Editing State function

{{Short description|Function describing equilibrium states of a system}}
{{Thermodynamics|expanded=sysprop}}
In the [[Thermodynamics#Equilibrium thermodynamics|thermodynamics of equilibrium]], a '''state function''', '''function of state''', or '''point function''' for a [[thermodynamic system]] is a [[Function (mathematics)|mathematical function]] relating several [[state variables]] or state quantities (that describe [[Thermodynamic equilibrium|equilibrium states]] of a system) that depend only on the current equilibrium [[thermodynamic state]] of the system<ref>{{harvnb|Callen|1985|pages=5,37}}</ref> (e.g. gas, liquid, solid, crystal, or [[emulsion]]), not the [[Thermodynamic process path|path]] which the system has taken to reach that state. A state function describes equilibrium states of a system, thus also describing the type of system. A state variable is typically a state function so the determination of other state variable values at an equilibrium state also determines the value of the state variable as the state function at that state. The [[ideal gas law]] is a good example. In this law, one state variable (e.g., pressure, volume, temperature, or the amount of substance in a gaseous equilibrium system) is a function of other state variables so is regarded as a state function. A state function could also describe the number of a certain type of atoms or molecules in a gaseous, liquid, or solid form in a [[Heterogeneous mixture|heterogeneous]] or [[homogeneous mixture]], or the amount of energy required to create such a system or change the system into a different equilibrium state.

[[Internal energy]], [[enthalpy]], and [[entropy]] are examples of state quantities or state functions because they quantitatively describe an equilibrium state of a [[thermodynamic system]], regardless of how the system has arrived in that state. They are expressed by [[exact differential]]s. In contrast, [[mechanical work]] and [[heat]] are [[process quantities]] or path functions because their values depend on a specific "transition" (or "path") between two equilibrium states that a system has taken to reach the final equilibrium state, being expressed by [[inexact differential]]s. Exchanged heat (in certain discrete amounts) can be associated with changes of state function such as enthalpy. The description of the system heat exchange is done by a state function, and thus enthalpy changes point to an amount of heat. This can also apply to entropy when heat is compared to [[temperature]]. The description breaks down for quantities exhibiting [[hysteresis]].<ref>{{harvnb|Mandl|1988|p=7}}</ref>

==History==
It is likely that the term "functions of state" was used in a loose sense during the 1850s and 1860s by those such as [[Rudolf Clausius]], [[William John Macquorn Rankine|William Rankine]], [[Peter Tait (physicist)|Peter Tait]], and [[William Thomson, 1st Baron Kelvin|William Thomson]]. By the 1870s, the term had acquired a use of its own. In his 1873 paper "Graphical Methods in the Thermodynamics of Fluids", [[Willard Gibbs]] states: "The quantities ''v'', ''p'', ''t'', ''ε'', and ''η'' are determined when the state of the body is given, and it may be permitted to call them ''functions of the state of the body''."<ref>{{harvnb|Gibbs|1873|pages=309&ndash;342}}</ref>

==Overview==
A thermodynamic system is described by a number of thermodynamic parameters (e.g. temperature, [[Volume (thermodynamics)|volume]], or [[pressure]]) which are not necessarily independent. The number of parameters needed to describe the system is the dimension of the [[state space]] of the system ({{math|''D''}}). For example, a [[monatomic gas]] with a fixed number of particles is a simple case of a two-dimensional system ({{math|1=''D'' = 2}}). Any two-dimensional system is uniquely specified by two parameters. Choosing a different pair of parameters, such as pressure and volume instead of pressure and temperature, creates a different coordinate system in two-dimensional thermodynamic state space but is otherwise equivalent. Pressure and temperature can be used to find volume, pressure and volume can be used to find temperature, and temperature and volume can be used to find pressure. An analogous statement holds for [[higher-dimensional space]]s, as described by the [[state postulate]].

Generally, a state space is defined by an equation of the form <math>F(P, V, T, \ldots) = 0</math>, where {{mvar|P}} denotes pressure, {{mvar|T}} denotes temperature, {{mvar|V}} denotes volume, and the ellipsis denotes other possible state variables like particle number {{mvar|N}} and entropy {{mvar|S}}. If the state space is two-dimensional as in the above example, it can be visualized as a three-dimensional graph (a surface in three-dimensional space). However, the labels of the axes are not unique (since there are more than three state variables in this case), and only two independent variables are necessary to define the state.

When a system changes state continuously, it traces out a "path" in the state space. The path can be specified by noting the values of the state parameters as the system traces out the path, whether as a function of time or a function of some other external variable. For example, having the pressure {{math|''P''(''t'')}} and volume {{math|''V''(''t'')}} as functions of time from time {{math|''t''<sub>0</sub>}} to {{math|''t''<sub>1</sub>}} will specify a path in two-dimensional state space. Any function of time can then be [[Integral|integrated]] over the path. For example, to calculate the [[work (physics)|work]] done by the system from time {{math|''t''<sub>0</sub>}} to time {{math|''t''<sub>1</sub>}}, calculate <math display="inline">W(t_0,t_1) = \int_0^1 P \, dV = \int_{t_0}^{t_1} P(t) \frac{dV(t)}{dt} \, dt</math>. In order to calculate the work {{mvar|W}} in the above integral, the functions {{math|''P''(''t'')}} and {{math|''V''(''t'')}} must be known at each time {{mvar|t}} over the entire path. In contrast, a state function only depends upon the system parameters' values at the endpoints of the path. For example, the following equation can be used to calculate the work plus the integral of {{math|''V'' ''dP''}} over the path:

:<math>\begin{align}
\Phi(t_0,t_1) &= \int_{t_0}^{t_1}P\frac{dV}{dt}\,dt
+ \int_{t_0}^{t_1}V\frac{dP}{dt}\,dt \\
&= \int_{t_0}^{t_1}\frac{d(PV)}{dt}\,dt = P(t_1)V(t_1)-P(t_0)V(t_0).
\end{align}</math>

In the equation, <math>\frac{d(PV)}{dt}dt = d(PV)</math> can be expressed as the [[exact differential]] of the function {{math|''P''(''t'')''V''(''t'')}}. Therefore, the integral can be expressed as the difference in the value of {{math|''P''(''t'')''V''(''t'')}} at the end points of the integration. The product {{mvar|PV}} is therefore a state function of the system.

The notation {{mvar|d}} will be used for an exact differential. In other words, the integral of {{math|''d''Φ}} will be equal to {{math|Φ(''t''<sub>1</sub>) − Φ(''t''<sub>0</sub>)}}. The symbol {{mvar|δ}} will be reserved for an [[inexact differential]], which cannot be integrated without full knowledge of the path. For example, {{math|1=''δW'' = ''PdV''}} will be used to denote an infinitesimal increment of work.

State functions represent quantities or properties of a thermodynamic system, while non-state functions represent a process during which the state functions change. For example, the state function {{math|''PV''}} is proportional to the [[internal energy]] of an ideal gas, but the work {{mvar|W}} is the amount of energy transferred as the system performs work. Internal energy is identifiable; it is a particular form of energy. Work is the amount of energy that has changed its form or location.

==List of state functions==
{{See also|List of thermodynamic properties}}
The following are considered to be state functions in thermodynamics:

{{div col|colwidth=35em}}
* [[Mass]]
* [[Energy]] ({{mvar|E}})
** [[Enthalpy]] ({{mvar|H}})
** [[Internal energy]] ({{mvar|U}})
** [[Gibbs free energy]] ({{mvar|G}})
** [[Helmholtz free energy]] ({{mvar|F}})
** [[Exergy]] ({{mvar|B}})
* [[Entropy]] ({{mvar|S}})
* [[Pressure]] ({{mvar|P}})
* [[Thermodynamic temperature|Temperature]] ({{mvar|T}})
* [[Volume (thermodynamics)|Volume]] ({{mvar|V}})
* [[Chemical composition]]
* [[Pressure altitude]]
* [[Specific volume]] ({{mvar|v}}) or its reciprocal [[density]] ({{mvar|ρ}})
* [[Particle number]] ({{mvar|n<sub>i</sub>}})
{{div col end}}

==See also==
*[[Markov property]]
*[[Conservative vector field]]
*[[Nonholonomic system]]
*[[Equation of state]]
*[[State variable]]

== Notes ==
{{Reflist}}

==References==
*{{cite book|last=Callen|first=Herbert B.|author-link=Herbert Callen|year=1985|title=Thermodynamics and an Introduction to Thermostatistics|publisher=[[Wiley & Sons]]|isbn=978-0-471-86256-7}}
*{{cite journal|last=Gibbs|first=Josiah Willard|author-link=Willard Gibbs|year=1873|title=Graphical Methods in the Thermodynamics of Fluids|journal=Transactions of the Connecticut Academy|volume=II|asin=B00088UXBK|url=https://en.wikisource.org/wiki/Scientific_Papers_of_Josiah_Willard_Gibbs,_Volume_1/Chapter_I|via=[[WikiSource]]}}
*{{cite book|first=F.|last=Mandl|title=Statistical physics|edition=2nd|date=May 1988|publisher=[[Wiley & Sons]]|isbn=978-0-471-91533-1}}

==External links==
*{{Commons category-inline|State functions}}

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[[Category:State functions| ]]
[[Category:Thermodynamic properties]]
[[Category:Continuum mechanics]]