Published on *PNG 520: Phase Behavior of Natural Gas and Condensate Fluids* (https://www.e-education.psu.edu/png520)

It is important to point out that the PCS that we have just discussed was originally outlined by van der Waals. In reality, it is the simplest version of the principle of corresponding states, and it is referred to as the* two-parameter PCS*. This is because it relies on two parameters (reduced pressure and temperature) for defining a “corresponding state.”

With the passing of time, more accurate PCS formulations have made use of more than two parameters. For instance, the three-parameter PCS affirms that two substances are in corresponding states not only when they are at the same reduced conditions (reduced pressure and temperature), but also when they have the same “acentric factor” value. In any case, the general statement of PCS remains untouched:

“Substances at corresponding states behave alike.”

What makes the difference is the definition of “what a corresponding state is.”

The acentric factor “w” is a concept that was introduced by Pitzer in 1955, and has proven to be very useful in the characterization of substances. It has become a standard for the proper characterization of any single pure component, along with other common properties, such as molecular weight, critical temperature, critical pressure, and critical volume.

Pitzer came up with this factor by analyzing the vapor pressure curves of various pure substances. From thermodynamic considerations, the vapor pressure curve that we studied in our first modules for pure components can be mathematically described by the Clausius Clapeyron equation:

$$\frac{1}{P}\frac{dP}{dT}=\frac{\Delta {\tilde{H}}_{vap}}{RT\Delta Z}$$ (8.7)

The use of the integrated version of equation (8.7) is very common for the mathematical fitting of vapor pressure data. The integrated version of equation (8.7) shows that the relationship between the logarithm of vapor pressure and the reciprocal of absolute temperature is *approximately* linear. That is, in terms of reduced conditions, vapor pressure data approximately follows a straight line when plotted in terms of “logP_{r}” versus “1/Tr”, or, equivalently:

$${\mathrm{log}}_{10}{P}_{r}=a\left(\frac{1}{{T}_{r}}\right)+b$$ (8.8)

If the two-parameter corresponding state principle were to hold true for all substances, the parameters “a” and “b” should be the same for all substances. That is, all vapor pressure curves of all imaginable substances should lie on top of each when plotted in terms of reduced conditions. Stated in another way, if the plot is of the form “logP_{r}” versus “1/T_{r}”, all lines should show the same slope (a) and intercept (b).

The bad news is that, as you may imagine, this is not always true. Vapor pressure data for different substances do follow different trends. The good news is that *some gases* follow the expected trend. Which are they? The noble gases. Noble gases (such as Ar, Kr and Xe) happen to follow the two-parameter corresponding states theory very closely. Hence, they yield themselves amenable to acting as a reference to evaluate “compliance” with the two-parameter equation of state.

Pitzer wanted to come up with a reliable way of quantifying the deviation of substances with respect to two-parameter corresponding state predictions. He decided to use noble gases as the base for comparison. Analyzing vapor pressure data for noble gases, Pitzer showed that a value of logP_{r} = – 1 was achieved at approximately T_{r} = 0.7. So, BINGO! There you are! He thought: if the vapor pressure data of a substance show that logP_{r} = – 1 at T_{r} = 0.7, it behaves as the noble gases and thus complies with the two-parameter corresponding states. If not, we are to compute the difference:

$$Difference=-{\mathrm{log}}_{10}\left({P}_{r}\right){T}_{r=0.7}-1$$ (8.9)

Pitzer called this difference the “acentric factor, $\omega $ ” of the substance. Noble gases, being the reference themselves, have an acentric factor value of zero ($\omega $=0). Substances with an acentric factor of zero are called “simple” substances. The acentric factor is said to be a measure of the non-sphericity (acentricity) of the molecules. Therefore, the three-parameter corresponding state theory of Pitzer reads: “Fluids that have the same value of $\omega $ will behave alike at the same conditions of reduced pressure and temperature.”