Integrating factor

\frac{\delta W}{P} = dV
\frac{\delta Q}{T} = dS
1/P is the integrating factor for the inexact differential δW.
(formulated by Emile Clapeyron in 1834)
1/T is the integrating factor for the inexact differential δQ.
(formulated by Rudolf Clausius in 1865)
In mathematics, an integrating factor is a function, μ(x,y), that, in some cases, can be multiplied into an inexact differential equation, converting it into an exact equation or complete differential. [1] The resulting exact equation, such as:

μM(x, y) dx + μN(x, y) dy = 0

however, may not be equivalent to the original equation in the sense that a solution of one is also a solution of the other. It is possible, for instance, for a solution to be lost or gained as a result of the multiplication.

Example
The following inexact equation: [1]

(x + y) dx + x ln x dy = 0

Can be made exact using the integrating factor μ(x,y) = 1/x, on (0, ∞). We first confirm that the equation is indeed inexact using the condition for an exact differential, whereby we note that M(x, y) = x + y and N(x, y) = x ln x so that ∂M/∂y = 1 and ∂N/∂x = 1 + ln x, whence ∂M/∂y ≠ ∂N/∂x. If, however, we multiply the equation by μ(x,y) = 1/x, we obtain:

(1 + y/x) dx + ln x dy = 0

whereby:

M(x, y) = 1 + y/x, N(x,y) = ln x, ∂M/∂y = 1/x = ∂M/∂y

Pressure volume work
The logic of the integrating factor can be applied to the inexact differentials đQ (or δQ) and đW (or δW). [3] In ideal gas systems, for instance, the following is the expression for pressure volume work in a steam engine, as was introduced by Emile Clapeyron in 1834:

δW = PdV

therefore 1/P is the integrating factor for đW. Thus, by rearrangement, the expression:

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

is an exact differential and V is a state variable. [3]

Entropy
This same logic was followed, beginning in 1850, by German physicist Rudolf Clausius in the transformation of inexact differential đQ (or δQ) into the exact differential dS. In short, in formulating the expression for entropy, over a period of fifteen years (1850-1865), Clausius famously used the inverse of absolute temperature (1/T) of the body as the “integrating” factor to convert the nonexact differential of heat dQ into an exact differential or, in other words, a caloric particle into a state function dS. [2] In this case, T is called the being the "integrating denominator". In particular, into the 1840s it was becoming known that the integral of a differential unit of heat dQ:

\int dQ

involved in any reversible change in a given substance or system is not independent of the path followed, or in other words of the intermediate conditions passed through. It follows that dQ is not a complete differential and that the integral of dQ cannot be expressed as a function of the initial and terminal conditions:

\int_i^f dQ \ne Q(f) - Q(i)

It is well known, however, that it is possible by means of an integrating factor to reduce dQ to a complete differential, and hence to express its integral thus transformed, as a new function of the initial and end conditions. In other words, a differential dQ that is not exact is said to be integrable when there is a function 1/τ such that the new differential dQ/τ is exact. The function 1/τ is called the integrating factor, τ being the integrating denominator. The factor that Clausius used for this purpose is the reciprocal of the absolute temperature T, such that a new function of heat can be said to exist:

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

in which dS is a complete differential. This expression, in effect, replaced the caloric theory. The integral of dS is then termed the entropy, symbol S, and convenient set of initial conditions being taken from which to measure its value. [2]

References
1. (a) Zill, Dennis G. (1993). A First Course in Differential Equations (section 2.4: Exact equations, pgs. 53-60; subsection: Integrating factor, pg. 59-60). PWS-Kent Publishing Co.
(b) Zill, Dennis G. (2008). A First Course in Differential Equations (section: Integrating Factors, pgs. 66-68). Cengage Learning.
2. (a) Durand, William F. (1897). “Note on Different Forms of the Entropy Function”. Physical Review (pgs. 343-47). American Institute of Physics.
(b) William F. Durand – Wikipedia.
3. Cheng, Yi-chen. (2006). Macroscopic and Statistical Thermodynamics (pg. 31). World Scientific.

Further reading
● Eu, Byung C. (1998). Nonequilibrium Statistical Mechanics (section: 2.7.1: The Inverse Temperature as an Integrating Factor, pgs. 40-41). Springer.
● Linder, Bruno. (2004). Thermodynamics and Introduction to Statistical Mechanics (entropy, integrating factor, pgs. 45-46). John Wiley & Sons.
● O’Connell, John P. and Haile, J.M. (2005). Thermodynamics: Fundamentals for Applications (entropy, integrating factor, pg. 49). Cambridge University Press.

External links
Integrating factor – Wikipedia.

TDics icon ns