On the Chemically Impossible "Other" Roots in Equilibrium Calculations, II Oliver G. Ludwig Villanova University, Villanova, PA 19085 Every instructor in general chemistry has been asked "How do you know?" that an equilibrium expression has no more than one "chemically possible" root, and that any other roots are bevond the stoichiometrv of the reaction. ( I )the author described, usingmathI n a previous ematics accessible to students. how an eauilibrium calculation leading to a quadratic &ation may be shown to have but one "chemical" root. The present work extends this demonstration to some reactions leading to cubic equations, using very easily comprehended algebra. (The method may not be general, in that some reactions may not yield to the technique, hut it does work for many of the reactions encountered in freshman chemistry.) Other authors (2, 3) have shown the existence of a unique chemical root for an ideal single phase system using the theory of algebraic equations, but such a technique is lost on freshmen whose mathematics is still a t the high school level. A demonstration using such advanced mathematics is little better that havine the instructor sav "take my word on this one"! Asimpler method has recently aooeared in this Journal 141.but is still more ahstract than &st freshmen are ready'f;; Let us take a concrete example, the dissociation of a moles of gas Ain a l-L container into three other gases:
.,
mol/l at equil: a - x
x
x
From eq 2 we see that, since Ka is positive, the product a h is positive. Since @ and y are positive by hypothesis, then a must be positive as well. But the sum of three positive numbers cannot equal zero as eq 3 demands; thus, we have a wntradiction and the assumption that both P and y are chemically possible must be incorrect. Q.E.D. As another example, we again start with a moles of gas Ain a l-L container and consider the reaction 2A mol/l at equil:
=
a - 2s
B+2C x2.z
with K = 4x3/(a- 2 ~ ) ~
and this time 0 < x < a12 for x to be chemically pos~ible.~ Thus the cubic equation becomes:
Again equating coefficients of powers of x in eq 1we obtain
x
Now let p and y both be !'chemically possible" by hypothesis, that is, both are positive and less than aI2. Then replacing both @ and y in eq 4 by a&, which exceeds both, we obtain the inequality:
where
K=x3/(a- x) and 0 < x
