Inexact differential



In 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} df \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 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.[2] Continuing with the same instance of ΔQ, for example, removing the Δ, 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

  1. ^ a b Laider, Keith, J. (1993). The World of Physical Chemistry. Oxford University Press. ISBN 0-19-855919-4. 
  2. ^ Baierlein, Ralph (2003). Thermal Physics. Cambridge University Press. ISBN 0-521-65838-1. 
 
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Inexact_differential". A list of authors is available in Wikipedia.