Inexact differentials

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.


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.


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.

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