Inexact differential

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Inexact differential

In thermodynamics, an inexact differential or imperfect differential is any quantity, particularly heat Q and work W, that are not state functions, in that their values depend on how the process is carried out.cite book|author= Laider, Keith, J.|title=The World of Physical Chemistry|publisher=Oxford University Press|year=1993|id=ISBN 0-19-855919-4] The symbol is thus expressed as:

:$mathrm\left\{d\right\}U=delta Q-delta W,$

where δQ and δW are "inexact", i.e. path-dependent, and dU is "exact", i.e. path-independent.

Overview

In general, an inexact differential, as contrasted with an exact differential, of a function "f" is denoted: $delta f,$

$int_\left\{a\right\}^\left\{b\right\} delta f e F\left(b\right) - F\left(a\right)$; as is true of point functions. In fact, F(b) and F(a), in general, are not defined.

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

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.

Differentials which are not exact are often denoted with a δ rather than a "d". For example, in thermodynamics, δ"Q" and δ"W" denote infinitesimal amounts of heat energy and work, respectively.

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 thus path dependent. In thermodynamic calculations, the use of the symbol $Delta Q$ is a mistake, since heat is not a state function having initial and final values. It would, however, be correct to use lower case $delta Q$ in the "inexact differential" expression for heat. The offending $Delta$ 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 $U$. [cite book|author= Baierlein, Ralph|title=Thermal Physics|publisher=Cambridge University Press|year=2003|id=ISBN 0-521-65838-1] Continuing with the same instance of $Delta Q$, for example, removing the $Delta$, the equation ::::$Q = int_\left\{T_0\right\}^\left\{T_f\right\}C_p,dT ,!$is true for constant pressure.

*Closed and exact differential forms for a higher-level treatment
*Differential
*Exact differential
*Integrating factor for solving non-exact differential equations by making them exact

References

* [http://mathworld.wolfram.com/InexactDifferential.html Inexact Differential] – from Wolfram MathWorld
* [http://www.chem.arizona.edu/~salzmanr/480a/480ants/e&idiff/e&idiff.html Exact and Inexact Differentials] – University of Arizona
* [http://farside.ph.utexas.edu/teaching/sm1/lectures/node36.html Exact and Inexact Differentials] – University of Texas
* [http://mathworld.wolfram.com/ExactDifferential.html Exact Differential] – from Wolfram MathWorld

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