A Question of Balancing ARTHUR PORGES Western Military Academy, Altom, Illinois FTER examining a great many chemical equations, one concludes that most of them are of the type in which the number of compounds involved exceeds the number of elements by unity. Mathematically this unit difference implies a unique balancing of the equation, since if the number of compounds be taken as the number of variables, and the number of elements (or radicals) as that of h e a r equations in the variables, the mathematical system of N homogeneous linear equations in N 1 variables has a unique solution except for multiples of the values of the variables. The following familiar equation will serve to clarify this point:
A
+
+
FeCllf K2Cr201 HC1- FeCL
+ KC1 + CrCIa + Hz0
(1)
If we let X , equal the number of molecules of the n-th com~ound.counting from left to ripht, - . and eauate the number of atoms of each element appearing in the left member of (1) to the number of atoms of the same element appearing in the right member, we have the following system of six homogeneous equations in seven variables.
-
Fe: CI: K Cr: 0: Hi
+ XS = 3x4 + Xs + 3x6 1
XI = X4
ZX,
2x1= XS 7x* = xi XI = 2x1
If we solve the system (2) for each variable in terms of, say, X2 (any choice is possible), we readily find
It is evident that the general integral solution of (2) is derived from X z = K, where K is any positive integer; and consequently although (2) has an infinite number of integral solutions, all of them derive from XP= K, and consist merely of multiples of values for the respective variables established by (3). All solutions other than that for K = 1 are therefore trivial chemically as well as mathematically. A second type of chemical equation, less numerous than the first variety, is that the balancing of which leads to R equations in N variables, where R 2 N. An example of this second type is the following: Here the linear equations derived from (4) are As: 2x1 = X3 S: 3x, x, = 3x, NHI: 2Xn = 3x3
+
1
The determinant, D, of the system ( 5 ) , given by
is easily shown to vanish, .and that consequently equations (5) are not independent. Once again, although infinitely many integral solutions of (5) are possible, they are all multiples of the simplest: XI = I, X* = 3, and Xo = 2
A differentsituation, vastly more intriguing than the two briefly treated above, arises when we encounter the rarest variety of chemical equation-the type in which the number of compounds exceeds that of the elements by two or more. An interesting equation of this third variety is the
following, which the writer has had among his notes for many years without being able to recall anything more definite of its origin than that i t was copied from a chemistry text (author unknown) years old a t the time and now, one presumes, long out of print.
+
-+
+
+
HAuClr Kde(CN)s KAu(CN), KAu(CN)S KAu(CN)zCL KC1 HCI [4Fe(CN)n,3Fe(CN)4 (7)
+
+
limit XI t o values established by 12K
> XI
S (28/3)K
Since by (9.4) X, must be even, its least value satisfying (11) must be XI = 10 when K = 1. By similar r'asoning it is not difficult t o show that all SOlutionS Of (8) are given by
Here the mathematical system consists of six homogenous equations in eight variables, the complex iron salt being regarded as a single molecule. Specifically the eouations are H: XI = X7 Au: XI = X n + X 4 + X s CI: 3X, = 2Xs XS X7 K: 4x,=x,+x,+xs+xs Fe: X9 = 7X8 (CN): 6Xn = 4Xa ZX. 2Xa
+ + + +
XS = 4K Xe = 16K X7 = 12K Xa = K
+ 18Xs
The existence of a t least one integral solution of (8) is established by the chemical equation itself. Mathematically, the nonvanishing of a sixth order determinant of the matrix of (8) establishes the existence of infinitely many solutions without indicating, however, how many of these are integral. If we solve the system (8) for six of the variables in terms of the other two, say, X I and XZ,we easily obtain X, = X, X, = X, xa = (c/,)(lzx*- 7Xd x 4 = ('/,,)(7X, 4x4 Xb = ('/d(3Xt - 4x2) X. = 4X%- X, X , = XI x 8 = ('/7)(Xd
+
(9.0) (9.1)
:(9.4)]
(9.5)
, ]
From (9.7), we see that Xz must be a multiple of 7. Let Xz = 7K, where K is any positive integer. The obvious inequalities 12x, 3x1
>
7x1
> 4x3
(10.1)
(11)
+- 322
where K takes on all positive integral values. Here it is apparent that the solutions are not multiples of that by K = The of the whole matter is *is: Is there, for equations of this third type which admits infinitely many distinct, nonmul~iplesolutions, a principle similar to that in mechanics? Of the unlimited number of ways of balancing such an equation as (7), is that corresponding to the solution in least i,t,gers of the ]inear equationsthe one inva~ably indicated by the laboratory work, which is, after all, the real criterion? Further, what would a minimum solution b e t h a t for which the square root of the arithmetic mean of the squares of the variables is less than for any other solution? I n the very few equations of the third type encountered by the writer, the balancing in least integers was always that for which K was also least, and was identical with the laboratory balancing: but given a general solution not minimized by the smallest admissible value of K-theoretically not impossible-then what?
.,,
