The numerical solution of equations in chemistry

Abstract

Very frequently in the solving of chemical problems, an equation or a system of simultaneous equations cannot be solved explicitly for the desired unknown quantity. Occasionally it is possible to make some simplifying approximations in the equation or equations which then permit an approximate explicit solution. However, such approximations may introduce more error than is tolerable, and some other technique must then be used. It is the purpose of this paper to summarize some of the important equationsolving techniques of numerical analysis and to illus-

Conductimetric detection of anions of very weak acids by incomplete suppressed ion chromatography

“…(9), one the signal-to-noise ratio of the system. In addition, if obtains the background conductance is too low, I will vary Raphson procedure [19]. By including the value of I S and detection limits, a computer-simulated experiin Eq.…”

Solving equations by successive substitution: The problems of divergence and slow convergence

“…Successive substitution can be applied either to a single equation (1)(2)(3)(4)(5)(6) or to systems of simultaneous equations (6,7). Only the single equation case will be discussed here.…”

“…The conditions of convergence for a sequence generated in this fashion can be obtained by examining the error of each approximation in the sequence 'n = rxn (6) From eq 5 it follows that dxn+i = F'(xn)dx", where F'(x) = dF/dx. As n becomes large and the errors e" become small, these differentials provide a good approximation to the errors, and thus e"+i = EM*,,…”

“…(1) 0.0 (2) 0.098200 (3) 0.034363 (4) 0.074432 (5) 0.048778 (6) The sequence is seen to converge slowly, with alternation, to a chemically realistic value of £, yielding the results £ -0.058655 mol a -£ = 0.441345 mol b -3£ = 0.624036 mol c + 2£ = 0.417309 mol as a solution to eq 14.…”

“…The better known of these techniques include successive substitution, the Newton-Raphson method, linear interpolation (regula falsi), and trial-and-error searches (such as interval bisection). One of the easiest to learn, because familiarity with calculus is not required, is the method of successive substitution (1)(2)(3)(4)(5)(6)(7). (This approach is occasionally called the method of iteration.)…”