History of Entropy - 1854 Definition

1854 Definition

In his 1854 memoir, Clausius first develops the concepts of interior work, i.e. that "which the atoms of the body exert upon each other", and exterior work, i.e. that "which arise from foreign influences which the body may be exposed", which may act on a working body of fluid or gas, typically functioning to work a piston. He then discusses the three categories into which heat Q may be divided:

1. Heat employed in increasing the heat actually existing in the body.
2. Heat employed in producing the interior work.
3. Heat employed in producing the exterior work.

Building on this logic, and following a mathematical presentation of the first fundamental theorem, Clausius then presented the first-ever mathematical formulation of entropy, although at this point in the development of his theories he called it "equivalence-value", perhaps referring to the concept of the mechanical equivalent of heat which was developing at the time rather than entropy, a term which was to come into use later. He stated:

the second fundamental theorem in the mechanical theory of heat may thus be enunciated: If two transformations which, without necessitating any other permanent change, can mutually replace one another, be called equivalent, then the generations of the quantity of heat Q from work at the temperature T, has the equivalence-value:

and the passage of the quantity of heat Q from the temperature T₁ to the temperature T₂, has the equivalence-value:

wherein T is a function of the temperature, independent of the nature of the process by which the transformation is effected.

In modern terminology, we think of this equivalence-value as "entropy", symbolized by S. Thus, using the above description, we can calculate the entropy change ΔS for the passage of the quantity of heat Q from the temperature T₁, through the "working body" of fluid (see heat engine), which was typically a body of steam, to the temperature T₂ as shown below:

If we make the assignment:

Then, the entropy change or "equivalence-value" for this transformation is:

which equals:

and by factoring out Q, we have the following form, as was derived by Clausius: