Research: Science and Education
Updated Principle of Corresponding States
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Dor Ben-Amotz* and Alan D. Gift Department of Chemistry, Purdue University, West Lafayette, Indiana 47907-1393; *
[email protected] R. D. Levine Fritz-Haber Research Center, Hebrew University, Jerusalem 91904, Israel
The principle of corresponding states, which reveals global correlations between the thermodynamic properties of different fluids, has broad appeal as a manifestation of fundamental links between diverse chemical phenomena. However, the discussion of this topic in current physical chemistry textbooks is quite outdated. On one hand, the experimental data most often used to illustrate this principle were compiled over a half a century ago (1–5). Even more significantly, little if any connection is made to more recent theoretical and computer simulation results or to the link between thermodynamic properties and intermolecular interaction potential parameters. Thus we have chosen to revisit this venerable subject by using recently compiled experimental and theoretical equation of state data to further explore fundamental correlations between key physical chemistry concepts. Traditionally, corresponding states scaling is accomplished by reducing the temperature, T, pressure, P, or number density, ρ, of a fluid to dimensionless form using two of the fluid’s critical point constants (TC, PC, or ρC). Note that the number density ρ = n兾V, may be expressed in mol兾L units, as it represents the number of moles, n, in a container of volume, V. We have recently demonstrated that the traditional corresponding states scaling procedure is not as effective as an alternative method (6) based on globally correlating the equations of state of simple fluids with each other and with that of a model Lennard-Jones (LJ) fluid (7), composed of spherical particles with the following intermolecular interaction potential energy function,
u LJ (r ) = 4 ε
σ r
12
−
σ r
6
(1)
where r is the distance between two molecules, σ is the corediameter (distance at which the potential crosses zero), and ε is the cohesive well-depth (energy at the potential minimum). In particular, the new LJ corresponding state (LJCS) method relies on the two parameters, σ and ε, rather than critical constants, as corresponding states scaling parameters. The experimental data used to illustrate LJ-CS scaling behavior derives from a definitive compilation published by the National Institute of Standards and Technology (NIST; refs 8, 9). Pedagogical advantages of LJ-CS scaling include demonstrating the link between microscopic (molecular interaction potentials) and macroscopic (thermodynamic) properties, as well as introducing gas non-idealities, fluid critical properties, and computer simulation methods. Furthermore, the fact that some fluids such as He, CO2, H2O, and n-butane do not globally correlate with a LJ fluid, clearly implies that these molecules have intermolecular interaction potentials that are fundamentally different in form. The key 142
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concepts introduced using LJ-CS scaling can be reinforced with the aid of Web-based tables and graphs of fluid thermodynamic data downloaded from the NIST Chemistry WebBook (9). Demonstration of L J-CS Scaling The global equation of state of fluids may be represented in a variety of formats. Most often this is accomplished by plotting isotherms of Z (= PV兾nRT ) or P as a function of either P, ρ, or V, where R is the gas constant. Several such representations of the equation of state of argon are shown in Figure 1. The solid curves in each plot represent isotherms (the six isotherms shown in Figure 1 correspond to T兾TC = 0.7, 1, 1.2, 1.5, 2.0, and 4.0). The dashed curve in each plot marks the vapor–liquid coexistence curve and an open circle marks the critical point. Although the four panels in Figure 1 all contain equivalent information, they each emphasize different aspects of the equation of state. For example, Figure 1A, which is the most common representation used in physical chemistry textbooks (2–5), clearly illustrates the deviation from ideal gas behavior (Z = 1) as a function of pressure along different isotherms. Figure 1B shows the same data plotted as a function of density rather than pressure, which has the advantage of very nicely untangling the isotherms. Figures 1C and 1D show the dependence of pressure on density and volume, respectively. These latter representations do not as clearly illustrate the deviations from ideal gas behavior, although the nearly linear dependence of P on ρ (or inverse dependence of P on V ) at high temperatures is reminiscent of ideal gas behavior. The equations of state of other fluids have the same general form as the argon results shown in Figure 1. In fact, the equations of state of many fluids are so similar in form that they can be nearly superimposed on each other by scaling the corresponding vertical and horizontal axis. Traditionally this is accomplished by using two critical constants to scale two experimental intensive variables (T and P or T and ρ). However, such a classical corresponding states scaling procedure does not reveal the true similarity between the equations of state of simple fluids as well as the LJ-CS procedure, which amounts to adjusting σ and ε of a LJ fluid so as to obtain the best overlap with the equation of state of an experimental fluid (6). In particular, the equations of state of various fluids were fit to that of a LJ fluid using a closed form expression for Z(ρ,T ), which very accurately represents LJ fluid computer simulation results (7). This LJ equation of state is incorporated into a fitting program implemented using IgorPro (version 4.0 by WaveMetrics Inc.), which compares LJ and experimental fluid Z values as a function of T and ρ, with σ
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Research: Science and Education
A
C
60
1.5
106 K
603 K
603 K
40
Z
P / MPa
1.0
20
0.5
critical point
106 K critical point
0.0
0 0
20
40
0
60
10
20
30
ρ / (mol/L)
P / MPa
B
D 1.5 60
603 K
1.0
Z
P / MPa
40
0.5
603 K
20
106 K
critical point
106 K
0.0
critical point
0
0
10
20
30
0.0
0.1
ρ / (mol/L)
0.2
0.3
V / (L/mol)
Figure 1. The equation of state of argon is represented as a function of either pressure, density, or volume along five isotherms (at approximately T = 106 K, 151 K, 181 K, 226 K, 301 K, and 603 K, corresponding to Tr = T/Tc = 0.7, 1.0, 1.2, 1.5, 2.0, and 4.0). The dashed curves mark the vapor–liquid coexistence curve and the critical point is marked by an open circle.
and ε treated as the only adjustable parameters. The resulting fits invariably converged successfully when using reasonable initial guess values for σ and ε (such as σ ≈ 300 pm, ε ≈ 1 kJ兾mol) to produce the best fit LJ constants for each fluid (see Table 1). The resulting global equation of state behavior of eight fluids obtained using this LJ-CS scaling procedure are illustrated in Figure 2. The five isotherms in this figure range from approximately the critical temperature to about twice the critical temperature (or T兾TC ≈ 1.0, 1.1, 1.2, 1.5, and 2.0). However, the densities and temperatures in Figure 2 are reduced using the LJ potential constants rather than the critical constants. Thus, the density is rendered dimensionless by multiplying by the cube of the LJ diameter, σ3 (with ρ expressed in molecules per unit volume), and the isotherms shown in Figure 2 are related to the LJ well depth, ε, by RT兾ε = 1.29, 1.42, 1.55, 1.94, and 2.58 (with the lowest isotherm in Figwww.JCE.DivCHED.org
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ure 2 corresponding to the lowest temperature). Any deviation of Z from the ideal gas equation of state value of Z = 1 represents non-idealities resulting from intermolecular interactions. At low densities, such non-idealities are often quantified using a virial expansion (Taylor expansion) of Z as a function of number density, ρ,
Z =
PV P = = 1 + B (T ) ρ + C (T ) ρ2 + ... nRT ρkT
(2)
where k is the Boltzmann's constant. Thus the second virial coefficient, B(T ) = (∂Z兾∂ρ)ρ=0, represents the first-order density correction to the ideal gas equation of state. This correction alone is sufficient to accurately predict the dependence of P on T and ρ for gases at pressures of 1 atm, or densities up to 1% of the density of a liquid. The second virial coefficient derives from interaction between pairs of molecules. If
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Research: Science and Education Table 1. Lennard-Jones Parameters LJ Parameters
Fluid
σ/pm
ε/(kJ/mol)
Ne
280.5
0.289
Ar
339.4
0.973
Xe
394.7
1.875
N2
361.2
0.811
O2
337.2
1.110
F2
326.4
1.037
CO
363.0
0.950
CH4
372.6
1.379
these interactions are assumed to be spherically symmetric, then the temperature dependence of B(T ) may be calculated directly from the interaction potential function, u(r) (10): B (T ) =
(1 − e
∞ 0
−u(r ) /(kT )
) 2πr dr 2
(3)
For a LJ fluid, eqs 1 and 3 may thus be combined to predict B(T ) at all temperatures. Since B(T ) has units of volume, it may be reduced to a dimensionless form using σ3. Thus one can compare the second virial coefficients of various fluids with each other and with a LJ fluid, using a corresponding states plot such as that shown in Figure 3. The data points in Figure 3 represent B(T ) values obtained from the NIST fluid database (8), which can be purchase through the NIST WebBook Internet site (9). These results clearly illustrate the good correspondence between second virial coefficients of various nonideal gases when plotted in this LJ-CS form. Note that the agreement between the B(T ) values is best at relatively high temperatures, near the Boyle temperature, TBoyle, at which B(T ) crosses zero. This is also the temperature at which a precise balance of competing attractive and repulsive intermolecular forces (represented by the LJ constants ε and σ, respectively) leads to nearly ideal (Boyle’s Law) equation of state behavior (to first order in ρ).
The statistical mechanical link between thermodynamics and intermolecular interactions implies that scaling an intermolecular interaction potential will propagate to scale all thermodynamic properties. This also means that fluids, which obey two-parameter corresponding states scaling, have interaction potentials that can be made to look the same by scaling two independent variables, such as the energy and distance axes used to plot the potentials. This scaling procedure is equivalent to determining the values of the two LJ parameters that best represent the equation of state of a given fluid. On the other hand, the equations of state of some fluids, such as He, CO2, H2O, n-butane (and longer chain hydrocarbons), differ significantly from a LJ fluid, in a way that cannot be corrected by two-parameter scaling. This implies that the intermolecular interaction potentials for these fluids differ fundamentally from a LJ form. In the case of He, this difference has been demonstrated to derive from its more quantum mechanical behavior, resulting from its low mass (11). In the case of CO2, an exceptionally large quadrupole moment (and nonspherical shape) undoubtedly play a significant role. For water, as most students would probably guess, it is hydrogen bonding that produces an intermolecular interaction potential of markedly non-LJ form. Chain molecules (such as n-alkanes) are clearly very nonspherical and thus their interactions are expected to differ from a spherical LJ form. Alternative scaling procedures have been proposed to extend the principle of corresponding states to some of these and other fluids, by invoking one or more additional scaling parameters (12–15). A commonly cited (and extensively tabulated) third scaling parameter is the so-called acentric factor, ω, defined as the experimentally measured value of ᎑log(P兾PC ) at a temperature of T兾TC = 0.7 (15). However, it should be noted that neither this procedure nor
1.5
Ar Ne Xe O2 LJ fit
N2 F2 CO CH4
1.0
Z
Topics for Discussion and Homework Lennard-Jones corresponding states (LJ-CS) scaling may be viewed as a theme that unites several central concepts in physical chemistry. The results shown in Figures 2 and 3 clearly illustrate the global similarities between the thermodynamic properties of different fluids and nonideal gases. The correlation between experimental and LJ fluid equations of state further implies that the intermolecular interaction potentials of simple atoms and molecules bear a close resemblance in shape to a LJ potential (eq 1). Numerical values of the LJ parameters of various fluids (Table 1) reveal differences in the repulsive core diameters and cohesive interaction energies of different molecules. These LJ-CS parameters are assured to be thermodynamically self-consistent with both fluid and vapor properties (as shown in Figures 2 and 3), and thus are more generally applicable than LJ parameters derived from only vapor phase data (6, 11). 144
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0.5
critical point 0.0 0.0
0.2
0.4
0.6
ρσ 3 Figure 2. Lennard-Jones corresponding states (LJ-CS) graph of the compressibility factor, Z = PV/nRT, of eight different fluids, plotted as a function of density along five isotherms. Both density and temperature are expressed in dimensionless Lennard-Jones units, ρσ3 and RT/ε = 1.29, 1.42, 1.55, 1.94, and 2.58, which correspond approximately to T/Tc = 1.0, 1.1, 1.2, 1.5, and 2.0. A color version of this figure is available in the Supplemental Material.W
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Research: Science and Education
any other proposed generalization of the principle of corresponding states is applicable to all of the above fluids. Even fluids that very accurately follow LJ-CS scaling behavior are not expected to have intermolecular interaction potentials that are strictly identical to that of a LJ fluid, since eq 1 is only an approximate model potential. The (1兾r)6 form of the attractive tail of the LJ potential is consistent with that expected for London dispersion, as well as dipole–induced– dipole and thermally angle averaged dipole–dipole interactions. However, the (1兾r)12 form of the repulsive portion of the LJ potential is dictated in large part by mathematical convenience, since changing variables to x = (σ兾r)6, reduces eq 1 to a quadratic equation. Furthermore, a LJ fluid is composed of molecules whose potential energies are strictly twobody additive, while the interaction potentials of real fluids in general contain higher-order multibody interaction contributions (16, 17). Thus the LJ-CS results presented in this work imply that, in spite of such multibody contributions, the interaction potential of a wide variety of simple atoms and molecules may be approximated by a two-body additive LJ function. Classroom and textbook applications of these concepts can take a wide variety of forms. Figures 1–3 (and Table 1) may be used as the basis for numerical examples and homework problems, such as predicting the LJ constants of a fluid from its critical properties, or prediction of the density, second virial coefficient, or Boyle temperature of a fluid (at a given pressure or temperature) from its LJ constants. Additional numerical examples and problems can make use of data available from the NIST Chemistry WebBook (9). This site contains a wide range of fluid thermodynamic data that can be used by students to produce graphs and tables of equation of state (and other) properties of over 30 fluids. Such
data could be used, for instance, to calculate fluid non-idealities by determining Z at a particular temperature and pressure. Alternatively, data from the WebBook could be used to calculate B(T ) at a particular temperature (by fitting Z to a linear function of ρ at low pressure), or even to predict higher virial coefficients by fitting Z to a polynomial function of ρ. A particularly appealing class of examples and homework problems could build on the connection between LJ and critical parameters. As a starting point, comparison of the LJ parameters of argon with its critical constants downloaded from the WebBook (TC = 150.69 K and ρC = 1兾VC = 13.4074 M, PC = 4.863 MPa, where VC is the critical volume expressed in L兾mol units) implies the following relations. RTC σ3 ≈ 1.29, ρCσ 3 ≈ 0.316 , and PC ε ε
(4)
(Note that the second of the above relation was obtained by first converting ρ C from molar to molecular units of molecules兾m3). The approximate validity of these relations for all of the fluids listed in Table 1 may readily be demonstrated using the corresponding critical constants (either downloaded from the WebBook or obtained from other standard reference tables). Furthermore, the critical constants of additional fluids may be used to estimate LJ parameters, as well as predict thermodynamic properties or test the degree to which LJ-CS scaling applies to different fluids. For example, Figures 2 or 3 combined with LJ parameters estimated with the aid of eq 4 can be used to predict P or B(T )as a function of ρ and/or T for fluids other than those listed in Table 1. Alternatively, experimental equation of state data downloaded from the WebBook can be converted to LJ reduced form (by converting T and ρ to RT兾ε and ρσ3, respectively) and then graphically compared with the results shown in Figures 2 and 3 to determine how well the fluid of interest conforms to LJCS scaling. W
Supplemental Material
Additional material, including more specific instructions for using the NIST Chemistry WebBook and sample homework problems, are available in this issue of JCE Online.
0
-2
B /σ3
= 0.117
Literature Cited Ar Ne Xe O2
-4
-6
N2 F2 CO CH4
LJ fit -8 1
2
3
4
5
RT / ε Figure 3. Lennard-Jones corresponding states (LJ-CS) graph of the second virial coefficient, B, of eight different fluids, plotted as a function of temperature, T, both expressed in dimensionless LennardJones units. A color version of this figure is available in the Supplemental Material.W
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1. Su, G.-S. Ind. Eng. Chem. 1946, 38, 803. 2. Berry, R. S.; Rice, S. A.; Ross, J. Physical Chemistry, 2nd ed.; Oxford University Press: New York, 2000. 3. Mortimer, R. G. Physical Chemistry, 2nd ed.; Harcourt Academic Press: New York, 2000. 4. McQuarrie, D. A.; Simon, J. D. Physical Chemistry: A Molecular Approach; University Science Books: Sausalito, CA, 1997. 5. Barrow, G. M. Physical Chemistry, 6th ed.; McGraw Hill: New York, 1996. 6. Ben-Amotz, D; Gift, A. D.; Levine, R. D. J. Chem. Phys. 2002, 117, 4632. 7. Johnson, J. K.; Zollweg, J. A.; Gubbins, K. E. Mol. Phys. 1993, 78, 591. 8. Lemmon, E. W.; Peskin, A. P.; McLinden, M. O.; Friend, D. G. NIST Standard Reference Database 12: Thermodynamic and
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9. 10. 11. 12.
Transport Properties of Pure Fluids Database, Version 5.0; NIST Standard Reference Data Program, Gaithersburg, 2000. NIST Chemistry WebBook. http://webbook.nist.gov/chemistry/ fluid (accessed Aug 2003). Hill, T. L. An Introduction to Statistical Thermodynamics; Dover Publications, Inc.: New York, 1986; p 267. Hirschfelder, J. O.; Curtiss C. F.; Bird, R. B. Molecular Theory of Liquids and Gases; Wiley: New York, 1965. Lee, B. I.; Kesler, M. G. AIChE J. 1975, 12, 510.
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13. Donohue, M. O.; Prausnitz, J. M. AIChE J. 1978, 24, 849. 14. Hacker, B. A.; Hall, C. K.. Ind. Eng. Chem. Fundam. 1985, 24, 262. 15. Kyle, B. G. Chemical and Process Thermodynamics, 3rd ed.; Prentice-Hall, Inc.: Upper Saddle River, NJ, 1999. 16. Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevado, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; Prentice Hall Inc.: Englewood Cliffs, NJ, 1986. 17. Barker J. A.; Henderson, D. Rev. Mod. Phys. 1976, 48, 587.
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