9248
J. Phys. Chem. 1995,99, 9248-9252
Investigation of Some Regularities for Dense Fluids Using a Simple Equation of State Bijan Najafi, Gholamabbas Pamafar,” and Saman Alavi? Department of Chemistry, Isfahan University of Technology, Isfahan, Iran 84156 Received: July 28, 1994; In Final Form: March 15, 1995@
A simple equation of state for dense fluids is used to investigate some empirically known regularities and also to explore some new ones. The known regularities predicted by the EOS are (i) there is a common bulk modulus point at which bulk modulus versus density of different isotherms of any fluid intersect at that single point and (ii) for each isochore with density greater than the Boyle density pressure versus temperature is linear. The new regularities explored by the EOS are (iii) there is a common compression point at which the compression factor against density of different isotherms of any dense fluid intersects at that single point and (iv) (B, - l ) v 2 varies linearly with 4’ for each isotherm of any dense fluid, where B, is the reduced bulk modulus and e = l / v is the molar density of the fluid. Experimental data were used to show the validity of the new regularities. A physical interpretation for each regularity is given.
1. Introduction Recently, a simple equation of state (regularity) has been derived for dense fluids‘s2 which has been found to be valid for nonpolar, polar, hydrogen-bonded and quantum fluids, as well as mixture^.^ The purpose of this work is to derive some known and a number of new regularities for dense fluids using the equation of state and also to give a physical interpretation for each regularity. This equation of state predicts that ( Z - l)v2 is linear with e2 for each isotherm of any dense fluid, where Z = pv/RT is the compression factor and e = l / v is the molar density. Experimentally, this equation of state (EOS) has been found to be valid for densities greater than the Boyle density and for temperatures below twice the Boyle temperature. We shall refer to this EOS as “linear isotherm regularity” or simply LIR from now on. The LIR is written as (2- l ) v Z = A + B e 2
(1) Using a simple model, the temperature dependence of A and B
parameters were obtained as A = A , - Al/RT and
Pa)
B = B,/RT (2b) where the constants A I and B1 are related to the intermolecular attraction and repulsion, respectively, and the constant A2 is related to the nonideal contribution of thermal pressures2 Although dense fluids are usually considered to be complicated on the molecular scale, all of them show a number of simple regularities, some of which have been known for years without any theoretical basis. However, the Ihm-Song-Mason EOS which is based on statistical mechanical perturbation the0ry~3~ has been used to give a theoretical basis for some experimentally known regularities (common bulk modulus point6 and linearity of p versus T for isochores’). 2. Common Compression Factor Point
The three-shell modification of the Lennard-Jones and Devonshire equation of state predicts a common intersection
lo
I 0
0
4
12
Figure 1. Searching for the common compression point of methane, using the experimental data of 600 K (W), 540 K (O), 500 K (A),450 K (A), and 400 K (0)isotherms.
point for Z versus e for different isotherms of a fluid.8 This intersection point may be called a “common compression point”. This common point is shown for a number of experimental isotherms of methane9,’0in Figure 1. Parsafar and Mason2 showed that for each isotherm ( Z - l)v2 varies linearly with e2,and this regularity (LIR) holds experimentally for densities greater than the Boyle density, @B = l / V B , and for temperatures below about twice the Boyle temperature, TB,where V B and TB are defined in terms of the second vinal coefficient B2 as follows: V B = TBBi(TB)and B2(TB) = 0 and B; = dBz/dT. The LIR may be used to derive the common compression point. The density at this point, eoz, can be obtained by setting (aZ/aT),,, equal to zero. An EOS can predict such a regularity if 402 given by that EOS is independent of temperature. Using the LIR, we obtain (aZ/aT), = ( A , - Blg2)e2/R?
By setting this partial derivative equal to zero at obtain
+Current address: Department of Chemistry, University of British Columbia, Vancouver, BC V6T 1Z1. Abstract published in Advance ACS Absrracts, May 1, 1995. @
0022-365419512099-9248$09.00/0
8
( P /P,)2
0 1995 American Chemical Society
e = eoz, we
J. Phys. Chem., Vol. 99, No. 22, 1995 9249
Regularities for Dense Fluids
r-
I 2'5
.$
N
0.1 I 0
I
4
8
(P
1.5
0.5
-0.5
12
I
'
8
8
/ Pc l2
Figure 2. Same as Figure 1, but plotting (Z - l)(v/v,)* against
( P
(e/
@,I2. Since A1 and B1 are temperature independent, all isotherms of a fluid will intersect at the common density of eoz. Experimental Test. The density at the common compression point, eoz, of a fluid can be obtained by using experimental p-v-T data for that fluid and plotting Z against e2for different isotherms and looking for a common intersection point. It is possible to illustrate that the intersection of isotherms occurs at the same density for Z versus e, Z versus Q2, and (2- l)v2 versus e2curves. The experimental p-v-Tdata of Younglove and Ely9 and Setzmann and Wagner'O were used to plot Z versus e*for CH4, as was shown in Figure 1. To obtain the density at the common point (Z - l)v2 may also be plotted against e2, which according to the LIR will give a straight line for each isotherm. In this case eoz will be obtained from the intersection point of the straight lines. It is clear that the latter approach for obtaining the density at the intersection point is more precise (and convenient). Experimental p-v-T data9.I0 have been used to plot (2l)(v/vc)2versus (e/ec)2for different isotherms of supercritical C& in Figure 2. Methane serves as our primary test material because of the abundance of p-v-T data for both e < eoz and e =. eoz. The value of eoz (reduced by gc)is found to be 7.4'/*, which corresponds to 27.61 mol L-' (ec= 10.150 mol L-' for Ch9). A similar plot is given for isotherms of liquid dimethylbutane (DMB) in Figure 3, using experimental data of Baonza et aL1' (by liquid we mean T < Tc). Even though the isotherms do not intersect in the liquid density range, their extrapolations give a common point at a higher density. Figure 3 gives eoz = 9.78 mol L-' (ec2.798 mol L-I, see ref 11). To compare these experimental values with the predictions of LIR, the intercept and slope of each isotherm in Figure 2 are used to obtain reduced values of the A and B parameters for the reduced LIR, (Z- l)(v/v,)* = A B(e/ec)2. The resulted values can be used to plot A and B versus 1/T to obtain A I and B1 (see eqs 2). The reduced values of A,/R = 209.3 K and BJR = 28.56 K were obtained for methane, which gives eoz = 27.48 mol L-' from = A I / B I . With a similar argument for DMB using the reduced values stated in Table 1, we obtain eoz = 9.78 mol L-I. Comparison with the Ihm-Song-Mason EOS. The IhmSong-Mason EOS4s5may be used to investigate the common compression point. This equation as given as
+
10
14
12
/ Pc12
Figure 3. Using experimental data (of ref 11) of 278 K (m), 258 K (0),238 K (A),and 208 K (0)isotherms of dimethylbutane to obtain the common compression point.
z=1+
(4-a)@ I 1+dbe
aQ 1 -ebA
where B2 is the second virial coefficient, a is the scaling factor for the softness of the intermolecular repulsive forces, b is the analogue of the van der Waals covolume, and 2 is a characteristic constant of the substance, equal to 0.454 for noble gas fluids and becoming smaller for complex fluids. Again, we may find eoz by setting (aZ/ar), equal to zero. The result isi2 bA@o,= -
[
1.22(db)(-db/dT) a2/dT
[l
+ ...I
(4)
If arguments similar to those given by Boushehri et a1.6 for presenting a physical interpretation of the common bulk modulus point are used, we find Qoz = llAb(0) where b(0)is the molecular covolume at Fer0 temperature. The value of b(0) gives the size of the molecules in terms of the range of the repulsive forces, r,, as b(0) = ( 2 d 3 ) r ; and therefore
We may then conclude that the Ihm-Song-Mason EOS also predicts a common compression point. We may use eq 5 to find eoz. For example, for argon," ;1 = 0.454,b(0) = 67.43 cm3 mol-', and eoz = 32.67 mol L-I, and for methane (using the necessary data from ref 6), Qoz = 25.62 mol L-I. The experimental p-v-T data9-''.'3-'6 of different fluids have been used to find eoz empirically. The results are compared with the calculated values given by LIR and Ihm.Song-Mason EOS (ISM) in Table 1. Also, the parameters of eqs 2 are included in the table to calculate eoz from LIR. Physical Interpretation. If the temperature dependences of the A and B parameters are explicitly expressed, LIR can be written as
Z=1
+ A,@' - A,e2/RT+ BIe4/RT
(6)
9250 J. Phys. Chem., Vol. 99, No. 22, 1995
Najafi et al.
TABLE 1: Density at the Common Compression Point, eoz, Obtained by Experiment (Exp), Compared with Those Given by LIR and Ihm-Song-Mason EOS (ISM), for Selected Fluids; Parameters of Eqs 2 Are Also Included
eoz,mol L-l fluid Ar(1)" Ar(g)b Wl)'
N2Wd CH'l(g)e DMB(1Y
EXP
LIR
ISM
AT, K
AtIR, K
BIIR,K
44.1 f 0.2 33.6 f 0.3 37.5 f 0.2 26.0 f 0.2 27.6 f 0.2 9.78 =L 0.04
44.58 f 0.29 33.41 f 0.57 37.29 f 1.14 27.32 f 0.68 27.48 f 0.49 9.78 f 0.18
32.67 f 0.98 32.67 f 0.98 25.84 f 0.78 27.01 f 0.81 25.62 f 0.76
100-140 400-600 170-209 308-673 400-600 208 -278
208.3 155.4 254.5 117.5 209.3 1787
18.51 24.59 21.50 19.67 28.56 146.2
Reference 13. Reference 14. Reference 15. Reference 16. e References 9 and 10. f Reference 11.
where the first to fourth terms represent the ideal, nonideal thermal pressure, attraction, and repulsion contribution on Z, I @a respectively. According to eq 6 at Q 2 = eoz2 = A,/Bl, the last I -I two terms cancel each other our exactly and Z = 1 A ~ Q o z . ~ *v l2 / .' Therefore, we may conclude that at eoz the average inter/ molecular separation is such that the intermolecular attraction W n 8-and repulsion forces cancel each other exactly; Le., the average 4 intermolecular separation is equal to .,, (F, is the separation of I the minimum of the effective pair potential well). In order to m" 4 - justify this proposition, an example will be given. For gaseous argon, eoz = 33.62 mol L-' corresponds to .,, = 3.67 A, 0 -which is a few percent less than r, = 3.82 A, the separation at , the minimum of the Lennard-Jones pair potential (see Table I11 in ref 2). This difference is due to the fact that the attraction -4 0 2 4 8 8 10 12 has a longer range than the repulsion, and hence the minimum 4 of the effective pair potential of the dense fluid is expected to ( P / PCl2 be located at a shorter separation than the minimum of the actual pair potential. Figure 4. Search for the linearity of (B, - l)(v/vc)* versus (e/@c)z for This conclusion may also be justified by using the IhmAr at 120 K. The experimental points are compared with the LIR (solid line) and ISM (dashed curve). Song-Mason EOS, for which eoz is given by eq 5. For most fluids the 2 d / 3 factor is around unity, and therefore eoz eo: = 0.6(A1/B1) corresponds to the density of the fluid in which the average (9) molecular separation is around r,. Since A1 and B1 are temperature independent, LIR predicts the existence of a common bulk modulus for each liquid. 3. Common Bulk Modulus Point Equation 8 may be rearranged into The near linearity of the bulk modulus (reciprocal compressibility) for isotherms of a liquid as a function of pressure was (B, - l)v2 = 3A 5Be2 (10) first noticed by Tait over 100 years ago. This linearity is the basis for a number of successful empirical equations of which predicts another regularity; that is, (B, - l)v2 versus g2 state.I7-l9 Recently, a new regularity has been reported in the is linear for all isotherms of a dense fluid. The experimental behavior of the reduced bulk modulus, B, = (l/RT)(ap/a@)Tfor data for B, of liquid argon are given by Street,I3 and the LIR liquids, where p is the pressure, e is the molar density, and RT and Ihm-Song-Mason EOS results are given in Figure 4, in has its usual meaning. Huang and O'Connellzo found that B, which the linearity is fairly good. versus molar volume isotherms intersect at a common point, The common bulk modulus point density of a fluid, @OB, can called the common bulk modulus point. B, is independent of be obtained experimentally by plotting B, versus e for different temperature at this point. They checked the experimental isotherms. However, according to LIR, it is more convenient existence of this point for more than 250 different liquids. to plot (B, - l)v2 versus e2,which is expected to be linear (eq A theoretical basis for the existence of such a point was 10) and to obtain @OB from the intersection of the lines. The recently given by Boushehri et aL6 using the Ihm-Song-Mason experimental data of Baonza et a1.I' for DMB are used to plot EOS. They found that the density of the common bulk modulus Figure 5, in which excellent linearities can be seen which point, @OB, is given by intersect at (@OB/@c)2 = 9.13, which corresponds to @OB = 8.45
I
+
h
> W
t
+
= l/Ab(O)
(7)
where I is the characteristic constant of the substance and b(0) is the molecular covolume at zero temperature. LIR may be used to reveal the existence of the common bulk modulus point. Using LIR, the reduced bulk modulus is obtained as
B, = 1
+ 3Ae2 + 5Be4
In order to find @OB, we may set (aB,/aT), equal to zero, for which the following result is obtained:
mol L-I. The experimental value of @OB for liquid argon, which is found to be 34.40 mol L-I, may be compared with the values obtained by LIR (AID? = 208.3 K and BIIR = 18.51 K, ec = 13.29 mol L-I) and ISM (I = 0.454 and b(0) = 67.43 cm3 mol-'), which are found to be 34.53 and 32.67 mol L-I, respectively. If the temperature dependencies of A and B parameters of eq 8 are stated explicitly, the reduced bulk modulus may be written as
B, = 1
+ 3Ae2 +e2(-3A1 + 5B1g2)/RT
(1 1)
J. Phys. Chem., Vol. 99, No. 22, 1995 9251
Regularities for Dense Fluids 800
800
200
,
I
0 0
200
400
800
800
1000
1200
T,K Figure 5. Search for the common bulk modulus point for DMB using experimental data of ref 11 for 278 K (W), 258 K (O), 238 K (A),and 208 K (0)isotherms.
where 1 and 3Ag2 are the contributions of ideal gas and nonideal thermal pressure in B,. The last two terms in eq 11 are the contributions of the intermolecular interactions. At e = @OB (considering that @OB2 = 3A1/5B1), the quantity inside parentheses cancels out, and B, is given by
Therefore, at the common bulk modulus point, the average intermolecular separations are such that the attraction and repulsion have equal contributions in B,, but with opposite signs.
e = 5 (A), 17 (0),20 (A),27 (0),and 29 mol L-'(W) isochores of N2 (experimental data are taken from refs 16 and 22).
Figure 6. p versus T for
(%Lo,
p = (const)T (at e = eo,)
p = e3(B,e2 - A,)
+ (Re + A2Re3)T
(13)
This equation shows that p versus T i s linear for each isochore of dense fluids. Experimental data of Robertson et a1.I6 and the summary of Jacobsen et al.21for N2 were used to plot p versus T for different isochores. The result is shown in Figure 6, in which for e > @B % 20 mol L-' the curves are quite linear. (It must be noted that LIR is valid for densities greater than the Boyle density,2 @B.) Because of the perfect gas behavior of the fluid, low-density isochores also become linear with respect to T . However, a negative curvature (concave downward) is observed for intermediate densities (see e = 17 mol L-' isochore in Figure 6). The isochore with e = eoz = 27 mol L-' is included in this figure. (For N2, AIIR = 117.5 K and BIIR = 19.67 K and ec = 11.177 mol L-' and therefore Qoz = 27.3 mol L-'.) It can be observed that there is a specific distinction between this isochore and the others in that it goes through the origin. The reason for this behavior can be explained as follows. At e = eoz, the partial derivative (i3ZBr), must be equal to zero; thus, =
(&)[? (%L0,l= O +
(14)
It is interesting to see whether LIR can predict such zero intercept for this isochore. By substituting @oz2= AJBI, eq 3, in eq 13, the following result is obtained
P = Reo,( 1 + AzeoZ3T
Another regularity observed experimentally is the near linearity of p versus T at constant density (isochores) over the entire range from the perfect gas to the dense fluid.' Any sensible equation of state must thus give nearly linear isochores. If we substitute pv/RT for Z, and state the explicit temperature dependencies of A and B, eq 1 may be written as
Consequently,
er
and therefore
4. p - T Isochores
(%)Qoz
=
(15)
which is a straight line that goes through the origin. 5. Conclusion
The LIR has been used in this work to predict two new regularities: first, existence of a (nearly) common intersection point for the compression factor versus density for all isotherms at the density eoz. At this point, the average intermolecular separation is such that the attraction and repulsion forces cancel out exactly. Second, (B, - l)v2 versus e2 is linear for each isotherm. These new regularities have been confirmed with the experimental observations (see Figures 1-5). Also LIR has been shown to be consistent with two experimentally known regularities: (i) existence of the common intersection point for the reduced bulk modulus versus density for all isotherms at @OB (see eq 9). At this point the average intermolecular separation is such that the effects of attraction and repulsion on the bulk modulus cancel out exactly. (ii) The linearity of p versus T for isochores with densities greater than the Boyle density (see eq 13). LIR shows that only for one unique isochore (specifically e = eoz) p versus T goes through the origin (see eq 15), which is consistent with experiment (see Figure 6). The LIR gives the ratio of @OB to @ozas (see eqs 3 and 9)
This ratio is very close to that found experimentally; see Table 2. The Ihm-Song-Mason EOS gives the value of unity for the ratio. This unrealistic result given by this EOS may be due to the fact that it is not accurate in the very dense supercritical region,2 in which the common compression point exists.
9252 J. Phys. Chem., Vol. 99, No. 22, 1995
TABLE 2: Density at the Common Bulk Modulus Point, Obtained by Experiment (Exp), Compared with Those Given by LIR and ISM; Experimental Ratio of @O$@o&p2 Is Also Given To Be Compared with the Value of 0.6 Obtained by LIR (Eq 9) @OB, mol L-] (@0B/@Od2
@OB,
Exp LIR ISM T, K exP 34.4 =k 0.2 34.53 f 0.23 32.67 f 0.98 100-140 0.61 f 0.01 27.5 i.0.2 28.89 f 0.88 25.84 f 0.78 170-209 0.54 f 0.01 208-278 0.75 f 0.01 DMB‘ 8.45 f 0.04 7.577 f 0.16 fluid
AP
a
Reference 13. Reference 15. Reference 11.
The common bulk modulus point occurs at liquid region between the triple point and the critical temperature. However, the common compression point exists at a density greater than liquid density region (see Figure 3). As pointed out by Boushehri and Mason: at lower temperatures the common bulk modulus occurs at higher densities. Also, the same behavior was observed for the common compression point; for instance, see the experimental results for argon in Table 1. Acknowledgment. This work has been financially supported by a Isfahan University of Technology research grant. The authors thank Professor E. A. Mason of Brown University for his very helpful comments. References and Notes (1) Parsafar, G. A. J . Sci. Zslamic Republic Iran 1991, 2, 111. (2) Parsafar, G. A.; Mason, E. A. J . Phys. Chem. 1993, 97, 9048.
Najafi et al. (3) Parsafar, G. A.; Mason, E. A. J . Phys. Chem. 1994, 98, 1962. (4) Ihm, G.; Song, Y.; Mason, E. A. J . Chem. Phys. 1991, 94, 3839. ( 5 ) Ihm, G.; Song, Y.; Mason, E. A. Fluid Phase Equilib. 1992, 75, 117. (6) Boushehri, A.; Tao, F.-M.; Mason, E. A. J . P hys. Chem. 1993,97, 2711. (7) Song, Y.; Mason, E. A. J . Chem. Phys. 1989, 91, 7840. (8) Hirschfelder, J. 0.;Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids, 2nd printing; Wiley: New York, 1964; pp 296-8. (9) Younglove, B. A.; Ely, J. F. J . Phys. Chem. Ref. Data 1987, 16, 577. (10) Setzmann, U.; Wagner, W. J . Phys. Chem. Ref. Data 1991, 20, 1061. (1 1) Baonza, V. G.;Alonso, M. C.; Delgado, J. N. J. Phys. Chem. 1993, 97, 2002. (12) Mason, E. A. Private communication. (13) Street, W. B. Physica 1974, 76, 59. (14) Stewart, R. B.; Jacobsen, R. T. J. Phys. Chem. Ref. Data 1989,18, 639. (15) Street, W. B.; Ringermacher, H. I.; Burch, J. L. J. Chem. Phys. 1972, 57, 3829. (16) Robertson, S . L.; Babb, S. E., Jr. J . Chem. Phys. 1969, 50, 4560. (17) Hayward, A. T. J. Br. J . Appl. Phys. 1967, 18, 965. (18) Macdonald, J. R. Rev. Mod. Phys. 1969, 40, 316. (19) Dymond, J. H.; Malhotra, R. lnr. J . Thermophys. 1988, 9, 941. (20) Huang, Y.-H.; O’Connell, J. P. Fluid Phase Equilibr. 1987, 37, 75. (21) Jacobsen, R. T.; Stewart, R. B.; Jahangiri, M. J . Phys. Chem. Ref. Dara 1986, 15, 735.
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