# Equation of State

In thermodynamics, an equation of state is a thermodynamic equation relating state variables that characterize 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 the physics of solids, an equation of state is used to depict how the volume or equivalently density of a material varies as a function of depth, i.e., 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 expressions 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 volume and pressure at a constant temperature. 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 a 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 of the gas laws completely developed at the end of the 18th century. These laws or statements preceded the ideal gas law since individually, and 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 equations 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. There are no intermolecular attractive forces, and 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 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 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 completely fails to predict phase changes, such as condensation from a gas to a liquid. Therefore, many more accurate equations of state have been developed for gases and liquids. For example, the Van der Waals equation of state was 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).
2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
5. Todreas Neil E., Kazimi Mujid S. Nuclear Systems Volume I: Thermal Hydraulic Fundamentals, Second Edition. CRC Press; 2 edition, 2012, ISBN: 978-0415802871
6. Zohuri B., McDaniel P. Thermodynamics in Nuclear Power Plant Systems. Springer; 2015, ISBN: 978-3-319-13419-2
7. Moran Michal J., Shapiro Howard N. Fundamentals of Engineering Thermodynamics, Fifth Edition, John Wiley & Sons, 2006, ISBN: 978-0-470-03037-0
8. Kleinstreuer C. Modern Fluid Dynamics. Springer, 2010, ISBN 978-1-4020-8670-0.
9. U.S. Department of Energy, THERMODYNAMICS, HEAT TRANSFER, AND FLUID FLOW. DOE Fundamentals Handbook, Volume 1, 2, and 3. June 1992.

Ideal Gas Law