Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

On Balancing Acidic and Basic Reduction/Oxidation Reactions with a Calculator

Matthew S. Fox

Department of Chemistry, Cheyenne Mountain High School, Colorado Springs, Colorado, United States

Abstract: Balancing chemical equations is a fundamental skill every chemist—from novice to professional—must have. In this report we present a calculator-based, linear algebraic method that balances even the most rigorous acidic and basic conditioned redox equations in an expeditious manner.

Keywords: calculator-based learning, redox equations

Received May 03, 2015; Revised May 20, 2015; Accepted June 08, 2015

Citation: Matthew S. Fox, On Balancing Acidic and Basic Reduction/Oxidation Reactions with a Calculator. World Journal of Chemical Education, 3 (3). 74-77. January 2015. doi:10.12691/wjce-3-3-4

1. Introduction

The algebraic method for balancing chemical equations is traditionally less popular than alternative methods, as the corresponding sets of linear equations are often tedious to equate. Likewise, using algebra to balance redox equations has proven to be even more difficult, allowing other methods such as inspection or half-reactions to be more commonly taught. Balancing chemical equations with linear algebra simplifies the algebraic method; however, according to McCoy, [1] “Linear algebra [will] not help balance [redox equations] properly.” Many authors proceeded to disprove this proposition, [2,3,4] while others introduced the necessary reformulations of both chemical and redox equations to derive effective linear algebraic methods [5,6,7,8,9].

It is the purpose of this work to establish a calculator-based procedure for balancing acidic and basic conditioned redox equations. The proposed method is appropriate for undergraduate chemistry classes and perhaps Advanced Placement courses, provided scientific calculators are available. The major contribution in this report is the linear algebraic representation of the acidic and basic half-reaction procedure. This contribution introduces a new, calculator-based method, which has potential to alter how balancing redox equations is taught.

2. Necessary Review

This manuscript will continually refer to the following chemical equation Ac, where

where x₁ through xr denote both the term and unknown coefficient for each compound reacting, xr+1 through xp the term and unknown coefficient for each compound being produced, and β the net charge associated with each compound.

The first condition for balancing any chemical equation asserts the law of conservation of mass. That is,

where both sides of the above equation refer to the same element. [4] For element A in Ac specifically

[picture] | (1)

where a denotes the subscript for element A in each term of Ac. Reformulating eq 1 for each element in Ac yields the linear homogeneous system

[picture] | (2)

which is equivalent to the matrix setting

[10]

If the coefficient matrix above is denoted by A and the solution vector , then the system follows the homogeneous equation A. [10,11] Solving the equation for is known as computing the null space, or kernel, of the coefficient matrix A. [10] Thus, the kernel of a chemical equation’s coefficient matrix is also the solution to the chemical equation, provided the stoichiometric coefficients are in their simplest, whole number ratios [1].

The next condition asserts the net reactant charge must be equal to the net product charge. [4] This introduces the charge equation, denoted as β(Ac), which is nearly identical to eq 1. Formally,

[picture] | (3)

If β(Ac) is appended to eq 2, then the corresponding matrix setting changes accordingly. Effectively, A gains a new row vector equivalent to the net charges associated with each compound. Thus,

[picture] | (4)

which resembles the general formula to balance any redox reaction.

For example, consider the equation

Here, x₁ and x₂ represent the set of , while and the set of xp. Eq 3 produces β(Bc), where β(Bc) = -+ 3 + 0- 2= 0, with the complete homogeneous system equivalent to

Therefore,

which is synonymous with eq 4. (Notice how the coefficients in the above equations correspond directly to the entries in the matrix.)

The kernel of B is most commonly equated by first calculating the reduced row echelon form (rref) of matrix B. Inputting B into a scientific calculator and selecting the matrix function ‘rref’ yields

(For a TI-84 calculator, input the following commands to obtain the rref: 2ND, , ENTER, and input the determined coefficient matrix. Once complete, input: 2ND, MODE, 2ND, ALPHA, APPS, 2ND, , ENTER, ENTER.)

To calculate the kernel of B, take the right-most column in the rref(B) and append an additional ‘−1’ to the bottommost row. (Note that if the matrix is square, meaning the same number of rows as columns, appending the ‘−1’ is not necessary) This obtains

From here, multiply each entry by a common factor to obtain whole number ratios and the greatest amount of positive coefficients possible. For B specifically, multiplying by a scalar of −2 yields the kernel of B, where

or

3. Methods

In this section we introduce the linear algebraic representation of the acidic and basic half-reaction procedure. Such representations are the main contributions of this article.

3.1. Reactions in Acidic Solutions

Remark. In this paper we denote redox equations balanced in acidic conditions as A⁺, where A is the coefficient matrix for the chemical equation Ac.

According to the acidic half-reaction procedure, H⁺ and H₂O are appended to separate sides of the initial redox equation, and then balanced accordingly. The former implies the unbalanced redox equation will have two terms appended similar to H₂O +…→…+ H⁺. The linear algebraic representation is then two column vectors appended to opposite sides of the coefficient matrix identical to

[picture] | (5)

Note the above row vectors will not necessarily be applied as cited above. That is, depending on how the coefficient matrix A was originally derived, β(Ac), Hydrogen, and Oxygen will not necessarily be listed in the first, second, and third rows, respectively. For sake of understanding, however, the row vectors in the examples of this paper do follow the order as shown in eq 5.

Consider the chemical equation Cc, where

The linear homogeneous system is obtained by following eq 2. Consequently,

Eq 4 yields

which represents the matrix system for Cc. Appending eq 5 yields the system or

Notice with the addition of eq 5, gains two entries due to Cc gaining two terms: H₂O and H⁺. Computing the null space of C⁺ asserts

or

To finish, the compounds with negative coefficients are transferred to the opposing side of the equation. [4] Ergo,

3.2. Reactions in Basic Solutions

In contradistinction to eq 5, the basic half-reaction procedure implies that both OH− and H₂O are added to the initial redox equation. Chemically, H₂O +… →…+ OH−. Linear algebraically,

[picture] | (6)

(Analogous to eq 5, the implementation of eq 6 will yield A− for chemical equation Ac.) Consider the chemical equation Dc, where

Utilizing eq 2, the linear homogeneous system is established. That is,

Thus,

Affixing eq 6 yields the system , where

The rref() produces

or

4. The Mathematical Procedure

The following list enumerates the procedure to balance acidic or basic redox equations in an efficient, computational process.

1. Be sure the equation is in its respective net ionic representation.

2. From the information given in the redox reaction, produce homogeneous equations for each element.

3. Establish the homogeneous charge equation from the unique charges associated with each compound.

4. Transform the equations into a matrix system, where each row represents each element and each column represents each term in the system’s coefficient matrix.

5. Depending on the information in the redox equation and whether the equation is to be balanced in either acidic or basic conditions, append either eq 5 or eq 6, respectively.

6. Equate the kernel of the coefficient matrix developed in the preceding step by utilizing a calculator’s reduced row echelon form (rref) matrix function.

7. Check that the kernel of A has entries in the lowest possible whole number ratios with the greatest number of positive coefficients possible.

8. Insert the kernel into the chemical equation and check that the equation is balanced. If negative coefficients appear in the final equation, transfer these coefficients and their corresponding compounds to the opposing side of the equation.

4.1. Limitations

The rref(A) is only effective if the order of A is , where m = n, or m < n by only a difference of 1. Fortunately, most redox equations follow these parameters, especially after eq 5 or eq 6 is appended. In principle, however, if an unbalanced redox equation has a number of different elements > the number of unknown coefficients (m > n), the linear algebraic method cannot be used.

5. Conclusion

The linear algebraic representation of the acidic and basic half-reaction procedure was introduced in this paper. The representation established a calculator-based method to balance redox equations in either acidic or basic conditions. The efficiency and intuitiveness associated with the proposed method allows students to balance rigorous redox reactions in an expeditious fashion, while not undermining the essential laws governing balanced reactions.

A potential criticism regarding the proposed method is intuition on the linear algebra used to produce the chemical result. Linear algebra is a complex subset of mathematics traditionally offered to sophomores and juniors in college. It is for this reason that the method is not rigorously proven to yield the chemical solution. Albeit a proof does exist, it is unnecessary to show as the procedure is calculator based; and as such neither the student nor the teacher is responsible for linear algebraic interpretation—only chemical.

Competing Interests

The author declares no competing financial interest.

Acknowledgements

This manuscript was supported in part by Janie Mueller, Zack Casias, Dave Hale, Ben Jakeman, and Logan Patterson.

References

[1]   | McCoy, B. Application of Linear Algebra: Balancing Chemical Equations. The University of North Carolina t Chapel Hill. http://www.unc.edu/_marzuola/Math547 S13/Math547S13 Projects/B McCoy Section001 BalancingChemicalEquations.pdf (accessed August 2014).
[2]   | Kolb, B. More on Balancing Redox Equations. J. hem. Educ. 1979, 56, 181.
[3]   | Carrano, S.A. Balancing an Atypical Redox Equation. J. Chem. Educ. 1978, 55, 382.
[4]   | Olson, J.A. An Analysis of the Algebraic Method for Balancing Chemical Reactions. J. Chem. Educ. 1997, 74 (5), 538-542.
[5]   | Kennedy, J.H. Balancing Chemical Equations with a Calculator. J. Chem. Educ. 1982, 59, 523.
[6]   | Blakeley, G.R. Chemical Equation Balancing: A General Method which is Quick, Simple, and has Unexpected Applications. J. Chem. Educ. 1982, 59, 728.
[7]   | Porter, S.K. How Should Equation Balancing be Taught?. J. Chem. Educ. 1985, 62, 507.
[8]   | Alberty, R.A. Chemical Equations are Actually Matrix Equations. J. Chem. Educ. 1991, 68, 984.
[9]   | Alberty, R.A. Conversion of Chemical Equations to Biochemical Equations. J. Chem. Educ. 1992, 69 (6), 493.
[10]   | Strang, G. Linear Algebra and Its Applications, 4th ed.; Academic Press, INC.: London, 2006; pp 73.
[11]   | Risteski, I.B. A New Singular Method for Balancing Chemical Equations and Their Stability. J. of the Chinese Chem. Society. 2009, 56, 68.