State function

Junior intermediate ABRAHAM LINCOLN'S LETTER TO HIS SONS TEACHER

                     ABRAHAM LINCOLN

Abraham Lincoln was the 16th president of the United States.He is widely regarded as one of the greatest president of America . "Abraham Lincoln Letter to his sons teacher" extols his conviction in values . The letter has a universal appeal

Lincoln appeals to his sons teacher to be soft kind and gentle . The new environment may startle him as it is his first day at school . The teacher should instill in his sons faith , love , and encourage . He urges the teacher to teach his son to treat success and failure with eqaunmity . The teacher should inculcate in his son the power of discrimination and the sense of appreciation . The child should know the value of honesty,diligence, integrity and sincere tears . He should learn to be gentle with good people and rude with rude people . The teacher should develop in his son sublime faith in himself and thereby in mankind and in God According to Lincoln , education is incomplete without pratical application . Education should replace indecision with confirmation and confidence . It should develop the child's character

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."^[1]

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

^ Gibbs 1873, pp. 309–342

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.

External links

Media related to State functions at Wikimedia Commons

[1] Gibbs 1873, pp. 309–342

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