Compressibility factor
[1] In general, deviation from ideal behaviour becomes more significant the closer a gas is to a phase change, the lower the temperature or the larger the pressure.
This allows repulsive forces between molecules to have a noticeable effect, making the molar volume of the real gas (
, was first recognized by Johannes Diderik van der Waals in 1873 and is known as the two-parameter principle of corresponding states.
As for the compressibility of gases, the principle of corresponding states indicates that any pure gas at the same reduced temperature,
The pressure-volume-temperature (PVT) data for real gases varies from one pure gas to another.
In order to obtain a generalized graph that can be used for many different gases, the reduced pressure and temperature,
Figure 2 is an example of a generalized compressibility factor graph derived from hundreds of experimental PVT data points of 10 pure gases, namely methane, ethane, ethylene, propane, n-butane, i-pentane, n-hexane, nitrogen, carbon dioxide and steam.
The quantum gases hydrogen, helium, and neon do not conform to the corresponding-states behavior.
Rao recommended that the reduced pressure and temperature for those three gases should be redefined in the following manner to improve the accuracy of predicting their compressibility factors when using the generalized graphs: where the temperatures are in kelvins and the pressures are in atmospheres.
These observations are: The virial equation is especially useful to describe the causes of non-ideality at a molecular level (very few gases are mono-atomic) as it is derived directly from statistical mechanics: Where the coefficients in the numerator are known as virial coefficients and are functions of temperature.
The virial coefficients account for interactions between successively larger groups of molecules.
Because interactions between large numbers of molecules are rare, the virial equation is usually truncated after the third term.
Deviations of the compressibility factor, Z, from unity are due to attractive and repulsive intermolecular forces.
The relative importance of attractive forces decreases as temperature increases (see effect on gases).
To better understand these curves, a closer look at the behavior for low temperature and pressure is given in the second figure.
N2 is a gas under these conditions, so the distance between molecules is large, but becomes smaller as pressure increases.
Higher temperature reduces the effect of the attractive interactions and the gas behaves in a more nearly ideal manner.
As the pressure increases, the gas eventually reaches the gas-liquid coexistence curve, shown by the dashed line in the figure.
When that happens, the attractive interactions have become strong enough to overcome the tendency of thermal motion to cause the molecules to spread out; so the gas condenses to form a liquid.
Just above the critical point there is a range of pressure for which Z drops quite rapidly (see the 130 K curve), but at higher temperatures the process is entirely gradual.
As temperature increases, the initial slope becomes less negative, the pressure at which Z is a minimum gets smaller, and the pressure at which repulsive interactions start to dominate, i.e. where Z goes from less than unity to greater than unity, gets smaller.
At the Boyle temperature (327 K for N2), the attractive and repulsive effects cancel each other at low pressure.
Then Z remains at the ideal gas value of unity up to pressures of several tens of bar.
It is extremely difficult to generalize at what pressures or temperatures the deviation from the ideal gas becomes important.
As a rule of thumb, the ideal gas law is reasonably accurate up to a pressure of about 2 atm, and even higher for small non-associating molecules.
For example, methyl chloride, a highly polar molecule and therefore with significant intermolecular forces, the experimental value for the compressibility factor is
[9] For air (small non-polar molecules) at approximately the same conditions, the compressibility factor is only
We can therefore expect that the behaviour of air within broad temperature and pressure ranges can be approximated as an ideal gas with reasonable accuracy.
values are calculated from values of pressure, volume (or density), and temperature in Vasserman, Kazavchinskii, and Rabinovich, "Thermophysical Properties of Air and Air Components;' Moscow, Nauka, 1966, and NBS-NSF Trans.
TT 70-50095, 1971: and Vasserman and Rabinovich, "Thermophysical Properties of Liquid Air and Its Component, "Moscow, 1968, and NBS-NSF Trans.
