Movatterモバイル変換

[0]ホーム

Jump to content

State function

Edit links

From Wikipedia, the free encyclopedia

Function describing equilibrium states of a system

Thermodynamics

The classicalCarnot heat engine

Branches

Equilibrium / Non-equilibrium

Laws

Systems

State
Equation of state Ideal gas Real gas State of matter Phase (matter) Equilibrium Control volume Instruments
Processes
Isobaric Isochoric Isothermal Adiabatic Isentropic Isenthalpic Quasistatic Polytropic Free expansion Reversibility Irreversibility Endoreversibility
Cycles
Heat engines Heat pumps Thermal efficiency

System properties

Note:Conjugate variables initalics

Process functions
Property diagrams Intensive and extensive properties
Work Heat
Functions of state
Temperature / Entropy (introduction) Pressure / Volume Chemical potential / Particle number Vapor quality Reduced properties

Material properties

Property databases

Specific heat capacity

c=

$T {\displaystyle T}$	$\partial S$
$N {\displaystyle N}$	$\partial T$

Compressibility

\beta =-

$1 {\displaystyle 1}$	$\partial V$
$V {\displaystyle V}$	$\partial p$

Thermal expansion

\alpha =

$1 {\displaystyle 1}$	$\partial V$
$V {\displaystyle V}$	$\partial T$

Equations

Potentials

Internal energy
$U(S,V)$
Enthalpy
$H(S,p)=U+pV$
Helmholtz free energy
$A(T,V)=U-TS$
Gibbs free energy
$G(T,p)=H-TS$

History
Culture

History
General Entropy Gas laws "Perpetual motion" machines
Philosophy
Entropy and time Entropy and life Brownian ratchet Maxwell's demon Heat death paradox Loschmidt's paradox Synergetics
Theories
Caloric theory Vis viva("living force") Mechanical equivalent of heat Motive power
Key publications
An Inquiry Concerning the Source ... Friction On the Equilibrium of Heterogeneous Substances Reflections on the Motive Power of Fire
Timelines
Thermodynamics Heat engines
Art Education
Maxwell's thermodynamic surface Entropy as energy dispersal

Scientists

Other

Category

In thethermodynamics of equilibrium, astate function,function of state, orpoint function for athermodynamic system is amathematical function relating severalstate variables or state quantities (that describeequilibrium states of a system) that depend only on the current equilibriumthermodynamic state of the system^[1] (e.g. gas, liquid, solid, crystal, oremulsion), not thepath 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. Theideal 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 aheterogeneous orhomogeneous mixture, or the amount of energy required to create such a system or change the system into a different equilibrium state.

Internal energy,enthalpy, andentropy are examples of state quantities or state functions because they quantitatively describe an equilibrium state of athermodynamic system, regardless of how the system has arrived in that state. They are expressed byexact differentials. In contrast,mechanical work andheat areprocess 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 byinexact differentials. 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 totemperature. The description breaks down for quantities exhibitinghysteresis.^[2]

History

[edit]

It is likely that the term "functions of state" was used in a loose sense during the 1850s and 1860s by those such asRudolf Clausius,William Rankine,Peter Tait, andWilliam 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 quantitiesv,p,t,ε, andη are determined when the state of the body is given, and it may be permitted to call themfunctions of the state of the body."^[3]

Overview

[edit]

A thermodynamic system is described by a number of thermodynamic parameters (e.g. temperature,volume, orpressure) which are not necessarily independent. The number of parameters needed to describe the system is the dimension of thestate space of the system (D). For example, amonatomic gas with a fixed number of particles is a simple case of a two-dimensional system (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 forhigher-dimensional spaces, as described by thestate postulate.

Generally, a state space is defined by an equation of the form $F(P,V,T,\ldots )=0$ , whereP denotes pressure,T denotes temperature,V denotes volume, and the ellipsis denotes other possible state variables like particle numberN and entropyS. 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 pressureP(t) and volumeV(t) as functions of time from timet₀ tot₁ will specify a path in two-dimensional state space. Any function of time can then beintegrated over the path. For example, to calculate thework done by the system from timet₀ to timet₁, calculate ${\textstyle W(t_{0},t_{1})=\int _{0}^{1}P\,dV=\int _{t_{0}}^{t_{1}}P(t){\frac {dV(t)}{dt}}\,dt}$ . In order to calculate the workW in the above integral, the functionsP(t) andV(t) must be known at each timet 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 ofVdP over the path:

{\begin{aligned}\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{aligned}}

In the equation, ${\frac {d(PV)}{dt}}dt=d(PV)$ can be expressed as theexact differential of the functionP(t)V(t). Therefore, the integral can be expressed as the difference in the value ofP(t)V(t) at the end points of the integration. The productPV is therefore a state function of the system.

The notationd will be used for an exact differential. In other words, the integral ofdΦ will be equal toΦ(t₁) − Φ(t₀). The symbolδ will be reserved for aninexact differential, which cannot be integrated without full knowledge of the path. For example,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 functionPV is proportional to theinternal energy of an ideal gas, but the workW 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.