State function

Last updated September 29, 2024

In the thermodynamics of equilibrium, a state function, function of state, or point function for a thermodynamic system is a mathematical function relating several state variables or state quantities (that describe equilibrium states of a system) that depend only on the current equilibrium thermodynamic state of the system^[1] (e.g. gas, liquid, solid, crystal, or emulsion), not the 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 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. 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. 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.^[2]

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 Rankine, Peter Tait, and 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."^[3]

Overview

A thermodynamic system is described by a number of thermodynamic parameters (e.g. temperature, 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 ( $D$ ). For example, a monatomic 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 for higher-dimensional spaces, as described by the state postulate.

Generally, a state space is defined by an equation of the form $F(P,V,T,\ldots )=0$ , where $P$ denotes pressure, $T$ denotes temperature, $V$ denotes volume, and the ellipsis denotes other possible state variables like particle number $N$ and entropy $S$ . 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 $P (t)$ and volume $V (t)$ as functions of time from time $t 0$ to $t 1$ will specify a path in two-dimensional state space. Any function of time can then be integrated over the path. For example, to calculate the work done by the system from time $t 0$ to time $t 1$ , 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 work $W$ in the above integral, the functions $P (t)$ and $V (t)$ must be known at each time $t$ 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 $V dP$ 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 the exact differential of the function $P (t) V (t)$ . Therefore, the integral can be expressed as the difference in the value of $P (t) V (t)$ at the end points of the integration. The product $PV$ is therefore a state function of the system.

The notation $d$ will be used for an exact differential. In other words, the integral of $d Φ$ will be equal to $Φ(t 1) - Φ(t 0)$ . The symbol $δ$ will be reserved for an inexact 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 function $PV$ is proportional to the internal energy of an ideal gas, but the work $W$ 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

The following are considered to be state functions in thermodynamics:

Mass
Energy (E)
- Enthalpy ( $H$ )
- Internal energy ( $U$ )
- Gibbs free energy ( $G$ )
- Helmholtz free energy ( $F$ )
- Exergy ( $B$ )
Entropy ( $S$ )
Pressure ( $P$ )
Temperature ( $T$ )
Volume ( $V$ )
Chemical composition
Pressure altitude
Specific volume ( $v$ ) or its reciprocal density ( $ρ$ )
Particle number ( $n i$ )

Notes

↑ Callen 1985 , pp. 5, 37
↑ Mandl 1988 , p. 7
↑ Gibbs 1873 , pp. 309–342

Related Research Articles

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Enthalpy is the sum of a thermodynamic system's internal energy and the product of its pressure and volume. It is a state function in thermodynamics used in many measurements in chemical, biological, and physical systems at a constant external pressure, which is conveniently provided by the large ambient atmosphere. The pressure–volume term expresses the work $that was done against constant external pressure to establish the system's physical dimensions from to some final volume, i.e. to make room for it by displacing its surroundings. The pressure-volume term is very small for solids and liquids at common conditions, and fairly small for gases. Therefore, enthalpy is a stand-in for energy in chemical systems; bond, lattice, solvation, and other chemical "energies" are actually enthalpy differences. As a state function, enthalpy depends only on the final configuration of internal energy, pressure, and volume, not on the path taken to achieve it.$

An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics. The requirement of zero interaction can often be relaxed if, for example, the interaction is perfectly elastic or regarded as point-like collisions.

The first law of thermodynamics is a formulation of the law of conservation of energy in the context of thermodynamic processes. The law distinguishes two principal forms of energy transfer, heat and thermodynamic work, that modify a thermodynamic system containing a constant amount of matter. The law also defines the internal energy of a system, an extensive property for taking account of the balance of heat and work in the system. Energy cannot be created or destroyed, but it can be transformed from one form to another. In an isolated system the sum of all forms of energy is constant.

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A thermodynamic potential is a scalar quantity used to represent the thermodynamic state of a system. Just as in mechanics, where potential energy is defined as capacity to do work, similarly different potentials have different meanings. The concept of thermodynamic potentials was introduced by Pierre Duhem in 1886. Josiah Willard Gibbs in his papers used the term fundamental functions. While thermodynamic potentials cannot be measured directly, they can be predicted using computational chemistry.

The internal energy of a thermodynamic system is the energy of the system as a state function, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accounting for the gains and losses of energy due to changes in its internal state, including such quantities as magnetization. It excludes the kinetic energy of motion of the system as a whole and the potential energy of position of the system as a whole, with respect to its surroundings and external force fields. It includes the thermal energy, i.e., the constituent particles' kinetic energies of motion relative to the motion of the system as a whole. The internal energy of an isolated system cannot change, as expressed in the law of conservation of energy, a foundation of the first law of thermodynamics. The notion has been introduced to describe the systems characterized by temperature variations, temperature being added to the set of state parameters, the position variables known in mechanics, in a similar way to potential energy of the conservative fields of force, gravitational and electrostatic. Internal energy changes equal the algebraic sum of the heat transfered and the work done. In systems without temperature changes, potential energy changes equal the work done by/on the system.

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Thermodynamics is expressed by a mathematical framework of thermodynamic equations which relate various thermodynamic quantities and physical properties measured in a laboratory or production process. Thermodynamics is based on a fundamental set of postulates, that became the laws of thermodynamics.

In thermodynamics, a thermodynamic state of a system is its condition at a specific time; that is, fully identified by values of a suitable set of parameters known as state variables, state parameters or thermodynamic variables. Once such a set of values of thermodynamic variables has been specified for a system, the values of all thermodynamic properties of the system are uniquely determined. Usually, by default, a thermodynamic state is taken to be one of thermodynamic equilibrium. This means that the state is not merely the condition of the system at a specific time, but that the condition is the same, unchanging, over an indefinitely long duration of time.

Classical thermodynamics considers three main kinds of thermodynamic processes: (1) changes in a system, (2) cycles in a system, and (3) flow processes.

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Thermodynamic work is one of the principal processes by which a thermodynamic system can interact with its surroundings and exchange energy. This exchange results in externally measurable macroscopic forces on the system's surroundings, which can cause mechanical work, to lift a weight, for example, or cause changes in electromagnetic, or gravitational variables. The surroundings also can perform work on a thermodynamic system, which is measured by an opposite sign convention.

The principle of minimum energy is essentially a restatement of the second law of thermodynamics. It states that for a closed system, with constant external parameters and entropy, the internal energy will decrease and approach a minimum value at equilibrium. External parameters generally means the volume, but may include other parameters which are specified externally, such as a constant magnetic field.

In classical thermodynamics, entropy is a property of a thermodynamic system that expresses the direction or outcome of spontaneous changes in the system. The term was introduced by Rudolf Clausius in the mid-19th century to explain the relationship of the internal energy that is available or unavailable for transformations in form of heat and work. Entropy predicts that certain processes are irreversible or impossible, despite not violating the conservation of energy. The definition of entropy is central to the establishment of the second law of thermodynamics, which states that the entropy of isolated systems cannot decrease with time, as they always tend to arrive at a state of thermodynamic equilibrium, where the entropy is highest. Entropy is therefore also considered to be a measure of disorder in the system.

In thermodynamics, heat is the thermal energy transferred between systems due to a temperature difference. In colloquial use, heat sometimes refers to thermal energy itself. Thermal energy is the kinetic energy of vibrating and colliding atoms in a substance.

In thermodynamics, the volume of a system is an important extensive parameter for describing its thermodynamic state. The specific volume, an intensive property, is the system's volume per unit mass. Volume is a function of state and is interdependent with other thermodynamic properties such as pressure and temperature. For example, volume is related to the pressure and temperature of an ideal gas by the ideal gas law. The physical region covered by a system may or may not coincide with a control volume used to analyze the system.

References

Callen, Herbert B. (1985). Thermodynamics and an Introduction to Thermostatistics. Wiley & Sons. ISBN 978-0-471-86256-7.
Gibbs, Josiah Willard (1873). "Graphical Methods in the Thermodynamics of Fluids". Transactions of the Connecticut Academy. II. ASIN B00088UXBK – via WikiSource.
Mandl, F. (May 1988). Statistical physics (2nd ed.). Wiley & Sons. ISBN 978-0-471-91533-1.

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[1] Callen 1985 , pp. 5, 37

[2] Mandl 1988 , p. 7

[3] Gibbs 1873 , pp. 309–342

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