Chemical reactions // Adjustment of its equation through equivalent systems, by applying the Gaussian method in matrix notation
In this opportunity I will present you with a publication related to the adjustment or equalization of chemical reactions by means of triangular equivalent systems using the Gauss method in matrix form, very important for the development of our students, since they will know the implementation of one of the tools of algebra to adjust a chemical reaction, before entering into the development of such adjustment, necessary is to make a brief review on systems of linear equations, remember a linear equation, identify also the systems of linear equations 2x2 and 3x3, that is, systems with two and three equations and / or unknowns respectively, as well as some methods used to solve them, likewise systems with high numbers of both equations and unknowns, the forms of the s triangular systems both lower and higher in matrix notation, the methods used to reach this type of triangular systems.
Also remember or know some important concepts about chemical reactions such as atoms and molecules which are the elements and compounds involved in these reactions, they require an adjustment or adequate balance of their equation for an optimal result in terms of composition of the final products, all this will allow us to properly calibrate our algebraic tool of matrix notation, always starting from the elementary principle for the resolution of complex problems in any field, starting from the simplest to consolidate the solutions for the most complex, therefore , we can start our interesting journey remembering the following:
Therefore, we can say that a system of linear equations is represented by a set of equations which share the same unknowns, it is important to be very clear about some essential fundamentals such as clearing an unknown, knowing how to recognize a first degree expression, as well as remember that an equation is not more than an equality which is only fulfilled for a certain value which satisfies said equality, therefore, these equalities that are always met will represent identities.
When we are in the presence of a certain system of two first-degree equations with two unknowns, to solve it we have to find two numbers, because we have two independent conditions (unknowns), which these numbers must fulfill, in the same way it happens to have to solve a system of three equations with three unknowns and so on, what will usually vary will be the methods to use and this will be one of our main objectives in this publication, to identify some of these important methods, and apply them to the reactions Chemicals to achieve your atomic adjustment.
For now we are going to describe the different methods that we can use to solve a system of linear 2x2 equations, they are the same:
Substitution
Once the variable is cleared, we proceed to the next step, which consists of substituting this value of (y) in the other equation, that is, in equation 2.
Then we can only check substituting the two values found in each of the equations of the system to see if the respective equalities are met:
Evening
This algebraic method consists of clearing one of the variables of the equations that make up the system, where the unknown to be cleared must be the same in both equations, for example:
The verification is performed in the previous method, we only use another type of method which allowed us to obtain the same results, therefore, we verify that the algebraic method to be used to solve a certain system of equations does not influence the solution.
Reduction Method
This method consists of multiplying some member of each equation by a certain convenient number, all with the firm purpose that the unknowns have equal coefficients but with different signs in the two related equations, in this way we will obtain an equation with only one unknown. For example:
The verification will be the same as those made with the other methods, what we can highlight again is that we have verified that whatever method we select, the result or solution will always be the same.
It is important to highlight the types of solutions that we can have from a certain system of first-degree equations, since it can have only one solution, therefore, we will say that we are in the presence of a determined compatible system such as the one we have just solved. ; but it can also happen that it has no solution, and we will give it the name of incompatible system, but we can also be in the presence of a system with infinite solutions, which we will call an undetermined compatible system.
Ecuation systems 3x3
Now we have a system of three equations with three unknowns, to solve the same we will be implementing the substitution method, since we will clear one of the unknowns of any of the equations present in that system, to then replace the expression we have obtained in the other two equations, therefore, we will obtain two equations with the remaining unknowns, that is, a new system that for this case will be 2x2, two equations and two unknowns, and this type of equations we already know how to solve, applying any method algebraic before developed two solutions will be obtained (if it has them), to later substitute in the equation obtained by means of the clearance of one of its variables, we will obtain the value of the incognito eliminated initially, for example:
We observe that when performing the corresponding operations, we obtain our new system now formed by equations 4 and 5, which are two unknowns, and since we have developed this type of system it is possible to do it through the methods already studied (substitution, equalization and reduction). You can choose the method of your choice for this case we will choose the reduction method:
The previously developed methods can be applied generally in linear systems that have few equations and unknowns, therefore, it is important to know the application of other methods which allow us to solve systems of linear equations many more complex, that is, with a high number Both equations and unknowns, then, we will rely on knowing more suitable methods to solve these types of systems with greater degree of difficulty, these methods are those that achieve the construction of a system equivalent to the given linear system, in a few words with the same solution, the main objective that is much easier to solve, in short words the conformation of triangular systems, both superior and inferior, due to its practical step structure, it is possible to realize that we can calculate these unknowns one by one, starting from top to bottom or from bottom to top depending on the case of the system we want to conform, in our case we will study or apply the superior triangular systems in matrix notation, first we will know these triangular systems and then locate the method that allows us to reach these types of systems, which in our case would be in matrix notation.
Triangular systems
To be able to structure any system of equations in a triangular equivalent (if possible), whether it is lower or higher, this system will be easier to solve, because each of the unknowns we clear them one by one to replace them in the other equations , for example:
Lower triangular system
The procedure carried out we call the descent algorithm, because the unknowns are looking for recurrence, for this case from top to bottom, this structure can be found in any system with a high number of equations and unknowns, but the inconvenience occurs when we have to take this system to the triangular form since it does not have a certain structure, that is the objective of this publication, to be able to find the methods as long as the system allows to convert it into a triangular inferior or superior, for our particular case we will apply the superior triangular structure.
Superior triangular system
In this procedure carried out we call it the upload algorithm, because the unknowns are looking for recurrence, for this case it would then be from bottom to top.
With these triangular systems (upper and lower), we can reach the structuring of our equivalent systems, with the main purpose of being able to solve it as practical and easy as possible.
Equivalent systems
We can say that two systems can be considered equivalent if both have the same solutions, therefore, certain operations or properties allow us to transform a certain system into another equivalent, for this we must consider the following:
1.- We can change the order of the equations of a given given system.
2.- We can also multiply two members of one of the linear equations by the same number which must be different from zero.
3.- We can eliminate a certain equation from our system, which we see as a linear combination of the others.
4.- Within the same system we can change one of the equations by a linear combination of itself, as well as some of the other.
We have then identified the properties to be able to transform a certain linear system into another equivalent but with greater ease of resolution, but it is important to highlight the way or method to be used to achieve this purpose, that is, convert the linear system into an equivalent one. superior triangular shape that is our purpose, therefore, the recognized Gaussian method allows us to obtain the above-mentioned objective making use of the properties mentioned above to obtain equivalent systems of superior triangular shape which we will apply in matrix notation for the adjustment of chemical reactions.
Gaussian method
This method consists of successive procedures such as:
We will have a series of stages which will allow us to reach our goal, therefore, in each of the stages, we must try to substitute zero for each coefficient that is below the diagonal of the system.
When performing the above, ie elementary transformations, we are replacing each corresponding equation of the system by another which makes the equivalent system and that in turn has zero such coefficients below the main diagonal.
In this way we will arrive at our equivalent triangular system, which will be solved by means of the upload algorithm.
It is important to refer to the transformations of this method, which are carried out directly on each equation of the system, but it is possible to make these transformations in a more practical way in matrix notation of the determined system, which is our case to study it from that point of seen, therefore, we must proceed as follows:
1.- We must write our system in matrix form, that is, Ax = b, where A is the matrix, the literal x will be the vector of our unknowns and finally b will be the vector of the independent terms of said system.
2.- We construct the corresponding extended matrix, which will be a matrix that we will denote [AIb], we will form it with the inclusion of vector b, which will be the last column of matrix A.
3.- We must apply all the elementary operations that allow us to build the superior triangular matrix.
Having already calibrated the algebraic method in notation or matrix form, which we will use for our adjustment in chemical reactions, we will then introduce ourselves to the brief journey in the area of chemistry, specifically in its reactions, to know some important concepts within this area, by So, we have:
Atom
This atom represents the smallest or smallest unit of a certain chemical element which has its own existence and which in turn can enter in combination, emphasizing that it is composed of a nucleus of protons and neutrons with an electron cover, the atom is maintains electrically neutral, because the number of protons of its nucleus is equal to the number of electrons present in the cortex of the same, therefore, we call atomic number the number of protons that are present in a certain nucleus of an atom which is part of an element.
A certain atom can gain or lose one or more electrons, turning its charge into positive or negative, giving rise to the appearance of an ion, these ions are what we call cations if their charge is positive and if their charge is negative we call them anions.
Molecule
It is constituted by a certain group of two or more atoms which is linked by chemical bonds, the molecule represents for a substance what the atom represents for any element, that is, a molecule is the minimum amount present in a certain substance which has the ability to exist in free state, while retaining all its chemical properties, The molecules present in the elements (substance which can not be divided into other simpler by chemical means) are constituted by one or more identical atoms, is to say, of the same class, but the molecules present in the chemical compounds are constituted by at least two atoms of different class.
The matter which constitutes or gives origin to any thing or entity of our environment can be found in two different forms, homogeneous and heterogeneous, it will depend on its properties and composition, be the same at any point (homogeneity) or change them when passing from one point to another (heterogeneity).
Chemical compounds:
When we speak of chemical compounds we are referring to pure substances (homogeneous phase of constant composition), formed by more than one type of atoms, these compounds can generally be decomposed, through the action of electric current (Electrolysis), through of the action of heat and of course by chemical means, highlighting that in none of the cases by physical means only, there are also pure substances that can not be decomposed, they are represented by the chemical elements that are present in our nature, there are some elements with unstable artificial characteristics, as we know these elements are divided or classified in metals; aluminum (Al), iron (Fe), copper (Cu), nickel (Ni), silver (Ag) among others, and non-metals; hydrogen (H), fluorine (F), chlorine (Cl), iodine (I), oxygen (O) among others, leaving a few with characteristics of intermediates known as semimetals or metalloids; arsenic (As), teluriu (Te), antimoniu (Sb), among others.
Chemical reaction
When we are in the presence of new substances from the transformation of one or more substances we say that this process is called chemical reaction, in which the initial substances are called reactants, and those that are obtained are called reaction products, whose formulation general is the following:
Recalling that A and B are the reactants, C and D represent the products of the reaction.
As we have already expressed in chemical reactions, different types of elements are related as compounds giving rise to new ones of different composition, this relation must carry an adequate proportion to both sides of said reaction, therefore, these reactions are represented through equations which inform us of all the conditions in which a certain chemical change occurs, so, in order to have the required success we need to adjust, equalize or balance any equation that represents any chemical reaction.
Now we can proceed to make the adjustment of a chemical reaction, remembering that in appropriate proportions a set of substances which we call reagents are converted or transformed into different ones that we call products, these reactions must be adjusted, both the reactants and those that originate from them, generally these quantities are expressed in whole numbers.
We proceed to adjust the coefficients (a, b, c, d, e, f) of the next chemical reaction by applying an equivalent system of superior triangular shape constituted by the Gaussian method in matrix notation:
Now we proceed to verify if the values of each of the found coefficients allow us to adjust or balance the equation of our chemical reaction, we proceed:
Conclusión
In conclusion, we can say that the equation of the chemical reaction is adequately balanced, that is, adjusted or equalized, through an equivalent system of superior triangular shape by applying the Gaussian method in matrix notation, a very useful tool. useful to reinforce pleasantly the existing methods in this area as they are; method by trial and error, arithmetic and redox method.
We also prove that chemistry is a science with strong ties of brotherhood with many scientific branches, in this case a sub-branch of mathematics, algebra, the abstract language par excellence, therefore, more universal and general that counts humanity in all its aspects.
Thank you in advance for your support in this pleasant and well-traveled journey, until the next publication friends of steemit.com
Note: All the images were made in paint and PowerPoint 2017 for Windows and the gifs with the PhotoScape application.
Bibliographic References Consulted
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[3] Stanley I. Grossman( 1983) . Álgebra Lineal. Grupo Editorial Iberoamérica.
[4] A. Baldor, Algebra. Edición 1992. Editora y distribuidora de textos americanos, copyright compañía cultural S.A.
[5] Universidad de Sevilla. Departamento de Ecuaciones diferenciales y Análisis Numérico. Curso Año 2015-2016.
