Collision Diameters, Interaction Potentials, and Virial Coefficients of

9104

J. Phys. Chem. 1996, 100, 9104-9115

Collision Diameters, Interaction Potentials, and Virial Coefficients of Small Quasi-Spherical Molecules Fernando del Rıó,* J. Eloy Ramos, and Alejandro Gil-Villegas† Departamento de Fı´sica, UniVersidad Auto´ noma Metropolitana, Iztapalapa, Apdo 55 534, Me´ xico DF, 09340 Me´ xico

Ian A. McLure Department of Chemistry, The UniVersity of Sheffield, Sheffield S3 7HF, U.K. ReceiVed: NoVember 7, 1995; In Final Form: February 5, 1996X

A recently introduced mapping approach to the equation of state of classical fluids is used to study the dilute gas phase. The approach introduces mean collision diameters σ(T) and R(T), which reflect the contributions to the pressure from the repulsive and attractive forces, respectively. The mean diameters are analyzed for a variety of molecules in the gas phase. The temperature dependence of σ(T) and R(T) is shown to be essentially dominated by two shape factors, SR and SA, characterizing the form of the repulsive and attractive parts of a the interaction, respectively. The method gives insight into the collision diameters and second virial coefficients, B(T), of model molecules, both spherical and nonspherical, including three-parameter potential functions and diatomic Lennard-Jones molecules. As a practical bonus, the theory provides a compact and highly accurate model for B(T) of model and real molecules. The theory also provides a route to reliable information on the effective intermolecular potential from a knowledge of B(T). The theory is applied to gaseous neon, argon, krypton, dinitrogen, dioxygen, difluorine, methane, and tetrafluoromethane, and their mean collision diameters and potential parameters are determined and analyzed.

1. Introduction Gas properties have been historically a leading source of information into the way molecules interact.1 In particular, the analysis of the second virial coefficients of gases, B(T), has been used to obtain information about the pair intermolecular potential. Reliable information about this potential is of fundamental importance in the study of real fluid properties, especially of fluid mixtures.2 Here we present a novel approach to this problem based on a recent theory of the equation of state (EOS) of fluids. The approach is then applied to selected pure gases of spherical and small molecules which play an important role in the field of simple fluid mixtures.2 In this theory, collision properties are introduced to obtain the EOS by means of a mapping procedure into a square-well (SW) system used as reference.3 The collision properties introduced are the mean diameters σ(F,T) and R(F,T) defined in terms of collision frequencies for the repulsive and attractive parts of the intermolecular potential.4,5 The diameter σ(F,T) is the effective size of the repulsive core of the molecule, and R(F,T) is the mean range of the attractive forces. The pressure of the fluid is then determined by the SW EOS and three quantities characterizing the fluid of interest: an effective energy, , and the diameters σ(F,T) and R(F,T). In section 2 we recall the definition of these collision diameters and obtain the low-density limits, σ(T) and R(T), appropriate for the dilute gas phase. The reader should take into account the fact that, due to the density dependence of the collision diameters, they vary on going from the dilute gas to the liquid.3 Hence, the results presented in this article are applicable only in the gaseous state and should not be extrapolated directly to dense fluids. Section 3 is used to † Present address: Department of Chemistry, The University of Sheffield, Sheffield, U.K. X Abstract published in AdVance ACS Abstracts, April 15, 1996.

S0022-3654(95)03295-3 CCC: $12.00

analyze three potential families commonly used as effective potential functions for spherical and nonspherical molecules: the Lennard-Jones (LJ) n-6, the Kihara, and the exp-6 potentials. From the analysis of the mean diameters for these potentials we introduce two constant quantities, SR and S0R, which characterize the softness of the molecular core, and another pair of constants, SA and SA0, which characterize the effective broadness of the shell where attractive forces act. It is further shown that σ(T) and R(T) can be expressed in terms of these constants over wide intervals of temperature with very high precision. It is further shown that our model provides very accurate virial coefficients B(T) not only for the spherical potentials considered but also for diatomic LJ molecules. Lastly, the ease and reliability of the use of the model to invert B(T) to obtain effective intermolecular potentials are demonstrated. Section 4 contains the results of the application of the theory developed in section 2 to real gases of spherical and small quasispherical molecules. The main conclusions of the work are summarized in section 5. 2. Mean Collision Diameters in Gases We derive first the collision properties for dilute gases. The general theory is fully developed in refs 3-5. Consider a fluid whose molecules interact through the pair potential u(r,Ω), where r is the distance between the centers of mass and Ω represents the relative orientation of the two molecules. At fixed Ω, the potential is assumed to have the general form shown in Figure 1. It has a well-defined minimum at rmΩ of depth u(rmΩ,Ω) ) -m. The point rmΩ separates the potential into repulsive (0 < r e rmΩ) and attractive (rmΩ < r e ∞) regions. We also assume that u(r,Ω) f ∞ when r f 0 and u(r,Ω) f 0 when r f ∞. The mean repulsive collision diameter σ(F,T) is given in terms of the background correlation function y(r,Ω) ) g(r,Ω) exp© 1996 American Chemical Society

Collision Diameters of Small Quasi-Spherical Molecules

J. Phys. Chem., Vol. 100, No. 21, 1996 9105 (7) gives B(T) for the fluid with intermolecular potential u(r,Ω):

B(T) ) -2π∫dΩ∫0 drr2(e-βu(r,Ω) - 1) ∞

Figure 1. Typical form of the pair intermolecular potential at fixed orientation. The repulsive and attractive collision diameters s(EK) and l(EK) depend on the kinetic energy EK.

[βu(r,Ω)], where g(r,Ω) is the radial distribution function,

(8)

Thus far the SW depth has not been chosen. Equation 7 corresponds to the virial coefficient of a spherical effective potential ueff(r) of depth and minimum at rm. In the limit of zero temperature, a pair of particles will be separated by the distance rm, and both collision diameters must tend to the same value: σ(Tf0) ) rm and R(Tf0) ) rm. By imposing this condition on (5) and (6)ssee Appendix A for detailssone finds that rm and the effective SW depth are given by 3 rm3 ) ∫dΩrmΩ

(9)

by3

3 eβΩ rm3 eβ ) ∫dΩrmΩ

(1)

We will use and rm as units of energy and distance, respectively. The volume of a sphere of diameter rm is bm ) π rm3 /6, so that the reduced mean volume of a molecule is b* ) b/bm with b ) πσ3/6 and also b* ) σ*3, where σ* ) σ/rm is the reduced mean diameter. The mean volume Λ covered by the attractive well is Λ ) πR3/6, and in reduced units Λ* ) Λ/bm ) R*3, where R* ) R/rm is the reduced mean range of the attractive forces. Final expressions for the reduced mean volumes are

(10)

-βu(r,Ω)

rmΩ ∂e σ3ysw(σ)eβ ) ∫dΩ∫0 drr3y(r,Ω)

∂r

where the superscript SW refers to a square-well system of diameter σ and well depth -. The angles Ω describing the orientation between the two molecules are normalized so that ∫dΩ ) 1. The mean attractive collision diameter, or collision range, R(F,T) is defined by ∞ ∂e-βu(r,Ω) (2) R3ysw(R)(eβ - 1) ) -∫dΩ∫r drr3y(r,Ω) mΩ ∂r

These equations guarantee that a SW fluid with parameters , σ, and R has exactly the same pressure as the fluid of interest.3 The collision diameter σ(F,T) contains all the information about the repulsive forces in the system, whereas the collision range R(F,T) incorporates the effect of the attractive forces. In the limit of low density F f 0, y(r,Ω) f 1 so that the mean diameters in (1) and (2) become functions of T only and simplify into

σ3eβ ) ∫dΩ∫0 drr3 rmΩ

∂e-βu(r,Ω) ∂r

(3)

and ∞ ∂e-βu(r,Ω) R3(eβ - 1) ) -∫dΩ∫r drr3 mΩ ∂r

(4)

Integrating by parts (3) and (4), one obtains the alternative forms 3 eβΩ - 3∫0 drr2e-βu(r,Ω)] σ3eβ ) ∫dΩ[rmΩ rmΩ

(5)

3 (eβΩ - 1) + R3(eβ - 1) ) ∫dΩ[rmΩ

3∫r drr3(e-βu(r,Ω) - 1)] (6) ∞

mΩ

The second virial coefficient B(T) takes the SW form

B(T) )

2π 3 [σ (T)eβ - R3(T)(eβ - 1)] 3

(7)

and one can readily verify that substitution of (5) and (6) into

b*(T*) ) 1 - 3∫dΩ∫0 dzz2e-β*[u*(z,Ω)1] zmΩ

Λ*(T*) ) 1 +

(11)

∞ 3 dΩ∫z dzz3[e-β*u*(z,Ω) - 1] (12) ∫ mΩ e -1 β*

where z ) r/rm, T* ) kT/ ) 1/β*, and u* ) u/. The second virial coefficient can be finally written in reduced form as

B*(T*) ) B(T)/rm3 )

2π [b*(T*)eβ - Λ*(T*)(eβ - 1)] (13) 3

In short, the properties of a given gas are determined by four quantities: the scale factors and rm (or bm) and the functions b*(T) and Λ*(T). Any two fluids with conformal potentials, i.e., with the same reduced potential u*(z,Ω), will satisfy the principle of corresponding states. In that case, from (11) and (12), b*(T) and Λ*(T) will be the same for both systems. For this reason, the volumes b*(T) and Λ*(T) may be called “shape functions”. Deviations from the law of corresponding states are manifest as differences in the temperature behavior of b* and Λ*. In the next section we will study the dependence of the shape functions on the main features of u(r,Ω). 3. Potential Functions and Virial Coefficients A. Spherical Molecules. To investigate the properties of the shape functions we first consider spherical potentials which depend on three parameters: u(r;,rm,ξ), so that variation in the dimensionless third parameter ξ changes the form of u(r). The potentials chosen are very commonly used to model spherical and quasi-spherical molecules. The first is the Lennard-Jones n-6 potential:6 n 6 uLJ n (r) ) n[(d/r) - (d/r) ]

with n ) [(6/n)6/n-6 - (6/n)n/n-6]-1, minimum at rm )

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Figure 2. Comparison of potential functions of different form, u(z), with z ) r/rm. The Kihara potential with a* ) 0.111 (solid line) has a softer core and broader attractive well than the Kihara with a* ) 0.666 (dashed line).

Figure 3. Reduced mean repulsive volume b*(T*) for various Kihara potentials with different core softness. Shown from the bottom up are the cases with a* ) 0, 0.2, 0.4, 0.6, 1.0, 3.0, 10, and 20. A hard sphere corresponds to the horizontal line b*(T*) ) 1.

d(n/6)1/n-6, and form parameter ξ ) n. The second is the spherical Kihara potential of hard-core diameter a:7,8

[(dr -- aa) - (dr -- aa) ]

uKa (r) ) 4

12

6

with minimum at rm ) a + 21/6(d - a) and form parameter ξ ) a* ) a/(d - a). Finally, we consider the exp-6 potential

uγexp(r) )

{

( )}

6 r exp[γ(1 - r/rm)] 1 - 6/γ γ rm

6

which has form parameter ξ ) γ. Each of these expressions really defines a class of potentials of different form depending upon the variation of ξ. These potentials all have depth and uLJ(r)d) ) uK(r)d) ) 0. The parameter ξ was chosen so that, as shown in Figure 2, for small ξ the potentials are “soft”, with rather shallow wells, but for large ξ the repulsive part becomes very steep and the well narrows. The attractive part is also steeper for large ξ: for uK(r) it becomes infinitely steep when ξ ) a* f ∞, but in this limit uLJ(r) and uexp(r) become equal to the Sutherland potential with a pure 1/r6 attraction. The functions b*(T) and Λ*(T) have been calculated numerically for different values of ξ, and the T dependence was found to be very similar for the three potential families. As illustration, Figure 3 shows the mean repulsive volume b*(T*) for several Kihara potentials with 0 e a* e 20. The behavior of b* with T is determined by the steepness of the repulsive part of u(r), i.e., by the softness of the molecular core. Molecules with a hard core have constant mean volume b* ) 1, but if their core is soft, any two particles will approach closer to each other at higher kinetic energies and hence the mean collision volume b* will decrease with increasing T. This effect is more pronounced the softer the core, as seen in Figure 3 for the smaller values of a*. When T* ) 0, the collision diameter σ ) rm and hence b*(T* ) 0) ) 1. This behavior is entirely similar in the LJ and exp-6 cases and will also be so for any other potential u(r) with a monotonic and smooth repulsive part. The behavior of the attractive mean volume Λ*(T*) is depicted in Figure 4 for various Kihara potentials with 0 e a* e 20. On average, attractive forces act on a shell of radius equal to the collision range R(T), and Λ*(T) is the volume of such an attractive shell. The behavior of Λ*(T*) is determined by the steepness of the attractive part of the potential or, in

Figure 4. Reduced mean attractive volume Λ*(T*) for various Kihara potentials with attractive shells of different broadness. Shown are the same cases as in Figure 3 from a* ) 0 (top) to a* ) 20 (bottom).

other words, by how broad the attractive shell is. For molecules with an infinitely steep attractive potential, such as that in the SW model, the mean volume of the attractive shell is constant: Λ* ) 1. But for potentials with a finite attractive steepness, as the temperature is raised, the collision range R(T) and Λ*(T*) will increase, as shown in Figure 4. As the attractive shell broadens, the temperature dependence of Λ*(T*) is stronger. For a given potential, Λ* increases monotonically from its lowtemperature limit, Λ* ) 1, to a well-defined high-T limit, Λ*∞, which is reached asymptotically. The value of Λ*∞ is larger the broader the attractive part of u(r). Again, the behavior for other potential classes is qualitatively similar to that shown in Figure 4. It is convenient at this point to introduce a standard to measure the steepnes of each part of the potential, i.e., of the softness of the repulsive core and of the broadness of the attractive shell. This choice is for the most part arbitrary. In this work we will take a very shallow potential, the uexp(r) with γ ) 12, as the standard. With this choice, most realistic potentials will be steeper in their repulsive and attractive parts; that is, compared with the standard, most molecules will have harder repulsive cores and narrower attractive shells. The shape functions of the standard potential, denoted by b* 0 and Λ*0, were calculated from the quadratures in (11) and (12)


J. Phys. Chem., Vol. 100, No. 21, 1996 9107

TABLE 1: Coefficients for the Standard Volumes b* 0(T*) and Λ* 0(T*) in (14) and (15) m cm lm

0

1

2

3

4

5

6

0.714 118 2.255 46

-0.089 682 3 0.395 651

-0.006 879 6 -0.140 63

0.000 913 448 0.005 031 77

0.015 989 8

-0.005 732 65

0.000 620 915

for 0.2 e T* e 10.0. This temperature interval goes from below the triple point to well above the Boyle temperature. The standard shape functions are parametrized in a form similar to that of LJ virial coefficients9 as polynomials in ln T*:

a

3

b* 0(T*) )

∑ cm(ln T*)m m)0

(14)

6

Λ*0(T*) )

lm(ln T*)m ∑ m)0

(15)

The constants cm and lm are given in Table 1. We will next analyze how the shape functions for uLJ(r), uK(r), and uexp(r) are expressed in terms of the standards (14) and (15). B. Shape Functions b*(T*) and Λ*(T*). As shown in Figure 3 for the Kihara case, for a given value of ξ (and ξ ) n, a*, or γ according to the potential family) b*(T*,ξ) decreases monotonically with T*, just as the standard b* 0(T*) does. One can thus expect the ratio b*(T,ξ)/b*0(T*) to be a slowly varying function of T for constant ξ. But since b*(T*,ξ) cannot be obtained from (11) in closed analytic form for any of these potentials, it is quite difficult to establish analytically its relationship with b*0(T*). Nevertheless, a remarkable simple relation is exhibited in Figure 5a, which shows plots of b*(T*,a*) with b* 0(T*) in the Kihara case for different values of a*. The plots are surprisingly linear for all values 0 e a* e 20 and over the temperature interval considered. Furthermore, this simple behavior is not restricted to the Kihara case; equally linear plots are obtained for uγexp(r), with 12 e γ e 100, and for uLJ n (r) with 9 e n e 200. Some of these are shown in Figure 5b. On the basis of this simple behavior of b*(T*,ξ), we write as an approximation

b*(T*,ξ) ) SR0(ξ) + SR(ξ) b* 0(T*)

(16)

where SR(ξ) is the slope and S0R(ξ) the intercept of the straight line obtained in plotting b*(T*,ξ) with b* 0(T*). We must stress two main features of approximation (16). The first is its simplicity and accuracy: From Figure 5a,b, one sees that the linear behavior in (16) is very accuratesthe small error introduced by (16) will be discussed belowsand that it is valid from rather low, T* ≈ 0.2, to very high, T* ≈ 10, temperatures. The second feature is that b*(T*,ξ) is determined by only two constants: SR(ξ) and S0R(ξ). Moreover, there is one further regularity, apparent in Figure 6, which shows S0R(ξ) plotted against SR(ξ). Such a plot should make explicit the differences in the repulsive parts of the potential families, but as is clearly seen in Figure 6, the relation S0R ) S0R(SR) is almost the same for the three cases and follows closely

SR0 = 1 - SR

(17)

Therefore, the constant SR characterizes the repulsive part of all potentials consideredsand presumably of others similar to themsand hence b*(T*) is fully parametrized by SR. Two

b

Figure 5. Mean repulsive volume b*(T*,ξ) in terms of the standard b* 0(T*). The points correspond to numerical values of b*(T*,ξ), and the straight lines are best fits to the points. (a) Kihara potentials with a* ) 0 (bottom), 0.2, 0.4, 0.6, 0.8, 1.0, 3.0, 10.0, and 20 (top). (b) Lennard-Jones potentials (0) with n ) 9 (bottom), 12, 24, and 200 (top); exp-6 potentials (4) with γ ) 14 (second from bottom) and γ ) 100 (second from top).

different potentials with the same value of SR will have identical shape functions b*(T*). The parameter SR measures effectively the softness of the molecular core and determines the contribution of the repulsive forces to the thermodynamic properties. The core softness SR ) 1 is that of the standard uexp(r,γ)12) and means a molecule with quite a soft core (SR ) 1 in (17) gives S0R ) 0, in agreement with (16)). A hard-core potential has SR ) 0 (and S0R ) 1), which from (16) gives a temperatureindependent mean size. The cores of realistic molecules will be harder than the standard but softer than a real hard-sphere; that is, in most cases 0 < SR < 1. As can be seen in Figure 5b, the effective volumes b*(T*) of the Lennard-Jones n-6 and the exp-6 potentials follow the linear behavior of approximation (16) within the temperature range considered. This means that up to T* ≈ 10 there is little difference in the mean collision diameters of an inverse power potential and an exp-6 with the same SR. The better adequacy of the exp-6 potential over an inverse power potential, demon-

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Figure 6. Behavior of SR and SR0, the parameters measuring the softness of the molecular core. The point at SR ) 0 and SR0 ) 1 corresponds to a hard-core potential, such as the Sutherland potential. The point SR ) 1 and SR0 ) 0 corresponds to the exp-6 potential with γ ) 12, here used as standard. The points show the Kihara (+), Lennard-Jones (b), and exp-6 (4) cases. The line corresponds to (17) in the text.

strated by high-energy molecular beam experiments, should manifest at T* . 10. The effects of the repulsive forces in the three potential families are strongly similar. This similarity, patent in Figure 6, signifies that they can model the effect of those forces without significant differences between them. This is an important fact because these potentials have been widely used to model molecular interactions. Nevertheless, as we will show now, they show major differences in modeling attractive forces. The analysis of the attractive contributions parallels closely that of the repulsive ones. Figure 7a shows the plots of Λ*(T*,ξ) against Λ* 0(T*) for the LJ case (ξ ) n) and Figure 7b for a few Kihara and exp-6 cases. Again, for the three families and ξ ) constant, the plots are very linear in the interval considered. In similarity with (16) we write the linear approximation

Λ*(T*,ξ) ) SA0(ξ) + SA(ξ) Λ* 0(T*) SA0

(18)

are the slopes and intercepts of the straight where SA and lines in Figure 7a,b. The constants SA and SA0 characterize the effect of the attractive forces for any potential for which (18) is a good approximation. SA measures the broadness of the attractive shell. Any two potentials with the same values of SA and SA0 will have the same temperature dependence of the attractive pressure. A plot of SA0 with SA shows the differences between the three potential families. From this plot, depicted in Figure 8, one sees that the Kihara potential has a quite different behavior from the LJ and exp-6 families. The important fact is that the ξ f ∞ limit of the latter is the Sutherland potential, which has a finite broadness SA = 0.66, whereas the Kihara potential tends to a sharp attractive shell: SA ) 0. In contrast to the general equivalence of their repulsive parts, the attractive parts of three potential families are approximately equivalent only for rather broad attractive shells, SA ≈ 1, where they have the same values of SA0 for a given value of SA. The Kihara potential is the only one of the three functions examined here to possess very sharp attractive shells, i.e., correspondingly steep attractive potentials.

Figure 7. Mean attractive volume Λ*(T*,ξ) in terms of the standard shape function Λ* 0(T*). The points correspond to numerical values of Λ*(T*,ξ), and the straight lines are best fits to the points. (a) LennardJones potentials with n ) 9 (top), 12, 15, 18, 24, 50, and 200 (bottom). (b) Kihara potentials (0) with a* ) 0 (top), 0.4, 1.0, 3.0, 10.0, and 20 (bottom); also shown are the exp-6 cases (4) with γ ) 14 and 100 (second and third from top).

Also from Figure 8 one can see that the approximation equivalent to (18)

SA0 = 1 - SA

(19)

is not very good here, except as a rough estimate in the Kihara case, or for molecules with rather broad attractive shells, when the three potential families are again equivalent. C. Virial Coefficients. Equations 16 and 18 give b*(T*) and Λ*(T*) for the spherical potentials here considered. Substitution of (16) and (18) in (13) gives a explicit expression for B(T) in terms of the scale factors and rm and the shape parameters SR, S0R, SA, and SA0. The capacity of this theory to reproduce the values of B*(T*) can be tested from the deviations: ∆B*(T*) ) B*(T*) - B*theo(T*), between the values calculated directly from (7), B*(T*), and those obtained from (13), (16), and (18). The values of the shape parameters in (16) and (18) were calculated with the equations in Appendix B for each potential as functions of the appropriate ξ. The deviations ∆B*(T*) were calculated and analyzed for a selection of potentials of different forms from the three families (Kihara with a* ) 0.2, 1.0, and 3.0; Lennard-Jones with n ) 9, 12, and 100, and exp-6 with γ ) 12, 18, and 200). All deviations are of the same order of magnitude and follow similar patterns in


J. Phys. Chem., Vol. 100, No. 21, 1996 9109 TABLE 2: Results of Inversion of B(T) for Model Potentialsa model

ξin

out rm

out

ξout

Q

Kihara, ξ ) a*

0.0 0.5 1.0 1.5 3.0 9 12 24 40 100 12 18 30 60 100

1.002 0.999 0.999 0.999 0.999 1.001 1.002 1.001 1.002 1.005 1.000 0.999 1.000 1.003 1.005

0.995 1.007 1.012 1.017 1.025 0.994 0.995 0.994 0.988 0.976 1.001 1.000 0.996 0.984 0.976

-0.002 0.506 1.022 1.544 3.088 8.89 11.95 23.9 38.5 87.2 11.9 18.0 29.7 56.0 88.4

0.001 0.002 0.002 0.003 0.002 0.003 0.001 0.000 0.001 0.002 0.001 0.001 0.000 0.001 0.002

LJ n-6, ξ ) n

exp-6, ξ ) γ

Figure 8. Behavior of SA and SA0, the parameters measuring the broadness of the molecular attractive shell. The lines correspond to (16) in the text: Kihara (solid line), Lennard-Jones (dashed line), and exp-6 (dotted line). In contrast with Figure 6, the Kihara attractive parameters have very different behavior from the Lennard-Jones and exp-6 cases. The point at SA ) 0 and SA0 ) 1, reached by the Kihara potential, corresponds to a potential with an infinitely steep attractive part. The LJ and exp-6 potentials have as sharpest limit the Sutherland potential. The point SA ) 1 and SA0 ) 0 corresponds to the exp-6 potential with γ ) 12, here used as standard.

a The shape functions b* and Λ*, in (16) and (18), were fitted to 39 values of B*(T*) in 0.5 e T* e 10.0. B*(T*) was calculated directly in with rm ) 1, in ) 1, and ξin as given below. The results of the out inversion are rm , out, and ξout. Q is the root-mean-square deviation.

and (18) can be considered as excellent in the interval 0.2 e T* e 10.0. A further and more demanding test was made by inverting B*(T*) in order to obtain the potential u(r). A least-squares procedure was used to find the best values of , rm, and ξ. These values were found by minimizing the root-mean-square (rms) deviation Q: in 3 3 Q2(,rm,ξ) ) -∑[(rm ) B*(kTi/in,ξin) - rm B*(kTi/,ξ)]2 i

Figure 9. Deviation plot in the second virial coefficients for testing approximations (14)-(16) and (18). The results are for Lennard-Jones potentials with n as labeled. Deviations for the Kihara and exp-6 potentials are similar in behavior and magnitude.

the temperature interval 0.2 e T* e 10. Figure 9 shows the typical case of the LJ n-6 potential. At lower T*, 0.2 e T* e 0.3, the deviations are larger and of both signs, where |∆B*(T*)| ≈ 1. Nevertheless, at these low temperatures B*(T*) is very large in absolute value and the error in B*theo(T*), introduced by approximations (16) and (18), is below 0.1%. At higher temperatures, where B*(T*) is small in absolute value, |∆B*(T*)| , 0.04, which corresponds to about 1 cm3/mol for argon. In comparison, measurements with precision better than 2% or 1 cm3/mol, whichever is greater, are considered of the best quality (class I) according to critical evaluation reviews of the field.10 Hence, except for some theoretical purposes that would require higher precision, the approximation in (16)

in Here in, rin m, and ξ are the values used in the calculation of B*(T*). In all cases the original potential was recovered with good accuracy. Typical results of this inversion are shown in Table 2. The maximum errors were found for potentials with steeper cores. The energy is recovered with errors smaller than 3%, the maximum error in the diameter rm was 0.5%, and in the shape parameter ξ was 3% for a* e 3 (Kihara), 5% for n e 40 (LJ), and 6% for γ e 60 (exp-6). For the steeper potentials the error in the shape parameter increased up to 13%. Further, the small mean deviations (Q e 0.003) in B* for all cases tested guarantee that the overall form of B*(T*) is faithfully recovered. From this test one can conclude that approximations (16) and (18) can be used confidently not only to reproduce B(T) for a potential with known parameters but also to extract information about the intermolecular potential when B(T) is known. In these inversions, as in Figure 9, the trial potential family used to determine the best parameters was assumed to contain the test potential used to calculate B*(T*). If one uses a potential form that does not include the particular test potential, the mean deviations are 1 order of magnitude greater than in Table 2 and the deviations ∆B*(T*) show systematic errors. D. Diatomic Lennard-Jones Molecules. Some of the potentials discussed above have been often used as effective potential functions to model nonspherical molecules. Nevertheless, the well-known insensitivity of the virial coefficient to changes in the intermolecular interaction makes it difficult to assess whether these potential models are appropriate. Most of this insensitivity arises from the partial cancelation of the effects of repulsive and attractive forces, which for the dilute gas appears as the difference of the two mean volumes in the SW form for B(T), eq 13. Therefore one expects that the mean collision diameters, and their associated mean volumes, are more

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TABLE 3: Effective Potential Parameters for the 2CLJ Diatomics L*

/40

rm/rm0

a*

SR

SA

Q

0.0 0.1 0.2 0.3 0.4 0.5 0.6

1.0000 0.9521 0.8645 0.7794 0.7086 0.6523 0.6067

1.0000 1.0145 1.0437 1.0785 1.1143 1.1490 1.1823

0.0000 0.0095 0.0390 0.0809 0.1279 0.1764 0.2230

0.8714 0.8691 0.8521 0.8290 0.8048 0.7813 0.7600

0.9900 0.9761 0.9357 0.8829 0.8298 0.7806 0.7379

0.002 0.002 0.002 0.002 0.001 0.001 0.003

sensitive to the shape of the potential function. Here we will show that the theory developed above is useful in the analysis of the virial coefficient of model diatomic molecules. We turn now to the two-center Lennard-Jones (2CLJ) potential, which is often used to model the interaction of two homonuclear diatomic molecules;11 this potential is written 2

2

u2CLJ(r) ) 0∑ ∑ [(d/rlm)12 - (d/rlm)6]

(20)

l)1 m)1

where l ) 1, 2 denotes the two interaction sites in one of the molecules and m ) 1, 2 the sites in the other molecule. Any pair of sites on different molecules interact with a spherical LJ 12-6 potential of depth 0 at the distance rm0 ) 21/6d between the sites. The two sites on a molecule are at a distance L ) L*d apart, which is commonly taken as the internuclear distance. The second virial coefficient of the interaction in (20) has been calculated by several authors,1,12,13 and approximate semiempirical expressions given for it.14 Here we have used (13), (16), and (18) with the data tabulated in ref 1 to obtain the shape parameter ξ, the effective depth , and the effective distance rm. To do so, one of the threeparameter effective potentials analyzed in the last section was assumed and , rm, and ξ were fitted to represent the tabulated values of B(T) by a least-squares procedure. The inversion test discussed above leads us to expect that, if the effective spherical interaction corresponding to the 2CLJ interaction is well modeled by the potential chosen, the values obtained for the parameters will be trustworthy. For 2CLJ molecules with elongations 0 e L* e 0.6, the spherical Kihara potential gave an excellent reproduction of B(T). The rms deviation in the temperature interval 0.4 e T* e 20.0 varies from Q ) 0.001 for L* ) 0.5 to Q ) 0.003 for L* ) 0.6. In comparison, Boublik’s model has Q between 0.122 and 0.064 in a shorter interval 0.5 e T* e 5.0.14 One can conclude that (16) and (18) plus the shape functions of the spherical Kihara potential give a simple and very accurate model for B(T) of 2CLJ molecules of moderate elongation. The values of * ) /40, r*m ) rm/rm0, and a* for the 2CLJ diatomic are shown in Table 3, together with the corresponding SR, SA, and rms deviation Q. As the elongation increases, the effective energy * diminishes from the L* ) 0 value of * ) 1 (equal to 4 times the site-site energy depth 0). The volume scale factor b*m ) (rm/rm0)3 also increases with L* and hence with the excluded volume of the diatomic. This behavior is shown in Figure 10. Increasing the elongation also produces a hardening of the effective molecular core (i.e. SR decreases) and a corresponding narrowing of the attractive shell (SA also decreases), as can be seen from Figure 11 and Table 3. Figure 11 also shows that SA decreases with L* faster than SR, and SA < SR for L* > 0.5. The rather fast narrowing of the attractive shell with L* can be seen as the reason for the success of the Kihara model, for, as can be seen in Figure 12, the LJ and exp-6 potentials always have SA > SR and are therefore unable to model accurately this effect.

Figure 10. Effective energy * ) /40 (bottom) and size b*m ) (rm/ rm0)3 (top) for the 2CLJ diatomics as a function of elongation L* ) L/d, where L is the distance between the atomic centers and d is the LJ parameter. The lines are fits to guide the eye.

Figure 11. Shape parameters SR (solid line, 0) and SA (dashed line, [) for the 2CLJ diatomics as a function of elongation L*. The lines are fits to guide the eye. SR(L*) and SA(L*) decrease from their LJ 12-6 values SR ) 0.8714 and SA ) 0.99.

4. Results for Spherical and Small Quasi-Spherical Molecules We have applied the theory developed in the first part of this paper to a selection of real gases. The gases selected are (a) the heavier noble gases, with spherical interactions and negligible quantum corrections to B(T); (b) diatomics, for which the 2CLJ model can be considered realistic; and (c) several gases with small and nearly spherical molecules for which departures from sphericity can be hopefully incorporated by the effective spherical potentials. For any given gas we have used reported experimental values Bexp(T) and, by the direct inversion procedure already described, determined the best values of the parameters , rm, and ξ for each potential family here considered. A. Spherical Molecules. We consider first the results obtained for the noble gases Ne, Ar, Kr, and Xe, which are


J. Phys. Chem., Vol. 100, No. 21, 1996 9111

Figure 13. Deviation plot for argon (+) and krypton (×). The dotted line shows a precision of 2% or (1 cm3/mol, whichever is greatest.

TABLE 5: Effective Corresponding States Parameters for the Noble Gases Figure 12. Plot of SA against SR for the potential models. The lines are fits to guide the eye. Lennard-Jones (dashed line, O), exp-6 (dotted line, 4), and Kihara (solid line, +). The LJ and exp-6 cases have the Sutherland limit SR ) 0, SA = 0.66.

Kihara

TABLE 4: Effective Potential Parameters for the Noble Gases

exp-6

substance [no. points] model /k (K) rm (Å) He [132] Ne [46] Ar [162] Kr [53] Xe [62]

Kihara exp-6 LJ/n-6 Kihara exp-6 LJ/n-6 Kihara exp-6 LJ/n-6 Kihara exp-6 LJ/n-6 Kihara exp-6 LJ/n-6

4.067 4.777 4.264 38.542 42.498 38.116 150.44 153.52 154.91 220.74 226.12 227.20 286.63 292.82 294.81

2.995 2.888 2.955 3.071 3.025 3.072 3.658 3.637 3.605 3.864 3.832 3.807 4.348 4.319 4.287

ξ

SR

SA

Q (cm3/mol)

-0.049 15.05 11.38 0.0325 16.17 12.72 0.1397 18.35 17.25 0.1634 19.45 18.31 0.1493 18.78 17.68

0.9052 0.8827 0.8897 0.8558 0.8482 0.8508 0.7989 0.7894 0.7413 0.7874 0.7632 0.7198 0.7942 0.7789 0.7324

1.066 0.8931 1.020 0.9443 0.8680 0.9607 0.8173 0.8309 0.8485 0.7932 0.8165 0.8327 0.8074 0.8250 0.8418

3.56 3.78 3.58 0.57 0.57 0.57 0.95 0.95 0.94 1.26 1.25 1.24 2.06 2.06 2.06

shown in Table 4. This table also includes the rms deviation of the final fit from the set of Bexp(T) and the number of experimental points used. The source of Bexp(T) was the comprehensive tabulation by Dymond and Smith.10 Of the results there reported for a particular gas we chose those ranked by the authors of class I or better. Class I data are those with estimated precision better than 1 cm3/mol or 2%, whichever is greater. We discuss first the goodness of the fit. For the three heavier gases there is a respectable collection of experimental results. The three potential families here considered give essentially a mean deviation of 1 or 2 cm3/mol, so that all three potentials reproduce B(T) of these gases within experimental error. This is clearly seen in the plots of the deviations ∆B(T) ) Bexp(T) Btheo(T) in Figure 13. The shape parameter ξ ) n, a*, or γ, of any potential family is very nearly the same for the three gases, which therefore also have very close values of SR and SA. This means that they are close to corresponding states behavior, which is no surprise at all. Taking as a common measure the average shape factors, 3

ξh ) ∑wiξi i)1

model

LJ n-6

shape parameters

substance

/k (K)

rm (Å)

Q (cm3/mol)

a* ) 0.145 SR ) 0.7963 SA ) 0.8118 γ ) 18.60 SR ) 0.7834 SA ) 0.8275 n ) 17.49 SR ) 0.7363 SA ) 0.8447

Ar Kr Xe Ar Kr Xe Ar Kr Xe

151.63 215.80 284.44 154.95 220.35 290.48 156.25 221.97 292.59

3.650 3.890 4.360 3.625 3.866 4.332 3.595 3.836 4.299

0.95 1.40 2.06 0.95 1.38 2.06 0.94 1.36 2.06

where i ) (1, 2, 3) ) (Ar, Kr, Xe) and the weight wi ) nxi/Qi, with nxi ) the number of experimental points for each gas. With these values of ξh, shown in Table 5, we calculated the virial coefficients and their deviations from experimental data. As shown in Table 5, assuming the same “average shape” for the three gases leads to theoretical values of B(T) well within experimental error. The lighter gases do not conform to the same principle of corresponding states. The results for neon, also in Table 4, show that its properties as a dilute gas are well taken into account by this theory, giving B(T) with a precision of 0.5 cm3/mol. Neon is seen to have a core softer than the heavier noble gases, and closer to that of the LJ 12-6 potential. The results for helium are also shown in order to mark the limits of the theory. In particular, the depth of the well obtained here is too small, /k ≈ 4 K, compared with the true depth for this gas: /k ≈ 10 K.15 An analysis of the deviation plot for this gas shows the inadequacy of this (classical) theory to account for B(T) at the lower temperatures. The three potential families seem equally able to reproduce the experimental data on the virial coefficients of the noble gases. This fact can be traced back to the relatively high softness of the cores of these molecules (although they are noticeably harder than the LJ 12-6 potential, which has SR ) 0.87 and SA ) 0.99), which puts them in the top-right corner of Figure 12, where all these families have very similar values of SA and SR. One should compare these results with those obtained by other methods. The pair potentials of Ar and Kr have been studied by many authors. The simplest potentials, like the LJ 12-6, give only fair agreement with experiment, with mean deviations of 2.4 cm3/mol for Ar and 6 cm3/mol for Kr. One of the potentials considered best for Ar is that of Barker, Fisher, and Watts.16 Considering the main features of the potential, the

9112 J. Phys. Chem., Vol. 100, No. 21, 1996

del Rıó et al.

TABLE 6: Intermolecular Distances at T ) 0 substance

V (cm3/mol)

r0/Åa

rm/Å Kihara

rm/Å exp-6

rm/Å LJ/n-6

Ne Ar Kr Xe

13.06 22.3 26.8 35.5

3.13 3.74 3.98 4.37

3.071 3.650 3.890 4.360

3.025 3.625 3.866 4.332

3.072 3.595 3.836 4.299

a T. Kihara, Intermolecular Forces, John Wiley: New York, 1978; p 2, table 1.1.

TABLE 7: Effective Potential Parameters for the Polyatomic Gases substance [no. points] model /k (K) rm (Å) N2 [96] F2 [45] O2 [47] CH4 [115] CF4 [51]

Kihara exp-6 LJ/n-6 Kihara exp-6 LJ/n-6 Kihara exp-6 LJ/n-6 Kihara exp-6 LJ/n-6 Kihara exp-6 LJ/n-6

139.26 149.57 150.37 148.65 154.31 154.83 172.53 182.29 182.98 205.51 213.69 214.56 270.83 320.45 321.10

3.859 3.749 3.729 3.599 3.546 3.527 3.593 3.514 3.497 3.985 3.923 3.901 4.765 4.464 4.457

ξ

SR

SA

Q (cm3/mol)

0.2474 25.69 24.77 0.2085 22.06 20.98 0.2365 24.33 23.33 0.2031 21.80 20.73 0.4309 59.71 58.43

0.7494 0.6450 0.6119 0.7665 0.7082 0.6707 0.7541 0.6672 0.6330 0.7690 0.7133 0.6750 0.6783 0.3593 0.3484

0.7173 0.7638 0.7720 0.7507 0.7897 0.8019 0.7264 0.7724 0.7819 0.7556 0.7920 0.8043 0.5902 0.6887 0.6911

1.13 1.12 1.12 0.20 0.21 0.22 0.21 0.30 0.21 1.48 1.47 1.46 1.71 1.73 1.73

energy and the position of the minimum, the BFW values are /k ) 142.10 K and rm ) 3.7612 Å. These values are very close to those of the Aziz potential:17 /k ) 143.22 K and rm ) 3.759 Å. Compared with the values of this work in Table 5, one finds that the Kihara parameters ) 151.63 K and rm ) 3.650 Å are the closest to the BFW and Aziz values. One possible origin of the discrepancy is that none of the potential families here considered are flexible enough to reproduce the true pair potential of the noble gases. Nevertheless, the deviation of the BFW model from the same set of experimental results is Q ) 1.94 cm3/mol, which is within experimental error, but is twice as big as the deviation for our model: Q ) 0.95. Further, the deviations of the BFW model from the same set of experimental points are very similar to those in Figure 15, but are seen to have a slight systematic tendency. So, another possible source of the discrepancy would be a small systematic error in the measured values of B(T). Since both the BFW and the Aziz models rely on other sources of information about the pair potential besides the second virial coefficients, this extra information could help cancel a bias in the B(T) data. Further evidence that at least part of the discrepancy may be due to the scatter of the experimental data on virial coefficients can be adduced from the results of the very sophisticated method of inversion of B(T) data developed by E. B. Smith and co-workers18 and applied to obtain the pair potential for argon. This method involves only virial coefficient data and gives /k ) 150 ( 5 K and d ) 3.31 Å; these values compare well with /k ) 151.63 K and d ) 3.30 Å from our approach. Hence the results of the theory described here may be considered consistent with the best potential derived exclusively from virial coefficient data. Finally, the values of rm obtained in this work for the noble gases agree with the values obtained from the solid state at T ) 015 and shown in Table 6. B. Diatomic Molecules. The results for three diatomic molecules (N2, O2, and F2) are presented in Table 7. Again, in all three cases the mean deviation of theory from experiment is very small and even more so for O2 and F2. The latter point seems to be caused by the correspondingly small internal scatter of the experimental values of B(T) selected for these two gases.

Figure 14. Deviation plot for nitrogen (+) and oxygen (×). The dotted line as in Figure 13.

The deviation plots for the virial coefficients of these gases (see Figure 14) show very good agreement of the theory with experiment. On the basis of the analysis of the 2CLJ diatomic in the previous section (see Figure 11), we would predict that molecules such as N2 or O2 should have harder cores and narrower attractive shells. The values of SR and SA for the diatomics in Table 6 are indeed lower than those for the spherical molecules. This difference points to a small but clear departure from corresponding states behavior. We should point out that the values of SR and SA for diatomic and linear molecules result from at least two sources. In the context of a site-site picture, the first ingredient is the softness SR0 of the cores of the site-site interaction. The second is the elongation L* of the molecule. In this respect the 2CLJ model is restricted to the second effect, because it does not take into account variations in SR0, which are taken as those of the LJ 12-6 interaction. In the case of N2, assuming that SR0 ) 0.80 (which is the value for argon), the molecular value of SR ) 0.75 and Figure 11 allow one to predict an elongation L* ) 0.36 and a site-site diameter d0 ) 3.3 Å, which agree well with values obtained by other methods (see ref 14). Nevertheless, to give full account of the core softness of O2 and F2, one is required to assume that their site-site interactions are less steep than the argon-argon interaction. This effect is probably due to the fact that SR0 depends on the electron distribution around each site, but its detailed investigation falls outside the scope of the present work. C. Tetrahedral Molecules. The procedure has also been applied to a pair of tetrahedral molecules, CH4 and CF4, and the results are also shown in Table 7 and the corresponding deviations in Figure 15. The agreement with the experimental virial coefficients is also very good here. These two molecules, together with the diatomics and noble gases already discussed, constitute a standard set of molecules for the study of simple fluid mixtures,2 which is one of the main reasons for their inclusion in this work. In its deviation from sphericity, a tetrahedral molecule can be expected to have properties similar to those of diatomics of moderate elongation. In the study by Hamann and Lambert of tetrahedral molecules with LJ 12-6 interactions,19 it was shown that their effective spherical potential corresponded to a LJ 28-7 potential. This potential is steeper than the LJ 12-7 both in its repulsive and in its attractive parts. Indeed, from Table 6 one sees that CH4 has core softness SR ) 0.77, smaller than that of argon (SR ) 0.80) and similar to the diatomics (SR ) 0.75-0.77). The core softness of CH4


J. Phys. Chem., Vol. 100, No. 21, 1996 9113 obtained For CH4 (LJ, Kihara, and exp-6) are depicted in order to show their great similarity.

5. Conclusions

Figure 15. Deviation plot for methane (+) and perfluoromethane (×). The dotted line as in Figure 13.

Figure 16. Effective interaction potentials for methane and perfluoromethane. The potentials at the left are the effective Lennard-Jones (solid line), Kihara (long dashes), and exp-6 (short dashes) for methane. At the right is the Kihara effective potential for perfluoromethane.

corresponds to a LJ exponent n ) 20.73, which is very close to the value n ) 21 considered as optimum for this gas by Ahlert et al.20 The change from SR ) 0.80 (argon) to SR ) 0.77 (CH4) is equivalent to that from the LJ 12-6 to the LJ 28-7 potential of Hamann and Lambert and, according to the theory developed here, is due to the nonsphericity of the CH4 molecule. As for the size and energy of CH4, we find a well depth /k ) 205.51 K, smaller than /k ) 232.2 K obtained by Tee et al.21 and used by Dantzler-Siebert and Knobler,22 and a size rm ) 3.985 Å, slightly larger than the result rm ) 3.840 Å.21,22 By contrast, CF4 is seen to have a core much harder than CH4. We find a deeper well, /k ) 270.83 K, than that obtained by Dantzler-Siebert and Knobler, /k ) 229.0 K,22 but close to that of Douslin et al: /k ) 291.1 K.23 The size rm ) 4.765 Å is close both to that of Douslin et al., rm ) 4.66 Å, and to that found by Hamman and Lambert by electron diffraction, rm ) 4.7 Å,19 but is smaller than that of Dantzler-Siebert and Knobler: rm ) 5.087. Figure 16 shows a comparison of the potentials found for these two tetrahedral gases. The three effective potentials

The use of mean collision diameters provides a novel route to study effective molecular interactions. The main conceptual gain relies perhaps in the clear separation between the effects of repulsive and attractive forces. The repulsive forces are characterized by a core diameter rm and a shape function b*(T*) incorporating the effects of their form and shape. Attractive forces are similarly characterized by an effective well depth and shape function Λ*(T*). We have shown that b*(T*) is essentially determined by the softness of the molecular core, SR, and Λ*(T*) by the broadness of the attractive shell, SA. The three potential families studied here show almost identical behavior in SR but with significant variation in SA. The introduction of the shape parameters provides a simple and accurate expression for the virial coefficients of spherical, quasispherical, and 2CLJ molecules which can be used to obtain information about effective intermolecular potentials. Application of the theory to selected real gases gives insight into the form of their effective interactions, and the potential shapes and parameters obtained agree well with previous determinations from the literature. The approach here proposed should be of greater value for gases for which limited experimental information does not allow the use of the more sophisticated inversion methods developed by E. B. Smith and co-workers.18 As an extra point of practical advantage, the theory was also shown to provide simple and accurate expressions for the virial coefficients of both model potentials and spherical and small real molecules. Extension of the theory to more complex models and its application to gases of linear molecules and mixtures is currently under way. Acknowledgment. F.d.R. and I.A.M. wish to acknowledge support from the European Commission through Contract No. CI1*CT94-0132 and from Conacyt (Me´xico) through Grant No. 0611 E9110. J.E.R. and A.G.V. also acknowledge support of the Sistema Nacional de Investigadores (Me´xico), and F.d.R. wishes to thank the University of Sheffield for its hospitality. Appendix A. Low-Temperature Behavior of the Collision Diameters Here we obtain the low-temperature limit of the mean repulsive and attractive diameters σ(T) and R(T). The approach follows that used for the entire virial coefficient,9 and hence we only point out the main steps. Perhaps the most significant difference is that while B(T) f ∞ at low temperatures, both σ(T) and R(T) should remain finite in the same limit. We first consider σ(T) as given by (5) in the main text. The integral over r on the right-hand side of (5) is

IR(T,Ω) ) ∫0 drr2e-βu(r,Ω) rmΩ

(A1)

where Ω is kept fixed. At low temperatures, β f ∞ and the main contribution to IR arises from points close to r ) rmΩ, where u(r,Ω) has its minimum -Ω. Expanding u(r,Ω) around rmΩ, one finds

βu(r,Ω) ) -βΩ + νΩ(x - 1)2 + O((x - 1)3)

(A2)

9114 J. Phys. Chem., Vol. 100, No. 21, 1996

del Rıó et al.

TABLE 8: Coefficients of (B1), (B2), and (B3) in Appendix B model Kihara

Lennard-Jones

exp-6

m

ζmR

ζmA

χmR

ξmA

0 1 2 3 4 5 0 1 2 3 4 5 6 0 1 2 3 4 5

0.874 773 -0.598 816 0.145 547 -0.012 045 3

0.989 811 -1.458 44 1.130 87 -0.529 479 0.135 487 -0.014 161 8 20.8904 -27.966 16.5827 -5.3246 0.968 007 -0.093 973 1 0.003 793 46 15.3816 -17.5388 8.609 87 -2.151 96 0.271 761 -0.013 805 2

0.998 863 -0.954 108 -0.058 231 6

1.002 62 -0.789 345 -0.375 382 0.141 006

0.605 552 1.014 70 -0.585 544 0.104 027 -0.006 201 97 2.762 77 -0.901 336 0.076 685 8

where βΩ ) βΩ, x ) r/rmΩ, and

νΩ )

-12.0486 65.0581 -134.814 138.046 -70.7163 14.4541

0.982 211 -0.945 102 -0.048 814 3

-6.756 47 -17.5388 8.609 87 -2.151 96 0.271 761 -0.013 805 2

0.978 988 -0.937 001 -0.040 499

way to find that

[

]

2 β ∂ u(x,Ω) 2 ∂x2

3 R3(T)eβ ) ∫dΩrmΩ eβΩ[1 + 3(R/2 + R2/π + R3/4π) + ...]

x)1

(A8)

One sees that VΩ f ∞ when T f 0. From (A2), and neglecting terms of third order and higher, (A1) becomes 3 IR(T,Ω) ) rmΩ eβΩ∫0 dxx2e-νΩ(x-1) 1

2

and again, since R f 0, 3 βΩ R3(Tf0)eβ ) ∫dΩrΩ e

(A3)

Further neglecting the contributions to the integrand except for x = 1, the lower limit may be extended to -∞. Then (A3) can be integrated to get

and so with (A6) we get R3(Tf0) ) r3m, as it should for the effective potential to have its minimum at rm. Therefore (A8) becomes

R3(T) ) rm3 + 3

3 IR(T,Ω) ) rmΩ eβΩ(R/2 - R2/π + R3/4π)

∫

where R(Ω) ) xπ/νΩ. Using this result in (5), the mean collision diameter is 3 [1 - 3(R/2 - R2/π + R3/4π) + ...] σ3(T)eβ ) ∫dΩeβΩrmΩ

rm3

∫dΩeβ rΩ3 (R/2 + R2/π + Ω

3 dΩeβΩrΩ

R3/4π + ...) (A9) Equations A7 and A8 simplify by introducing weighted angle averages for R and its powers:

(A4)

j /2 - R2/π + R3/4π + ...) (σ/rm)3(T) ) 1 - 3(R

(A10)

Now, when T f 0, R f 0 and the limiting value will be given by

(R/rm)3(T) ) 1 + 3(R j /2 + R2/π + R3/4π + ...)

(A11)

∫

-β

σ (Tf0) ) e 3

3 dΩeβΩrmΩ

(A5)

We now impose the condition that the effective spherical potential has a well-defined minimum at rm, and this distance will be equal to the collision diameter at T ) 0. Hence σ3(Tf0) ) rm3 and is finite. From (A5) this is obtained by setting 3 rm3 eβ ) ∫dΩrmΩ eβΩ

(A6)

With this choice of and rm (A4) is written as 3 -3 σ3(T) ) rm

∫

rm3

∫dΩeβΩrΩ3 (R/2 - R2/π +

3 dΩeβΩrΩ

R3/4π + ...) (A7) The low-temperature behavior of the attractive collision diameter R(T) in (6) of the main text can be obtained in a similar

From these equations and (8) in the text one recovers the low-T form of B(T). Appendix B. Shape Parameters for Kihara, Lennard-Jones, and exp-6 Potentials The parameters SR(ξ) and S0R(ξ), in (16) and SA(ξ) and in (18) determine the T dependence of the mean collision diameters and of the second virial coefficient. Polynomial fits were found in terms of appropriate variables. These fits are adequate for 0 e a* e 20 (Kihara), 9 e n e 200 (LJ), and 12 e γ e 100 (exp-6). Then SR(ξ) and SA(ξ) can be expressed by

SA0(ξ)

MR

SR(ξ) )

∑ ζmR[ln f(ξ)]m, SA(ξ) )

m)0

MA

ζmA[ln f(ξ)]m ∑ m)0

(B1)

where f(ξ) ) n for LJ, f(ξ) ) γ for exp-6, and f(ξ) ) 1 + a* for the Kihara cases. The coefficients vary with the potential and are given in Table 8. The associated parameters S′R and S′A are better expressed in terms of the corresponding SR and


J. Phys. Chem., Vol. 100, No. 21, 1996 9115

SA as MR

SR0(SR)

)

∑

MA

χmR SRm,

SA0(SA)

m)0

)

∑ χmASAm

(B2)

m)0

with coefficients also given in Table 8. References and Notes (1) Maitland, G. C.; Rigby, M.; Smith, E. B.; Wakeham, W. A. Intermolecular Forces; Oxford: New York, 1981. (2) Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures, 3rd ed.; Butterworths: London, 1980. (3) Gil-Villegas, A.; Vega, C.; del Rıó, F. Phys. ReV. E 1996, 53, 2326. (4) Del Rıó, F. Mol. Phys. 1992, 76, 21. Del Rıó, F.; Gil-Villegas, A. Mol. Phys. 1992, 77, 223. (5) Gil-Villegas, A. Ph.D. Thesis, UAM Me´xico, 1994. (6) Jones, J. E. Proc. R. Soc. 1924, A106, 463. (7) Kihara, T. ReV. Mod. Phys. 1953, 25, 831. (8) Kihara, T. AdV. Chem. Phys. 1963, 5, 147. (9) Mason, E. A.; Spurling, J. H. The Virial Equation of State; Pergamon Press: Oxford, 1969. (10) Dymond, J. H.; Smith, E. B. Virial Coefficients of Pure Gases and Mixtures; Clarendon Press: Oxford, 1980.

(11) Sweet, J. R.; Steele, W. A. J. Chem. Phys. 1967, 47, 3022; 1967, 47, 3029. (12) Wojcik, M.; Gubbins, M. E.; Powells, J. G. Mol. Phys. 1982, 45, 1209. (13) Kohler, F.; Quirke, N. AdV. Chem. Ser. 1983, 204, 209. (14) Boublı´k, T. Mol. Phys. 1983, 49, 675. (15) Kihara, T. Intermolecular Forces; John Wiley: New York, 1978. (16) Barker, J. A. In Rare Gas Solids; Klein, M. L., Venables, J. A., Eds.; Academic Press: New York, 1975. (17) Aziz, R. A.; Chen, H. H. J. Chem. Phys. 1977, 67, 5719. (18) Smith, E. B.; Tindell, A. R.; Wells, B. H.; Crawford, F. W. Mol. Phys. 1981, 42, 937. (19) Hamann, S. D.; Lambert, J. A. Austr. J. Chem. 1954, 7, 1. (20) Ahlert, R. C.; Biguria, G.; Gaston, J. W., Jr. J. Phys. Chem. 1970, 74, 1639. (21) Tee, L. S.; Gotoh, S.; Stewart, W. E. Ind. Eng. Chem. Fundam. 1966, 5, 356. (22) Dantzler-Siebert, E. M.; Knobler, C. M. J. Phys. Chem. 1971, 75, 3863. (23) Douslin, D. R.; Harrison, R. H.; Moore, R. T.; McCullough, J. P. J. Chem. Phys. 1961, 35, 1357.

JP953295P

Collision Diameters, Interaction Potentials, and Virial Coefficients of

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