Peter J. Nassiff Boston University Boston. Massachusetts 02215 and E. R. Boyko Providence College Providence, Rhode Island 02918
I
Teaching the Method of Successive Approximations
Evew beeinnine chemistrv student is introduced to the method of &ccesi;e approximations as one way of handling certain calculations. Usually this iterative procedure is presented without explanation of its mathematical foundations ( I ) ; however, we have found that a student who has had only the' basic elements of calculus can understand and appreciate the mathematics forming the basis of the method and thus determine when it is appropriate to his calculations. T o demonstrate this, we will first consider some simple examples using this method, and then discuss the mathematical conditions necessary for employing it in more complex calculations. ----~ The method is basically this ( 2 , 3 ) :An equation, g(x) = 0, is rewritten as x = f(x). When a reasonable first approximation, xo, is made, the equation becomes x1 = f(xo). The resultine value for X I is then resuhstituted. and the eauation. xz = f(xl), now provides the next approximation, xz. The process is repeated n times, x, = f(x,-I), until convergence to sufficient accuracy occurs. Although the method of successive approximations is usually applied to cubic or higher order equations, first-year chemistry students use i t for simple (quadratic) acid-hase equilibria problems (4). For example, H X is a weak acid with K, = [H+][X-]/[HX], and c is the initial overall acid concentration. Solving for [H+] gives [Ht] = ( K d c - [H+])I1". The first approximation, xo, is [H+] = 0, so that [H+] = (K.. c)"Z. Since the initial overall acid concentration of the weak acid in these prohlems is usually large compared to the [H+], the problem is usually terminated after the first approximation
-
[HX] = c
- [H+] z c
(1)
But the process can be carried one step further-the first calculated value of [H+] (now XI)can he substituted into eqn. (I), and a new [H+],now xp, can he found. Nevertheless, the process usually converges in one step for the values of K. encountered in a freshman course. A slightly more complicated example is the van der Wads equation
which can he written
V, =
nRT
+nb
Ix --l"+ll
I z -x,l
< Ix - 4
= lf(x) -f(xo)l
(5)
But we have the Mean Value Theorem
-
If(%) f(f~o)l= IP((), Ix - XOI
