Muhammad Mujtaba Shaikh,Mumtaz Yousaf,




Mathematical methods,Chemical equations,Linear Algebra,Linear Programming,Integer Linear Programming,FLOPs, Mathematical Chemistry,


In this study, a comparative analysis is to be conducted between different mathematical techniques to find out the best one which can be more suitable from all perspectives to balance equations of chemical reactions and to provide case-to-case recommendations for the practitioners. The linear algebra approach, linear programming approach, and integer linear programming approach have been successfully utilized for chemical equation balancing.  Some chemical equations have been taken from the literature to see the performance of the above approaches. After highlighting the advantages and disadvantages of the existing approaches, some proposals for modification are presented. The proposed modifications have been worked out on all problems, and the integer solution is attained for all problems; even in cases where existing methods failed. The final recommendations on easier and better techniques have been provided. The two modified methods achieved top ratings among the existing and proposed methods.


I. Abdelrahim M. Zabadi and RamizAssaf, (2017), “Balancing a Chemical Reaction Equation Using Algebra Approach”, International Journal of Advanced Biotechnology and Research(IJBR), vol 8, Issue 1, pp: 24-33.
II. Arcesio Garcia, (1987), “A new method to balance chemical equations”, Journal of chemical equation, vol 64, pp: 247-248.
III. Charnock, N. L, (2016), “Teaching Method for Balancing Chemical Equations: An Inspection versus an Algebraic Approach”. American Journal of Educational Research, vol 4, pp: 507-511
IV. Dr. Kakde Rameshkumar Vishwambharrao, Sant Gadage Maharaj Mahavidyalaya, Loha (Maharashtra), (September 2013), “Balancing Chemical Equations by Using Mathematical Model”, International Journal of Mathematical Research & Science (IJMRS), vol. 1, Issue 4, pp: 129-132.
V. E.V. Krishnamurthy, (1978), “Generalized matrix inverse approach to automatic balancing of chemical equation”, International Journal of Mathematical Education in Science and Technology, vol 9, no 3,
pp: 323–328.
VI. Ice B Risteski, (2012), “New very Hard Problems of Balancing Chemical Reactions”, Bulgarian Journal of Science Education, vol 21, No 4, pp: 574
VII. Ice.B Ristaki, (2014), “A new generalized algebra for the balancing of chemical reactions”, Materiali technologije/materials and technology, vol 48, Issue 2, pp: 215-219.
VIII. Lochte, H.L, (1997), “Pole reaction method (ion electron/half reaction)”, Journalof chemical education, vol 74, Issue 11, pp: 146-157.
IX. Mansoor Niaz and Anton E Lawson, (1985), “Balancing Chemical Equations: The Role of Developmental level and Mental Capacity”, Journal of Research in Science Teaching, vol 22, No 1, pp: 41-51.
X. Mumtaz Yousaf, Muhammad Mujtaba Shaikh, Abdul Wasim Shaikh (2020). Efficient Mathematical Programming Techninques For Balancing Equations of Complex Chemical Reactions. Journal of Maecahnics of Continua and Mathemtical Sciences. 15 (10): 53-66.
XI. Paul M. Treichel John C. Kotz, at https://www.britannica.com/science/chemical-reaction/Photolysis-reactions
XII. R. David Jones, A. Paul Schwab, (1989), “Balancer: A computer program for balancing chemical equations”, Journal of agronomic education, vol 18, Issue 1, pp: 29-32.
XIII. R.O Akinola, S.Y. Kutchin, I.A Nyam, (2016), “using row reduced echelon form in balancing chemical equations”, Advance in linear algebra and Matrix Theorey, vol 6, Issue 4, pp: 429-445.
XIV. Risteki I.B, (2016b), “A new generalzed matrix inverse method for balancing chemical equations and their stability”, Journal of Chinese chemical Soc, vol 56, Issue 4, pp: 65-79.
XV. S K Sen, Hans Agarwal, Sagar Sen, (Feburary 2006), “Chemical Equation Balancing: An integer programming appoach”, Mathematical and Computer Modelling, vol 44, pp: 678–691
XVI. William C. Herndon, (November 1997), “On Balancing Chemical Equations: Past and Present”, Journal of Chemical Education, vol 74, No 11, pp: 1359-1362.

View Download