# 11.3: Zc as a Measure of Goodness of an EOS

Some values for critical compressibility factors are shown below:

### Experimental Zc for Various Substances

CO_{2} = 0.2744

CH_{4} = 0.2862

C_{2}H_{6} = 0.2793

nC_{5} = 0.2693

nC_{6} = 0.2659

The values of critical compressibility factors shown here are relatively close to each other, but, in actuality, they are different. They are, in fact, substance-dependent. This is a striking finding if we recall our highly-praised Principle of Corresponding States. Didn’t we say that at the same reduced conditions, all substances “must” have, at the very least, the same compressibility factor, Z? Here is a case where we have different substances at the same corresponding states (P_{r} = T_{r} = 1, right at the critical point) but different “Z” values. Afterall the Corresponding States Principle is not infallible (as was stated by Pitzer). As we recall, he proposed the introduction of a third parameter (acentric factor) into the corresponding state definition to alleviate these kinds of “problems.”

At the very least, we may say that the values of Z_{c} (compressibility factor at the critical point) of different substances are “close enough” among themselves. That is, they are not “grossly different,” so as to say that the application of the two-parameter corresponding states principle would be outrageous at the critical point. The fact of the matter is, as a consequence of the Corresponding States Principle, all cubic EOS predict a “unique” and “universal” value of Z at the critical point, regardless of the substance. The list below tells us how they perform.

### “Universal” Zc predicted by different EOS

Ideal EOS = 1.000

vdW EOS = 0.375

RK EOS = 0.333

SRK EOS = 0.333

PR EOS = 0.301

From the list above, we would have been expecting some kind of “average” Z_{c} of 0.27 or so. But none of the equations of state we have studied is capable of predicting a value that low. The “best” job is done by PR EOS, which provides the “closest” match to the real values observed for most substances. This illustrates why the PR EOS performs somewhat better near critical conditions.

### Contributors

Prof. Michael Adewumi (The Pennsylvania State University). Some or all of the content of this module was taken from Penn State's College of Earth and Mineral Sciences' OER Initiative.