# What are the limitations of the Theorem of Corresponding states?

I recently came across the theorem of corresponding states, which states that all gases exhibit the same compressibility factor, under same reduced pressure and temperature (and also the critical compressibility factor). Thus:

$$Z=f(P_r,T_r,Zc)$$

where

$$Z=\frac{PV}{RT}$$

$$P_r=\frac{P}{P_c}$$ $$T_r=\frac{T}{T_c}$$

My question is, what are the limitations of this theorem? Can I consider it as an absolute truth, or is it just like any other gas law, being valid only within certain ranges and for certain acccuracies?

• Not "how valid" but ask "what are the limitations" from the start. – Mithoron Jun 26 '19 at 15:08
• The van-der-Waals and Redlich-Kwong equations (and perhaps others) can both be recast in terms of reduced quantities therefore, in so far as an actual gas is described by these equations, the the law of corresponding states should apply. Certainly gases/vapours as diverse water, CO2, N2 and isopentane, among many others, follow this law very well. Your question is thus about the applicability of, for example, the vdW equation. – porphyrin Jun 26 '19 at 21:53
• Corresponding states very nearly causes the curves describing the state behavior of all gases to collapse to a single set of curves. In 1955, Pitzer added the ascentric factor as an additional parameter, which improves the correlation even more. The ascentric factor of each individual gas accounts for differences in polarity of molecules. In practice, corresponding states does a superbly accurate job. – Chet Miller Jun 27 '19 at 0:00
• @ChetMiller thanks for this info, so does this mean that I can write $Z$ as $$Z=f(P_r,T_r,Z_c,\omega)$$ where $\omega$ is the acentric factor? – Pritt says Reinstate Monica Jun 27 '19 at 5:00
• The usual form is $$Z=Z^{0}(P_r,T_r)+\omega Z^{1}(P_r,T_r)$$See Smith and van Ness, Introduction to Chemical Engineering Thermodynamics. They provide the plots of $Z^0$ and $Z^{1}$ – Chet Miller Jun 27 '19 at 12:06