
Opening header section to German physicist Rudolf Clausius' 1854 "theorem of equivalence of transformations", consisting of about twenty pages of logic reasoning as to the transformations of heat and work that occur in the working body during one heat cycle, which lays out the foundation for what would later become the Clausius inequality. [1]

Opening header section to German physicist Rudolf Clausius' 1854 "theorem of equivalence of transformations", consisting of about twenty pages of logic reasoning as to the transformations of heat and work that occur in the working body during one heat cycle, which lays out the foundation for what would later become the Clausius inequality. [1]

In thermodynamics, the theorem of equivalence of transformations is defined by the following statement:

“In all cases where a quantity of heat is converted into work, and where the body effecting this transformation ultimately returns to its original condition, another quantity of heat must necessarily be transferred from a warmer to a colder body; and the magnitude of the last quantity of heat, in relation to the first, depends only upon the temperatures of the bodies between which heat passes, and not upon the nature of the body effecting the transformation.”

The theorem of the equivalence of transformations, as it is called, is based on or rather an extrapolation of the mechanical equivalent of heat, was stated by German physicist Rudolf Clausius in 1854, and assumes there to be two ‘kinds’ of transformations:

(a) negative transformation: the transformation of heat into work.
(b) positive transformation: the passage of heat from a warmer body to a colder body, which may be regarded as the transformation of heat at a higher temperature, into heat at a lower temperature.

Moreover it assumes that when a quantity of heat Q is passed from the hot body into the working body, during the expansion stroke, and then into the cold body, during the contraction stroke, that this is called a “double transformation” and that the magnitude of this one transformation process (positive transformation or negative transformation) can be quantified by an extensive state variable called the “equivalence-value”, defined mathematically by the ratio:

$\frac{Q}{T} \,$

where Q is an inexact differential quantity of heat, and T is the integrating denominator, such that the inverse of the absolute temperature:

$\frac{1}{T}\,$

is called the integrating factor, and is the temperature of either the hot body or the cold body, depending on the transformation.

It is assumed that in the "perfect thermodynamic engine", the equivalence-values exactly compensate each other, such that the positive transformation cancels out the magnitude of the energy change of the negative transformation, such that the mathematical value of the double transformation, given by the following expression:

$\frac{Q}{T_H} - \frac{Q}{T_C} \,$

is zero. This we call a "reversible" process or transformation.

If in the course of the double transformation, the equivalence value changes are positive, and not zero, as are all processes which are in any way possible, the sum of which we call the equivalence vales of all uncompensated transformations, symbol N, we then call this type of double transformation an "irreversible" process or transformation.

This set of logic, of reversible and irreversible transformations, as embodied in the "theorem of the equivalence of transformations", is the foundation to the now-famous Clausius inequality (1865):

$\oint \frac{dQ}{T} \leq 0$

which in turn is the foundation to the Lewis inequality (1923)

which is the central governing equation of human existence as well as for all natural processes freely occurring on the surface of the earth.

References
1. Clausius, Rudolf (1865). The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies (theorem of the equivalence of transformations, pgs. 116-35). London: John van Voorst.