Nomenclature Molecular core radius. cm Molecular core radius. dimensionless Second virial coefficient, cclgm mole Second virial coefficient, dimensionless lntegrand, eqns. (9) and (10). dimensionless Boltrmann's constant, ergslK Avogadro's number, g mole-' Pressure, atm Critical pressure, atm Ideal gas constant, cc-atmlg mole1K Distance between centers of molecules, cm Temperature. OK Critical temperature. 'K TemDerature. dimensionless Spec Ic uoldms cc g mole Inlermolec.8 ar paor polen! R nncrg), ergs oepm of tne polenltal energ) mlnmum, ergs Dimen~i~nle core ~ s separation Distance between molecular centers for = 0, cm Distance between molecular centers for I' = 0. A Pitzer accentric factor
TB= 70.76 XVflP= 0.535 XLIQ= 0.299 RETURNP
r
Figure 2. Fit of boiling temperature versus mole fraction for cyclohexanepropanol using van Laar's equation.
changed. However, the simple routines used to store the data on disk and retrieve the data from the disk are unchanged and the simple Turtlegraphics of Apple Pascal can he used. A sample display of a binary system, and a van Laar fit of this data is given in Figwe 2. The results of a two-stage distillation with an initial 0.1 mole fraction of cyclohexane is included. Thew vanor nressure orograms are availahle on asinele disk for an ~ ~1lplus ~ witii PASCAL. e There is a $8.00 charge to cover the exnense of duolicatine the disks and mailing. Send the requestsLtoDr. am& GT~W;N.J.I.T.,Chemistry ~ivision, 323 High Street, Newark, Nd 07102.
The Kihara model is probably the best for fitting thermodynamic data, and this has been done for a large number of nonpolar molecules. Furthermore, the parameters of this potential function can he readily estimated given temperature, critical temperature, critical pressure, and Pitzer's accentric factor. T o facilitate the integration of eqn. (2), the following din~ensionlessvariables are defined hT T* = -
Second Virial Coefficients of Nonpolar Gases Using the Kihara Potential
r
- 2a
2: = 21!""'
Clayton P. Kerr
Tennessee Technoiogical University Cookeville. TN 38505
(5)
- 2a)
2a =n' - 2a
The virial equation of state with only the second virial coefficient retained can be written as
An important advantage of the virial equation of state is that there is a theoretical relationship between the virial coefficients and the intermolecular ootential enerrv .. function of the molecules. Mixtures of gases can also he handled in a rigorous fashion. For the second virial coefficient of nonoolar molecules this relationship is
(4)
,q* =
(6)
R
(7)
2/3rN,d3
Next, eqn. (2) and eqn. (3) are combined, and the dimensionless variables are substituted into this result. This result is shown as eqn. (8). =
(A)" (&) (- +($ i)) -
L-(lX (2%
*2
+ 2%
*[
+ 21!zF2)dt
(8)
Next, the integral is broken into five parts: from E = 0 to 0.7, from 0.7 to 0.89, from 0.89 to 1.0, from 1.0 to 2.5, and from 2.5 to 4.5. At ( = 0.89.1' is zero, and a t = 1,r is a minimum. For values of T* > 0.25, which covers most applications, the exponential term in the first integral is negligible and can be dropped. Furthermore, the upper limit on the last integral can he changed to 4.5 because, for greater values of E, the integrand is essentially zero. With these simplifications, the value of the first integral can be determined analytically and eqn. (8) becomes
