# Introduction to Chemical Engineering Processes/Generalized Correlations

## Critical Constants

At room temperature (about 298K), it is possible to add enough pressure to carbon dioxide to get it to liquify (some fire extinguishers work by keeping liquid carbon dioxide in them under very high pressure, which rapidly vaporizes when the pressure is relieved [1]. However, if the temperature is raised to higher than 304.2 K, it will be impossible to keep carbon dioxide in a liquid form, because it has too much kinetic energy to remain in the liquid phase. No amount of pressure can turn carbon dioxide into a liquid if the temperature is too high.

This threshold temperature is called a critical temperature. Any pure stable substance (not just carbon dioxide) will have a single characteristic critical temperature. Pure stable substances will also have a single characteristic critical pressure, which is the pressure needed to achieve a phase transition at the critical temperature, and a critical specific volume which is the specific volume (volume per mass) of the fluid at this temperature and pressure.

Critical pressures are typically extremely large, ranging from 2.26 atm for helium to 218.3 atm for water [2], and about 40 atm on average. Critical temperatures typically range from 5.26 K (for helium) to the high 600s K for some aromatic compounds.

A substance which is at a temperature higher than the critical temperature and a pressure higher than its critical pressure is called a supercritical fluid. Supercritical fluids have some properties in common with gasses and some in common with liquid, as may be expected since it they are not observed to be liquid but would be expected to be liquefied at extreme pressures.

## Law of Corresponding States

Recall from the last section that the compressibility of any substance (but most useful for gasses) is defined as:

${\displaystyle Z={\frac {P*{\hat {V}}}{RT}}}$

The compressibility of a gas is a measure of how non-ideal it is; an ideal gas has a compressibility of 1. At the critical point, in particular, the compressibility is:

 ${\displaystyle Z_{C}={\frac {P_{c}*{\hat {V}}_{c}}{R*T_{c}}}}$

Critical constants are important because it has been found experimentally that the following rule is true for many substances:

 The Law of Corresponding States Many substances behave in similar manners to each other depending on how far the system conditions are from the critical temperature and pressure of the substance. In particular, the compressibility of a substance is strongly correlated to its variance from the critical conditions.

It has been found experimentally that many substances have very similar compressibility at their critical point. [1]. Most nonpolar substances in particular have a critical compressibility of about 0.27. The similarity of the critical compressibility between substances is what gives some weight to the law of corresponding states. However, the fact that the critical compressibility is not exactly the same for all substances leads to potential estimation errors if this method is used.

The critical constants are able to effectively predict the properties of a substance without gathering a large amount of data. However, it is necessary to define how the properties of the substance change as the system variables become closer to or farther from the critical point of the substance. These methods are discussed in the following sections.

### Compressibility Charts

Recall that many substances have similar critical compressibility values near 0.27. Therefore, charts have been developed which relate compressibility at other conditions to those at the critical point. In order to use these charts, the system parameters are normalized by dividing by the critical constants to yield reduced temperature, pressure, and volume:

 Reduced parameters ${\displaystyle T_{r}={\frac {T}{T_{c}}}}$, ${\displaystyle P_{r}={\frac {P}{P_{c}}}}$, ${\displaystyle {\hat {V}}_{r}={\frac {\hat {V}}{{\hat {V}}_{c}}}}$
1. how fire extinguishers work
2. see Wikipedia article on critical properties