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) |

**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:

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*:

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:

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:

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.