Fundamentals of Chemical Kinetics: Ordinary and Sequential Reactions of the First Order

_{(License: Public Domain, Author: geralt) Source: Pixabay}

Fundamentals of Chemical Kinetics: Ordinary and Sequential Reactions of the First Order

Millions of chemical reactions are constantly occurring in the universe, each of which is indispensable for the functioning of different systems, both physical and biological. Commonly in chemistry we observe reactions of great impact that impress our senses, many of them even frighten, due to the dangerousness, either by release of energy or by the toxicology of the reagents. Although we know that all these reactions can occur, using thermodynamic data such as Gibbs' free energy, it is the kinetics that really provide us with the necessary information to quantify theoretically and experimentally how fast a reaction is, how fast its intermediaries are, and how long it could be said that the reaction has "finished". In this article, we will highlight the basic aspects of this branch of physicochemistry that models the temporal dependence of this type of phenomena.

Given a general chemical reaction, you have to:

_{Own Authorship}

The reaction speed can be defined as the rate of change of the moles of a spice involved in the system as a function of time. Since chemical reactions have an associated stoichiometry, the higher the stoichiometric number with respect to the other chemical species, the lower its speed of "disappearance" or "appearance":

_{Own Authorship}

An example of this is the following reaction:

_{Own Authorship}

It is important to emphasize that a negative sign is placed, as a convention, to indicate that this chemical species is being consumed in time, that is to say that the number of moles will decline with the passing of time. According to the previous example, it can be interpreted that the rate of disappearance of methane is twice that of oxygen, or otherwise, the rate of disappearance of oxygen is twice that of methane, and this happens precisely because of the stoichiometric requirement of the 1:2 reaction. Similarly, the positive rate is interpreted as that of those chemical species (products), whose concentrations or number of moles increase with the course of the reaction.

In other words, the reaction speed is an extensive property, which depends on the amount of matter (moles). It can be considered an intensive reaction speed, which depends on the concentration of the species in question, the latter form is much more useful at the experimental level, since spectroscopic methods can be used to evaluate the concentrations of the chemical species involved as a function of time.

_{Own Authorship}

Therefore, we can also associate the reaction speed as the change of concentration of a reagent as a function of time, relative to its stoichiometric number.

In all fields of chemistry, the study of kinetics is extremely important to study the behavior of reactions, in medical or industrial applications, it is convenient to know the dependence of one or all reagents on the reaction speed, or the contribution of each of the initial concentrations, to the value of the total speed. I consider this section one of the daily utilities of chemical engineering, where it is intended to optimize a known reaction (and knowing that it is thermodynamically favorable), to occur in the shortest possible time, thus saving economic costs, energy, and of course, saving time.

Not only in the industrial part it is important to study this, for pharmaceutical chemistry and/or biochemistry, to study the advance of the bioavailability of some active principle, which is acting as a drug in some organism, is of crucial importance to obtain conclusions congruent with its pharmacological effect.

The law of velocities is expressed as follows:

_{Own Authorship}

The product of the concentrations of the reactants, elevated to a constant, and another constant k, which is a proportionality constant characteristic of a reaction at given temperature and pressure values, is observed. The exponents of each of the concentrations are determined experimentally. It is said that a reaction is of order "𝛼" with respect to "A", of order "𝛽" with respect to "B", and of global order "𝛼 + 𝛽", if the above described is fulfilled.

As mentioned above, these constants are determined experimentally. Usually the isolation method is used: In which all species are kept at a high concentration with respect to a spice that must be very diluted. In this case the variation of the concentrations of the concentrated species could be assumed as negligible, obtaining an initial velocity almost completely dependent on the diluted species, in this way the order of this reagent can be calculated in an approximate way.

Another convenient method to obtain it is to measure the variation in the concentration of a species in time twice, in which the only difference will be the initial concentration of only one of the reactants, by means of a relation of the values of R, it is also possible to obtain a fairly approximate value of the order for this compound.

An example of this method is shown below:

_{Own Authorship}

In the previous table it is observed that in replicas 1 and 2, the concentration of the spice two is the same, therefore, when making a relation with the experimental data the following expression can be obtained:

_{Own Authorship}

In this way, the order of reaction can be calculated with respect to A, and since it is a simple reaction of only 2 reactants, the calculation of the order of reaction with respect to B can be carried out without problem, in an analogous way. Once both orders are obtained, the global order of the reaction is finally known and the associated velocity expression can be raised.

Usually it is possible to obtain non-differential expressions of the concentrations of the different species. This mathematical approach allows us to know the evolution of these variables as a function of time, including the velocity constants associated with each step of the proposed mechanism. In this way, for an ordinary reaction of first order we have the following:

_{Own Authorship}

Being its expression of velocity

_{Own Authorship}

which can be interpreted as a differential equation and solved by the method of separation of variables

_{Own Authorship}

From this expression is obtained the behavior of the initial concentration of a reactant X, in the form of exponential decay, behavior that is congruent with what was observed experimentally for reactions of order 1. If one wanted to know the evolution, in other words, the appearance of the product, the following can be raised:

_{Own Authorship}

In this case the expression is obtained that mathematically models the appearance of product Y as a function of time.

In some cases, reactions may be sequential. This happens when a multi-step mechanism is proposed or includes intermediaries associated with each of the steps. In order to study this type of reaction, it is necessary to start from the law of velocity for each of the species.

_{Own Authorship}

_{Own Authorship}

The differential equation presented above can be solved to find the following solution for the intermediate concentration I

_{Own Authorship}

As intermediaries of great chemical interest for the study or experimental confirmation of a proposed reaction mechanism, the function describing the behaviour of its concentration over time will be deduced:

Demonstration:

Being an ordinary differential equation of the first order, the solution to the homogeneous equation will be found.

_{Own Authorship}

To do this, we will look for the roots of the characteristic polynomial associated with the equation.

_{Own Authorship}

Once the roots are obtained, the following expression is proposed as a solution to the homogeneous differential equation

_{Own Authorship}

Once the homogeneous solution is found, we need to propose a general solution that satisfies the original equality.

_{Own Authorship}

This proposed solution will be derived as many times as necessary to be able to replace it in the initial expression, and to be able to find the coefficient A.

_{Own Authorship}

Substituting:

_{Own Authorship}

Therefore the coefficient A is

_{Own Authorship}

And the general solution is the concentration of I ([I]), the following family of functions

_{Own Authorship}

However, it is necessary to know the particular solution to this problem, for this it is necessary to determine the value of constant C in the general solution. To achieve this, the initial value of the concentration of I will be used in a time equal to 0, that is, when the reaction has not yet begun... Being I, an intermediate species, we have the following:

_{Own Authorship}

By substituting this condition, C can be found easily.

_{Own Authorship}

Therefore the overall solution of the proposed equation is

_{Own Authorship}

If you want to know the concentration of the product as a function of time, it is enough to replace the necessary expressions in the following equality

_{Own Authorship}

In this sense, a first-order differential equation has been solved, which allows us to know the concentration of the intermediary (or intermediaries) as a function of time, knowing each of the velocity constants associated with each step.

_{Characteristic graph of a first-order sequential reaction - (License: CC BY-SA 3.0, Author: Knights who say ni) Source:Wikimedia Commons}

This has been all for today, I hope you enjoyed the content of the post, I personally believe that kinetics is a fundamental area for many disciplines beyond chemistry, any comments will be welcome (doubts, suggestions, etc.). Soon I hope to do the second part about second order reactions and balances.

We are still waiting for the great developments that the future will bring us, you decide whether to be a spectator or a doer. Every day we can learn something new.

Thank you for reading.

---

References:

All the images and dividers of my authorship were edited and processed using the software PowerPoint 2016.

• Theodore Brown, Eugene LeMay, Bruce Bursten, Julia Burdge, (2004), Chemistry. The central science. (9th Edition). Pearson Education, S.A., Chapter 04.

• Douglas Skoog, Donald West and James Holler (2006) Fundamentals of Chemistry Analytical (4th Edition) Editorial Reverté.

• CRC HandBook of Chemistry and Physics, 86th Edition, CRC Press 2005

• Arthur I. Vogel et al. (1989) Textbook of Quantitative Chemical Analysis, 5th edition, Longman Scientific & Technical.

• Engel, Thomas, and Philip Reid. 2006. Physical chemistry. San Francisco: Pearson Benjamin Cummings.

---

---

Posted from my blog with SteemPress : http://aleestra.vornix.blog/fundamentals-of-chemical-kinetics-ordinary-and-sequential-reactions-of-the-first-order/

---