The Clausius Equation of State and a Two-Parameter Modification J. G. Eberhart University of Colorado, Colorado Springs, CO 80933 Not long aRer van der Waals presented his historic, twoparameter equation of state to the scientific world ( I ) , Clausius suggested a three-parameter equation of state ( 2 ) of the form
In a previous paper (3) the equations for calculating b, c, and a from the critical constants u,, T,, andp, were derived for eq 1. They are
In eqs % Z, I =p,uJRTc , is the experimental value of the critical compressibility factor. By contrast, two-parameter equations of state have a ' k i l t in" or theoretical critical compressibility factor. For the van der Waals equation, for example, this is 2,' = 318 = 0.375 (3, 4). A close examination of eq 2 reveals a rather unusual feature of the Clausius equation of state. The constant b represents the minimum molar volume for the fluid represented by eq 1. As a consequence, b should be a positive constant. However, eq 2 indicates that b is only positive if 1- 1/42,is positive or ifZ, > 114 = 0.25. For most fluids the range of values of Z, is 0.25-0.30 (4-6), but for a few fluids (like water, methanol, ammonia, acetone, etc.) Z, is less than 114. In such cases the calculated value of b is actually negative, which is, of course, a physical impossibility. Such a bizarre result suggests that the calculated value of b for all fluids may be much less than the experimental value. Indeed for the simple fluid argon, where 2, = 0.291, ea 2 vields . b = 0.141 v,. or blu, = 0.141. This result is indeed much lower than the expenmental value ofvllu, = 0.281 fur areon (7, and also the van dcr Waals predict~onof h L,, = 1 3 = 0.333 (8). The sign of the constant c can similarly be determined from eq 3, which indicates that c > 0 if Z, < 318, while c < 0 if Z, > 318. There appear to be no fluids with a value of Z, beyond 318 (5, 61,so it can safely be assumed that c is positive. Note that ifZ, has the Berthelot (3)value of 318, then c = 0. This is to be expected because eq 1also reduces to Berthelot's equation of state when c = 0.
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There is a modification (and simplification) of the Clausius equation of state that avoids the excessively small (and occasionally negative) values of b described above. This modification involves the replacement of c in eq 1with b. The result is the two-parameter equation of state
I t is easily shown that the two parameters in eq 5 can be obtained from the critical constants via the relationships
while the critical compressibility factor has a theoretical value of
This simplified version of the Clausius equation provides a fixed value of blu, of 115 = 0.200, which is superior to the blu, value for eq 1of 0.141 for argon. In addition, this modification of the Clausius equation avoids the negative values of b that are possible with the original Clausius equation. In summary, then, the original Clausius equation has the advantage that its three parameters can be adjusted so that the critical compressibility factor agrees with experiment. However, it has the disadvantage that it predicts liquid-state molar volumes that are much too small. In contrast, the simplified Clausius equation provides much better liquid-state molar volumes without a great sacrifice of accuracy in the critical compressibility factor. Literature Cited 1. VM der Waah, J. 0. Over de Conlinuiteit van den Gas-en Vlceisfot2oest and, Thesis. University of Leiden, 1873. 2. Clausius, R.Ann. Physik 1880,169,337357. 3. Eberhalt. J. G.J. Chem.Ed. 1989.66.990493. 4. Eberhalt, J. G. J. Chem Ed. 1989,66,906-909. 5 . Mathews,J. F Chem.Reu. 1972,72,71-100. 6. Kudchadker,A.P.;Ahi, G. H.;Zwol?nski,B.J. Chrm Rev. 1988,68,65%735. 7. McGlashan. M.L. Chemieul Thermalvnomics: AcademiePrers: London, 1979;p205. density of liquid argon st absblute zero was calculated hom eqs 13.3.10and 133.11and used to estimate the minimum molar volume ofthe fluid.
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Volume 69 Number 2 February 1992
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