Inexact differential

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{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 e 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 "de"pendent. 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} e 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, 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_{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

* [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


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Exact differential — In mathematics, a differential dQ is said to be exact , as contrasted with an inexact differential, if the differentiable function Q exists. However, if dQ is arbitrarily chosen, a corresponding Q might not exist. OverviewIn one dimension, a… …   Wikipedia

  • Heat — In physics, heat, symbolized by Q , is energy transferred from one body or system to another due to a difference in temperature. [cite book|author= Daintith, John |title=Oxford Dictionary of Physics|publisher=Oxford University… …   Wikipedia

  • Différentielle exacte — En analyse, une forme différentielle est dite exacte s il existe une forme différentielle dont elle est la dérivée extérieure, c est à dire s il est possible de l intégrer. En particulier, une forme différentielle ω de degré 1 définie sur un… …   Wikipédia en Français

  • First law of thermodynamics — In thermodynamics, the first law of thermodynamics is an expression of the more universal physical law of the conservation of energy. Succinctly, the first law of thermodynamics states: DescriptionThe first law of thermodynamics basically states… …   Wikipedia

  • Differentielle exacte — Différentielle exacte Une différentielle est dite exacte s il existe une fonction dont elle dérive, c est à dire s il est possible de l intégrer. La différentielle ω définie sur un ouvert U est dite exacte s il existe une fonction f… …   Wikipédia en Français

  • Différentielle Exacte — Une différentielle est dite exacte s il existe une fonction dont elle dérive, c est à dire s il est possible de l intégrer. La différentielle ω définie sur un ouvert U est dite exacte s il existe une fonction f différentiable sur U telle… …   Wikipédia en Français

  • Enthalpy — Thermodynamics …   Wikipedia

  • Integrating factor — In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus …   Wikipedia

  • List of mathematics articles (I) — NOTOC Ia IA automorphism ICER Icosagon Icosahedral 120 cell Icosahedral prism Icosahedral symmetry Icosahedron Icosian Calculus Icosian game Icosidodecadodecahedron Icosidodecahedron Icositetrachoric honeycomb Icositruncated dodecadodecahedron… …   Wikipedia

  • Work (thermodynamics) — In thermodynamics, work is the quantity of energy transferred from one system to another without an accompanying transfer of entropy. It is a generalization of the concept of mechanical work in mechanics. In the SI system of measurement, work is… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”