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{{about|the mathematical concept used in thermodynamics|the engineering indicator|process variable}}
{{Thermodynamics|cTopic=[[List of thermodynamic properties|System properties]]}}
{{see also|List of thermodynamic properties}}

In [[thermodynamics]], a [[quantity]] that is well defined so as to describe the path of a process through the [[equilibrium state]] space of a [[thermodynamic system]] is termed a '''process function''',<ref name="Sychev1991">{{cite book |last=Sychev |first=V. V. |title=The Differential Equations of Thermodynamics |year=1991 |publisher=Taylor & Francis |isbn=978-1560321217}}</ref> or, alternatively, a '''process quantity''', or a '''path function'''. As an example, [[mechanical work]] and [[heat]] are process functions because they describe quantitatively the transition between equilibrium states of a thermodynamic system.

Path functions depend on the path taken to reach one state from another. Different routes give different quantities. Examples of path functions include [[work (thermodynamics)|work]], [[heat]] and [[arc length]]. In contrast to path functions, [[state function]]s are independent of the path taken. Thermodynamic [[State function|state variables]] are point functions, differing from path functions. For a given state, considered as a point, there is a definite value for each state variable and state function.

Infinitesimal changes in a process function {{mvar|X}} are often indicated by {{mvar|δX}} to distinguish them from infinitesimal changes in a state function {{mvar|Y}} which is written {{mvar|dY}}. The quantity {{mvar|dY}} is an [[exact differential]], while {{mvar|δX}} is not, it is an [[inexact differential]]. Infinitesimal changes in a process function may be integrated, but the integral between two states depends on the particular path taken between the two states, whereas the integral of a state function is simply the difference of the state functions at the two points, independent of the path taken.

In general, a process function {{mvar|X}} may be either [[Holonomic constraints|holonomic]] or non-holonomic. For a holonomic process function, an auxiliary state function (or integrating factor) {{mvar|λ}} may be defined such that {{math|''Y'' {{=}} ''λX''}} is a state function. For a non-holonomic process function, no such function may be defined. In other words, for a holonomic process function, {{mvar|λ}} may be defined such that {{math|''dY'' {{=}} ''λδX''}} is an exact differential. For example, thermodynamic work is a holonomic process function since the integrating factor {{math|''λ'' {{=}} {{sfrac|1|''p''}}}} (where {{mvar|p}} is pressure) will yield exact differential of the volume state function {{math|''dV'' {{=}} {{sfrac|''δW''|''p''}}}}. The [[second law of thermodynamics]] as stated by [[Carathéodory]] essentially amounts to the statement that heat is a holonomic process function since the integrating factor {{math|''λ'' {{=}} {{sfrac|1|''T''}}}} (where {{mvar|T}} is temperature) will yield the exact differential of an entropy state function {{mvar|''dS'' {{=}} {{sfrac|''δQ''|''T''}}}}.<ref name="Sychev1991" />

==References==
<references />

==See also==
*[[Thermodynamics]]

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[[Category:Thermodynamics]]

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Process function - Revision history

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