
2010 tattoo of the 1856 formulation of the principle of the equivalence of transformations, using the 1875 inexact differential notation δ of German physicist Carl Neumann, on the forearm of a newly graduated philosophy student Ivanka. [6]

2010 tattoo of the 1856 formulation of the principle of the equivalence of transformations, using the 1875 inexact differential notation δ of German physicist Carl Neumann, on the forearm of a newly graduated philosophy student Ivanka. [6]

In thermodynamics, the principle of the equivalence of transformations states that the combined value of all the transformations of a given body (or system), symbolized by a number N, is equal to the negative of the integral of the equivalence-values entering or leaving the body:

$N = -\int \frac{dQ}{T} \,$

where dQ is an element of heat, whereby any heat withdrawn from the body is to be considered as an imparted negative quantity of heat; the integral is extended over the whole quantity Q; T is the absolute temperature, and N is the equivalence-value of all uncompensated transformations involved in a cyclical process. [1] The principle of the equivalence of transformations is an alternative formulation of the second main principle or second law of thermodynamics.

N | Symbol
Clausius, to note, gives a lengthy asterisked footnote concerning the ‘uncompensated transformations’ to which N refers, but never gives N a name. [2] The basic rule, he presents, is that N equates to zero for a reversible process and has a numerical value for non reversible processes.

Two fundamental theorems
In his followup 1856 fifth memoir “On the Application of the Mechanical Theory of Heat to the Steam Engine”, Clausius gave N the name "equivalence-value of all uncompensated transformations" and would defined the two fundamental theorems "which hold good in every cyclical process”, i.e. the first and second laws of thermodynamics (first main principle and second main principle), by the following two equations: [4]

$Q = A \cdot W \,$

$\int \frac{dQ}{T} = - N \,$

where A is the thermal equivalent of the unit of work, and W is the external work performed during the cyclical process, Q signifies the heat imparted to the changeable body during a cyclical process, and dQ an element of the same heat, whereby any heat withdrawn from the body is to be considered as an imparted negative quantity of heat, and the integral is to be extended over the whole quantity of Q.

Temperature
In its first formulation, as presented in his 1854 article, to note, Clausius defined his equivalence of transformation principle as:

$N = -\int \frac{dQ}{\tau} \,$

wherein after he went into discussion to show that τ (tau) must be the absolute temperature T.

History
The principle of the equivalence of transformations was first formulated by German physicist Rudolf Clausius in his 1854 article “On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat”. [1] The article was reprinted verbatim as ‘memoir four’ to his first edition 1865 textbook The Mechanical Theory of Heat and then rewritten as ‘chapter four’ to his second edition 1875 textbook. [3]

See also
● Transformational content
● Transformation-equivalents

References
1. Clausius, Rudolf. (1854). “On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat”, Poggendoff’s Annalen, Dec., Vol. xciii, pg. 481; translated in the Journal de Mathematiques, Vol. xx. Paris, 1855, and in the Philosophical Magazine, Aug. 1856, S. 4. Vol xii, pg. 81.
(b) Clausius, Rudolf (1865). The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies (Fourth Memoir: On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat, pgs. 111-35, definition of N, pgs. 141-42) (trans. Thomas Hirst, 1867). London: John van Voorst.
2. ibid, Clausius. (1865). definition of N, pgs. 141-42.
3. Clausius, Rudolf. (1875). The Mechanical Theory of Heat (Chapter IV: The Second Main Principle Under Another Form, or the Principle of the Equivalence of Transformations, pgs. 91-109) (trans. Walter Browne, 1979) London: Macmillan & Co.
4. Clausius, Rudolf (1856). “On the Application of the Mechanical Theory of Heat to the Steam Engine”, Published in Poggendorff’s Annalen, March and April, Vol. xcvii. Pgs. 441 and 513; translated in the Philosophical Magazine, s. 4. Vol. xii, pp. 241, 338, and 426; and in Silliman’s Journal, S. 2. Vol. xxii, pp. 180 and 364, Vol. xxiii. P. 25.
5. (a) Quote: “I got this last week and I’m sure you know, it’s the second law of thermodynamics (the original equation, by Clausius) before –N even represented entropy. I’m not a physics or math major, I’m a female philosophy person. I really do love physics, though, and I’m about to leave my home country and all my undergrad friends behind and go to do my MSc at the London School of Economics. So the sentiment behind this is that now, after undergrad, we begin to disseminate. Entropy. It’s also a great boyfriend filtration system: ‘Can I have your number?, Wait, what’s this mean?, Um … I don’t know?, Too bad, you were cute (walk off).” (Ivanka, circa 02 May 2010).
(b) Zimmer, Carl. (2010). “Graduating into Entropy”, Discover Magazine Blog, May 02.