# Integrating denominator

In mathematical thermodynamics, an integrating denominator is the denominator of an integrating factor, the denominator being the part of the fraction below the line signifying division that functions as the divisor of the numerator, part above the line, and in fractions with 1 as the numerator indicates into how many parts the unit is divided.

Integrating denominators are very important in thermodynamics, although the underlying historical origin, proof, and justification for these various numerators is a bit elusive. The exact pre-Clapeyron era mathematics (1675-1834) to this proof remain to be tracked down.

Clapeyron
The idea of an integrating denominator seems to have been first introduced in the 1834 "Memoir on the Motive Power of Heat" by French physicist Emile Clapeyron, although this is not at all stated directly. Clapeyron speaks of "latent caloric" and "absolute quantities of heat" dQ. [1]

The modern-day perspective of Clapeyron-mathematics, although he does not seem to directly give this derivation, is that the variable of pressure P of the body is the integrating denominator of the integrating factor:

$\frac{1}{P} \,$

such that when this integrating factor is multiplied into inexact differential of a quantity of work δW of the body the resulting function:

$\frac{\delta W}{P} = dV$

is said to be an exact differential.

Clausius
In German physicist Rudolf Clausius' 1854 theorem of the equivalence of transformations, as found in his "On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat", an "unknown function" of the temperature, symbol T, is introduced as follows:

$f(t) = \frac{1}{T} \,$

Through argument, Clausius then goes on to conclude that this "unknown function" of temperature must be the absolute temperature of William Thomson's temperature scale. He then concludes that when this function, or integrating factor as it seems to be called in modern parlance, is multiplied into an inexact differential quantity of heat Q, the resulting function:

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

which Clausius calls the "equivalence value" is an exact differential. In modern parlance, when Clausius' "unknown function" of the inverse if the absolute temperature:

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

or integrating factor as it is called in modern terms, is multiplied into inexact differential of a quantity of heat δQ of the body the resulting function:

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

is said to be an exact differential. In 1865, Clausius gave this so-called exact differential the symbol "dS" defined as follows, in modern symbol notation:

$\frac{\delta Q}{T} = dS$

along with now-famous name entropy or transformational content.

Caratheodory
In 1908, Greek mathematician Constantin Caratheodory published "Studies in the Foundation of Thermodynamics" in which he introduced what has come to be known as theorem of Caratheodory. [3] In modern parlance, the so-called Caratheodory theorem says some to the effect of the following conclusion: [4]

“If a Pfaffian expression has the property that, in every neighborhood of a point P, there are points which cannot be connected to P along curves which satisfy the Pfaffian equation, dQ = 0, then the Pfaffian expression must admit an integrating denominator.”

This seems to be the first usage of the term "integrating denominator" in thermodynamics. In the decades to follow, this version of thermodynamics began to creep into the various thermodynamics books and textbooks.

American chemist Howard Reiss, in his "Mathematical Apparatus" chapter gives a fairly decent overview of the so-called relation between the exact or complete differential, the Pfaff differential expression, and the integrating denominator, all of which he indicates as italicized terms, and devotes a five-page section to attempt to answer the question: "when does a Pfaff differential expression possess an integrating denominator?" Reiss then gives an overview of the theorem of Caratheodory to answer this question. [5]

References
1. Clapeyron, Émile. (1834). “Memoir on the Motive Power of Heat”, Journal de l’Ecole Polytechnique. XIV, 153 (and Poggendorff's Annalender Physick, LIX, [1843] 446, 566).
2. 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.
3. Caratheodory, Constantin. (1908). "Studies in the Foundation of Thermodynamics" (Untersuchungen uber die Grundlagen der Thermodynamik). Bonn; published in: Math. Ann., 67: 355-386, 1909.
4. Kirkwood, J.G. and Oppenheim, Irwin. (1961). Chemical Thermodynamics (pg. 36). McGraw-Hill Book Co. Inc.
5. Reiss, Howard. (1965). Methods of Thermodynamics (integrating denominator, 9+ pgs; section: Theorem of Caratheodory, pgs. 22-26). Dover.

● Kestin, Joseph. (1966). A Course in Thermodynamics (integrating, pgs. 440, 461, 465). London: Blaisdell Publishing Co.
● Kestin, Joseph. (1979). A Course in Thermodynamics, Volume 1 (integrating denominator, 12+ pgs). Taylor & Francis.