal constant of the gases, and T the absolute temperature. Therefore, a slow reaction can be due to a low-frequency factor value or a high activation energy value.

The zero-order kinetics

When the speed of a reaction is independent of the concentration of the reagents, it is called zero-order kinetics. This is typical, for example, of heterogeneous reactions, where the exchange surface controls the speed of the reaction. Consider, again, the simple reaction:

A+B -àP  (4)

Its reaction law for P always holds:

D[p]/dt=k_r[A]^nA [B]^ns    (4)

In this case, since the exponents of (4) are all null, the reaction law becomes:

-d[A]/dt= -d[B]/dt= d[P]/dt= k_r  (6)

Therefore, if one refers to one of the reactants, the zero-order kinetics establishes that this reactant decreases over time with constant speed. Therefore, for example, if the field data of the concentration of a pollutant are arranged, plotted against time, according to a straight line, it can be deduced that the reaction that determines the progressive disappearance of the contaminant follows a zero-order kinetics, which is constant andS exactly equal to the inclination of the straight line.

First-order kinetics

When the speed of a reaction depends on the concentration of only one of the reagents, it is called first-order kinetics. Consider, again, the simple reaction:

A+B -àP  (4)

If, in this case, for example, the speed of the reaction depends only on A, the reaction law for A becomes:

d[A]/dt=k_r[A]^nA =k_r[A]    (7)

Or

R_A=k_r[A]   (8)

Where the unknown k r that is the reaction constant can be determined by different methods

The (7) can be expressed in a similar way to the transport equations examined so far, in terms of concentration, if C is the concentration of the generic contaminant, such as:

dC/dt=± k_rC (9a)

Where the negative sign describes the case in which the chemical reaction tends to make the contaminant progressively disappear, while the positive sign refers to the opposite case in which the reaction produces this contaminant.

First-order kinetics

This equation can be easily solved by assigning an initial condition, represented by the initial value of the contaminant at a certain preset time, usually t = 0. In fact, let this condition:

C(t=0)=Co   (10)

By separating the variables, (10) takes the form:

dC/C=± k_r, dt  (9b)

Which integrating both members, provides, after defining the reaction constant through the initial condition:

C(t)=Co exp(± k_r,t)   (11)

for k r negative, represents an exponential decay of the concentration over time. For the case of C 0 = 1 and k r = -1, where the concentration tends asymptotically to the null value.

The second-order kinetics

When the speed of a reaction depends on the concentration of two of the reagents, this is called second-order kinetics. Consider, once again, the simple reaction:

A+B -àP  (4)

in this case, the speed of the reaction depends on both A that B. The reaction law is:

-d[A]/dt= d[P]/dt =k_r[A]^nA[B]^nB  (12)

Also (12) can be expressed in terms of the concentration C of a generic contaminant such as:

dC/dt=±k_r, C²   (13)

Here the negative sign refers to a reaction that tends to make the contaminant progressively disappear, and k r has dimensions of [L 3 · M -1 · T -1]. Here, too, an initial condition like the first one must be assigned. The (13) has the form:

C(t)= 1/ ∓krt+1/Co (14)

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Author: Vicki Lezama