Mathematics in chemistry: indeterminate forms and their meaning

Abstract

The mathematical language and its tools are complementary to the formalism in chemistry, in particular at an advanced level. It is thus crucial, for its understanding, that students acquire a solid knowledge in Calculus and that they know how to apply it. The frequent occurrence of indeterminate forms in multiple areas, particularly in Physical Chemistry, justifies the need to properly understand the limiting process in such cases. This article emphasizes the importance of the L’Hôpital's rule as a practical tool, although often neglected, to obtain the more common indeterminate limits, through the use of some specific examples as the radioactive decay, spectrophotometric error, Planck's radiation law, second-order kinetics, or consecutive reactions.

Keywords:

• applied mathematics
• physical chemistry
• indeterminate forms
• L’Hôpital's rule
• radioactive decay
• spectrophotometric error
• Planck's radiation law
• chemical kinetics

Notes

Notes

1. Although in practice L’Hôpital's rule is useful for calculating this limit, in theoretical terms, it is not acceptable as this limit is indispensable for establishing the sine function derivative rule, , that one uses in the differentiation of the numerator of the ratio. Such a limit, nevertheless, might be found in other ways. In fact, departing from the condition , obtained from simple geometric considerations, valid for x ≠ 0, and realizing that

it results from the squeezing principle that

Another obvious process consists of using the analytical definition of the sine function itself through the series and divide it by x.

2. Note that from equality (9) results, by differentiation, , from which we take (apart the sign), .

3. Since (with ΔT = constant > 0)

and

with

4. In this study, we want to retain a very simple approach of this theme. Much more sophisticated treatments of this subject may be found in the literature, focusing on the theoretical and experimental study of the various factors affecting the precision of molecular absorption spectrophotometric measurements 13.