Definitions

# 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:

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

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: $delta f,$

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

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, 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
$Q = int_\left\{T_0\right\}^\left\{T_f\right\}C_p,dT ,!$
is true for constant pressure.