J. Phys. Chem. B 2006, 110, 3271-3275
3271
A New Regularity for Internal Pressure of Dense Fluids Vahid Moeini* Payame Noor UniVersity, Behshahr Center, Behshahr, Iran ReceiVed: August 24, 2005; In Final Form: December 28, 2005
In this paper, we derive a new regularity for dense fluids, both compressed liquids and dense supercritical fluids based on the Lennard-Jones (12-6) potential function and using speed of sound results. By considering the internal pressure by modeling the average configurational potential energy, and then taking its derivative with respect to volume, we predict that isotherm [(∂E/∂V)T/FRT]V2 is a linear function of F2, where E is the internal energy, (∂E/∂V)T is the internal pressure, and F ) 1/V is the molar density. The regularity is tested with experimental data for ten fluids including Ar, N2, CO, CO2, CH4, C2H6, C3H8, C4H10, C6H6, and C6H5CH3. These problems have led us to try to establish a function for the accurate calculation of the internal pressure based on speed of sound theory for different fluids. The results of the fitting show limited success of the pure substances. The linear relationship appears to hold from the lower density limit at the Boyle density and from the triple temperature up to about double the Boyle temperature. The upper density limit appears to be reached at 1.4 times the Boyle density. The results are likely to be useful, although they are limited.
each isotherm1
Introduction The authors report their findings of testing literature results for pVT for pure dense fluids, according to which (Z - 1)V2 is linear with respect to F2 for each isotherm, where Z ≡ pV/RT is the compression factor.1 This equation of state works very well for all types of dense fluids, for densities greater than the Boyle density but for temperatures below twice the Boyle temperature. The regularity was originally suggested on the basis of a simple lattice-type model applied to a Lennard-Jones (12,6) fluid. The purpose of this paper is to point out a new regularity that applies both to compressed liquids and to dense supercritical fluids. The regularity is that the quantity [(∂E/∂V)T/FRT]V2is linear in F2, where F ) 1/V is the molar density; the quantity (∂E/∂V)T is known as the internal pressure. A typical isotherm for Ar at 300 K is shown in Figure 1. The intercept A and the slope B are significantly related to attraction and repulsion, respectively, and both depend on temperature. In general, the term (∂E/∂V)T represents the contribution due to effective pressure arising from the attractive or repulsive forces between the molecules. In an ideal gas, the internal pressure is zero, and in real gases, the term is usually small compared to the external pressure P. Liquids and solids, on the other hand, have strong attractive or cohesive forces, and the internal pressure (∂E/∂V)T may therefore be large compared to the external pressure.2,3 In this paper, the accurate thermal pressure is used for the evaluation of the internal pressure of dense fluids using the speed of sound theory.4 Agreement with experimental thermal pressure coefficient data is shown to be, in general, quite good. The results of these calculations are also compared to those obtained using the statistical mechanical theory to predict such a linear relation in the dense-fluid region.
(Z - 1)V2 ) A + BF2
where Z ≡ pV/RT is the compression factor, F ) 1/V is the molar density, and A and B are the temperature-dependent parameters as follows:
A ) A′′ B)
A′ RT
B′ RT
A general regularity was reported for pure dense fluids, according to which (Z - 1)V2 is linear with respect to F2, for * E-mail:
[email protected]. Fax: +98-152643-2231.
(2) (3)
Here, A′ and B′ are related to the intermolecular attractive and repulsive forces, respectively, while A′′ is related to the nonideal thermal pressure, and RT has its usual meaning. This regularity holds for densities greater than the Boyle density and temperatures lower than twice that of the Boyle temperature. In the present work, we begin with the exact thermodynamic relation5-7
(∂E/∂V)T ) T(∂p/∂T)F - p
(4)
in which T(∂p/∂T)F is usually called the thermal pressure and (∂E/∂V)T is called the internal pressure. For example, for a van der Waals fluid, the thermal pressure coefficient can be written as
(∂p/∂T)F ) R/(V - b)
(5)
In practice, internal pressure for a van der Waals fluid is obtained by substitution of eq 5 in eq 4
(∂E/∂V)T ) aF2
Theory
(1)
(6)
This result for (∂E/∂V)T implies that internal energy of a van der Waals gas increases when it expands isothermally (that is, (∂E/∂V)T > 0) and that the increase is related to the parameter
10.1021/jp0547764 CCC: $33.50 © 2006 American Chemical Society Published on Web 01/27/2006
3272 J. Phys. Chem. B, Vol. 110, No. 7, 2006
Moeini
Figure 1. Typical isotherm of [(∂E/∂V)T/FRT] V2 vs F2 for Ar at 300 K.
a, which models the attractive interactions between the particles. A larger molar volume, corresponding to a greater average separation between molecules, implies weaker mean intermolecular attractions. Calculation of (∂E/∂V)T for a gas that obeys the virial equation of state can be written as
(∂E/∂V)T ) RTAF2 + ‚‚‚
(7)
where A ) T(∂B2/∂T)V and B2 is second virial coefficient. Linear isotherm regularity (LIR), which was originally devised for normal dense fluids, is based on the cell theory and considers only nearest adjacent interaction. LIR is applied well to Lennard-Jones fluids for which the interaction potential can be modeled rather accurately by the (12-6) powers of inverse intermolecular distance. The Lennard-Jones potential function suitably describes the interactions between the molecules of a fluid under the condition that it behaves as a normal fluid. In LIR, we attempted to calculate the internal pressure by modeling the average configurationally potential energy and then taking its derivative with respect to volume.8 In the present work, we assume that any kinetic energy contribution to the internal energy E will vanish on taking the derivative, since the temperature is held constant. We approximate the average potential energy by summing the contribution from nearest neighbors only, assuming single inverse powers for the effective repulsion and attraction, i.e.
U)
(
)
Cn C m N z(F) n - m 2 jr jr
(8)
where U is the total potential energy among N molecules, z(F) is the average number of nearest neighbors, jr is average distance between nearest neighbors, and Cn and Cm are constants. It is well-known that z(F) is proportional to F, as is the case for liquid argon, rubidium, cesium, and V ≈ jr3, so that U can be written as1
Kn Km U ) n/3+1 - m/3/+1 N V V
(9)
where Kn and Km are constants. Carrying out the differentiation, we obtain the internal pressure for dense fluids
pint )
( ) [ ∂E ∂V
T
≈
]
∂(U/N) ∂V
T
) A1Fm/3+2 - B1Fn/3+2 (10)
Figure 2. (a) Plot of A vs inverse temperature. Solid line is the linear fit to the A data points, for Ar. (b) Plot of B vs inverse temperature. Solid line is the linear fit, for Ar.
where A1 and B1 are constants and in order to model the experimental results, we should take m ) 3 and n ) 9. Combining the foregoing results, we find the internal pressure for dense fluids based on the Lennard-Jones (12-6) potential function
(∂V∂E) ) A F - B F 3
1
T
5
1
(11)
If we divide both sides of eq 11 by F3, we obtain
[
]
(∂E/∂V)T F3
) A 1 - B1 F 2
(12)
and from this simple model, we predicted that the isotherm [(∂E/∂V)T/FRT]V2 is a linear function of F2 and so predicted the temperature dependence of its parameters. The final result is therefore of the form
[
]
(∂E/∂V)T 2 V ) A + BF2 FRT
(13)
where the intercept A and the slope B are significantly related to attraction and repulsion, respectively, and both depend on temperature. In practice, we show that A ) A1/RT and B ) B2 - B1/RT (see Figure 2a,b). Thus, the intercept A is proportional to 1/T, and the slope B is linear in 1/T. In this paper, we applied argon fluid for our primary test because of the abundance of available pVT data.9
Internal Pressure of Dense Fluids
J. Phys. Chem. B, Vol. 110, No. 7, 2006 3273
Internal Pressure From classical thermodynamics, the pressure can be written
p)T
(∂T∂p) - (∂V∂E) F
T
(14)
where (∂E/∂V)T is a contribution to the pressure from intermolecular forces and is zero for hard spheres. Equation of state data for the real fluid are used to obtain T(∂p/∂T)V.10 One of the most difficult problems within the context of thermodynamics lies in the shortage of experimental data for some basic quantities such as thermal pressure coefficients (TPC), which are tabulated for extremely narrow temperature ranges, normally around the ambient temperature for several types of liquids. Furthermore, the measurements of the thermal pressure coefficients made by different researchers often reveal systematic differences between their estimates. In this paper, accurate equation of state and speed of sound were used to obtain the TPC of Ar, N2, CO, CO2, CH4, C2H6, C3H8, C4H10, C6H6, C6H5CH3, HFC125, and R152a at indefinite range. Calculations were performed at different density, and values obtained at wide density are compared to experimental values. The applicability of these two methods to fluids was examined in this study. Initially, we present here a simple method of the ideas and use thermal pressure coefficient directly in place of equation of state to analyze experimental pVT data. The equation of state described in this paper is explicit in Helmholtz energy A with the two independent variables density F and T. At a given temperature, the pressure can be determined by Helmholtz energy
∂A p(T, F) ) ∂V T
( )
(15)
Using the general expression of the relation to the reduced Helmholtz energy φ ) A/(RT) and its derivatives gives
p(δ,τ) ) 1 + δφδr FRT
(16)
where δ ) F/FC is the reduced density and τ ) TC/T is the inverse reduced temperature. Both the density F and the temperature T are reduced with their critical values FC and TC, respectively. Since the Helmholtz energy as a function of density and temperature is one of the four fundamental forms of an equation of state, all the thermodynamic properties of a pure substance can be obtained by combining derivatives of the reduced Helmholtz energy φ. Where φδ and φδr are defined as
φδ )
(∂φ∂δ)
φδr )
( )
(17)
τ
and
∂φr ∂δ
τ
(18)
then, finally, to derive thermal pressure coefficient, we need only convert the dτ to dT
dτ ) -
Tc T2
dT
(19)
Figure 3. Experimental values of thermal pressure coefficient (Goodwin 1985) vs density for CO fluid are compared with thermal pressure coefficient using the speed of sound at 200 K.
Figure 4. Thermal pressure coefficient as a function of density for different fluids (HFC125 ref 17, R152a ref 18).
The above procedure leads to obtaining the thermal pressure coefficient from the reduced Helmholtz energy for real fluids
() [ ∂p ∂T
δ
) FR (1 + δφδr) - δτ
( )] ∂φδr ∂τ
δ
(20)
The present calculations for thermal pressure coefficient show equal results from using the accurate equation of state and using the speed of sound. The speed of sound has presented a simple method for calculating the thermal pressure coefficient, and it shows that the results obtained are very accurate in relation to experimental data. In this paper, accurate thermal pressure coefficients have been calculated the using speed of sound. For comparison, the same thermal pressure coefficients calculated using the speed of sound and using experimental thermal pressure coefficient data for CO are shown in Figure 3. In this paper, we also show thermal pressure coefficients calculated via speed of sound plotted as a function of density for different fluids in Figure 4. The resultant values of thermal pressure coefficient are presented along with the values obtained from the literature.4 Experimental Tests In examining the linearity of [(∂E/∂V)T/FRT]V2 versus F2, we wish to address the following specific questions: (i) Over what range of density does the linearity hold for these isotherms? (ii) Over what temperature range does it hold? (iii) For what types of fluids is the linearity valid? (iv) Does [(∂E/∂V)T/FRT]Vn for n ) 2 vary linearly with F2? How about for n ) 0 and n ) 1?
3274 J. Phys. Chem. B, Vol. 110, No. 7, 2006
Moeini TABLE 1: Equation of [(DE/DV)T/GRT](W/WC)2 ) A + B(G/GC)2
Figure 5. Typical isotherm of [(∂E/∂V)T/FRT] V2 vs F2 for supercritical C6H5CH3 at 700 K.
T/K
A
B
R2
∆p/MPa
200 300 400 500 600 700 800 900 1000 1100 1200
0.9343 0.5763 0.3866 0.2814 0.2145 0.1709 0.1362 0.1093 0.0886 0.0717 0.0572
-0.0953 -0.0689 -0.0536 -0.0447 -0.0386 -0.0345 -0.0308 -0.0277 -0.0253 -0.0231 -0.0211
0.9990 0.9996 0.9998 0.9996 0.9993 0.9986 0.9986 0.9989 0.9990 0.9994 0.9992
50-600 100-1000 200-1000 250-1000 300-1000 300-1000 350-1000 400-1000 450-1000 500-1000 550-1000
a Reduced intercept (A) and slope (B) of the coefficient of determination (R2); the pressure range (∆p) of the data for Ar (data from ref 9).
TABLE 2: Same as Table 1, for Different Fluids fluid Ara N2b N2b COc CO2d CH4e C2H6e C3H8e C4H10e C6H6f C6H5CH3g C6H5CH3g a
Figure 6. Same as Figure 5, for liquid C6H5CH3 at 300 K.
We begin with C6H5CH3 in its supercritical range.11 A typical isotherm (700 K) is shown in Figure 5, where it can be seen that the points at the low densities deviate significantly from the linear relation. For convenience only, the molar densities and volumes have been reduced by the critical value FC ) 3.15 mol L-1. It turns out that all the C6H5CH3 isotherms become linear for F g 5.53 mol L-1 and that this is nearly equal to the Boyle density of about FB ) 5.67 mol L-1. The Boyle temperature TB and the Boyle volume VB are defined in terms of the second virial coefficient B2 as follows: B2(TB) ) 0 and VB ) TBB2′(TB), where B2′ ) dB2/dT. To investigate the liquid region, we turn first to C6H5CH3 in its liquid range.11 (By liquid, we mean T < TC.) Except near the critical point, all the isotherms are found to be quite linear over almost the entire density range from the vaporization line to the freezing line. A typical isotherm (300 K) is shown in Figure 6. The general conclusion, which is borne out for the other substances examined, is that the linear relation holds in the liquid region for F g FB. In summary, we can tell that the upper density limit for linearity in the liquid region is the freezing line. Thus, the general rule in both the liquid and supercritical regions is that the linear relation holds for F g FB. To find the temperature range over which the linearity holds when F g FB, we have plotted [(∂E/∂V)T/FRT]V2 against F2 for Ar at different temperatures. The results are summarized in Table 1, including the intercept and slope of the fitted straight line and the pressure range for Ar fluid. Because both [(∂E/ ∂V)T/FRT]V2 and F2 are subject to experimental error, we also show the coefficient of determination R2, which is simply the square of the correlation coefficient. From these values, it appears that the linearity is good. Therefore, the lower temper-
T/K
A
B
R2
∆p/MPa
200 70 150 200 400 250 400 400 500 640 300 700
0.9343 2.841373 1.2126 0.8460 1.0995 0.9904 1.05723 1.2486 1.1896 1.1561 5.8742 1.3081
-0.0953 -0.2319179 -0.1265 -0.0884 -0.1019 -0.1046 -0.1010 -0.0910 -0.0824 -0.0747 -0.4648 -0.1116
0.9990 0.9968 0.9920 0.9990 0.9972 0.9996 0.9999 0.9918 0.9972 0.9997 0.9982 0.9946
50-600 0.06-30 20-500 45-100 40-100 50-200 30-70 20-100 30-70 30-90 0.1013-100 30-100
Ref 9. b Ref 12. c Ref 13. d Ref 14. e Ref 15. f Ref 16. g Ref 11.
ature limit is of course the triple-point temperature, the lowest temperature at which a stable liquid phase exists. To see whether the regularity is limited to a certain type of fluid or is generally true, we have plotted [(∂E/∂V)T/FRT]V2 against F2 for a number of different fluids at different temperatures. The linearity seems to hold as long as F g FB. The various fluids tested are listed in Table 2, together with the intercept, the slope, the pressure range of the experimental data, and the coefficient of determination for a typical isotherm. We conclude that the linearity is a property of many types of dense fluids. Finally, we test how sensitive the linearity of [(∂E/∂V)T/FRT]Vn against F2 is with respect to the power of n. The reason for this concern is that in the original lattice-type model the power of n depended explicitly on the 12th power repulsion of the Lennard-Jones potential. In Figure 7,, we have shown a typical plot of [(∂E/∂V)T/FRT]Vn against F2 for n ) 0, 1, and 2, for the 150 K isotherm of supercritical nitrogen.12 It is apparent that the isotherm turns downward for n ) 0 and n ) 1, but is linear only for n ) 2. Discussion The cohesive forces, which are the result of forces of attraction and forces of repulsion between liquid molecules, holding a liquid together create a pressure within the liquid which has been termed the internal pressure. Internal pressure, a fundamental liquid property, is closely related to the different properties of liquids such as ultrasonic velocity, free volume, viscosity, surface tension, solubility parameter, and latent heat of vaporization in the liquid phase.19 The precise meaning of the internal pressure is contained in a generalized manner in the well-known thermodynamic eq 4. United forces of external and internal pressure equalize the thermal pressure which tries to expand the matter. If the thermal
Internal Pressure of Dense Fluids
J. Phys. Chem. B, Vol. 110, No. 7, 2006 3275
Figure 7. Test of the linearity of [(∂E/∂V)T/FRT] Vn against F2 for n ) 0, 1, and 2, for the 150 K isotherm of supercritical nitrogen.
pressure of a fluid is available, then the internal pressure can be calculated easily. The main objective of the present research is to compute the internal pressure. Therefore, the speed of sound was used to calculated the thermal pressure of quite different fluids for which reliable experimental values are available. The calculated and experimental values of thermal pressure are found to present a close agreement with each other. By careful examination of the present results, it seems to us that this work contains a point of particular interest, that internal pressure decreases as the density increases at constant temperature. The physical interpretation in terms of molecular forces is as follows: If the shape of the intermolecular potential function, U(r), for molecules in fluids is considered, it is seen that U(r) can be divided into two parts, attractive UA and repulsive UR
U(r) ) UA + UR
(21)
Differentiation at constant temperature with respect to volume gives
[∂U(r)/∂V]T ) (∂UA/∂V)T + (∂UR/∂V)T
(22)
Pi ) Pi,A + Pi,R
(23)
namely
where attractive internal pressure Pi,A is seen to be positive, and repulsive internal pressure Pi,R is negative. As the density is raised at constant temperature, the repulsive pressure becomes predominant, and therefore, internal pressure decreases. On the other hand, the reason is that the resultant forces under lowpressure conditions are attractive, and as the pressure increases, the repulsive forces become predominant.19 Conclusions The linearity of [(∂E/∂V)T/FRT]V2 versus F2 has been checked with experimental measurements. A simple model that mimics the linearity has been used to predict the temperature dependence of the intercept and slope parameters and shown to agree with
experiment. The calculations all suggest that the linear region is due to a balance between intermolecular attractive and repulsive forces. The range of density and temperature over which the linear relation is valid can be approximately specified as follows. The lower density limit is approximately the Boyle density FB, for all temperatures from the triple point up to about 2TB, which is far into the supercritical region, roughly 5TC. The upper density limit is less certain1. In the true liquid state (T < TC), the limit seems to be the freezing line. In the supercritical region (T > TC), the calculation indicates that the limit of validity is about 1.4FB (about 2.5FC). The present results have several obvious applications. At the simplest level, we now have a new and useful way of plotting data on the isotherms of compressed dense fluids. Moreover, the temperature dependence of two constants that characterize each linear isotherm are known: The intercept A is proportional to 1/T, and the slope B is linear in 1/T (Figure 2a,b). Finally, we expect the linearity of [(∂E/∂V)T/FRT]V2 versus F2 to hold for dense mixtures as well as for single substances. The reason for this expectation is the well-known success of so-called one-fluid approximations for mixtures. Such approximations assume that mixtures obey the same equation of state (EOS) as single substances, but with parameters that depend on the composition of the mixture. Since the present linearity has been shown to be consistent with an EOS, adjusting the parameters of the EOS for compositions will not alter this result. Testing of this expectation and investigation of the composition dependences of parameters A and B remain for future work. Acknowledgment. We acknowledge the Payame Noor University for the financial support, and also Professor B. Najafi and Professor G. A. Parsafar for their useful comments. References and Notes (1) Parsafar, G.; Mason, E. A. J. Phys. Chem. 1993, 97, 9048. (2) Parsafar, G.; Moeini, V.; Najafi, B. Iran J. Chem. Chem. Eng. 2001, 20, 37. (3) Laidler, K. J.; Meiser, J. H. Physical Chemistry; The Benjamin/ Cummings Publishing Company, Inc.: San Francisco, 1982; pp 742. (4) Ghayeb, Y.; Najafi, B.; Moeini, V.; Parsafar, G. A. High Temp. High Pressures 2003/2004, 35/36, 217. (5) Parsafar, G.; Mason, E. A. Phys. ReV B. 1994, 49, 3049. (6) Ely, J. F.; McQuarrie, D. A. J. Chem. Phys. 1974, 60, 4105. (7) Atkins, P. W. Physical Chemistry, 6th ed.; Oxford University Press: New York, 2001; pp 127. (8) Ghatee, M. H.; Bahadori, M. J. Phys. Chem. B 2001, 105, 11256. (9) Stewart, R. B.; Jacobsen, T. J. Phys. Chem. Ref. Data 1989, 18, 639. (10) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Ttheory of Gases and Liquids, 2nd printing; Wiley: New York, 1964; pp 647652. (11) Goodwin, R. D. J. Phys. Chem. Ref. Data 1989, 18, 1565. (12) Jacobsen, R. T.; Stewart, R. B.; Jahangiri, M. J. Phys. Chem. Ref. Data 1986, 15, 736. (13) Goodwin, R. D. J. Phys. Chem. Ref. Data 1985, 14, 849. (14) Span, R.; Wagner, W. J. Phys. Chem. Ref. Data 1996, 25, 1509. (15) Younglove, B. A.; Ely, J. F. J. Phys. Chem. Ref. Data 1987, 16, 577. (16) Goodwin, R. D. J. Phys. Chem. Ref. Data 1988, 17, 1541. (17) Piao, C. C.; Noguchi, M. J. Phys. Chem. Ref. Data 1998, 27, 775. (18) Outcalt, S.; McLinden, M. O. J. Phys. Chem. Ref. Data 1996, 25, 605. (19) Goharshadi, E. K.; Nazari, F. Fluid Phase Equilib. 2001, 4749, 1-7.
