Some Simple Balancing ANTHONY STANDEN S t . John's College, Annapolis, Maryland EVERAL papers which have appeared recently ( 1 , 2 , 3 )on balancing equations seem to have made the matter somewhat more complicated than it need be. Porges (1) distinguishes three classes of chemical equations; first, those in which the number of variables exceeds the number of mathematical equations by one, second, those in which the number of mathematical equations is equal to or even greater than the number of variables, and thud, those in which the number of variables exceeds the number of mathematical equations by two or more. To the f i s t class belong very many chemical equations; they are easily balanced by solving for the mathematical variables in terms of one of them. As an example of the second class, Porges gives:
S
AS&
+ (NHdrS
-+
(NHJrAsS
When solved in terms of As, S, and NH,, three mathematical equations are obtained, but they are not independent. It can be seen by inspection that the product of the reaction can be written-for equation balancing purposes-3 [(NH4)%S]As&. The equation is then appropriately treated in terms of As& and (NH4)B. It is of the form
.
There is nothing for it but to set out the mathematical analysis, and this gives only six equations for eight unknowns. Numbering the coefficients of the molecular species XIXeXa. . .in the order in which they occur. it can be shown that solutions are given by:
where K can have any positive integral value. Thus this equation can be balanced in an infinite number of ways which are not all multiples of one, simplest way. Whenever it occurs that an equation can be balanced in diierent ways that are not related as multiples, this is a sign that the equation as originally written is not one equation but two equations, or possibly more. The way in which these equations are found can be shown from the example under discussion. Write the equation for K = 2, write underneath i t the equation for K = 1, and subtract. The result is
A +3B-AB$
and i t is unnecessary to break it up into the terms As, S, and NH4. True examples of the second class of equations abound: every metathesis is one, whether ionic or nonionic. Examples :
The "terms" of this last equation are CHs, 1, W B r , and CzHc It would seem that examples could not be found where the number of mathematical equations actually exceeds the number of variables; for if the mathematical equations were inconsistent, the whole thing would be an impossibility, while if they were consistent it would indicate that the chemical equation had not been appropriately broken down into its terms. For the third class, Porges gives : HAuCIs
-
+
+
which must be just about the most complicated chemical eauation ever written. The divalent cold is reduced .~ ~as well as oxidized, and furthermore the oxidized, trivalent gold shows up in two different molecular species.
.~~~~~ ~
~
~
-
Both of these equations are of the first type, the number of variables exceeding the number of mathematical equations by one. It is clear that these reactions might take place in any ratio to one anotherincluding nonintegral r a t i o s u n t i l all the KAU(CN)~CI~ from the first reaction was used up. We can now understand the bizarre "nonstoichiometequations of Steinbach (2). Every one of them really represents two chemical reactions, not one. To illustrate by one example, KMnOt
+ HpOs + HISO,
-
KHSOI
+ MnSO, + Hs0 + 01
analysis gives only five equations among chemiseven unknowns, Accordingly we can write cal equations : 4KMnO.
+
KSe(CN)s KAu(CN)& au(CN)a K A u ( c m c L + KC1 + + [4Fe(CN)a.3Fe(CN)*1
~
Now write the equation for K = 0. This is
+ 8HnS01
--
+
4KHS0, 4MnS04 2H20n 2Hn0 01
+
+ 6&0 + 502
-,-hiS is not to say that either of these chemical reactions can actually take place separately: it is only to say that the chemical eauations can be balanced se~arately. Again, there is no sioichiometric reason why these re&tions should not take place together in any ratio what-
461
ever. The ratio in which they do take place must be determined by experiment, since it piobably varies with concentration, temperature, etc. The "nonstoichiometric" equation given by McGavock (3)* CsHx + 2 CnH,
--
+ CsHl + CH'
can also be written as two reactions: C&s OHM
C7H14 2 CxH,
+ CH,
+ CiHs
Again, this is not to say that the reaction actually does proceed by these steps, but only that its stoichiometric properties are most easily understood in this way. The second equation, since it amounts merely to a partial depolymerization of "methylene," can be balanced in many different ways; we have only to take care of
* Note on erratum in the McGavock article (3) : Equations (1) and (3) should be interchanged.
the carbons, and the hydrogens will take care of themselves. It is balanced in smallest numbers with the coefficients 1, 2, 1, but there is nothing to prevent its being balanced with the coefficients2, 1,4, or 3, 9, 1, which would produce nonintegral coefficients for the original equation if one stuck to the coefficient 1 for C8H18and CH4. Before this reaction can be regarded as a genuine "nonstoichiometric" reaction, it must be shown that it is one reaction. It might be a combination of these two reactions: 2CsH18+ 7CzH. 3CaHa + 7C.&
+ ZCH, + 3CH'
LITERATURE CITED
(1) POROES, &RTHUR, "A question of balancing," J. CHEM. Eln x , 12, 266 (1945). (2) STEINBACH, 0. F., "Nonstoichiometric equations," ibid., z1 ,66 (1944). (3) McC~+vocn. C., "Nanstoichiarnetric equations," -- m WILLIAM ,.- . -, ibza., z, v- (1~43).
