Universal Thermodynamics at the Liquid–Vapor Critical Point - The

Article pubs.acs.org/JPCB

Universal Thermodynamics at the Liquid−Vapor Critical Point Isaac C. Sanchez* and Kevin L. Boening McKetta Chemical Engineering Department, University of Texas, Austin, Texas 78712, United States ABSTRACT: For 68 fluids that include hydrogen bonding and quantum fluids, the fugacity coefficient that defines the residual chemical potential adopts a near universal value of 2/3 at the critical point. More precisely, the reciprocal of the fugacity coefficient equals 1.52 ± 0.02 and includes fluids as diverse as helium (1.50), dodecafluoropentane (1.50), and water (1.53). For 65 classical fluids, a dimensionless thermal pressure coefficient and internal pressure attain critical values of 1.88 ± 0.11 and 1.61 ± 0.11, respectively. From equations of state, values of these new critical constants have been calculated and agree favorably with experimental values. Specifically, for the critical fugacity coefficient, the following results were obtained for its reciprocal: van der Waals (1.44), lattice gas (1.43), scaled particle theory (1.46), and the Redlich−Kwong eq (1.50). The semiempirical Redlich−Kwong equation is also the most accurate for the thermal pressure coefficient (1.86) and internal pressure (1.53). Physical interpretations of these results are discussed as well as their implications for other critical phenomena.

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an ideal gas, B = 1, but in a liquid where attractive interactions dominate, B ≫ 1. For example, B ≃ 4 × 104 for water at room temperature. In a hard sphere (HS) fluid B < 1 and physically represents the insertion probability for randomly inserting a HS into a HS fluid.2,3 In general, B is an ensemble average of a Boltzmann factor that involves the interaction energy of a single molecule with all other molecules in the fluid.2 The compressibility factor/equation of state is given by a reciprocal relationship to eq 3:

INTRODUCTION It is well-known that the certain thermodynamic properties vanish or diverge as the liquid−vapor critical point is approached with universal exponents associated with the Ising universality class.1 As an example, the chemical potential vanishes along the critical isotherm as (μ − μc ) ∼ |ρ − ρc |δ

(1)

where μc is the critical chemical potential, ρc is the critical density, and δ is a critical exponent whose generally accepted value is 4.2. A property that remains finite and nonzero at the critical point is the compressibility factor, Zc = PcVc/RTc, which for most classical fluids lies in the range, 0.2 < Zc < 0.3. Another finite property at the critical point is the chemical potential, which can be written in terms of a dimensionless fugacity coefficient ϕ = 1/ZB, relative to an ideal gas at the same temperature and pressure (a residual chemical potential):

Z = 1 − ln B +

γṼ ≡ (2)

where B is the “Widom insertion factor” given by2,3 ln B = 1 − Z +

∫0

ρ

1−Z dρ ρ

̃ ≡ Pint

(3)

ln B dρ

(5)

V ⎛⎜ ∂P ⎞⎟ ≡ VγV /R R ⎝ ∂T ⎠V

(6)

V ⎛⎜ ∂U ⎞⎟ = VγV /R − PV /RT ≡ γṼ − Z RT ⎝ ∂V ⎠T

(7)

where U is the configurational or internal energy and T, P, and V are the temperature, pressure and volume, respectively. A recent review of internal pressure and the closely related thermal pressure coefficient of liquids and solutions is available.5

(4)

which illustrates that −kT ln B represents the transfer free energy for transferring a molecule at fixed density from an ideal gas to the fluid state. Alternatively, it can be thought of as the isothermal free energy change when an ideal gas at a molar density ρ is converted to a real fluid at the same density ρ. For © XXXX American Chemical Society

ρ

and internal pressure

If the chemical potential is referenced to an ideal gas at the same number density ρ as the fluid, then μ = μig (T , ρ) − RT ln B

∫0

Integrations in eqs 3 and 5 must be carried out at a fixed temperature. Recently, the following 2 dimensionless properties were defined:4 thermal pressure coefficient,

R

μ ≡ μ − μig (T , P) = RT ln ϕ = −RT ln ZB

1 ρ

Received: October 6, 2014 Revised: October 29, 2014

A

dx.doi.org/10.1021/jp510096e | J. Phys. Chem. B XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry B

Article

The choice of V/R as a scale parameter to render these properties dimensionless is not arbitrary. With this unique choice, models in the van der Waals genre yield dimensionless properties for saturated liquids that, to an excellent approximation, are a function of reduced density, ρR = ρ/ρc = Vc/V, only.4

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CALCULATION PROTOCOLS All data were obtained from the NIST Chemistry WebBook6 except for acetone and dimethyl ether which were taken from the NIST/TRC Web Thermo Tables. The thermal pressure coefficient γV ≡ (∂P/∂T)V was calculated from the temperature coefficient of the saturated vapor pressure curve γσ ≡ (∂P/∂T)σ evaluated at the critical point. The subscript σ indicates that the derivative is taken along the saturation curve. These 2 derivatives become equal at the critical point:7 γVc̃ = γσc̃

(8)

Figure 2. Compressibility factor for ammonia along the critical isotherm (405.4 K, Zc = 0.255). Solid line is a quartic polynomial best fit of the experimental data.6 This equation is then used in eq 3 to calculate the insertion factor Bc at the critical point.

Close to the critical temperature the vapor pressure varies linearly with temperature and data points within 1 degree of the critical point were used to determine the slope. See, for example, Figure 1.

Redlich−Kwong (RK) equation.9 Equations used in the calculations are summarized in the Appendix. Note for the cohort of non-hydrogen (44) and hydrogen bonding fluids (21), variations in γ̃cV (1.88 ± 0.11) and P̃ cint(1.61 ± 0.11) are comparable to those in Zc (0.270 ± 0.015). Also notice that individual averages of γ̃cV and P̃cint for non-hydrogen and hydrogen bonding fluids do not differ statistically from one another, whereas they do for Zc. As an example, there is significant difference in Zc values for propane and water (0.276 vs 0.229), whereas there is negligible difference in their dimensionless internal pressures (1.56 vs 1.54). This implies that γ̃cV and P̃ cint as critical constants are more characteristic of the critical state than Zc because non-hydrogen and hydrogen bonding fluids are not differentiated. And some deviation in these properties can be attributed to experimental uncertainties in measured critical volumes, which can be as much as 3%. In contrast, the fugacity coefficient is not scaled by the critical volume and is much less sensitive to experimental uncertainties in Vc. Although the dimensionless critical internal pressure and thermal pressure coefficient of water and propane are nearly identical, the critical internal pressure of water is 6 times larger than propane and its thermal pressure coefficient is 3.5 times larger. There are other choices of scale parameters that render these properties dimensionless. For example, Pcint could be divided by the critical pressure Pc and γcV could be multiplied by Tc/Pc. Neither of these choices works as well as Vc/RTc = Zc/Pc for the internal pressure or Vc/R = ZcTc/Pc for the thermal pressure coefficient. As mentioned previously, this choice of scale parameters is not arbitrary. For subcritical liquids, the choices V/RT and V/R for the internal pressure and thermal pressure coefficient are the choices required by any VDW type EOS to yield saturated liquid properties that are only a function of reduced density.4 The present study illustrates that continuing to use these scale parameters all the way to the critical point yields the nearly universal values for classical fluids of 1.61 ± 0.11 for the internal pressure and 1.88 ± 0.11 for the thermal pressure coefficient. As can be seen in Table 3, the Redlich−Kwong equation best describes the new critical constants, especially the fugacity

Figure 1. Saturated vapor pressure of water6 very near the critical temperature (647.1 K). The slope of this curve (2.632 bar/K) yields the thermal pressure coefficient (γcV) at the critical point. The dimensionless value is obtained by multiplying γcV by Vc/R and yields γ̃cV = 1.77.

Tabulated densities and pressures along the critical isotherm were used to calculate compressibility factors, which were curve-fitted with a quadratic or quartic polynomial in density. This fitted equation was used in eq 3 to calculate the insertion factor at the critical point. Figure 2 illustrates the curve fit for ammonia.

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RESULTS AND DISCUSSION Calculated values of the new critical constants are tabulated in Table 1 for 44 non-hydrogen bonding fluids and in Table 2 for 21 fluids capable of hydrogen bonding (H−X interactions possible where X is O, N, Cl, or F, albeit some are very weak). In addition, 3 quantum fluids are included in Table 2. Table 3 summarizes the data which are compared with equation of state predictions: van der Waals (VDW), scaled particle theory (SPT),4 lattice fluid model (LF),8 and the semiempirical B



Article

Table 1. Critical Properties of Non-Hydrogen Bonding Fluids; PVT Data Source6 fluid

Zc

γ̃cV

P̃cint

Bc

ZcBc

krypton carbon monoxide argon nitrogen triflouride nitrogen propene xenon oxygen methane hydrogen sulfide ethane, hexafluoro ethene ethane methane, trichloro fluoro methane, tetrafluoro cyclobutane, octafluoro sulfur hexafluoride ethane, 1,1,2-Cl, 1,2,2-F methane, chloro trifluoro methane, dichloro difluoro propane fluorine isobutane propane, octafluoro pentane, dodecafluoro ethane, 1,2-Cl, 1,1,2,2-F propyne carbon dioxide dinitrogen monoxide butane cyclopropane carbonyl sulfide cyclohexane methane, tetramethyl pentane, 2-methyl isopentane sulfur dioxide pentane hexane benzene toluene heptane octane decane

0.292 0.291 0.290 0.290 0.289 0.289 0.289 0.288 0.286 0.285 0.282 0.281 0.279 0.279 0.279 0.278 0.278 0.277 0.277 0.277 0.276 0.276 0.276 0.276 0.275 0.275 0.275 0.275 0.274 0.274 0.274 0.273 0.273 0.271 0.271 0.270 0.268 0.268 0.266 0.265 0.265 0.263 0.257 0.250 0.276 ± 0.009

1.70 1.75 1.66 1.86 1.71 1.87 1.70 1.69 1.68 1.83 2.02 1.76 1.78 1.92 1.94 2.13 1.92 2.02 1.89 1.90 1.83 1.70 1.87 2.08 2.19 1.88 1.89 1.89 1.84 1.90 1.77 1.78 1.91 1.88 1.97 1.93 1.93 1.94 1.92 1.91 1.91 1.90 1.99 2.10 1.88 ± 0.012

1.41 1.46 1.38 1.57 1.42 1.58 1.41 1.40 1.40 1.54 1.74 1.48 1.50 1.64 1.66 1.85 1.64 1.74 1.61 1.62 1.56 1.43 1.59 1.81 1.92 1.61 1.61 1.61 1.57 1.63 1.50 1.51 1.64 1.60 1.70 1.65 1.66 1.68 1.66 1.64 1.65 1.64 1.73 1.85 1.61 ± 0.012

5.15 5.13 5.15 5.15 5.18 5.25 5.19 5.21 5.25 5.31 5.35 5.35 5.40 5.42 5.37 5.43 5.39 5.50 5.44 5.47 5.46 5.39 5.47 5.51 5.46 5.48 5.50 5.50 5.52 5.54 5.51 5.56 5.58 5.58 5.65 5.62 5.67 5.68 5.72 5.69 5.81 5.85 5.98 6.20

1.50 1.50 1.49 1.49 1.50 1.51 1.50 1.50 1.50 1.51 1.51 1.50 1.51 1.51 1.51 1.51 1.50 1.52 1.51 1.51 1.51 1.49 1.51 1.52 1.50 1.51 1.51 1.51 1.51 1.52 1.51 1.51 1.52 1.51 1.53 1.52 1.52 1.52 1.52 1.51 1.54 1.54 1.54 1.54 1.51 ± 0.01

fluids plus 3 quantum fluids. Even water, whose properties are usually anomalous relative to other fluids, looks very normal at 1.53. Equivalently, the fugacity coefficient to an excellent approximation equals 2/3. Clearly, the critical residual chemical potential represents a very strong corresponding state. Note that the fluids in the Tables 1 and 2 are ordered in decreasing values of the critical compressibility factor, Zc. Nonhydrogen bonders on average have larger values of Zc. As the strength of the hydrogen bonding interaction increases, Zc decreases with water and methanol having the smallest values. Associated with this decrease in Zc is an increase in the insertion factor, i.e., as Zc ↓, Bc ↑. These trends have a physical interpretation that can be gleaned from the thermodynamic cycle shown in Figure 3.

coefficient. However, the RK equation is semiempirical and severely underestimates the occupied critical volume fraction, ηc. As in the VDW model, the b parameter in the RK model equals 4 times the volume of a hard sphere (HS) of diameter σ (see the Appendix). And since Vc = b/c, ηc = c/4 = 0.0649, a value more than a factor of 2 smaller than determined from computer simulations for Lennard-Jones fluids (∼0.16).10 Given that the LJ fluid equation of state (EOS) describes the EOS of simple fluids such as the inert elements,11 suggests the RK EOS severely underestimates ηc. Nevertheless, the RK EOS works remarkably well in predicting the new critical constants. As constants characterizing the critical state, neither γ̃cV or P̃cint compare to the fugacity coefficient since it adopts a near universal reciprocal value of ZcBc = 1.52 ± 0.02 for 65 classical C



Article

Table 2. Critical Properties of Hydrogen Bonding and Quantum Fluids; PVT Data Source6 fluid

Zc

ethane, 1,1-Cl, 1-F methane, dichloro fluoro dimethyl ether ethane, 1-Cl, 1,2,2,2-F ethane, pentafluoro methane, chloro difluoro ethane, 2,2-Cl, 1,1,1-F ethane, 1-Cl, 1,1-F propane, 1,1,1,3,3,3-F propane, 1,1,1,3,3-F ethane, 1,1,1,2-F methane, trifluoro ethane, 1,1,1-F ammonia ethane, 1,1-F methane, difluoro methane, fluoro acetone water deuterium oxide methanol

0.271 0.270 0.270 0.269 0.268 0.268 0.268 0.268 0.267 0.266 0.260 0.258 0.255 0.255 0.252 0.243 0.240 0.236 0.229 0.226 0.221 0.255 ± 0.016

helium hydrogen neon

P̃ cint

Bc

ZcBc

1.66 1.60 1.59 1.70 1.72 1.60 1.71 1.65 1.77 1.77 1.70 1.61 1.53 1.55 1.62 1.55 1.46 1.53 1.54 1.56 1.74 1.63 ± 0.08

5.56 5.59 5.69 5.65 5.65 5.66 5.66 5.68 5.81 5.72 5.88 5.95 6.05 6.14 6.17 6.50 6.64 6.76 6.63 6.78 6.94

1.51 1.51 1.54 1.52 1.52 1.52 1.52 1.52 1.55 1.52 1.53 1.53 1.54 1.56 1.56 1.58 1.59 1.60 1.53 1.53 1.53 1.54 ± 0.03

0.58 1.07 1.31

4.94 4.91 4.95

1.50 1.49 1.50

γ̃cV 1.93 1.87 1.86 1.97 1.99 1.87 1.98 1.92 2.04 2.03 1.96 1.87 1.78 1.80 1.87 1.79 1.70 1.76 1.77 1.79 1.96 1.88 ± 0.09 Quantum Fluids 0.88 1.37 1.61

0.303 0.303 0.303

Table 3. Fluid critical property summary nodels critical property

non-hydrogen bonding (44)

hydrogen bonding (21)

RK

SPT

VDW

LF (r = 2)

Zc

0.276 ± 0.009

0.255 ± 0.016

0.333

0.360

0.375

0.375

1.88 ± 0.09

1.86

1.73

1.50

1.58

1.63 ± 0.08

1.53

1.37

1.13

1.21

5.56−6.94 1.54 ± 0.03

4.51 1.50

4.05 1.46

3.84 1.44

3.84 1.44

0.270 ± 0.015 γ̃cV P̃ cint

1.88 ± 0.012 1.88 ± 0.11 1.61 ± 0.012 1.61 ± 0.11

Bc ZcBc

5.13−5.98 1.51 ± 0.01 1.52 ± 0.02

From a physical viewpoint, the most important free energy change occurs in going from the ideal gas state at Tc, ρc to the critical fluid (−ln Bc). Since the density of both states is the same, all changes are associated with “self-solvation.”3,11,12 Experimentally, Bc > 1 implies that favorable attractive interactions offset the unfavorable entropic change that occurs on self-solvation. The larger B becomes, the more dominant is energetics over entropics. As has been emphasized previously,3,11,12 fluid thermodynamics are completely captured in the insertion factor, B. The fugacity product ZB is not the product of two independent factors. As seen clearly in eq 5, Z is a function of B so that

Figure 3. Thermodynamic cycle illustrates various dimensionless free energy contributions to the critical fluid relative to two ideal gas states.

In going from the ideal gas state at Tc, ρc to the critical state, the favorable change in free energy is −ln Bc. In going from the ideal gas state at Tc, ρc to the one at Tc, ρc increases the free energy by −ln Zc. The latter free energy change is purely entropic, whereas the −ln Bc change involves both energetic and entropic changes.

⎡ 1 ZB = ⎢1 − ln B + ⎣ ρ D

∫0

ρ

⎤ ln B dρ⎥B ⎦

(10)



Article

A possible place where the current findings might prove useful is in predicting critical point parameters for substances with high critical temperatures such as ionic liquids and metals.14 Additionally, corresponding state concepts such as Zeno lines and Batchinsky’s law need to be reinvestigated in light of the new critical constants reported here.15−17 It is well-known that the liquid−vapor and liquid−liquid critical states are isomorphic (Ising universality class). An interesting question is whether or not the strong corresponding state observed in the liquid−vapor critical chemical potential is also seen in liquid−liquid critical behavior. This will be a worthy subject for future investigation.

The residual chemical potential can be expressed in terms of a residual enthalpy and entropy: μ R = H R − TS R = −RT ln ZB

(11)

Although the EOS models all predict that the residual enthalpy and entropies are finite at the critical point, it appears that both diverge. For example, the residual enthalpy can be expressed as ⎛ ∂[ln Z B] ⎞ HR = − (1/Z − 1)γσ̃ + ⎜ ⎟ RT ⎝ ∂[ln T ] ⎠σ ⎛ ∂[ln Z B] ⎞ = − (1/Z − 1)γσ̃ − Tασ ⎜ ⎟ = − ln Z B + S R /R ⎝ ∂[ln ρ] ⎠σ

CONCLUSIONS

■

APPENDIX

The residual chemical potential at the liquid−vapor critical point is a very strong corresponding state where the fugacity coefficient adopts a near universal value of 2/3 for 68 fluids that include both quantum fluids and water. The reciprocal of the fugacity coefficient, ZcBc equals 1.52 ± 0.02. For 65 classical fluids, two other critical properties, the dimensionless thermal pressure coefficient (1.88 ± 0.11) and internal pressure (1.61 ± 0.11) have also been tabulated. Overall, no statistical difference exists between hydrogen and non-hydrogen bonding fluids for these constants. As a consequence, these new critical constants are more universal than critical compressibility factors because the latter differentiates between non-hydrogen and hydrogen bonding fluids, while the former does not. As an example, the critical internal pressure is 1.56 for propane and 1.54 for water, while the corresponding compressibility factors are 0.276 and 0.229. Calculated equation of state values for the fugacity coefficient and the two new critical constants from four different equations agree favorably with experimental values. The semiempirical Redlich−Kwong equation yields the most satisfactory results. Since the liquid−vapor critical point is a member of the Ising universality class, these results suggest that other Ising-like critical phenomena (liquid−liquid and ferromagnetic) might also exhibit universal critical free energies and other thermodynamic properties.

(12)

where ασ is the thermal expansion coefficient along the saturation curve. As the critical point is approached, ασ diverges.7 The relationship of ασ to the usual thermal expansion coefficient αP is given by ⎛ γ ⎞ ασ = αP ⎜⎜1 − σ ⎟⎟ γV ⎠ ⎝

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(13)

On the liquid side, γσ < γV so that, ασ > 0, whereas on the vapor side, γσ < γV and ασ > 0. At the critical point both ασ and αP diverge.7 As can be seen in the Tables, fluids capable of hydrogen bonding exhibit somewhat lower values of Zc than nonhydrogen bonding fluids. But also notice that the higher alkanes, such as decane, also have Zc values similar to those of hydrogen bonding fluids. In addition to strong intermolecular interactions that tend to reduce Zc, there is also a size effect that reduces Zc. The lattice fluid model8 qualitatively captures this size effect and yields Zc = 0.386 for a monomer fluid and 1/3 for infinitely long chain. The corresponding values for the reciprocal fugacity are ZcBc = 1.43 for the monomer fluid and e3/2/3 = 1.49 for the infinitely long chain. So the LF model predicts the fugacity should exhibit a small chain size effect, which can be observed in Table 1 in going from methane to decane. The LF model also predicts a size effect for γ̃cV and P̃cint, both increasing slowly with chain length (see Appendix) and diverging as r → ∞. Again this tendency can be detected in Table 1 from methane to decane. As can be seen in the Tables, none of the EOSs correctly predict Zc (overestimate) or Bc (underestimate) individually, but approximate the product ZcBc relatively well (compensating errors). The empirical EOSs of Lemmon and Span13 (LS), yield accurate values of Zc and Bc, but overestimates γ̃cV and P̃cint. For example, γ̃cV and P̃ cint for xenon (1.70, 1.41) vs LS (2.01, 1.72), neopentane (1.88, 1.60), vs LS (2.32, 2.05), and hexafluoroethane (2.02, 1.74) vs LS (2.40, 2.12). In general, the LS EOS overestimates these properties by 10 to 20% and is inferior to the RK EOS, which slightly underestimates these same properties (see Table 3). Additionally, the LS EOSs involve 12 adjustable parameters for every fluid even when the expressed in reduced variables; i.e., the EOSs do not satisfy a corresponding states principle. It should not come as a complete surprise that the critical properties and free energy are correctly predicted, but a second order property (second derivative on the free energy) such as the thermal pressure coefficient is not.

Equations of state: van der Waals (VDW), scale particle theory (SPT),4 Redlich−Kwong (RK),9 and lattice fluid (LF)8 van der Waals

Z=

9ρ 3 − R; 3 − ρR 8TR

ln B =

− ρR 3 − ρR

+

9ρR 4TR

ZC = 3/8

+ ln(1 − ρR /3)

Bc = (2/3) exp(7/4) = 3.836 Zc Bc = (1/4) exp(7/4) = 1.438

γṼ = E

3 ; 3 − ρR

̃ = γVc = 3/2Pint

9ρR 8TR

;

c

̃ = 9/8 Pint



Article

⎧ ρ (1 + r ) Z = 1 − ⎨r + R 2TR ⎩

Scale Particle Theory 2

z=

⎪

αρR

⎪

1+η+η ; − TR (1 − η)3

η = ηc ρR ;

+

73 − 7 = 0.1287 12

ηc =

1 + ηc + ηc 2 (1 − ηc )3

−α=

1975 + 157 73 = 0.360 768

1 ⎛ 2η 2 + 5η − 1 ⎞ 5 + ln(1 − η) + ⎜ ⎟ − 2Z 2 ⎝ (1 − η)3 ⎠ 2

ln B =

Bc = [1 − 1/(1 +

̃ = Pint

1 + ηc + ηc 2

γVc̃ =

=

(1 − ηc )3

TR

1063 + 109 73 = 1.731 1152

γṼ = Z +

Redlich−Kwong

Z=

ρR 1 − 1 − cρR 3c(1 + cρR )TR 3/2 1/3

c=2

− 1;

ln B = − +

Zc = 1/3;

cρR 1 − cρR

+

■

3c(1 + cρR )TR

3/2

r r r r

=2 =4 =9 →∞

;

1+ r ; 2

γVc̃ = Zc +

⎧1.21 ⎪ ⎪ 3/2 c ̃ Pint = ⎨ ⎪2 ⎪ ⎩∞

r=2 r=4 r=9 r→∞

AUTHOR INFORMATION

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS Financial support from the William J. Murray, Jr. Endowed Chair in Engineering is gratefully acknowledged.

3c 2TR 3/2

Zc Bc = 1.502 ρR 1 + 1 − cρR 6c(1 + cρR )TR 3/2 ρR ̃ = Pint 2c(1 + cρR )TR 3/2

γṼ =

̃ = Pint

REFERENCES

(1) Sengers, J. V., Levelt-Sengers, J. M. H. I. Progress in Liquid Physics; Croxton, C. A., Ed.; Wiley: Chichester, U.K., 1978: Ch. 4. (2) Widom, B. Potential-Distribution Theory and the Statistical Mechanics of Fluids. J. Phys. Chem. 1982, 86, 869−872. (3) Stone, M. T.; in’t Veld, P. J.; Lu, Y.; Sanchez, I. C. Hydrophobic/ Hydrophilic Solvation: Inferences from Monte Carlo Simulations and Experiments. Mol. Phys. 2002, 100, 2773−2792. (4) Sanchez, I. C. Dimensionless Thermodynamics: A New Paradigm for Liquid State Properties. J. Phys. Chem. B 2014, 118, 9386−9397. (5) Marcus, Y. Internal Pressure of Liquids and Solutions. Chem. Rev. 2013, 113, 6336−6551. (6) Afeefy, H. Y., Liebman, J. F., Stein, S. E. Thermophysical Properties of Fluid Systems. In NIST Chemistry WebBook, NIST Standard Reference Database Number 69, Lemmon, E. W., McLinden, M. O., Friend, D.G., Eds.; National Institute of Standards and Technology: Gaithersburg MD; http://webbook.nist.gov.

= 4.507

c

⎧1.58 ⎪ ⎪1.87 γVc̃ = ⎨ ⎪ 2.36 ⎪ ⎩∞

r)

Notes

+ ln(1 − cρR )

ln(1 + cρR )

1 1 + = 1.86 1−c 6c(1 + c)

r]

*(I.C.S.) E-mail: [email protected].

⎤ ⎡ 2 c 1 Bc = (1 − c)(1 + c)1/3c exp⎢ − + ⎥ ⎣ 1−c 3c(1 + c) ⎦

γ̃Vc =

r )]r exp[1 +

r )ρR /TR

Corresponding Author

Vc = b /c ρR

ρR (1 +

2TR 1+ r c ̃ Pint= 2

c Pint = α = 1.371

V a − V−b (V + b)T 3/2

r )] + (1 +

r }Zc

⎧1.43 r = 1 Zc Bc = ⎨ ⎩1.49 r → ∞

αρR

Z=

⎪

⎧ e 2 /2 = 3.694 r = 1 ⎪ Bc = ⎨ 3.84 r=2 ⎪ 3/2 ⎩ e = 4.48 r → ∞

⎡ 2863 − 59 73 ⎤ 181 + 7 73 exp⎢ ⎥ = 1.456 ⎣ ⎦ 1536 768

1 + η + η2 ; (1 + η)3

γṼ =

⎪

ρR

ln B = r ln[1 − ρR /(1 +

⎡ 2863 − 59 73 ⎤ Bc = (1 − ηc ) exp⎢ ⎥ = 4.047 ⎣ ⎦ 1536

Zc Bc =

r )] ⎫ ⎬ ⎭

r ) ln[1 − ρR /(1 +

ZC = (1 + r ){r ln(1 + 1/ r ) + 1/2 − ⎧ 0.386 r = 1 ⎪ = ⎨ 0.375 r = 2 ⎪ ⎩ 1/3 r → ∞

α = (6529 + 715 73 )/9216 = 1.371

Zc =

r(1 +

1 = 1.53 2c(1 + c)

Lattice Fluid

(r = 1 is the lattice gas model.) F



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Universal Thermodynamics at the Liquid–Vapor Critical Point - The

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