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10.2: Modern Cubic EOS

At this point, we have a couple of comments on the cubic behavior that the pioneer work of vdW introduced to the field of equations of state. First, we can say that the vdW cubic behavior is qualitatively reasonable; and second, we can say that it is capable of describing the continuity between liquid and vapor. Nevertheless, vdW cubic EOS has been proven not to bequantitatively suitable for most engineering purposes. Certainly, it yields unacceptable errors for the quantitative prediction of densities and any other related thermodynamic property. However, all of the development in the field of phase behavior that has been achieved today is due to the work of van der Waals. Although his own equation is seldom used because of its lack of accuracy, his principles are still the foundations of the current developments. vdW concepts were so far reaching that he won the Nobel Prize for his equation.

The truth is that van der Waals’ accomplishment in 1873 triggered a tremendous effort among scientists to make modifications to his EOS which would remove from it large disagreements with experimental data. This effort has not yet ceased today and is not likely to stop in the near future. Much of this endeavor has focused on how to better model the attractive parameter “a” and the repulsive term “b”, with the hope that we can get better quantitative predictions. Naturally, the qualitative cubic-nature of vdW’s original EOS is always preserved, and hence all subsequent refinements belong to the same family of modified-van-der-Waals equations of state. We refer to vdW EOS and all its descendents as cubic equations of state, because, as we have said, they take a cubic form when expressed in terms of volume or compressibility factor and are explicit in pressure.

It is fair to claim that modern cubic EOS started to make a difference when a temperature dependency was introduced to the attractive parameter “a”. Interestingly enough, van der Waals was convinced that the parameters “a” (and even “b”) of his equation of state were not necessarily constants and suggested that, indeed, some dependency on temperature could be found. A very interesting discussion on this, from van der Waals himself, is found in the lecture speech that he offered during his acceptance of the Nobel Prize in Physics, in 1910, for his work on the continuity of vapor and liquid. This speech and the biography of this great physicist, Johannes Diderik van der Waals (1837-1923), can be found in the web resources of the Nobel Prize organization.

The most popular cubic EOS, which time has proven to be most reliable, are:

  • Redlich-Kwong EOS,
  • Soave-Redlich-Kwong EOS (very popular among chemical engineers),
  • Peng-Robinson EOS (very popular among petroleum and natural gas engineers).

Keep in mind that, once you have an EOS, you can derive virtually any property of the fluid.


  • 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.