The Relationship between the Number of Elements and the Number of Independent Equations of Elemental Balance in Inorganic Chemical Equations R. ~ubramaniam,'N. K. Goh, and L. S. Chia Division of Chemistry, National Institute of Education, Nanyang Technological University, 469 Bukit Timah Road, Singapore 1025, ~epublicof Singapore In the algebraic method of balancing chemical equations, the stoichiometric coefficients of the reactants and products are denoted by algebraic symbols and are then determined by the use of conservation conditions for the elements in the chemical equation (1,2). Not all chemical equations can be balanced fully by the algebraic technique. The criterion for determining whether a chemical equation can be balanced fully by the algebraic technique is for the number of independent equations of elemental balance to be one less than the number of reactants and products ( I , 2). For example, the skeletal chemical equation
which has five independent equations of balance, corresponding to each of the elements, in seven unknowns: x=2a z=b z=a+b Zy +Zr=Zc 4x+2y+4.~=4a+4b+e+Zd
for K for Mn far S far H for0
-
cannot be balanced fullv " bv " the algebraic techniaue. The conventional interpretation of chemical equations, where the number of reactants and ~ r o d u c t sexceeds the number of independent equations of hemental balance by two or more is that the overall equation a s written does not represent a unique reaction but is the sum of two or more simultaneous competing reactions (2). For the equation xBF3 + yNH3 + cBF3 NH3 only two independent equations of balance are possible for the four elements
Clearly, the number of independent equations of elemental balance in a chemical equation must be less than or equal to the number of elements. I n a n effort to obtain some general information on these trends in chemical equations, we examined a database of 140 (nonionic) inorganic chemical equations culled from standard textbooks. A good mix of simple and complicated equations was ensured. Some of the interesting observations from this survey are summarized below:
.
For the majority of chemical equations, the number of independent equations of balance is equal to the number of ele-
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Journal of Chemical Education
ments and is one less than the number ofreactants and products (62.7%).For example, 4NH3 + 50, + 4N0 + 6H,O The second most common equation occurs when the number of independent equations of balance is one less than the number of elements and also one less than the number of reactants and products (21.6%).For example, 2KC10, + 2KCI + 30, Next is the case having the number of independent equations of balance two less than the number of elements and also one less than the number of reactants and products (9.7%).Far example, Fe2(S0& + 3Ba(OHI2+ 2Fe(OH)8+ SBaSO, Far 4.5% of the chemical equations, the number of independent equations of balance is equal to the number of elements and is two less than the number of reactants and products. Far example, 2KMn0,
+ 5H20, + 3H,S04 + &SO,
+ ZMnSO, + 8H20 + 50,
' 1 % ~ rnrrst c a w has r h r numbrr of indcpmdrnt c~uatiunrof
bnlnnrr three less than ihr number ofclcrnents and alsuune lcsa than r h r numbrr of'renclnnts ond pr~dwts 1 5'; . For example, (PNCl,), + 6KS02F+ (PNF,), + 6KC1+ 6S02 Apoint of interest emerging from this study is the existence of (at least) four categories to depict the relationship between the numher of elements and the number of independent equations of elemental balance in chemical equations. Though the percentages do not encompass a very large database of chemical equations, they, nevertheless, provide some indication of the preponderances of the various categories. Such categorizations, though deviating from the conventional classification of chemical reactions based on the nature of reactions such as redox, acid-base, etc., open up another avenue for looking into the system of chemical reactions. Acknowledgment We wish to thank the reviewer for his comments on our article. Literature Cited 1. Porges. A. J. Chem Educ 1945.22.266, 2. Standen, A. J. Chpm. Edrrr. 1945.22.461
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