3498
G. R. SOMAYAJULU AND B. J. ZWOLINSKI
Isomeric Variation Procedures for Physicochemical Properties of Alkanes
by G. R. Somayajulu and B. J. Zwolinski Department of Chemistry, Team A & M University, College Station, Texas
(Received April 16, 1966)
The basis of the Greenshields-Rossini equation for the calculation of the molal refraction of alkanes has been examined from the standpoint of contributions from bonds, pairs of bonds, and other structural features, and a treatment of molal refraction has been suggested. On the basis of this model, a two-parameter equation was derived for the isomeric variation in molal refraction. The Greenshields-Rossini isomeric variation procedures for properties such as standard enthalpies of formation, enthalpies of vaporization, and molal volumes were accordingly modified.
Introduction Partington’ reviewed several empirical procedures developed by Eisenlohr, Vogel, Denbigh, and others for the calculation of the molal refraction of hydrocarbons and related compounds. These procedures which treat the molal refraction in terms of contributions from atoms, bonds, or groups of atoms are often approximate and do not reveal the true nature of the problem. The procedure recently developed by Greenshields and Rossini* for the calculation of the molal refraction of isomeric alkanes differs from the rest in the basic formulation and is also the most successful one. We have recently treated* the energies of alkanes on a generalized basis and have shown that several of the terms of the generalized treatment follow from Brown’s4LCBO treatment of the alkanes. With this background we have examined the basis of the Greenshields-Rossini equation for molal refraction and derived a two-parameter equation for the isomeric variation in molal refraction. We have extended this equation with an additional parameter for the treatment of isomeric variation in other properties such as enthalpies of formation, enthalpies of vaporization, and molal volumes.
A New Treatment of Molal Refraction
We have found that interactions between pairs of bonds separated by one C-C-C angle are primarily responsible for steric repulsion. The generalized treatment without the steric terms has been found to follow from Brown’s LCBO treatment for alkanes. Recognizing that there has been no general treatment of molal refraction in the literature, we have examined the basis of the most successful Greenshields-Rossini equation for the isomeric alkanes. The steric repulsion treatment of isomeric alkanes was developed for molecular energies of isomeric alkanes in the ideal gaseous state. The application of such a treatment to properties of the liquid state may a t first sight appear to be a bit drastic. Practice and, to a certain degree, detailed statistical mechanical calculations have demonstrated that one can justify the use of intramolecular parameters to correlate certain microscopic bulk properties. Thus, the problem regarding hydrocarbon liquids is particularly simple since as with van der Waals liquids we need to consider only steric and polarizability effects of certain groups such as methyl and ethyl to describe the cohesive properties. Other effects such as hyperconjugation and inductive and dipole effects can be assumed to be negligible. Thus, certain intramolecular parameters important in defining the energies and steric repulsions
We have recently developed8 a generalized treatment for the enthalpies of formation of isomeric alkanes in the vapor phase, in which the energy of an alkane is treated as the sum of contributions from (i) bonds, (ii) pairs of adjacent bonds, (iii) trios of adjacent bonds, (iv) pairs of bonds separated by one C-C bond, and (v) pairs of bonds separated by one C-C-C angle.
(1) J. R. Partington, “An Advanced Treatise on Physical Chemistry,” Vol. I V , Longmans, Green and Co., London, 1953. (2) J. B. Greenshields and F. D. Rossini, J. Phys. Chem., 62, 271 (1958). (3) G. R. Somsyajulu and B. J. Zwolinski, Trans. Faraday Soc., 58. 2327 (1966). (4) R. D. Brown, J. Chem. Soc., 2615 (1953).
The Journal of Physical Chemistry
PHYSICOCHEMICAL PROPERTIES OF ALKANES
3499
of the gaseous molecules will be reflected in the intermolecular force fields in the condensed phases. Our analysis revealed that certain factors which are important for the treatment of energies are not as important for the calculation of molal refraction. Molal refraction may be treated simply as the sum of contributions from (i) bonds, (ii) pairs of bonds, and (iii) pairs of C-C bonds separated by one C-C bond. The number of pairs of C-C bonds separated by all the C r C, bonds in the molecule corresponds to the Platt number.5 The subscripts i and j denote whether the carbon atom is primary, secondary, tertiary, or quaternary. We have found that Pu, the number of pairs of C-C bonds separated by any single C r C , bond, is readily calculated from the following expression
Pfj = (i - l ) ( j - 1 )
(1)
The Platt number, P , in turn is given by the sum of the Puts of all the C&, bonds in the molecule. The Pu’s for the ten Cf-Cj bonds in saturated alkanes have the following values: PI1 = PIz = PI3 = P14 = 0, PZ2= 1 , P23 = 2, P24 = 3, P33 = 4, P34 = 6, Pa = 9 . The general expression for the molal refraction of an alkane in the liquid state can then be written in the following form
R(CnHzn+2) = (272
+ 2 ) a +~ ~(n - 1 ) a ~ c+ $. ZbHc + Zbcc + PR3
(2) where the a’s represent the contributions from the bonds, b’s represent the contributions from the pairs of bonds as in ZbHH
H
/ \
H
/
C
C
\
H
\
C
C
Ra is the contribution for a pair of C-C bonds separated by one C-C bond, and P is the Platt number. In order to simplify eq 2, we introduce the three new constants, R1,R2,and 6, defined as follows.
+ R2 a c ~+ 3bcc bHc + bcc)
RI = ~ C H 3 / 2 b ~ ~ =
6=
In general, for any CnHZn+2,the molal refraction is given by the following expression.
R(CnJ&n+Z) = (2n
+ 2)R1 + (n - 1)Rz + (3n1 + 4n2 + 3.~~316 + PR3
(3)
Greenshields and Rossini have developed the following threeparamet,er isomeric variation equation for molal refraction
AR = 0% 4- @n44- T A P (4) where a, @,and y are constants, AR = R(isomer) R(normal), and AP = P(isomer) - P(norma1). On the basis of our eq 3, the isomeric variation, AR, can be derived as follows.
+ 3n4)(26)+ APR8
AR = -(n3
(5)
Equation 5 contains only two parameters. A comparison of eq 4 and 5 with selected experimental values reveal that the two-parameter equation is as good as the three-parameter equation. The results of multiple regression analysis performed on a computer using the available Lorenz-Lorentz molal refraction data6 for the sodium d i n e at 25” for 53 isomeric alkanes in the CS to C S range are given below (the values are given in ml/mole) . For eq 4
AR = 0.021n3
+ 0.058n4 - 0.130AP
av dev = 0.024; std dev = 0.037
C
/
C
+ 3R2 + 146 + RI R(i-C4Hio) = lOR1 + 3R2 + 126
R(n-C4Hlo) = l0Rl
For eq 5
AR = 0.019(n3
+ 3n4) - 0.129AP
av dev = 0.023; std dev = 0.037 The above analysis justifies the reliability of the twoparameter equation. Also the constants of eq 5 can be given simple physical significance.
Tatevskii’s Equation According to Tatevskii,’ molal refraction of alkanes can be represented in terms of contributions from four types of CH bonds and ten types of CC bonds. On
‘/z(bHH
This permits us to express the molal refraction of alkanes in terms of R I ,R2, R3, and 6 as shown below.
R(CH4)
= 4R1
R(C$Hd = 6R1 R(C3Hs) = 8R2
+ R2 + 66
+ 2Rz + 106
(5) J. R. Platt, J . phy8. C h . ,56, 328 (1952). (6) “Selected Values of Properties of Hydrocarbons and Related Compounds,” American Petroleum Institute Research Project 44, Chemical Thermodynamic Properties Center, Texas A&M University, College Station, Texas (loose-leaf data sheets, extant 1964). (7) V. M. Tatevskii, V. A. Benderskii, and S. S. Yarovoi, ”Rules and Methods for Calculating the Physicochemical Properties of Paraffinic Hydrocarbons,” translation edited by B. P. Mullins, Pergamon Press, New York, N.Y., 1961,p 13.
Volume 70, Number 11
November 1966
3500
G. R. SOMAYAJULU AND B. J. ZWOLINSKI
the basis of our model, it is possible to define the contributions of the four types of CH bonds and ten types of CC bonds as follows.
R1 = molal refraction of the CH bond of methane R,
=
R1
+6
=
molal refraction of the primary CH
bond
R,
=
R1
Rt
=
R1
+
26 = molal refraction of the secondary CH bond
+ 36 = molal refraction of
the tertiary CH
bond The contribution Ri, of any C i C , bond may be defined as
Rij
i, j
=
R2
+ PijR3
= 1, 2, 3, 4,and
i
