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.Laider, Keith, J. (1993). The World of Physical Chemistry. Oxford University Press. ISBN 0-19-855919-4.  The symbol , or δ (in the modern sense), which originated from the work of German mathematician Carl Gottfried Neumann in his 1875 Vorlesungen uber die mechanische Theorie der Warme, indicates that Q and W are path dependent. In terms of infinitesimal quantities, the first law of thermodynamics is thus expressed as:

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

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: $\backslash delta\; f\backslash ,$

$\backslash int\_\{a\}^\{b\}\; \backslash delta\; f\; \backslash ne\; F(b)\; -\; F(a)$; 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 dependent. This may be expressed mathematically for a function of two variables as $\backslash \; \backslash mbox\{If\}\backslash \; df\; =\; P(x,y)\; dx\; \backslash ;\; +\; Q(x,y)\; dy,\backslash \; \backslash mbox\{then\}\backslash \; \backslash frac\{\backslash partial\; P\}\{\backslash partial\; y\}\; \backslash \; \backslash ne\; \backslash \; \backslash frac\{\backslash partial\; Q\}\{\backslash partial\; x\}.$

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.

Differentials which are not exact are often denoted with a δ rather than a d. For example, in thermodynamics, δQ and δW 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 $\backslash Delta\; Q$ is a mistake, since heat is not a state function having initial and final values. It would, however, be correct to use lower case $\backslash delta\; Q$ in the inexact differential expression for heat. The offending $\backslash Delta$ belongs further down in the Thermodynamics section in the equation $q\; =\; U\; -\; w\; \backslash$, which should be $q\; =\; \backslash Delta\; U\; -\; w\; \backslash$ (Baierlein, p. 10, equation 1.11, though he denotes internal energy by $E$ in place of $U$.Baierlein, Ralph (2003). Thermal Physics. Cambridge University Press. ISBN 0-521-65838-1.  Continuing with the same instance of $\backslash Delta\; Q$, for example, removing the $\backslash Delta$, the equation

$Q\; =\; \backslash int\_\{T\_0\}^\{T\_f\}C\_p\backslash ,dT\; \backslash ,\backslash !$

is true for constant pressure.

See also

• 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

External links

• Inexact Differential – from Wolfram MathWorld
• Exact and Inexact Differentials – University of Arizona
• Exact and Inexact Differentials – University of Texas
• Exact Differential – from Wolfram MathWorld

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