Nonstoichiometric Equations WILLIAM C. MCGAVOCK Trinity University, Sun Antonio, Texas HE above title is quoted from the article by 0. F. Steinbach which recently appeared in THIS JOURNAL ( I ) . Examples cited by him are oxidationreduction reactions for which a unique set of coefficients may be obtained by the usual method of 0-R balancing ( 2 ) . These reactions, therefore, are not "nonstoichiometric." Nevertheless, and in spite of the lack of examples, the term "nonstoichiometric" is thought provoking. I t has long been realized that a substance may participate in two different reactions simultaneously, in which case the relative amounts of the two products would bear no fixed ratio to one another. In handling such systems two different equations are written. Further, it has been realized that an intermediate may decompose a t such a rate that it remains among the analyzable components of the reaction mixture. No effort is made in such a case to set up a stoichiometric ratio between the amount of the intermediate and the amount of the final product. This would be a uonstoichiometric condition, but no need for a special classification is felt for such a relationship. Is there, then, such a thing as a "nonstoichiometric" reaction? The answer is believed to be affirmative. Definition: A nonstoichiometric reaction may be defined as one in which a given set of reactants yields end products whose molecular proportions are variable in a continuous sense. Examfile: An example has been found by McGavock (3) in the cracking reaction:
T
algebraic method, proposed by Bottomley (4) in 1878, to each species of atom in the reaction equation above yields the following algebraic equations. (Let a = 1.) 2b
+ 3c = 7d (Hydrogen)
8a = Zb
+ 3c + d (Carbon)
(1) (2)
From these two equations, i t is not possible to solve for the four molecular coefficients. A third equation is needed. This may be obtained by using the conservation law as applied to electrons. The oxidationnumbers are the instruments of this application, shown in the diagram above. It follows that:
which on simplifying becomes: From (1) and (2) it is apparent that a = d = 1. Then all three equations reduce to: 2b
+ 3c = 7
The only integral, positive values satisfying this equation are: b = 2; c = l. The equation then is written: lCsHls 2GH4 1C8H6 CHI
-
+
+
However, i t is not possible, either on an algebraic or empirical basis, to exclude nonintegral values for the coefficients. Each point on the straight line (Figure 1) in the accompanying graph represents a pair of positive, nonintegral values. An infinitude of such points, -zl/, -2 corresponding to an infinitude of pairs of coefficients in the above equation, exists. On the basis of this finding, it is asseried that this cracking reaction represents a CBH,. b C.HI c 8Hs d CH, a truly nonstoichiometric reaction. -Z1/4 6'1, -4 Finally, i t is necessary to explain the existence of the Application of the consenration law through the multiple sets of coefficients for the equations cited by
+!
-
+
+
2b
+ 2c = d + Zf c =d
(Hydrogen)
+ e (Sulfur)
(4)
(5)
There are five equations and seven unknowns. No unique solution is possible from this set of equations alone; an infinitude of sets of solutions exists. The fact that Steinbach has discovered a few of these sets is not significant in determining the nature of the reactions.
Steinbach. The algebraic method of Bottomley is again of assistance. Upon writing the algebraic equations for each of the reaction equations cited, it is ap~ a r e n tthat each set of coefficients which is correct (see note below) conforms to the algebraic requirements thus imposed. A single example is sufficient: a KMnO,
+ b HIOn + c &SO4
-t
d KHSOI
+ c MnSOl + fHnO f g o .
Applying the law of conservation to each species of atom, we obtain: a = d (Potassium)
(1)
Nok on Errata in the Steinbach article (1): Coefficients for 0% in the KMnO, equation: Change "18" t o "23." Change "23" t o "25." In the K ~ cequation: ~ ~ o ~ Change "K9Crr08" t o "KnCrsO,." Star the second set of coefficients rather than the third In the KOCI-KCLOn equation: The fifth and sixth sets of coefficients are incorrect.
LITERATURE CITED
( 1 ) STEINBAC~, O. F., "Nonstoichiometric equations," J. CHEM. EDUC.,21, 66 (1944). (2) MCALPINE, R. K., AND B. A. SOULE,''Plescott and Johnson's Qualitative Chemical Analysis," D. Van Nostrand Company, Inc., New Pork, 1933, pp. 62S56. (3) M c G ~ v o c ~W. , C., "Organic Oxidation-Reduction Reactions," San Antonio, Texas, 1945, p. 135. (4) BOTTOMLEY, JAMES,"Note on a method for determining the coefficients in chemical equations," Chem. Neus, 37, 110 (1878).
