Deviations from the principle of corresponding states By D. Cook and J. S. Rowlinson Department of Chemistry, The University of Manchester (Communicated by E. A. ,Guggenheim —Received 7 May 1953) * Certain intermoleeular potential energies which are functions of the relative orientation of the molecules may be reduced to a form similar to that of simple spherical molecules by taking a statistical average over all orientations. Such an average energy is an explicit function of the temperature. Two such intermolecular potentials are used to calculate the difference in the equations of state of assemblies of elliptical molecules (part I), of dipolar molecules (part II), and an assembly of equivalent spherical molecules, which conforms to the principle of corresponding states. These calculations are compared with the observed deviations from this principle of the vapour pressure and rectilinear diameter of thirteen liquids.
406 D. Cook and J. S. Rowlinson PART I. ELLIPTICAL MOLECULES 2. The intermolecular potential The most convenient potential of the form of (1*2) which represents adequately those properties of the inert gases for which theory of experiment may be accurately compared is j£(r) = 4e[(cr/r)12 - (<r/r)6]. (2-1) The assumption of a potential of this form is a greater restriction than that imposed by (1-2) but is necessary if the deviations from the principle of corresponding states are to be calculated without evaluating the partition function of the assembly. The potentials between molecules such as nitrogen and oxygen probably depend upon their separation in a way similar to (2* 1). These potentials both attractive and repulsive will, however, depend also on the orientation of the molecules. Here we make the simplifying assumption, necessary for the treatment used here, that only the attractive potentials vary with the orientation while the repulsive maintain spherical symmetry. If the molecules have axial symmetry such a potential may oe w ntten p- / . . \ 6 ™l l(f,91,» ^ ) = 46 - ( 7) (r+«ff(0iA.0))J> (2-2) where a is a constant for each species, where and dz are the angles between the axes of the molecules and the line joining their centres, and where 0 is the azimuthal angle between the planes containing these axes and the line of centres. The function of these angles must satisfy two conditions: Cn Par f*2n g(dx, d2, (j>)sin 6isn d d 6.<p = 0, (2-3) JoJ 0J0 - 1 < g(0lt 02, + L (2-4) Otherwise it may be of any form which is thought suitable to represent the variation of the total potential with the orientation. For oblate molecules with a centre of symmetry, such as the lower cyclic paraffins, convenient functions are obtained from combinations of Legendre functions of even order, P2n(cos and P2n(cos#2). The associated functions Pf™(cos 6),though they do not automatically satisfy (2-3), can be used to represent prolate molecules such as those discussed below. Functions of P|(cos0) and P|(cos 0)are not very suitable. If the former satisfies (2-3) its maximum and minimum are unequal (+ | and —1). The latter has two maxima and so represents a more complicated shape than necessary. This function of P\{cosd) is more suitable g{6x 6,>2, <f> =) if sin4 + j-£ sin4 02 -1. (2- 5) This satisfies (2-3) and (2-4) and has a maximum and minimum which are nearly equal (+ f at 6X — dz = \ tt, and — 1 at 6X — d% = 0, n).The variation of the attractive potential with the angle (f) is probably small and has been neglected. The ratios of the maximum (1) and minimum (2) values of the collision diameter and energy of (cri/cr ) = [(! + 7a/8)/(l-a)]*, = (2‘®) 2.
Principle of corresponding states 407 Comer (1948) has calculated the second virial coefficient of elliptical molecules using an intermolecular potential which was more complicated, and probably more accurate, than (2-2) and (2-5), but which cannot be used in the treatment below. Unlike (2*5) his potential depends to a small degree on the azimuthal angle.
408 D. Cook and J. S. Rowlinson 3. The second virial coefficient The second virial coefficient of the gas in the equation of state (2T1) may be written ^ B0 = (InNa*)S /?bt0-«2»+« (3-1) 71 = 0 where /?n = —2*(2?i_3) r{|(2?i—1 )}/?&!. (3-2) If cre and re of (2-9) are substituted for <r0 and r0 one obtains Be = B0+ (| r r N o * ) (8% ) 2 *(2» - 1) finiS****>. (3-3) 71=0 All the terms of the infinite series are negative, and so the virial coefficients of elliptical molecules are smaller at all temperatures than those of the equivalent spherical molecule. However if 8° < 1 and r > rc then the second sum is much smaller than the first. It is an observed fact that the second virial coefficient of gases such as oxygen, nitrogen, ethylene and ethane (for all of which 8° < 0-05, see below) do conform to the principle of corresponding states (Guggenheim & McGlashan 1951; Hamann & McManamey 1953). However, the virial coefficients of organic vapours at their boiling points are up to 50 % lower than would be expected for spherical molecules (Rowlinson & Townley 1953). A recent review by Guggenheim (1953) shows that negative deviations from the principle of corresponding states are just observable in propane and quite large in butane and the butenes. The virial coeffi cient of carbon dioxide, recently measured by MacCormack & Schneider (1950), shows similar deviations at low temperatures. Part of these deviations are probably due to the variation of the dispersion forces with orientation but the majority can more reasonably be accounted for by quadrupole-quadrupole forces (Pople, private communication).
Principle of corresponding states 409 phases on to the p, T plane. Therefore, from (4*2) and (4-3) the vapour pressure p* is related to p* at the same temperature by JP? = Pti 1 + 3<J) - 28T(dp*jdT) = p$(l + 38-28T(dlnp*l(4-4) The reduced vapour pressures II (= P*lP°) are related by In ne = In + 3(<J- 8°) - Inp^/dT). (4-5) 7T0 The variation of n0 with T can be adequately represented by ]nn0 = ~c(l-60)l60, (4-6) where c is a positive constant and 6 is the reduced temperature, TITC. Hence ln^e = — c(l — 60)jd0 + 3(8-8 C) - = -c (l-d e)lde-8*[(l-de)(2c-30e)ldl] (4-7) to the first order of small quantities. The first term on the right-hand side of (4*7) is the value of In ne predicted by the principle of corresponding states. The second term, which is denoted Ap, is the deviation from this value. The constant c is (2*346 In 10) for argon (Guggenheim 1945) and so Ap is negative at all temperatures. Figure 1 shows experimental values of Ap for ten substances plotted against the reduced temperature. The vapour pressure of argon is used as the standard of normal behaviour. The sources of the experimental values of the vapour pressures and critical const-ants are given in the appendix.
410 D. Cook and J. S. Rowlinson The tenth substance, ethane, is omitted as the results show a mean density lower than that expected from the principle of corresponding states. This is probably an experimental error.
Principle of corresponding states 411 of molecules. Thus the three lowest values are those of the three diatomic molecules, oxygen, nitrogen and carbon monoxide. In the hydrocarbons, ethane and ethylene are lower than propane and propylene, which, in turn, are lower than the linear molecule acetylene. The ratios of maximum and minimum collision diameters and energies can be calculated from (2-6). They increase from 1-09 for the diameter and 2*84 for the energy in oxygen to 1*30 and 23-1 in propane.
412 D. Cook and J. S. Rowlinson Corner (1948) analyzed the second virial coefficients with a different molecular model and obtained eccentricities which increased in almost the same order as those in table 1, though his ratios of the diameters were higher (1*43 for propane) and his ratios of the energies lower (3* 1 for propane) than those found here. The energy ratio is very sensitive to small changes in the molecular model and Comer’s figures might seem the more reasonable. However, quantal calculations of the various components of the potential of hydrogen (Hirschfelder, Curtiss, Bird & Spotz, private communi cation, to be published) give ratios of 1*5 for the diameter and 16 for the energy ratios which are surprisingly large for a diatomic molecule. This calculated potential for hydrogen, however, gives an average diameter and energy which compare well with those found from the virial coefficient.
Principle of corresponding states 413 The equivalent non-polar molecule used in these equations is that with values of cr and e obtained from (7-3) by putting t = 0. This will be a hypothetical molecule whose repulsive potential is the same as that of the polar molecule but whose attractive potential is equal to the sum of the dispersion potential and the dipole- induced dipole potential of the polar molecule. A pair of equations analogous to (2-11) and (2*12), but using the parameters of (7-5) and (7*6), are now used to calculate the deviations of dipolar molecules from the principle of corresponding states.
414 D. Cook and J. S. Rowlinson The parameter 8C is again determined from the deviations of the vapour pressures and rectilinear diameters from the principle of corresponding states. The equations which express these deviations are In; = -c(l-d d)ldd- 8%(l-6d)(4,c-59(-5) Td and ptlpl = (i-|^ ) + ^c[(l-^)(7 + 9^)/4 (9-6) These equations may be derived in the same way as (4*7) and (5-3).