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Theoretical Rationale for a Thermodynamic Glass State. Isaac C. Sanchez and Sean P. O'Keefe. The Journal of Physical Chemistry B 2016 120 (35), 9443-9...
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New Zeno-Like Liquid States Isaac C. Sanchez,* Sean O’Keefe, and Jeffrey F. Xu

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McKetta Department of Chemical Engineering, University of Texas, Austin, Texas 78712, United States ABSTRACT: A Zeno line is a straight line in a fluid’s temperature-density plane where the compressibility factor equals unity and stretches from supercritical to subcritical regimes. A second unexpected linearity also occurs in liquids in their normal liquid range (NLR) where the compressibility factor approaches zero. This Zeno-like line includes liquids as diverse as monatomic and diatomic elements to hydrogen bonding liquids, with water and helium being the only currently known exceptions. Remarkably, this second linearity is also present in polymeric liquids, molten metals, and salts, i.e., it is a generic characteristic of liquids with negligible or low vapor pressures. This observation yields the following new corresponding states principle: Saturated liquid densities are a linear f unction of temperature in the NLR for molecular, polymeric, inorganic, ionic, and metallic liquids and superpose to form a single master curve. Another Zeno-like linearity in the NLR has also been identified for the configurational energy of many liquids. Extension of this line to zero temperature defines the ground state configurational energy of the hypothetical disordered liquid (glass).

1. INTRODUCTION Although not one of the original paradoxes offered by the ancient Greek philosopher Zeno, it is an unexpected surprise that the compressibility factor Z = 1 locus in the temperaturedensity plane is linear for many fluids, not only in the supercritical region, but also well into the subcritical regime. This unique locus is also known as the “Boyle line”1 and was discovered over 100 years ago by Batschinski.2 This line intersects the temperature axis at the Boyle temperature T*B where the second virial coefficient vanishes. It serves as an index for the competition between attractive and repulsive forces: when Z > 1 repulsive forces dominate (hard fluid) the equation of state and when Z < 1 attractive forces dominate (soft fluid).3 On the Z = 1 line, these opposing forces balance and the fluid exhibits pseudo ideal gas behavior. The Zeno line starts at zero density at the Boyle temperature, which typically is 2 to 2.5 times the critical temperature, and when it reaches the critical temperature, the density is about 2 times greater than the critical density with a pressure about 8 times greater than the critical pressure. For many fluids, the linear Zeno line extends deep into the subcritical liquid regime, sometimes covering a span of several hundred degrees. It can also be shown that the Z = 1 line is the only iso-Z line that converges to a unique temperature as density approaches zero; Z > 1 lines diverge positively away from the temperature axis with decreasing density and Z < 1 lines diverge negatively and terminate on the coexistence (COEX) curve.4 As the temperature approaches absolute zero, the extrapolated Zeno line enters the metastable liquid regime and intersects the density axis at the Boyle density ρ*B that physically represents the hypothetical density of the disordered (glassy) liquid at absolute zero. With the exception of the van der Waals equation of state (VDW EOS), no other known nonempirical EOS captures the Zeno linearity all the way from supercritical to subcritical regimes.5 © 2016 American Chemical Society

Just as the Z = 1 locus defines the Zeno line, there is another locus in the temperature-density plane that defines another striking linearity in liquids. This new linear locus is defined by liquid states for which Z → 0. On the liquid−vapor coexistence curve, the saturated vapor compressibility factor Zvap → 1 with decreasing temperature, whereas for the saturated liquid, Zliq → 0 as the temperature decreases. The saturated vapor becomes more like an ideal gas because of decreasing vapor pressure (Psat → 0), whereas on the liquid side of the COEX curve, Zliq = Psat/kT ρliq → 0 for the same reason. For example, for many liquids at their normal boiling points, − ln Zliq = 5.5 ± 0.2 and represents a corresponding state.6 This compressibility factor becomes even smaller as the triple point is approached. Thus, in the normal liquid range (NLR ⇒ vapor pressures less than 1 bar), which for some liquids can extend over a hundred degrees, Zliq ≃ 0. Apparently unrecognized or overlooked in the past, saturated liquid densities in the NLR are linear-like in temperature and form a strong corresponding states principle (CSP). When this line is extrapolated into the metastable liquid regime all the way to absolute zero, it intersects, or comes very close to, the Boyle density ρ*B , i.e., it has an intercept in common with the first Zeno line. This new Zeno-like line includes liquids as diverse as monatomic and diatomic elements and hydrogen bonding ammonia and methanol, but with helium and water as the only known exceptions. This striking linearity is also observed in polymer liquids, molten metals, and salts, i.e., it is a common characteristic of liquids with negligible or very low vapor pressures. Another unexpected linearity of possible greater theoretical significance has also been identified. This second Zeno-like Received: February 8, 2016 Revised: March 30, 2016 Published: April 1, 2016 3705

DOI: 10.1021/acs.jpcb.6b01364 J. Phys. Chem. B 2016, 120, 3705−3712

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The Journal of Physical Chemistry B anomaly involves the configurational energy that is shown to vary linearly with density or temperature in the NLR. Extension of this line to zero temperature defines the ground state configurational energy of the hypothetical disordered liquid (glass). This linear dependence on density implies that the usual VDW approximation for the configurational energy is valid in the NLR for a wide variety of liquids that include some hydrogen bonding liquids such as ammonia.

2. NEW ZENO-LIKE STATES 2.1. Insights from the Van Der Waals Equation. The VDW EOS can be expressed in reduced variables as follows: 9ρ P 3 ≡Z= − R ρkT 3 − ρR 8TR

(1)

where density and temperature are reduced by their respective critical values, ρR = ρ/ρc and TR = T/Tc. Equating Z to unity yields the first linear Zeno line: ρ ρ TB T + B ≡ B + B =1 27/8 3 TB* ρB*

(2)

where the Boyle temperature T*B /Tc = 27/8 and the Boyle density ρB*/ρc = 3. When these Boyle parameters are used to reduce temperature and pressure the VDW EOS adopts a deceptively simple form:

Z=

ρ̃ 1 − 1 − ρ̃ T̃

(3)

where ρ ̃ = ρ /ρB* and T̃ = T /TB*. In the NLR where Zliq ≃ 0, setting eq 1 equal to zero in the NLR results in a parabolic equation and a corresponding states principle (CSP):

TR =

9 ρ (1 − ρR /3) 8 R

Figure 1. Reduced temperature-reduced density plot of saturated liquid densities in the normal liquid range (vapor pressures less than 1 bar) for the indicated liquids as suggested by the VDW equation, eq 4. Note the linearity, which is not predicted by the VDW equation.

(4)

The density intercept is designated as ρ*g , where the subscript g reminds us that this intercept is associated with the hypothetical glass properties of the liquid at absolute zero. The temperature intercept is designated as T*c because this represents the theoretical critical temperature of a fluid in which the liquid side of the liquid−vapor coexistence curve remains linear all the way to the critical temperature (see Figure 3). It is believed that ρ*g = ρ*B , i.e., the 2 Zeno lines share a common density intercept (see Discussion). Each liquid satisfies the following simple linear equation:

This equation suggests saturated liquids in the NLR should obey a CSP when temperature and density are reduced by their respective critical values. This idea is tested in Figure 1 for 11 liquids. As can be seen, the CSP holds to some degree, but not as well as might be expected. However, notice the apparent linearity of the density data with temperature, something not predicted by the VDW equation. This linear-like behavior may come as a mild surprise since water, our most familiar liquid, does not behave this way. Also note that some liquid reduced densities exceed 3, which is the maximum allowed by the VDW equation. It should also be mentioned that the extension of the 3-D Ising model to low temperatures does not predict linearlike behavior.7,8 Also to be noted, Guggenheim’s classic paper on corresponding states, empirically suggested that the liquid branch of the coexistence curve possesses a linear contribution at low temperatures.9 Now if eq 3 is set to zero, a different CSP is predicted: T̃ = ρ ̃(1 − ρ ̃)

ρ T + =1 ρg* Tc*

(6)

where ρ and T are, respectively, the density and temperature of a saturated liquid in the NLR. This equation also holds along low-pressure isobars where Z ≃ 0. Figure 3 schematically illustrates the 2 Zeno lines along with their temperature and density intercepts. Table 1 is a summary of both types of intercepts for various fluids, and as can be seen, for the most part ρg* ≃ ρB*. Although eq 5 incorrectly predicts a nonlinear dependence of temperature on density, it does predict that as T → 0, ρ → ρ*B , i.e., the saturated liquid line on the coexistence curve when extended to absolute zero intersects with the Boyle density intercept of the first Zeno line. Others

(5)

Rescaling the temperature and pressure with the Boyle parameters yields a much better CSP (not shown), but the outstanding feature of the data, i.e., the apparent linearity is not captured. This linearity can be exposed more clearly by defining intercepts for each liquid line and then replotting the data as illustrated in Figure 2 (see Table 1). 3706

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extent of the NLR depends on the triple point pressure; the lower it is, the greater the temperature rise required to reach atmospheric pressure. For decane, the triple point pressure is O(10−5) bar, whereas for xenon it is about 0.8 bar, and hence the big difference in the NLRs. 2.2. Polymeric, Inorganic, Ionic, and Metallic Liquids. Polymer liquids should have a large NLR since they have negligible vapor pressures above their melting or glass temperatures. And indeed, it is well-known that polymer liquid densities are a linear function of temperature without any known exceptions for both nonpolar and polar polymers.11 The universal equation for polymer liquid density is of the same form as eq 6 and the characteristic density and temperature parameters (ρ*,T*) have been tabulated for many polymers.11 The physical meaning of this polymer density parameter is the same as it is above for molecular liquids, i.e., it is the hypothetical density at absolute zero of the disordered liquid (glassy) state. Most organic polymers tend to thermally decompose by 300 C and this places an upper bound on the NLR. Pushing this concept even farther, molten metals and salts have very low or negligible vapor pressures and large NLRs, and once again, the density of liquid metals and salts obey a linear density law.12,13 In Table 3 density and temperature intercepts (ρ*g ,T*c ) have been determined for some metals and salts and have been used to construct the master plot shown in Figure 5. 2.3. Linearity in the Configurational Energy. In the NLR, the internal energy of vaporization ΔEvap to an excellent approximation equals the liquid configurational energy, Uliq. Thermal energy contributions to the internal energy cancel in forming ΔEvap, and the configurational energy of the dilute gas phase is to a good approximation negligible. Thus, |Uliq| = ΔEvap

Figure 2. Dimensionless density versus temperature plots of the data shown in Figure 1. Density and temperature are reduced by the parameters listed in Table 1. Typical deviations from the linear line are of the order of 0.5% and often within experimental uncertainties.

(7)

in the NLR. When ΔEvap is plotted against temperature or density, linear plots are obtained with slope m:

have come to the same conclusion that the extension of the VDW EOS to zero temperature intersects the density axis at ρB*, but use a different argument.10 Table 1 divides fluids into 3 different groups differentiated by their Trouton’s constants, the dimensionless entropy of vaporization at the normal boiling point:6

m=−

ρg* d|U | Tc* d|U | = >0 Ug* dT Ug* dρ

⎛ ⎞ ρ⎟ U T =1−m = 1 − m⎜⎜1 − Ug* Tc* ρg* ⎟⎠ ⎝

⎧ 8.5 to 9.5 Group I ⎪ ΔSvap/k = ⎨ 9.8 to 10.8 Group II ⎪ ⎩>11.2 Group III

(8)

Extension of either line defines the ground state energy Ug* of the disordered liquid (glass) at zero temperature. Some glass parameters are tabulated in Table 1. Although methanol exhibits linear density−temperature behavior, its configurational energy exhibits a slight negative curvature. The linearity is illustrated in Figure 6. An interesting implication of the configurational energy linearity is that the usual VDW approximation that configurational energy varies diectly with density is valid for liquids in the NLR. In the VDW approximation, − UVDW = 2aρ, where a is the usual VDW parameter that measures the attractive strength of the interaction. From eq 8 we see that the VDW parameter can be identified with the following:

Note that the ranges do not overlap; some liquids may not neatly fall into any group. Group I includes the inert elements, diatomic molecules, and methane, but not quantum liquids. Group II includes nonpolar and most polar organics as well as some weakly hydrogen bonding fluids, and Group III includes most strong hydrogen bonding fluids. Water does not fall into any of the Groups. In Table 2, Zeno line and saturated liquid data in the NLR are tabulated for decane and then plotted in Figure 4. The figure illustrates the striking linearity of the 2 lines and how they tend to merge at absolute zero in conformity with the VDW prediction as well as with other arguments.10 The linearity of the saturated density data in the NLR for decane extends over 200 K, whereas for xenon it is only about 4 K. The

a=

1 d |U | m |Ug*| = 2 dρ 2 ρg*

(9)

The VDW value for the “a” parameter is 3707

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The Journal of Physical Chemistry B Table 1. Summary of Parameters Associated with Zeno-Like Statesa liquids

TB* Tc

ρB* ρc

ρc

argonb kryptonb xenonb methaneb oxygenb nitrogenb CO average

2.71 2.72 2.74 2.67 2.64 2.57 2.56 2.66

3.52 3.45 3.47 3.50 3.51 3.63 3.62 3.53

3.60 3.66 3.66 3.55 3.58 3.64 3.73 3.63

ethylene ethaneb propane butaneb pentaneb hexane decane dodecane C6H12 C6H6 C6H5CH3 C(CH3)4 H2S CF4 CHF3 CH2F2 CH3F C2F5H C2F6 average

2.54 2.49 2.46 2.39 2.28 2.32 2.21 2.13 2.36 2.34 2.41 2.38 2.58 2.31 2.39 2.48 2.64 2.26 2.27 2.38

3.65 3.71 3.74 3.78 3.90 3.89 4.04 4.24 3.76 3.81 3.79 3.68 3.66 3.74 3.94 4.04 3.81 3.91 3.89 3.84

3.70 3.65 3.73 3.80 3.85 3.95 4.12 4.33 3.89 3.86 3.91 3.77 3.79 3.70 4.02 4.18 4.05 3.96 3.97 3.91

ammonia methanol

2.54

4.16

4.26 3.90

Tc* Tc

ρg*

|Ug*| RTc*

Group I 2.08 2.09 2.10 2.18 2.19 2.10 2.04 2.11 Group II 2.19 2.17 2.14 2.11 2.10 2.03 1.96 1.90 2.00 1.98 2.05 2.00 2.06 2.09 2.02 2.00 1.96 1.99 1.95 2.04 Group III 2.07 2.09

m=

ρg* d |U | Ug* dρ

Trouton’s constant

3.4 3.4 3.4 3.2 3.3 3.6 3.9 3.5

1.26 1.25 1.27 1.23 1.25 1.29 1.30 1.26

8.9 9.0 9.2 8.9 9.1 8.7 8.8 8.9

3.8 3.8 4.3 5.2 5.1 5.6 7.4 8.6 5.2 5.2 5.5 5.1 4.1 4.6 5.3 5.2 4.9 5.9 5.6 5.3

1.38 1.31 1.36 1.37 1.41 1.36 1.41 1.42 1.36 1.42 1.44 1.37 1.32 1.35 1.33 1.30 1.30 1.36 1.33 1.36

9.6 9.6 9.8 9.9 10.0 10.1 10.6 10.7 10.2 10.5 10.4 9.7 10.5 9.8 10.6 10.8 10.3 10.5 10.6 10.2

4.8

1.28

11.7 12.6

Temperatures and densities reduced by their respective critical values. Trouton constants (entropies of vaporization) define fluid Groups I, II, & III (see text). Zeno and configurational energy parameters omitted for methanol because the loci exhibit slight negative curvature, but saturated density−temperature linearity still holds. bZeno intercepts (T*B , ρ*B ) from ref 3. Present study used NIST Web Book database (http://webbook.nist. gov). a

a

ρc RTc

=

9 8

linearity implies that the liquid side of the liquid−vapor coexistence curve is linear-like in temperature in the NLR and represents a very strong corresponding states principle: saturated liquid densities are a linear f unction of temperature in the NLR for molecular, polymeric, inorganic, ionic, and metallic liquids and superpose to form a single master curve. Although others might exist, helium and water currently remain the only outstanding exceptions to this CSP. As Figure 3 schematically illustrates, if this linear behavior were to persist to low density, then the critical temperature would approach the theoretical temperature intercept Tc* at zero density. The only liquid that might exhibit this type of linear behavior all the way to the critical point is a polymeric liquid of high molecular weight. According to the lattice fluid model of chain molecules14 as chain length increases Tc*/Tc → 1 and the critical density and pressure approach zero so that ρg*/ρc → ∞. This general trend can already be discerned in Table 1 among the normal alkanes in going from ethane to dodecane. For very long chains, the saturated vapor pressure would remain negligibly small all the way to the critical point so that the requirement that Z ≃ 0 would be satisfied over the

(10)

Using the tabulated values for the seven monatomic and diatomic Group I liquids given in Table 1, the calculated VDW “a” parameter averages 1.28 as compared with 9/8. The effective a parameter for Group II liquids averages higher at 1.85.

3. DISCUSSION Liquids in their NLR (vapor pressures less than 1 bar) exhibit striking linear density behavior with temperature as illustrated in Figures 2 and 5; these data superpose to form a single master curve. Deviations from linearity are typically less than 0.5%, i.e., often within the experimental accuracy of the data. This linearity, which also holds along isobars where Z ≃ 0, has been noted before for specific liquids, but its broad generic character and implications have gone unrecognized until now. It includes hydrogen bonding liquids such as ammonia and methanol as well as polymeric, inorganic, ionic, and metallic liquids, whose vapor pressures are negligible or very low. This ubiquitous 3708

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Figure 3. Schematic temperature-density phase diagram that illustrates the intercepts for the Zeno line Z = 1 and the new Zeno-like line defined by Z ≃ 0. When both lines are extrapolated to zero temperature, they intercept the density axis at approximately at the same density, ρB* ≃ ρg*, which is the hypothetical liquid (glass) density at absolute zero. As seen in Table 1, this density is usually 3.5 to 4.0 times the critical density for small organics, but tends to be larger for chain molecules such as dodecane and for molten metals. The Z = 1 locus intercepts the temperature at the well-known Boyle temperature T*B , which is usually about 2 to 2.5 times the critical temperature. The extension of the Z ≃ 0 linear line to zero density intercepts the temperature axis at T*c , a hypothetical critical temperature. If the Z ≃ 0 condition persisted to low densities, then something theoretically possible for a polymer liquid, then Tc → T*c .

Figure 4. Zeno line (Z = 1) and saturated liquid data (Zsat ≃ 0) for decane that illustrate the striking linearity of the 2 lines and how they tend to merge near absolute zero in conformity with the VDW prediction. The convergence of the 2 lines is not perfect (ρB*/ρc = 4.04 and ρ*g /ρc = 4.12); usually ρ*g is 1 to 2% larger than ρ*B . See Table 2 for tabulated data.

weight polymer whose theoretical critical temperature is of the order of a thousand degrees. Note in Table 3 that the 5 alkali metals from Li to Cs, along with Al have very similar values of ρ*g /ρc (∼5), which are larger than any organic in Table 1, and all have similar values of T*c / Tc (∼1.7), which are smaller than any organic. On the basis of the heuristic arguments above, both ratios suggest that alkali metals should exhibit a larger NLR, and indeed they do. What comes as a surprise for sodium is that linearity persists to temperatures 450° above its normal boiling point of 1156 K and probably continues to even higher temperatures if ever measured. Overall, the measured linear behavior for Na

entire liquid range right up to the critical temperature. So even though the LF model does not predict a linear density dependence on temperature, it gives some insight into why the condition Z ≃ 0 that defines the linear range might increase with chain length. Of course, it will probably never be known whether organic polymers behave this way because most begin to thermally degrade before reaching 300 °C. A liquid−vapor critical point has never been observed for a high molecular

Table 2. Zeno Line and Saturated Liquid Data in the NLR are Tabulated for Normal Decanea Zsat ≃ 0

Z=1 temp. (K)

density (mol/L)

pressure (bar)

temp. (K)

density (mol/L)

675 650 625 600 550 500 450 400 350 300 275 260

3.35 3.47 3.60 3.72 3.96 4.21 4.44 4.68 4.92 5.16 5.29 5.36

188 188 187 186 182 175 167 156 144 129 121 116

447 444 424 404 384 364 344 324 304 284 264 244

4.25 4.27 4.39 4.51 4.62 4.74 4.85 4.96 5.07 5.18 5.29 5.40

Zsat

pressure (bar) 1 0.932 0.539 0.291 0.145 6.6 2.7 9.2 2.7 6.4 1.2 1.5

× × × × × × ×

10−2 10−2 10−3 10−3 10−4 10−4 10−5

6.3 5.9 3.5 1.9 9.8 5 2 7 2 5 1 6

× × × × × × × × × × × ×

10−3 10−3 10−3 10−3 10−4 10−4 10−4 10−5 10−5 10−6 10−6 10−7

a

Critical temperature is 617.7 K, critical density is 1.64 mol/L, and the normal boiling point is 447 K. Data plotted in Fig. 4. Data from NIST Web Book database (http://webbook.nist.gov). 3709

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The Journal of Physical Chemistry B Table 3. Density−Temperature Parameters for Some Molten Metals and Saltsa liquid

ρg* (g/cm3)

aluminum16 magnesium17 calcium18 strontium18 barium18 iridium18 lithium19 sodium19 potassium19 rubidium20 cesium20 sulfur SnCl4 TiCl4 SbCl3 AlCl3

2.59 1.80 1.56 2.54 3.70 21.8 0.563 1.015 0.919 1.62 2.02 2.02 2.98 2.24 3.48 2.57

ρg* ρc

4.6 4.4

4.7 4.9 4.7 5.5 5.3 3.7 4.0 3.9 3.9 5.0

Tc* (K)

Tc* Tc

11 130 7690 8615 11 330 10 940 25 600 5590 4240 3710 3470 3390 3490 1150 1290 1516 946

1.69 3.0 2.6 3.7 3.0 3.3 1.74 1.65 1.67 1.72 1.76 2.7 1.95 2.0 1.9 1.5

a

Unless otherwise indicated, density and critical point data from NIST Web Thermo Tables (http://wtt-pro.nist.gov/wtt-pro/). Critical temperatures in ref 18 are estimated values; critical densities not available.

stretches over 1200°! For cesium, linearity persists to at least 350° above its boiling point of 944 K and over 1000° overall. Density data for some of the other alkali metals above their normal boiling points are not available for comparison, but based on the behaviors of Na and Cs, one expects similar extended linear density ranges. This raises the general question about the extent of density linearity beyond the NLR. Negative deviations of the order of 1% from linear behavior for xenon begin about 50° above its normal boiling point or at Zliq ≃ 0.02. Sodium remains linear to 450 degrees above its boiling point where Zliq ≃ 0.04 but deviations are still less than 0.5%. All density/temperature parameters (ρg*Tc*) tabulated in Tables 1 and 3 were determined using only data in the NLR, where Zliq is usually less than 0.005. If a larger temperature range were to be employed in fitting the data, then somewhat different parameters would be obtained and the apparent range of linearity could be extended at the expense of overall accuracy. As a result, a general rule for linearity extent beyond the normal boiling point remains undefined. A key physical concept is the energy required to liberate an atom or molecule from the liquid to vapor states, i.e., the molar energy of vaporization. In the case of chain molecules, this increases with chain length because the chain molecule attractively interacts with many monomers that belong to other polymers. Although these monomer−monomer interactions are often weak VDW types, there are many of them per chain, and their number increases linearly with chain length. A consequence is that the equilibrium vapor pressure drops precipitously with increasing chain length, effectively increasing the NLR. In molten metals, the attractive forces between atoms are much stronger than the VDW forces among organic molecules. Tighter bonding manifests as higher critical and normal boiling temperatures, as well as much smaller thermal expansion coefficients. The latter translates into a larger temperature increase required to go from the triple point to the normal boiling point, i.e., a larger NLR. This explains how

Figure 5. The linear dependence of density on temperature in the NLR for some liquid metals and inorganics. Unlike organic liquids, density linearity can persist to well above the normal boiling point. For sodium, linearity persists to temperatures 450° above its normal boiling point of 1156 K and probably continues to even higher temperatures if ever measured. For cesium, linearity persists to at least 350° above its boiling point of 944 K. Note that this superposition of data is identical to Figure 2 and could have been included in Figure 2. All density data above were obtained from the NIST Web Thermo Tables (http://wtt-pro.nist.gov/wtt-pro/).

the NLR might be expanded, but it does not explain why the liquid might still exhibit linearity well beyond the NLR as observed in Na and Cs. Molten salts experience strong Coulombic interactions and might also be expected to exhibit similar expanded NLR ranges. However, the requisite density experimental data, as well as critical point data, are sparse for molten salts and this expectation remains untested. Intercepts for the Z = 1 locus, are the Boyle temperature, where the second virial coefficient vanishes, and the Boyle density is given by the following:15 dB

ρB* =

TB* dT2

T = TB*

B3(TB*)

(11)

where B2 and B3 are the second and third virial coefficients of the density virial expansion. In normal practice, neither the Boyle temperature nor density is determined from virial coefficients because the requisite virial coefficient data are often not available. Instead, PρT data are used to establish the Z = 1 locus, and if it is a straight line, it is extended to zero density to determine the Boyle temperature and to zero temperature to obtain the Boyle density. That was the method used to determine the intercepts reported in Table 1. Similarly, saturated PρT data used to determine the Z ≃ 0 locus in the 3710

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become completely dominant, so much so that molecules in the liquid state can no longer escape to the vapor state. Even at the normal boiling point only about 1 out of 250 molecules are able to escape from liquid to vapor states.6 The NLR is a regime of strong attractive force domination, and repulsive forces appear to serve as an excluded volume background force that prevents the liquid from collapsing. For many organics, this attractive interaction increases in strength linearly with density, or equivalently, increases linearly with decreasing temperature. These Zeno-like liquid properties place some imposing constraints on any theoretical equation of state. Simultaneous satisfaction of both linear requirements, Z = 1 and Z ≃ 0, plus a VDW type interaction energy at high densities represent severe constraints on any model based EOS.

4. CONCLUSIONS Saturated liquid densities are a linear f unction of temperature in the NLR for molecular, polymeric, inorganic, ionic, and metallic liquids and superpose to form a single master curve. Although others might exist, water and helium are the only known exceptions to this very strong corresponding states principle. Extrapolation of this line to absolute zero temperature defines a theoretical glass density. Many organics that include some hydrogen bonding liquids such as ammonia, possess a configurational energy in the NLR that varies linearly with density in agreement with the wellknown VDW approximation for the interaction energy. Extrapolation of this line to absolute zero temperature defines the ground state configurational energy of the hypothetical disordered liquid (glass). The requisite experimental data (energies of vaporization) are not yet available to test this property for molten metals and salts.

Figure 6. Master plot for saturated liquid configurational energy in the NLR. The dimensionless slope m does not vary much from one liquid to the other (see Table 1). Since the density varies linearly with temperature, a plot of U/Ug* versus mρ/ρg* is also linear. This result implies that the usual VDW approximation that configurational energy varies as density is valid for liquids in the NLR. Vaporization energies obtained from NIST Web Book database (http://webbook.nist.gov).

■

NLR were extended to zero temperature to determine the hypothetical glass density ρg*. According to the VDW model, ρ*g = ρ*B , but ρ*g tends in general to be about 1 to 2% greater than the Boyle density, as illustrated in Table 1. A second Zeno-like linearity has also been discovered that may prove to be more significant from a theoretical perspective. In the NLR, the internal energy of vaporization ΔEvap to an excellent approximation equals the liquid configurational energy, Uliq. Thermal energy contributions to the internal energy cancel in forming ΔEvap, and the configurational energy of the dilute gas phase is to a good approximation negligible. Configurational energies of simple and organic liquids exhibit linear behavior in both temperature and density in the NLR as illustrated in Figure 6. Extension of this line to zero temperature defines the ground state configurational energy U*g of the disordered liquid (glass). Some glass parameters are tabulated in Table 1. Water does not exhibit this behavior nor does methanol, but other hydrogen bonding liquids such as ammonia do display this linearity. It is unclear if molten metals and salts display this unique property because the requisite vaporization energies are not available. An important theoretical implication of this observation is that the usual VDW approximation that configurational energy varies diectly with density is valid for many liquids in their NLR. A fundamental and unexplained issue is this: why do diverse liquids in terms of interatomic interactions and structure behave so simply and similarly in their NLR? What is the underlying physics for these 2 very strong corresponding states principles that appear to be intimately related? The condition Z = 0 corresponds to the situation where attractive forces have

AUTHOR INFORMATION

Corresponding Author

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

The authors declare no competing financial interest.

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

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