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A precise, simple and general Basic Le Châtelier Principle based on elementary calculus: What Le Châtelier had in mind?

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Abstract

The Le Châtelier Principle is one of the most important concepts in chemistry, and it has been the topic of many publications over the years. However, its meaning and application are often fraught with misunderstanding and confusion. As a suggested replacement, we present herein a precise general statement that we call the Basic Le Châtelier Principle (BLCP), which is in keeping with a common thread in Le Châtelier’s original statements. The BLCP is formulated as a consequence of well-known properties of a simple but general optimization problem, which elevates its range of application beyond chemistry to any phenomenon governed by such an optimization principle. We show applications of the BLCP to simple example problems in economics and in physics, in addition to the usual chemistry problems,. Following a brief outline of Le Châtelier’s original statements, we formulate the BLCP, which incorporates the notion of “de signe contraire” (of opposite sign), common to all his statements. It arises by abstracting the chemical reaction equilibrium problem (CREP) in the single-reaction case to the general problem of minimizing a differentiable function \(f(x;\{p_j\})\), where x is the single independent variable and \(\{p_j\}\) is a set of parameters. The BLCP arises from an exact expression for the dependence of the sign of the incremental change in the optimal solution \(x^*\) on the sign of the incremental change in a parameter, which is derived using techniques taught in an early undergraduate calculus course. When translated back to the CREP, this yields unambiguous expressions for the sign of the incremental change in the equilibrium reaction extent, \(\xi ^*\), arising from an incremental change in each of T, P, the initial species amounts, \(\{n_i^0\}\), and the standard reaction free energy change, \(\varDelta G^{\Box }\). Special emphasis is placed on the requirement that f must satisfy a positive definite second derivative condition, for which we present a proof in the case of multiple reactions in an ideal solution model system. We also briefly consider the extension of the single-variable BLCP derived herein to the case of multiple independent variables and to finite parameter perturbations.

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Acknowledgements

The contributions to this paper of the late Professor R. W. Missen are gratefully acknowledged; the basic ideas herein were presented at the 2001 annual Can. Soc. Chem. Eng. (CSChE) conference in Halifax, NS, and at the 2016 Canadian Applied and Industrial Mathematics (CAIMS) conference in Edmonton, AB. I thank Professor Janet Elliott of the University of Alberta, Dr. Allan Leal of ETH Zurich, and Dr. Hermann Eberl of the University of Guelph for useful discussions. This work has been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) under Grant RGPIN–2018–0421.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Guelph, Guelph, ON, N1G 2W1, Canada

    William R. Smith

  2. Department of Chemical Engineering, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    William R. Smith

  3. Faculty of Science, University of Ontario Institute of Technology, Oshawa, ON, L1H 7K4, Canada

    William R. Smith

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Smith, W.R. A precise, simple and general Basic Le Châtelier Principle based on elementary calculus: What Le Châtelier had in mind?. J Math Chem 58, 1548–1570 (2020). https://doi.org/10.1007/s10910-020-01140-3

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