An Alternative Proof that Equilibrium Concentrations for a Chemical Reaction are Always Uniquely Determined by the Initial Concentrations P. Glaister
Department of Mathematics, University of Reading, Whiteknights, Reading, U.K I read with much interest the recent article "Are the Equilibrium Concentrations for a Chemical Reaction Always Uniquely Determined by the Initial Concentrations" by E. Weltin' in which i t was shown that, for a general chemical process, and for a given set of initial concentrations, only one of the roots of the function that determines the concentration of the various species a t equilibrium is chemically meaningful. This result generalized the wellknown, simplified cases, for example, when the function is a polynomial of degree two. Specifically, for a general process a A + bB + . . . + uU + uV + ...
with equilibrium constant K,, (for a given temperature), and for given initial concentrations a,, b., . . ., u,, u,, . . ., Weltin argued that the two functions P(x) and K,, R(x), where and R(x) = ( - a x +a,)' (-b.r:+ bdb...
(2)
intersect once a n d only once i n the "chemically allowed" interval
(N.B. Each of the terms in brackets i n P(x) and R(x) represents a concentration, and all of these are guaranteed to remain chemically meaningful (positive) if x is i n this interval.) The quantity x describes the extent of the reaction, and the point of intersection x = x, is then given by P(x,)iR(x,) = K,, representing the unique state of equilibrium. Weltin's proof appealed to the monotouicity properties of the functions P(x) and R(x), and the relative ordering R > P a t the lower end and R < P a t the upper end of the interval. The purpose of this note i s to demonstrate a n equally s i m p l e proof of t h i s r e s u l t t h a t i s m a t h e m a t i c a l l y equivalent, but different in implementation. The principle we use, which we elaborate on shortly, c a n easily be demonstrated with a graph (as can Weltin's proof provided monotonicity and relative order are both considered) and
says that, if a function changes sign in an interval, but its gradient does not change sign in that interval, then the function must vanish once and only once in the interval. I t i s certainly worth pointing out to students both these approaches to reinforce the idea and to show other simple ideas that can be invaluable for further study or research. Defining the function Q(x) by
then we wish to show that Q(x) has one and only one root i n the interval given by eq 3 .To achieve this, i t is sufficient to show that Q(x) changes sign and &'(XI does not change sign i n (m,M), where the prime denotes differentiation with respect to x. Essentially, if Q(x) changes sign in this interval, then the graph of Q(x) must cross the x-axis, meaning that Q(x) has a t least one root i n the interval; and if Q'(x) does not change sign, i t can not vanish in the interval, and hence Q(x) has no turning point i n the interval. Thus Q(x) can cross the x-axis only once, and hence Q(x) has only one root in the allowed interval, a t x = x,, say. Since
we obtain the unique value x, from From eqs 1-4 i t is easy to show that Q(m) < 0 and Q(M) > 0, that is, Q(x) vanishes a t least once in (m,M). Furthermore, on differentiating eq 4 using eqs 1-2 i t i s easily seen that, for x in (m,M), Q'(x) > 0 meaning that Q(x) has no turning point in (m,M) and hence one and only one root i n this chemically allowed interval. We stress, however, that although we have combined the functions P and R to form Q, i t is P and R sepamtely that establish the allowed interval and, likewise, that the positive nature of Q' and the sign change of Q follow only from the properties of P and R. This fact is evident in the proof given above since the individual properties of P and R are required. . 67, 548. 'Weltin, E. J Chem. ~ d u c1990,
Volume 69 Number 1 January 1992
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