In terminology, living chemical reaction, as opposed to a nonliving chemical reaction, refers, hypothetically, to a “chemical reaction” that is “alive”.

Overview
In 1925, Gilbert Lewis, in his §7 “Non-Mathematical Sciences”, of his Anatomy of Science, attempted to digress on the what he referred to as the “missing link connecting the animate with the inanimate”, i.e. connecting the evolution of “living organisms” originating from chemicals governed by physics, chemistry, and thermodynamics; after discussing how crystals have the power of reproduction, autocatalysis, and other standard “emergence” patch solutions, he ruminates on whether him writing a book was but a “chemical reaction”, the gist of which is as follows: [1]

“Suppose that this hypothetical experiment could be realized, which seems not unlikely, and suppose we could discover a whole chain of phenomena [evolution timeline], leading by imperceptible gradations form the simplest chemical molecule to the most highly developed organism [human molecule]. Would we then say that my preparation of this volume [Anatomy of Science] is only a chemical reaction [extrapolate up approach], or, conversely that a crystal is thinking [extrapolate down approach] about the concepts of science? Nothing could be more absurd, and I once more express the hope that in attacking the infallibility of categories I have not seemed to intimate that they are the less to be respected because they are not absolute. The interaction between two bodies is treated by methods of mechanics; the interaction of a billion such bodies must be treated by the statistical methods of thermodynamics.”

Here we see Lewis is vexed. In his §8 “Life, Body and Mind”, however, leaves the solution to await a future science:

“The science of physics rests on the postulate of determinism; the science of biology, unless it is to ignore deliberately the phenomenon of behavior, must abandon this postulate and substitute therefor a postulate of choice or freedom. Perhaps our genius for unity will some time produce a science so broad as to include the behavior of a group of electrons and the behavior of a university faculty, but such a possibility seems now so remote that I for one would hesitate to guess whether this wonderful science would be more like mechanics or like a psychology.”

In circa 2006 to 2009, Libb Thims, prior to arrival of the defunct theory of life solution, similar to Lewis, was stuck on the origin of life problem, with ideas such as Stuart Kauffman’s auto-catalytic closure floating around as tentatives; the crux issue that broke the mold, in Thims’ mind, was the idea of a “magical” so-to-say “living” or start-of-life chemical reaction, aka a magical Miller-Urey like "spark day", however complicated, that would have had to exist at one second (or day) some 3.85 billion years ago. All reactions, however, hydrogen atoms reacting together to eventually form humans reacting together, are of the following form:

A + B → C + D

or

Reactants → Products

or other variations along this basic model ; all connected together via one long continuous “mechanism” (or chemical mechanism), big bang to present, according to which if one were discerning enough, e.g. Laplacian demon, could write out the entire mechanism, step, by step, by step, etc., up to the present day. If “life”, as an actual real measureable physical concept—as opposed to metaphysical concept, e.g. ether, or religio-mythology concept, e.g. spirit—were to exist, then it would have to one of the mechanism “steps” would have had to quantified by some type of magical “free energy” change measurement, which is an untenable position of logic to hold, therefore life does not exist; something along the lines as follows:

Nonliving
NonlivingNonliving?
Living
Living
Living
Reactants → ProductsReactants → ProductsReactants → ProductsReactants → ProductsReactants → ProductsReactants → ProductsReactants → Products

which, when chemical thermodynamics is applied to the problem, leads to nonsensical results, absurdities, etc., and therefore must be jettisoned as defunct religio-mythology.

Quotes
The following are related quotes:

“The animal cell, which is called ‘alive,” is simply a living chemical reaction—or more exactly a living oxidation of carbon—that is to say, an oxidation of carbon which has organized itself by the creation of services of entry and exit of materials of reaction, precisely as we organize our factories industrially.”
Ernest Solvay (1910) “The Living Body as a Factory” (ΡΊ), Lecture to the Belgian Association of Engineer; in: Revue Scientifique, Paris Dec 3.

Nonliving chemical reactions are driven by thermodynamics (heat) explore the possibilities open to them in an ergodic fashion—that is, by a process in which every exploratory sequence is the same. Life, on the other hand, explores its possibilities through evolution. It accumulates information—first in genes, then in memory—to help guide its search down narrower and more productive paths. How and when did information come to dominate the energetic processes of the physical world, and in doing so give rise to life?”
Robert Frenay (2006), Pulse [2]

References
1. Lewis, Gilbert N. (1925). The Anatomy of Science. Silliman Lectures; Yale University Press, 1926.
2. Frenay, Robert. (2006). Pulse: the Coming Age of Systems and Machines Inspired by Living Things (gray goo, pg. 49; nonliving chemical reactions, pg. 142). MacMillan.

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