Editing State function (section)

==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.