# Equation of state

As equation of state, the functional relationship between thermodynamic state variables is referred to, with the help of which can describe the state of a thermodynamic system. In this case, choose one of the state variables as a function of state and the other, depending on their state variables as state variables. Condition equations are needed to describe the properties of fluids and fluid mixtures solids. Equations of state are bijective - so they allow that can be solved for each state variable.

## Introduction

The best-known equations of state are the state description of gases and liquids. The most important and at the same time simplest representative, which is used generally to explain the nature of a state equation is the general gas equation. Although this describes only an ideal gas precisely, but can be used at low pressures and high temperatures, as an approximation for real gases. At high pressures, low temperatures and in particular, phase transitions, however, fails so that other equations of state are required. Equations of state of real systems are always approximate solutions and can not accurately describe all conditions, the properties of a substance.

Equations of state are no deductions from the general laws of thermodynamics. You have to be found empirically or by means of statistical methods. All state equations of a thermodynamic system including a state equation and known, all the state variables of the system, as can be determined by means of the thermodynamics of the same, all thermodynamic properties.

In thermodynamics, a distinction is made between caloric and thermal equations of state. Based on the second law of thermodynamics, but these are dependent on one another.

## Thermodynamic background

State equations provide a material-specific relationship between the thermodynamic state variables represents a thermodynamic system, which consists of one or more gaseous, liquid or solid phases, is uniquely determined in the thermodynamic equilibrium by a number of state variables. State variables depend only on the current state but not on the previous history of the system. Two states are equal if and only if all appropriate state variables match. Such state variables are, for example, the temperature, pressure, volume, and the internal energy. In a mixture of several components, the amounts of substances are also state variables, wherein, instead of the individual substance amounts usually the total molar amount and the mole fractions are used for the description.

The state variables of a system are not all independent of each other. The number of independent variable state variables, ie the number of degrees of freedom, depends, according to the Gibbs phase rule of the number of components and the number of different phases of the thermodynamic system from:

- In a single-phase one-component system (such as liquid water, K = P = 1), therefore, sufficient for uniquely determining two variables of state of the state. For a given amount of substance are the state variables, and not independently. Are the temperature and the pressure specified for example, then automatically sets a specific volume that can not be varied without changing at the same time or not.
- Located in a single-component system, two phases are in equilibrium (K = 1, P = 2), then is sufficient to define a state variable. Is given, for example, the temperature then was raised in the phase equilibrium between liquid and vapor automatically a certain substance-specific pressure which is called the vapor pressure. The functional relationship between temperature and vapor pressure is an equation of state.

## The thermal equation of state

The thermal equation of state is the state variables pressure, volume, temperature and amount of substance related.

Most thermal equations of state, such as the general gas equation and the van der Waals equation included explicitly, ie as a state function, the print:

Is the molar volume, or density as a function of temperature and pressure is given, this corresponds to a thermal state equation that explicitly contains the volume:

The average molar mass of the system respectively.

All of these forms are equivalent, and include the same information.

For this results in the total differential:

This can be achieved by simplifying

- The volume expansion coefficient
- The compressibility
- The molar volume

With the result:

## The caloric equation of state

The properties of a thermodynamic system, ie the substance-specific interactions of all state variables are, but not fully determined by a thermal equation of state. The determination of the thermodynamic potential, all of which contain information about a single thermodynamic system, requires additional caloric state equation, also called the energy equation. It includes a state variable which is not dependent on the thermal condition of the equation, only the temperature.

Especially common is the simple measurable specific heat capacity at normal pressure bar. Is provided (eg, by a table of values for the spline interpolation or a polynomial of the 4th degree ), the specific enthalpy and the specific entropy at normal pressure can be calculated depending on the temperature:

With

- The Normalbildungsenthalpie
- The Normalentropie per mole,

Both under normal conditions ( K, bar). You are tabulated for many substances.

This results in the specific free enthalpy yields at atmospheric pressure depending on the temperature:

With the density as a function of temperature and pressure, that is, a thermal state equation, it can free the specific enthalpy can be calculated, not only for any temperature, but also for any pressures:

Since the Gibbs free energy with respect to the variables and a thermodynamic potential, so that all thermodynamic quantities of the system are determined and predictable.

In an alternative, but equivalent way, the caloric state equation describes the linking of two different thermodynamic potential, namely, the internal energy U and the enthalpy H with three thermodynamic state variables: the temperature T, the volume V (or the pressure p ), and amount of substance n

For and results in the total differentials:

With the assumption (constant amount of substance ) and the relations

Follows

And

## Examples

### Ideal gases

- Ideal gas equation (s), also approximation for real gases under certain conditions

### Real gases

- Van der Waals equation
- Virialgleichungen
- Clausius equation
- Equation of state of Berthelot
- Equation of state of Dieterici
- Redlich - Kwong equation of state
- Redlich - Kwong equation of state of Soave -
- Equation of state of Peng -Robinson
- Equation of State Benedict - Webb - Rubin
- Equation of state of Benedict - Webb - Rubin -Starling
- Equation of state of Martin- Hou

### Highly compressed matter

- Equation of state of Mie - Grüneisen
- Equation of state of white dwarfs
- Equation of state of neutron star (yet unknown)

### Explosives

- Equation of state of Becker- Kistiakowsky -Wilson
- Equation of state of Jones - Wilkins -Lee
- Equation of state of Jacobs - Cowperthwaite Zwisler

### Solids under hydrostatic pressure

- Equation of state of Birch - Murnaghan

### Water

- Taitsche equation

### More

- Equation of state Elliott - Suresh - Donohue