Equation of State

In thermodynamics, an equation of state is a thermodynamic equation relating state variables which characterizes the state of matter under a given set of physical conditions. Equations of state are used to describe gases, fluids, fluid mixtures, solids and the interior of stars. In physics of solids an equation of state is used to depict how the volume or equivalently density of a material vary as a function of depth, i.e., as a function of pressure and temperature.

The most prominent use of an equation of state is to correlate densities of gases and liquids to temperatures and pressures.

Perhaps one of the first expression of an equation of state is the Boyle-Mariotte Law. At the end of the 17th century, Robert William Boyle and Edme Mariotte independently studied the relationship between the volume and pressure of a gas at constant temperature. The results of certain experiments with gases at relatively low pressure led Robert Boyle to formulate a well-known law. It states that:

For a fixed mass of gas at constant temperature, the volume is inversely proportional to the pressure.

That means that, for example, if you increase the volume 10 times, the pressure will decrease 10 times. If you halve the volume, you will double the pressure.

You can express this mathematically as:

pV = constant

This law is one the gas laws that were completely developed at the end of the 18th century. These laws or statements preceded the ideal gas law, since individually these laws are considered as special cases of the ideal gas equation, with one or more of the variables held constant.

One of the best known equation of state is the one for ideal gases, the ideal gas law. Since ideal gas is defined as one in which all collisions between atoms or molecules are perfectly elastic and in which there are no intermolecular attractive forces, there is no such thing in nature as a truly ideal gas. On the other hand, all real gases approach the ideal state at low pressures (densities) and moderate temperatures. At low pressures molecules are far enough apart that they do not interact with one another.

pV = nRT

where:

• p is the absolute pressure of the gas
• n is the amount of substance
• T is the absolute temperature
• V is the volume
• R  is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant,

In this equation the symbol R is a constant called the universal gas constant that has the same value for all gases—namely, R =  8.31 J/mol K.

The power of the ideal gas law is in its simplicity. When any two of the thermodynamic variables, p, v, and T are given, the third can easily be found. However, the ideal gas law becomes inaccurate at higher pressures and lower temperatures, and completelly fails to predict phase changes, for example, a condensation from a gas to a liquid. Therefore, a number of more accurate equations of state have been developed for gases and liquids. For example, the Van der Waals equation of state formulated in 1877.

References:

Reactor Physics and Thermal Hydraulics:

1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
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6. Zohuri B., McDaniel P. Thermodynamics in Nuclear Power Plant Systems. Springer; 2015, ISBN: 978-3-319-13419-2
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