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Information about 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.<ref name="Laider" >Laider, Keith, J. (1993). The World of Physical Chemistry. Oxford University Press. ISBN 0-19-855919-4.  The symbol d (with a cross bar, as in: 'hbar'), or δ (in the modern sense), which originated from the work of German mathematician Carl Gottfried Neumann in his 1875 Vorlesungen uber die mechanische Theorie der Warme, indicates that Q and W are path dependent.<ref name="Laider" /> In terms of infinitesimal quantities, the first law of thermodynamics is thus expressed as:



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:

; 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

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 is a mistake, since heat is not a state function having initial and final values. It would, however, be correct to use lower case in the inexact differential expression for heat. The offending belongs further down in the Thermodynamics section in the equation :, which should be : (Baierlein, p. 10, equation 1.11, though he denotes internal energy by in place of .[1] Continuing with the same instance of , for example, removing the , the equation
:::
is true for constant pressure.

See also

References

1. ^ Baierlein, Ralph (2003). Thermal Physics. Cambridge University Press. ISBN 0-521-65838-1. 

External links

Thermodynamics (from the Greek θερμη, therme, meaning "heat" and δυναμις, dynamis, meaning "power") is a branch of physics that studies the effects of changes in temperature, pressure, and volume on
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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.
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In thermodynamics, a state function, or state quantity, is a property of a system that depends only on the current state of the system, not on the way in which the system got to that state. A state function describes the equilibrium state of a system.
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A thermodynamic process may be defined as the energetic evolution of a thermodynamic system proceeding from an initial state to a final state. Paths through the space of thermodynamic variables are often specified by holding certain thermodynamic variables constant.
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Carl Gottfried Neumann (May 7, 1832 - March 27, 1925) was a German mathematician, born in Königsberg (now Kaliningrad, Russia) and died in Leipzig.

Neumann worked on the Dirichlet principle, and can be considered one of the initiators of the theory of integral equations.
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The first law of thermodynamics is an expression of the universal law of conservation of energy, and identifies heat transfer as a form of energy transfer. The most common enunciation of the first law of thermodynamics is:


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;     ;  


These conditions, which are easy to generalize, arise from the independence of the order of differentiations in the calculation of the second derivatives.
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In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given ordinary differential equation.

Consider an ordinary differential equation of the form


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Thermodynamics (from the Greek θερμη, therme, meaning "heat" and δυναμις, dynamis, meaning "power") is a branch of physics that studies the effects of changes in temperature, pressure, and volume on
..... Click the link for more information.
In thermodynamics, a state function, or state quantity, is a property of a system that depends only on the current state of the system, not on the way in which the system got to that state. A state function describes the equilibrium state of a system.
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Path-dependence is a phrase used to mean one of two things (Pierson 2004). Some authors use path-dependence to mean simply "history matters" - a broad conception - while others use it to mean that institutions are self reinforcing - a narrow conception.
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In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose differential is zero (dα = 0), and an exact form
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Differential may refer to:

Mathematics

  • Differential (calculus), multiple related meanings in calculus and differential geometry, such as an infinitesimal change in the value of a function

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;     ;  


These conditions, which are easy to generalize, arise from the independence of the order of differentiations in the calculation of the second derivatives.
..... Click the link for more information.
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given ordinary differential equation.

Consider an ordinary differential equation of the form


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