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# Equations of State

## Introduction

An equation of state is a mathematical expression that describes a system at thermodynamic equilibrium. These equations are empirical relationships that have been developed based on experimental measurements.

Several equations of state for gases are described below for illustration.

## The Ideal Gas Law

The Ideal Gas Law is:

PV = nRT

where P is pressure, V is volume, n is the number of moles of gas, R is the Gas Constant (8.3144 J/mol·K or 0.083145 L bar/mol·K), and T is temperature (in Kelvin).

This equation of state describes gas-phase systems reasonably well at low pressure. The ideal gas model assumes that there is no interaction between gas species and it neglects the volume occupied by the gas species.

## The van der Waals equation of state

The van der Waals equation of state provides a better description of gases at higher pressures than does the ideal gas law. It introduces two empirical parameters to correct for the interaction between gas atoms or molecules and for the volume occupied by the gas molecules. The van der Waals equation of state is:

P, V, n, R, and T have the same meaning as in the ideal gas law. a and b are empirical parameters that correct for the interaction between gas atoms or molecules and for the volume the gas species occupy, respectively. The values of a and b can be found in reference tables or determined by fitting experimental data.

## The Redlich-Kwong equation of state

A more effective equation of state for gases is the Redlich-Kwong equation:

P, V, n, R, and T have the same meaning as in the ideal gas law. a and b are again empirical parameters that correct for the interaction between gas species and the volume they occupy, respectively. The values of a and b are not the same as in the van der Waals coefficients, and can be found in reference tables or determined by fitting experimental data.

The following plot shows the differences between the three equations of states for He. There is very little difference below approximately 100 atm.

The following plot shows the differences between the three equations of states for He at 298 K in a 1.0 L container.

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