J. Phys. Chem. 1993, 97, 271 1-2714
2711
Common Bulk Modulus Point for Compressed Liquids Ali Boushehri,+Fu-Ming Tao,S and E. A. Mason' Department of Chemistry, Brown University, Providence, Rhode Island 0291 2 Received: September 9, 1992
Almost all liquids show a number of simple regularities, such as rectilinear diameters, linear p-T isochores, and linear Zen0 contours. Another regularity, recently discovered by Huang and O'Connell, is a common intersection point for the isotherms of the reduced bulk modulus of a compressed liquid as a function of density. A theoretical analysis of this empirical regularity is presented in terms of a statistical-mechanical equation of state. The analysis indicates the limits of validity of the regularity, and shows what molecular characteristics of the liquid are revealed by the location of the common intersection point. The crucial feature is the precise temperature dependence of an equivalent van der Waals covolume, and the location of the point yields direct information on molecular size and shape.
1. Introduction
Although liquids and dense fluids are usually considered to be rather complicated on a molecular level, they nevertheless show a remarkable number of simple regularities. Five of the betterknown ones are the following: (1) Near linearity of a Clausius-Clapeyron plot of In pvap vs 1 / T from the triple point to the critical point, for all kinds of liquids, despite the fact that the assumptions made in obtaining this relation all fail catastrophically as the critical point is approached. I (2) Near linearity of the mean density of a saturated liquid and its equilibrium vapor as a function of temperature. This is the socalled "law of rectilinear diameters".* (3) Near linearity of p vs T a t constant density (isochores), over the entire range from the perfect gas to the compressed liquid. All the early investigators of fluid compressibility were impressed by this behavior.3 (4) Near linearity of the bulk modulus (reciprocal compressibility) of a liquid as a function of pressure. This regularity, first noticed by Tait over 100 years ago, is the basis for a number of successful empirical equations of state for liquids." ( 5 ) Linearity of the Zen0 contour and its correlation with the line of rectilinear diameters. The Zen0 contour is the locus of T vs p points at which the compression factor, Z pv/RT, is The Zen0 line lies largely in the supercritical region, whereas the line of rectilinear diameters of course lies in the subcritical region. The last three of these regularities, although discovered empirically, have recently been given some theoretical basis through an equation of state (EOS)obtained by statisticalmechanical perturbation theory.9-12 Recently, a new regularity has been found for liquids in the behavior of their reduced bulk modulus:
wherep is the pressure, p is the number density, and k T has its usual meaning. Huang and 0 C 0 n n e l l ' ~found that the isotherms of B as a function of molar volume intersected at a common point; at this point B is independent of temperature. They checked the existence of this common bulk modulus point on over 250 liquids, and used it as the basis of a correlation scheme for the volumetric
properties of compressed liquids and liquid mixtures. The location of the common modulus point of course depends on the particular liquid. What is remarkable is that the isotherms do not intersect all over the B-p region, but at just one point. The reduced bulk modulus has special theoretical interest because it is related to the spatial integral of the direct correlation function c(r) byI4 1 - B = pJc(r) dr
as emphasized by Huang and O'C0nne1l.l~ From this relation we may expect that the common bulk modulus point reveals some characteristic feature of the molecular fluid structure. The purpose of this paper is to present a theoretical analysis for the common bulk modulus point in terms of a statisticalmechanical EOS. We first show that, unlike the van der Waals EOS,the statistical-mechanical EOS does describe the common bulk modulus point. However, it also shows that the isotherms do not intersect at exactly one point but rather over a small range of density, and that this behavior is largely restricted to the rather limited temperature range between the triple and critical points. We then use the statistical-mechanical EOS to show what characteristics of the fluid are revealed by the location of the common bulk modulus point in the liquid regime. 2. Theoretical Prediction from the Equation of State
The statistical mechanical EOS has the following f0rm:~-I5 p/pkT = 1
+ B2p + a p [ G ( b p ) - 11
(3)
which contains three temperature-dependent parameters: E2(the second virial coefficient), a (a scaling factor for the softness of the intermolecular repulsive forces), and b (an analogue of the van der Waals covolume). The function G(bp) is an average pair distribution function at contact for equivalent hard convex bodies. The parameters Bz, a,and b are related to the intermolecular potential u(r)by integration; for central potentials the expressions are B2(T) = 2nJOm[1- e-'(r)/kT]r2dr
' Permanent address: Department ofChemistry,ShirazUniversity, Shiraz, Iran. t Present address: Department of Chemistry, Harvard University, Cambridge, MA 02138. 0022-3654/93/2097-27 11$04.00/0
(2)
0 1993 American Chemical Society
(4)
Boushehri et al.
2712 The Journal of Physical Chemistry, Vol. 97, No. 11, 1993
m I
04
06
05
07
05
06
06
07
1 /bp
Figure 1.
09
06
1
1/ P V s
Isotherms of 1 - B vs 1/ b p for a (12,6) fluid, as predicted from
Figure 2.
Same isotherms as in Figure I , plotted as 1 - B vs 1 / P U B .
eq 9.
Here uo(r) is the repulsive part of u(r)
uo(r) =
U(F)
o,
+
t,
r r
< rm > rm
(7)
where t is the depth of the potential well and rm is the position of its minimum. This is the Weeks-Chandler-Andersen16decomposition of u(r). Notice that a is just the contribution of the repulsive forces toB2. For nonspherical potentials theexpressions for B2,a,and 6also involve integrations over orientation angles,’s but we do not need the explicit formulas here. Let us first consider a fluid of particles that interact according to the Lennard. Jones ( ~ 6pair ) potential. For a (1 2,6) fluid we can use the CarnahanStarlingl7 expression for G(bp) in eq 3, as shown by Song and Masonz9
0) = 1-7/2 (8) (1 - TI3 where tl = bp/4 is the packing fraction, F~~~ the definition of B we then obtain 1 - E = 2(a - B2)p - 32ap (16 - b ) (’) (4 - b d 4 A Plot of 1 - B as a function of the reduced volume, 1/ b ~is, shown in Figure 1 for several isotherms. Numerical values of Bz, a,and b for the (12,6) potential have been tabulated by Song and Mason.9 Numerical simulations of the (12,6) fluid give a critical temperature of kTc/c = 1.35 and a triple-point temperature of kTtp/t = 0.68.‘* These isotherms do not intersect, and there seems to be nothing particularly noteworthy about Figure But the parameter is temperature dependent, and if we instead use a constant such as the Boy1e u B and plot - as a function Of '/'Bpi we obtain the results shown in Figure 2. Now the isotherms do intersect at (nearly) the same point, which must require a very special temperature dependence of b. Careful examination of Figures 1 and 2 leads to the following conclusions: (1) The EOS as given by eq 9 predicts a (nearly) common modulus point, consistent with the empirical findings of Huang and O’Connell.13 (2) The common modulus point is largely restricted to the liquid region between TtPand T,. At lower temperatures the
intersection occurs a t smaller volumes, and at higher temperatures it occurs at somewhat larger volumes. Thus the relation cannot be extended to the supercooled liquid, although it may extend a bit into the supercritical region. This was not observed by Huang and OConnell because they could search only the temperature rangebetween Tlpand T,,aslimitedby theavailableexperimental data. (3) The crucial feature of the EOS that leads to a common intersection is the temperature dependence of b. If b is temperature independent, then the isotherms will not intersst, as in Figure 1. This is implicit in the work of Huang and O’Connell, who noted that the van der Waals EOS does not lead to the intersection of the isotherms, and this must be primarily due to the fact that b is taken as a constant in the van der Waals
EOS. (4) The existence of the common modulus point requires not only that the covolume 6 be temperature dependent, but also that it have precisely the correct temperature dependence, at least in the limited range between TlPand Tc. From Figure 2 we see that this is thecase for thestatistical-mechanical EOS. It is interesting that b varies by only about 10% over this temperature range. The foregoing discussion is based on the EOS for the mythical (12,6) fluid. For real fluids a more accurate EOS has been obtained by applying a small correction to eq 3 for attractive forces and by invoking a principle of corresponding states to find G ( b p ) . The result is11 (B,-a)p+
ap
p = 1 + (10) 1 +66p 1 -hbp PkT This is formed from eq 3 by adding a small correction term, Sbp2(a- B2)/( 1 + Sbp), and taking G(bp) = 1 / ( 1 - Xbp). Here is a characteristic constant of the substance, equal to 0,454 for noble-gas fluids and becoming smaller for more complex fluids. The empirical fact that G-1 is linear in bp, rather than as predicted by the CarnahanStarling result of eq 8, has been interpreted as due to many-body forces in a dense fluid.” The constant 6 can be taken as 6 = 0.22h for practical purposes. The is reduced bulk modulus according to eq 1 - B = 2(a - B,)p
(1 + 6 b / 2 ) (1
+
(2 - A b ) - aP(1 - hbp)’
Plots of 1 - B vs I / v B p for two representative real fluids, methane and n-octane, are shown in Figures 3 and 4. Since no
The Journal of Physical Chemistry, Vol. 97, No. 11, 1993 2113
Common Bulk Modulus Point
basis for the physical interpretation of the common bulk modulus point, to which we now turn our attention.
0
3. Physical Interpretation - 20
Let us suppose that the isotherms intersect at a density po and common modulus BO. At this point we have the condition (aB/aT),o = 0
(12)
Applying this condition to eq 11 and neglecting the small 6 correction term, we obtain after a little algebra
(13) lfllll
-100 055
1 , 1 1 , 1 1 , , , 1 , 1 , , 1 1 1 1 , 1 / , , ,
06
065
07
075
08
085
l/P%
Figure 3. Isotherms of 1 - B vs 1 /PUB for methane, as predicted from eg 1 1. The temperature difference between two neighboring curves is 10
K: 0
-
-50
-
a I
which is the equation that determines PO. The corresponding equation for Bo is eq 1 1 with p = po. Equations 13 and 1 1 relate experimental values of po and Bo to a large number of molecular quantities-&, a,b, plus their temperature derivatives and A. This result appears almost uselessly complicated, but consideration of some numerical magnitudes allows substantial simplification. In eq 13, the quantity (dB2/dT) is over 1 order of magnitude larger than either (-db/dT) or (-da/dT) in the liquid temperature range. The only way that the equality can be achieved is thus for the quantity (1 - Xbpo) in the denominators on the right-hand side to be small, and then clearly the first term on the right is dominant. To a first approximation we can therefore write
in which we have set Xbpo 1 in the other factor. We can see by inspection that eq 14 has the correct qualitative behavior. As T decreases, (dB2/dT) increases more rapidly than does ( A b / dT), so that the right-hand sideof (14) increases (and approaches 1). Correspondingly, on the left-hand side b increases as T decreases, and so po remains nearly constant as T varies. This accounts for the existence of the common bulk modulus point. If the temperature dependence of b is not a t least approximately consistent with that of the right-hand side of (14), the common modulus point will no longer exist. Unfortunately, eq 14 still relates PO to too many molecular quantities, but we can circumvent this defect if we assume that the temperature dependence of b as given by the right-hand side can be extrapolated to lower temperatures. As T 0, (dBz/dr) 0 , and so
-100
-150
-200"
1 '
04
"
"
-.
-
I1 "
0 45
"
"
05
"
0 55
l/P%
K.
where b(0) is the molecular covolume a t zero temperature. The value of b(0) gives the 'size" of the molecules in terms of the range of the repulsive forces, r,:9
suitable pair potentials are available for these two substances, we have calculated B2 values from the correlation given by Tson o p o u l o ~ a, ~modified ~ version of the correlation of Pitzer and Curl.2o From B2(T)we can obtain the Boyle temperature T Band the Boyle volume, VB, which are the scale factors used to find a(T) and b(T) according to some effective pair potential model.9,11.15Since a and b are insensitive to the potential model, we have used the (12,6) potential. The characteristic constant A is then obtained from eq 10 for a few high-density points, such as saturated liquid densities. For methane we obtained TB = 503.2 K, vB = 57.3 cm3/mol, and A = 0.409; for n-octane we obtained TB= 1244.5 K,V B 391.9 cm3/mol, and X = 0.290. From Figures 3 and 4 we draw essentially the same conclusions as for the (12,6) fluid. In particular, there is a common modulus within a density spread of about 5%. It is therefore clear that the theoretical EOS should be capable of providing a molecular
b(0) = (27r/3)rm3 (16) The parameter X can be regarded as giving some measure of the molecular shape, since it generally decreases with increasing molecular complexity.lI Thus the density PO of the common modulus point rather directly reflects the size and shape of the molecule, at least to a first approximation. As a numerical example, consider the case of CHI, for which l/p& 0.68 from Figure 3. Using X = 0.409 and V B = 57.3 cm'/mol, obtained as described previously, we calculate from eqs 15 and 16 a value of rm = 4.23 A. This agrees to about 1% with the value of rm = 4.18 A deduced from the second virial coefficient (Table IV of ref 11). We conclude that an experimental value of PO can lead to a fairly reliable value of the quantity X1/3rm. The result from eq 11 for BOis less simple. Consideration of numerical magnitudes does not produce much obvious simpli-
Figure 4. Isotherms of 1 - B vs l/puB for n-octane, as predicted from eq 1 1. The temperature difference between two neighboring curves is 20
-
Boushehri et al.
2714 The Journal of Physical Chemistry, Vol. 97, No. 11, 1993
fication, and the value of 1 -BO still depends on the values of B2, a, b, and A. The two terms on the right-hand side of (1 1) are ofcomparablemagnitude but opposite sign. Moreover, each term is rather temperature dependent, but their difference must be nearly constant because BO is nearly constant in the liquid temperature range. We can exploit this behavior by substituting po from eq 15 into eq 11 for BO,which yields l-Bo=
2 b - 4 ) 11 +o.llblb(O)l 1 a P-blb(O)l (17) ---A b(O) [l + O.22b/b(0)l2 A b(O) [l - b/b(O)]’ Here it is not adequate to set 6 = 0 as we did in obtaining eqs 14 and 15 for po, because we are taking the difference between two quantities of comparable magnitude. Notice that all dependence on a “size” parameter such as U B or rmhas now been explicitly removed; the dimensionless ratios a/b(O),b/b(O),and Bz/b(O) depend only on a reduced temperature, say T / T B ,and the “shape” parameter A. However, the dependence of eq 17 on T / T Bmust essentially cancel out because BOis nearly constant. Thus the right-hand side of eq 17 depends only on the shape parameter A. We conclude that the experimental value of BOis determined by the molecular shape. Thus po and BOare measures of the molecular size and shape. Unfortunately, it does not seem to be possible to calculate reliable values of A from an experimental value of Bo. because eq 17 is too sensitive to the specific numerical values of Bz, a, and b. However, the reverse is not true, and BO can be obtained if A is already known. A somewhat more general interpretation of the physical significance of Bo and po can be obtained from eq 11 by recalling that the average pair distribution function a t contact is given by G(bp) = 1/( 1 - Abp), so that eq 11 can be rearranged into the form Go(Go+ 1)
a-B2
2(
(Bo-1)
7) +apo
(18)
where GO= G(bp0) is a weak function of temperature a t the fixed density po. For simplicity of discussion, we have here dropped the small terms involving 6, but they could easily be included. Since a is the contribution of just the repulsive forces to B2, the quantity (CY-&) must represent themagnitude of thecontribution of just the attractive forces to B2, and the first term on the right represents the ratio of the attractive contribution to B2 to the repulsivecontribution to Bz. Thesecond termon the right, which is comparable in magnitude to the first term, depends on the experimental quantity (BO- l)/po, which, like a,represents the effect of the molecular size and shape. Together the two terms determine the value of the average pair distribution function at contact in the liquid regime a t a densitypo. The first term depends on the interplay between intermolecular attraction and repulsion, and the second term depends on just the repulsion. This is curiously reminiscent of the behavior of C(bp) along the Zen0 contour, where it is approximately equal to ( a - B2)/a.’2 It is interesting that the temperature dependence of B is so weak over the whole range of liquid densities, as has been shown by the extensive survey of Huang and 0Connelll3 and is here illustrated by Figures 2-4. We can now see why this is so. If (aB/aT), is small, then eq 13 must approximately hold a t all densities, not just a t po, and therefore eq 14 for Abpo must also hold over the entire liquid range. From this we can conclude that b/b(O)does not vary much in the liquid region, which is consistent with our earlier observation that b varies by only about 10% over this temperature range. 4. Discussion
The essential conclusion is that the existence of the common bulk modulus point for compressed liquids can be given a
theoretical basis in terms of a statistical-mechanical EOS, valid for both spherical and nonspherical molecules. This theoretical basis shows that the values of the density and the modulus at the crossing point are determined by the molecular size and shape-that is, by the repulsive forces. Nevertheless, the attractive forces must have a suitable relation with the repulsive forces, or else the liquid state would be unstable; this relation appears here through the factor ( a - B z ) / a . These features are gratifyingly consistent with the relation of B to the direct correlation function given in eq 2: numerical calculations show that c(r) is largely determined by the repulsive forces, but also has a distinct shortrange contribution from the attractive forces.I4 In retrospect, it is perhaps not so surprising that PO and BOdo not furnish much information on the attractive forces, other than through a relation to the repulsive forces via the factor (a B z ) / a . The fact that we are dealing with a liquid already tells us the potential well depth roughly within a factor of 2, since the liquid regime only covers a temperature range of such a factor. (Note: Ttp< c/k < T,, and T,/ Ttp= 2.) It is also not so surprising that liquid densities depend on molecular size and shape, and that b is temperaturedependent (although not density dependent). What is surprising is that these dependences have to conspire in just such a way as to produce a common intersection point for the B - p isotherms, and that this seems to be a universal feature of all normal liquids. The values of po and BO,when combined with information on the second virial coefficient ( 8 2 and a ) , give direct information on G(bpo), the average pair distribution function a t contact, at the fixed density po and over the temperature range of the liquid state. Finally, the present results furnish an alternative method for using the Huang-O’Connell correlation scheme for the volumetric properties of compressed liquids, since the location of the common bulk modulus point can be obtained from the EOS, whose parameters can be found without the use of any measurements on compressed liquids. This should be possible because the calculated values of po and BO are not overly sensitive to the values of the EOS parameters. Acknowledgment. We thank Professor John P. O’Connell for his helpful comments, suggestions, and questions. A.B. thanks the Iranian authorities for granting him leave of absence to work on this project. This work was supported in part by NSF Grant NO. C H E 88-19370. References and Notes (1) Scott, R. L. J . Chem. Educ. 1953, 30, 542. (2) Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures, 3rd ed.;Butterworth: London, 1982. (3) Beattie, J. A.; Stockmayer, W. H. Rep. Prog. Phys. 1940, 7 , 195. (4) Hayward, A. T. J. Brit. J . Appl. Phys. 1967, 18, 965. ( 5 ) Macdonald, J. R. Rev. Mod. Phys. 1969, 40, 316. (6) Dymond, J. H.; Malhotra, R. I n f . J . Thermophys. 1988, 9, 941. (7) Ben-Amotz, D.; Herschbach, D. R. Isr. J . Chem. 1990, 30, 59. (8) Xu, J.; Herschbach, D. R. J , Phys. Chem. 1992, 96, 2307. (9) Song, Y.; Mason, E. A. J . Chem. Phys. 1989, 91, 7840. (10) Song, Y.; Caswell, B.; Mason, E. A. Inr. J . Thermophys. 1991, 12, 855. (11) Ihm, G.; Song, Y.; Mason, E. A. J. Chem. Phys. 1991, 94, 3839. (12) Song, Y.; Mason, E. A. J . Phys. Chem. 1992, 96, 6852. (13) Huang, Y.-H.; OConnell, J. P. Nuid Phase Equilib. 1987. 37, 75. (14) McQuarrie, D. A . Srorisfical Mechanics; Harper and Row: New York, 1976. (15) Song, Y.; Mason, E. A. Phys. Rev. A 1990, 42,4749. (16) Weeks, J. D.; Chandler, D.; Andersen, H. C. J . Chem. Phys. 1971, 54. 5237. (17) Carnahan, N . F.; Starling, K. E. J . Chem. Phys. 1969.51, 635. (18) Barker, J. A.; Henderson, D. Rev. Mod. Phys. 1976, 48, 587. (19) Tsonopoulos, C. AIChE J . 1974, 20, 263. (20) Pitzer, K. S.; Curl, R. F., Jr. J . Am. Chsm. Soc. 1957, 79, 2369.
