Inexact differential

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In thermodynamics, an inexact differential or imperfect differential is any quantity, particularly heat Q and work W, that are not state functions (a property of a system that depends only on the current state of the system, not on the way in which the system acquired that state), in that their values depend on how the process is performed.[1] The symbol Inexact differential symbol.PNG, or δ (in the modern sense), which originated from the work of German mathematician Carl Gottfried Neumann in his 1875 Vorlesungen über die mechanische Theorie der Wärme, indicates that Q and W are path dependent.[1] In terms of infinitesimal quantities, the first law of thermodynamics is thus expressed as:

\mathrm{d}U=\delta Q-\delta W\,

where δQ and δW are inexact (path-dependent), and dU is exact (path-independent).

Contents

[edit] Overview

For an exact differential df,

\int_a^b \mathrm{d}f = f(b) - f(a).

But for an inexact differential δf,

\int_a^b \delta f \ne f(b) - f(a),

and, in general, the values f(a) and f(b) are not even defined. Additionally, an inexact differential does not have an expression for the total derivative.[2]

An inexact differential is one whose integral is path dependent. This may be expressed mathematically for a function of two variables as \ \mbox{If}\ df = P(x,y) dx \; + Q(x,y) dy,\ \mbox{then}\ \frac{\partial P}{\partial y} \ \ne \ \frac{\partial Q}{\partial x}.

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.

[edit] Example

As an example, the use of the inexact differential in thermodynamics is a way to mathematically quantify functions that are not state functions and are thus path dependent. In thermodynamic calculations, the use of the symbol ΔQ for heat is a mistake, since heat is not a state function having initial and final values. It would, however, be correct to use lower case δQ in the inexact differential expression for heat. The offending Δ belongs further down in the Thermodynamics section in the equation q = U - w \ , which should be q = \Delta U - w \ (Baierlein, p. 10, equation 1.11, though he denotes internal energy by E in place of U).[3] Continuing with the same instance of ΔQ, for example, removing the Δ, the equation

Q = \int_{T_0}^{T_f}C_p\,\mathrm{d}T

is true for constant pressure.

[edit] See also

[edit] References

  1. ^ a b Laider, Keith, J. (1993). The World of Physical Chemistry. Oxford University Press. ISBN 0-19-855919-4. 
  2. ^ Weisstein, Eric. "Inexact Differential". MathWorld. http://mathworld.wolfram.com/InexactDifferential.html. Retrieved 13 August 2009. 
  3. ^ Baierlein, Ralph (2003). Thermal Physics. Cambridge University Press. ISBN 0-521-65838-1. 

[edit] External links

Retrieved from "http://en.wikipedia.org/wiki/Inexact_differential"
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