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. The symbol , 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. 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", 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: $delta\; f,$

$int\_\{a\}^\{b\}\; delta\; f\; ne\; F(b)\; -\; F(a)$; 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 $mbox\{If\}\; df\; =\; P(x,y)\; dx\; ;\; +\; Q(x,y)\; dy,\; mbox\{then\}\; frac\{partial\; P\}\{partial\; y\}\; ne\; frac\{partial\; Q\}\{partial\; x\}.$

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.

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 $Delta\; Q$ 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 $U$. Continuing with the same instance of $Delta\; Q$, for example, removing the $Delta$, the equation

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

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