Article pubs.acs.org/JPCB

Dimensionless Thermodynamics: A New Paradigm for Liquid State Properties Isaac C. Sanchez* McKetta Department of Chemical Engineering, University of Texas, Austin, Texas 78712, United States S Supporting Information *

ABSTRACT: Equations of state in the van der Waals genre suggest that saturated liquids should adhere to the following corresponding states principle (CSP): saturated liquids at the same reduced density (ρR = ρ/ρc) have comparable dimensionless thermodynamic properties. This CSP is shown to be applicable to a variety of thermodynamic properties that include entropy of vaporization, cohesive energy density, thermal expansion coefficient, isothermal compressibility, thermal pressure coefficient, compressibility factor, temperature coefficient of the vapor pressure, heat capacity difference, and surface tension. For two classes of liquids, all properties rendered dimensionless by the proper choice of scaling variables superpose to form “master curves” that illustrate the CSP. Using scaled particle theory, an improved van der Waals model is developed whose results are compared with existing experimental thermodynamic data in dimensionless form. Properly expressing thermodynamic properties in dimensionless form acts to consolidate and harmonize liquid state properties.

I. INTRODUCTION In a previous publication1 various contributions to the configurational entropy of several simple saturated liquids were shown to correlate well with reduced density ρR ρ/ρc and with the computed properties of a Lennard-Jones fluid. It was suggested that saturated liquids should have comparable thermodynamic properties at the same reduced density. But this corresponding states principle (CSP) comes with a caveat: thermodynamic properties that have dimensions do not obey this CSP. A good example is cohesive energy density (CED), which has units of pressure and can vary widely from one liquid to another even when compared at the same reduced density. Because CED is a measure of “molecular cohesiveness”, it might be expected that some critical value of the CED is reached at the normal boiling point or that low boiling liquids have smaller CEDs. Neither intuition is correct. For example, propene normally boils at 225 K at a CED of 239 MPa while perfluoropropane boils at a slightly higher temperature of 236 K, but with a lower CED of 153 MPa. What will be shown is that if CEDs are rendered dimensionless by dividing by the pressure variable ρkT, the rescaled CEDs start to become more intelligible. For example, the dimensionless CEDs are 8.8 for propene and 9.1 for perfluoropropane. Furthermore, many common organic solvents do indeed boil at a critical value of this new dimensionless CED (9.2 ± 0.6). One could have chosen the critical pressure to render the CED dimensionless, but it does not yield a CSP. The choice of the pressure ρkT is not arbitrary; it is what any generic van der Waals (VDW) model suggests to reduce the CED to a function of ρR only.

For liquids in the normal liquid range (NLR is from triple point to normal boiling point), the compressibility factor Zliq is of order 10−3 or smaller, whereas Hs(ρR) is of order 10. In other words, in the NLR, the dimensionless configurational energy should be completely dominated by the reduced density. The saturated vapor in equilibrium with a liquid in the NLR will behave to an excellent approximation as an ideal gas with no

II. DIMENSIONLESS THERMODYNAMIC PROPERTIES Understanding what scaling variables to employ can be gleaned from any VDW type model characterized by a configurational

Received: April 28, 2014 Revised: July 7, 2014 Published: July 11, 2014

© 2014 American Chemical Society

energy (−U) that is proportional to density (ρ) and in which the EOS has the general form Z = P /ρkT = 1 + HS(ρ) − U /kT = 1 + HS(ρR ) − ρR u/kT

(1)

where ρR is the reduced density (ρR ≡ ρ/ρc), ρc the critical density, and u a constant. The function HS(ρ) is the contribution to the EOS from hard sphere repulsion with Hs(0) = 0 and HS(ρ) > 0. For the classical VDW equation of state (EOS)

where the reduced temperature TR = T/Tc and Tc is the critical temperature. A. Cohesive Energy Density. Returning to the more general form of a VDW type EOS (eq 1), it is easily seen that the liquid configurational energy of a VDW fluid is given by Uliq /kT = 1 + HS(ρR ) − Z liq

9386

(3)

dx.doi.org/10.1021/jp504140z | J. Phys. Chem. B 2014, 118, 9386−9397

The Journal of Physical Chemistry B

Article

This new CED leads to an intuitive and appealing idea: as a liquid is heated from its freezing point, its CED begins to decrease and when it reaches a value of about 9, it is near its normal boiling point. B. Entropy of Vaporization and Trouton’s Rule. The entropy and enthalpy of vaporization are given by

configurational energy. Thus, the dimensionless energy of vaporization ΔUvap is given by ΔUvap/kT ≃ Uliq /kT ≃ 1 + HS(ρR )

(4)

It is noted parenthetically that ΔUvap = ΔEvap, where ΔEvap is the change in internal energy upon vaporization. The thermal contribution to the internal energy cancels when calculating ΔEvap, leaving only the contribution from configurational energy changes. A thermodynamic property closely associated to the vaporization energy is the cohesive energy density (CED or δ2), a property widely used in liquid mixture models and defined as 2

δ ≡ Uliq /V ≃ ρΔUvap

ΔH vap/kT = ΔSvap/k = ΔUvap/kT + Zvap − Z liq

But in the NLR, Zvap ≃ 1 and Zliq ≃ 0 so that 2 ΔSvap/k ≃ 1 + ΔUvap/kT = 1 + δ ̃ = 2 + HS(ρR )

(8)

Thus, for several nonpolar and slightly polar organic liquids ΔSvap/k = 1 + 9.2 ± 0.6 = 10.2 ± 0.6

(5)

(9)

which is Trouton’s rule at the normal boiling point. For the VDW model in the NLR

which has units of pressure (δ is the well-known Hildebrand solubility parameter). The CED can be rendered dimensionless by dividing by ρkT, which yields a CED that depends only on ρR in the NLR:

ΔSvap/k =

6 − ρR 3 − ρR

(10)

and ΔSvap/k = 9.1 for ρR = 2.63 and 10.2 for ρR = 2.67. There are two well-known exceptions to Trouton’s rule: one is for some hydrogen bonding liquids that yield larger values of Trouton’s constant and small monatomic and diatomic molecules that yield smaller values. For this reason, it is convenient to define three broad liquid groups by their vaporization entropies at their normal boiling points or Trouton constants (ΔSvap/k):

2

δ ̃ ≡ δ 2/ρkT = 1 + H(ρR ) − Z liq ≃ 1 + H(ρR )

(7)

(6)

This dimensionless CED decreases from the freezing point to the normal boiling, where it reaches a value of 9.2 ± 0.6 for a wide variety of nonpolar and slightly polar liquids that includes aliphatic and aromatic hydrocarbons and fully halogenated hydrocarbons, as illustrated in Figure 1. The corresponding experimental value of the reduced density at the normal boiling point for the organics shown in the tables is ρR = 2.63 ± 0.07.

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

Note that the ranges do not overlap; some liquids will not neatly fall into any group. Group I includes the inert elements and diatomic molecules given in Table 1. Group II includes nonpolar and slightly polar organic liquids given in Tables 2−4. Note that Group II includes several common organic solvents such as carbon tetrachloride, benzene, cyclohexane, and nheptane. Group III liquids include many polar and hydrogen bonding liquids such as alcohols. Water is in its own unique and singular group. The focus here is on Group I and II type liquids. As will be seen, both Groups I and II share some thermodynamic properties, whereas for other properties, a clear difference exists. A generalized Trouton’s rule is illustrated in Figure 2 for some Group II liquids. The entropy of vaporization has two contributions. Beginning with the saturated vapor approximated as an ideal gas at Psat/kT, condensation to the liquid state can be thought of as requiring two steps: first, the compression of the ideal gas to the same density as the liquid followed by “turning on the interactions” to convert the ideal gas to a liquid with the same density as the ideal gas from which it came. This second contribution to the entropy is the self-solvation entropy, ΔSp. The first contribution is easily calculated because it involves only an ideal gas: ΔScond /k = ln(ρig /ρliq ) = ln Z liq − ln Z ig = ln Z liq < 0 (11)

Because ΔSvap and Zliq are experimental measurables, the selfsolvation entropy ΔSp can be calculated from the thermodynamic cycle (see Figure 3):

Figure 1. Dimensionless cohesive energy density as a function of reduced density. The average CED for the 18 organic liquids listed in Tables 2−4 at their normal boiling points is 9.2 ± 0.6. 9387

dx.doi.org/10.1021/jp504140z | J. Phys. Chem. B 2014, 118, 9386−9397

The Journal of Physical Chemistry B

Article

freedom. It effectively isolates the entropic contribution of molecular interactions in the solvation process. C. Chemical Potential. The chemical potential can be written in terms of an excess absolute fugacity, ZB, relative to an ideal gas at the same temperature and pressure: μ = μig (T , P) − kT ln(ZB)

(13)

where B is the “Widom insertion factor” and is related to the EOS by4,5 ρR

1−Z dρ ρ ρR H (ρ) 2ρ S dρ = R − HS(ρR ) − 0 ρ kT

ln B = 1 − Z +

∫0

∫

(14)

Using the EOS (eq 1) to eliminate temperature yields ln B liq = 2(1 − Z liq) + Hs(ρR ) −

∫0

ρR

Hs(ρ) dρ ρ

(15)

In the NLR, Zliq can be set to zero so that Bliq becomes a function of ρR only. For the VDW EOS in the NLR Figure 2. Entropy of vaporization for some Group II molecules illustrating a generalization of Trouton’s rule. Many Group II liquids at their normal boiling points have reduced densities of 2.63 ± 0.07, which corresponds to a ΔSvap/k value of about 10 (Trouton’s constant). But at ρR = 2, Trouton’s constant would be about 5.

ΔSvap + ΔScond + ΔSp = 0

ln B liq =

6 − ρR 3 − ρR

+ ln(1 − ρR /3) > 0

(16)

Equality of chemical potentials between saturated liquid and vapor requires (Z B)liq = (Z B)vap = (Z B)ig = 1

(12)

(17)

where the last equality is obtained when the saturated vapor behaves ideally. Thus, in the NLR, it is expected that

Self-solvation entropy has been accurately described by others2,3 as the entropy loss when a molecule at a fixed position in an ideal gas is transferred to a fixed position in a liquid of density ρliq. Because the molecule is fixed in space before and after the transfer, it eliminates any entropic contributions associated with changes in molecular translational degrees of

ln B liq = −ln Z liq

(18)

Because Bliq should depend only on ρR, this leads to the conclusion that Zliq should also depend only on ρR. The validity of this conclusion is illustrated in Figure 4. The only surprise is

Figure 3. Thermodynamic cycle illustrating two important entropy contributions to the entropy of vaporization. This cycle assumes that the saturated vapor can be treated as an ideal gas. In the second cycle step where the interactions are “turned on”, the favorable chemical potential change is kT ln Zliq. The generalization of this cycle for a nonideal vapor phase can be found in the Supporting Information. 9388

dx.doi.org/10.1021/jp504140z | J. Phys. Chem. B 2014, 118, 9386−9397

The Journal of Physical Chemistry B

Article

assumed that the vapor state is ideal in the NLR. A dimensionless temperature coefficient γσ can be formed that should be a function only of reduced density by dividing by kρ:

that the superposition of data appears to continue all the way to the critical point.

γσ̃ ≡ γσ /kρ = Z liq(ΔSvap/k) = Z liq[2 + H(ρR )]

(23)

which for a classical VDW liquid reduces to γσ̃ =

⎡ 6−ρ ⎤ R ⎥ exp⎢ − ⎢⎣ 3 − ρR ⎥⎦ (3 − ρR )

3(6 − ρR ) 2

(24)

In Figure 5 the behavior of this newly defined coefficient illustrates two points: First, there is a clear difference between

Figure 4. Behavior of the logarithm of liquid compressibility factor, −k ln Zliq, that contributes to the entropy of vaporization. See Figure 3 and text. No appreciable difference seems to exist between Group I and II molecules. Even the prototypical Group III liquid methanol falls in line with Group I and II liquids.

Considering the entropy of vaporization and Figure 3, it is easy to show that the favorable change in chemical potential in the solvation step is given by Figure 5. Behavior of the temperature coefficient of the saturation pressure curve with liquid density. Group II liquids values of γ̃σ are 5− 15% larger than those for Group I. The critical value of this property varies little between diverse liquids and even includes water. See Supporting Information.

−Δμ/kT = ln B liq = ΔH vap/kT + ΔSp/k = (ΔSvap + ΔSp)/k

(19)

From eqs 10, 12, and 16, the solvation entropy for a VDW liquid is ΔSp/k = ln(1 − ρR /3) < 0

Group I and II liquids. Group I liquids have vaporization entropies smaller than those of organics. This difference is clearly manifested in the behavior of γ̃σ, which for Group I liquids is 5−15% smaller than that for Group II liquids. Second, the correlation extends all the way to the critical point. See Supporting Information for remarks on the behavior of γσ and the closely related thermal pressure coefficient, γV = (∂P/∂T)V and where γσ = γV at the critical point. E. Thermal Expansion, Compressibility, and Thermal Pressure. Thermodynamic properties that depend on second derivatives of the chemical potential are the thermal expansion coefficient (α), isothermal compressibility (κ), and the two heat capacities, CP and CV. The thermal expansion coefficient and isothermal compressibility can be calculated from the VDW EOS

(20)

D. Temperature Coefficient of the Saturation Pressure. The well-known Clausius−Clapeyron equation relates the behavior along the coexistence line between two pure phases:

dP ΔS = dT ΔV

(21)

Applying this equation specifically to liquid−vapor coexistence yields ΔSvap dP sat ∂P ⎞ ≃ ρliq Z liq ΔSvap ≡ ⎟ ≡ γσ = dT kT /P sat − Vliq ∂T ⎠σ (22)

where the σ subscript denotes that the temperature derivative is taken along the liquid−vapor coexistence line. It has also been

α̃ ≡ Tα = −T 9389

∂ln ρR ⎞ 3 − ρR ⎟= ∂T ⎠ 2ρR − 3

(25)

dx.doi.org/10.1021/jp504140z | J. Phys. Chem. B 2014, 118, 9386−9397

The Journal of Physical Chemistry B κ ̃ ≡ (kTρ)κ =

1 ∂(ρR Z) ∂ρR

)

=

Article

(3 − ρR )2 3(2ρR − 3)

T

(26)

In the above, the EOS was used to eliminate temperature and then Zliq was set equal to zero. Thus, a dimensionless compressibility (κ̃) has been defined that depends only on ρR. In Figure 6 the newly defined compressibility is plotted against ρR; for compressibility, both Group I and II molecules

Figure 7. Thermal expansion coefficient versus reduced density. Unlike compressibility, Group I and II liquids clearly differ in thermal expansion behaviors.

Figure 6. Isothermal compressibility versus reduced density. Little difference is seen between Group I and II liquids.

follow the same universal curve. But for Tα, there is a clear difference between Groups I and II, as illustrated in Figure 7. The thermal pressure coefficient is given by γV = (∂P /∂T )V = α /κ γṼ =

Tα α̃ 3 = = 1 + HS(ρR ) = (kTρ)κ κ̃ 3 − ρR

(27)

Therefore, in the NLR where Zliq is negligible, the new dimensionless CED and γ̃V are equal to one another (Figure 8):

α̃ 2 = δ̃ (28) κ̃ It should be noted that this result is obtained only if the attractive energy varies linearly with density (a VDW type energy); otherwise, this equality is not expected (see discussion below). Another related property is the internal pressure Pint defined as γṼ =

Pint =

⎛ ∂U ⎞ Tα ⎜ ⎟ = −P ⎝ ∂V ⎠T κ

̃ = Pint /kTρ = Pint

α̃ 3 − Z liq ≃ 3 − ρR k̃

Figure 8. Group I and II liquids differing in thermal pressure coefficient in the NLR. Critical values of this property, which have never been reported before, vary little among diverse liquids (see Supporting Information). (29) 9390

dx.doi.org/10.1021/jp504140z | J. Phys. Chem. B 2014, 118, 9386−9397

The Journal of Physical Chemistry B

Article

F. Heat Capacity. Two other second-order properties are the two heat capacities, CP and CV, which are related through the following well-known relationship: CP − C V (Tα)2 α̃ 2 3 = = = k kTρκ κ̃ 2ρR − 3

ηc =

(33)

Using this value of ηc in eq 32 to predict experimental entropies of vaporization consistently underestimates ΔSvap. Alternatively, ηc can be treated as an adjustable parameter chosen to yield the correct ΔSvap or to yield the correct value of one of its components, ΔScond or ΔSp. Although ΔSvap differs for Group I and II liquids, the ΔScond = k ln Zliq contribution to ΔSvap appears to be more universal, even including some Group III liquids such as methanol (see Figure 4). For this reason, ηc is chosen to yield the correct value of ΔScond at the normal boiling point. This adjusted value of ηc is then used to predict all other thermodynamic properties. Using ηc as an adjustable parameter amounts to rescaling the reduced density so that it has the correct critical density. Values of ηc determined in this way are shown in Tables 1−4. Focusing on Group I liquids in Table 1, note the constancy of the required ηc. Except for fluorine, the rescaled critical volume fraction is bounded very tightly

(30)

G. Surface Tension. Unlike the other thermodynamic properties, no general definition of surface tension (σ) exists to guide in the selection of an appropriate scaling variable. However, surface tension has units of energy per unit area, and as has been seen, energy scales as kT and molecular surface area should scale as V2/3. Thus, a dimensionless surface tension (σ̃) can be defined as

σ ̃ = σ /kTρ2/3

73 − 7 = 0.1287... 12

(31)

This newly defined surface tension is plotted against ρR in Figure 9, and good superposition of surface tensions is obtained

0.156 ≤ ηc ≤ 0.160

This critical value range lies in the same range as that of a Lennard-Jones (LJ) fluid. From computer simulations, the critical density ρcσ3 of a LJ fluid has been estimated to be between 0.29 and 0.31.8−10 As is customary, if σ is taken as the effective diameter of the LJ particle, then the fraction of space occupied by a LJ fluid at the critical occupied density is 0.152 ≤ ηc(LJ) ≤ 0.160

If traditional VDW molecular volumes are used, the calculated critical densities tend to be uniformly higher by 10−20%. Using these rescaled critical densities, other thermodynamic properties are predicted. As can be seen in Table 1, the agreement between calculated and experimental vaporization (ΔSvap) and self-solvation (ΔSp) entropies is very good. Because the dimensionless CED depends on ΔSvap, the agreement between experimental and calculated CEDs is also very good. Even fluorine’s experimental properties agree well with calculated values. For comparison, an LJ fluid has at its normal boiling point1 Figure 9. Surface tension versus reduced density. Group I liquids have values larger than those of Group II by about 10−20% in the normal liquid range.

ΔSvap/k =8.9 at ρR = 2.63

which agrees well with the 9.0 average for Group I liquids. The clear inference is that Group I liquids appear to behave much like LJ liquids. The volume fractions calculated at the normal boiling point ηB are also shown in Table 1. This value is calculated by multiplying the reduced density ρR at the normal boiling point by the respective value of ηc. The average is 0.416 and implies that these liquids will have vapor pressures near 1 atm at this unique occupied volume fraction. A more appealing view is to look at the experimental values of the cohesive energy densities δ̃2 shown in Table 1. The variation is less than 10% with an average of 8.0. So when Group I molecules reach a CED of about 8, their vapor pressures are about 1 atm. With respect to the ability of the SPT model to quantitatively correlate properties of Group I liquids, the first real disappointment is revealed in the calculation of the thermal expansion coefficient and isothermal compressibility. As seen in Table 1, the calculated values consistently overestimate both properties by sometimes as much as 50%. However, the ratio of

for a wide variety of nonpolar liquids. However, Group I liquids show surface tensions that are 10−20% greater than those of Group II liquids.

III. MODEL COMPARISONS WITH EXPERIMENTAL DATA In the Supporting Information, an improved VDW hard sphere model is described based on scale particle theory (SPT).6,7 Thermodynamic liquid properties depend only on a single variable, the occupied volume fraction (η). The connection to the experimental variable ρR is given by η = ηcρR (32) Critical properties are derived for the SPT model and the critical volume fraction (ηc) is given by 9391

dx.doi.org/10.1021/jp504140z | J. Phys. Chem. B 2014, 118, 9386−9397

The Journal of Physical Chemistry B

Article

Table 1. Experimental Comparisons of SPT Model with Group I Small Molecules at Their Normal Boiling Pointsa property crit. temp. crit. vol. boiling temp. volume @ TB red. density comp. factor ηc ηB ΔSvap/k −ΔSp/k CED, δ2̃ CED, δ2 (J/cm3) α̃ = Tα 104κ (bar−1) 102κ̃ γ̃V = α̃ /κ̃ (CP − CV)/k σ (dyn/cm) 103σ̃ 105κ̃σ̃ a

Tc (K) Vc (cm3/mol) TB (K) VB (cm3/mol) ρR = Vc/VB −ln Zliq adjusted calcd exptl calcd exptl calcd exptl calcd exptl exptl calcd exptl exptl calcd exptl calcd exptl calcd exptl exptl exptl

N2

CO

F2

Ar

O2

CH4

Kr

Xe

126.2 89.2 77.3 34.6 2.58 5.23 0.157 0.406 8.65 8.49 3.42 3.27 7.6 7.5 142 0.44 0.55 3.12 5.8 7.2 7.6 7.5 3.3 4.9 8.8 14.8 86

132.9 92.2 81.5 35.3 2.61 5.26 0.156 0.407 8.80 8.54 3.54 3.29 7.8 7.5 150 0.45 0.65 3.12 6.0 8.6 7.5 7.5 3.3 4.9 9.5 15.1 91

144.4 64.1 85 25.3 2.53 5.63 0.166 0.419 9.37 9.13 3.74 3.49 8.4 8.1 234 0.40 0.61 1.74 4.8 7.6 8.3 8.1 3.3 4.9 13.4 16.4 79

150.9 74.6 87.3 28.7 2.60 5.53 0.160 0.416 8.90 8.98 3.37 3.44 7.9 8.0 200 0.39 0.62 2.2 5.5 7.8 7.1 8.0 2.8 4.9 12.5 16.1 89

154.6 73.4 90.2 28 2.62 5.59 0.159 0.417 9.14 9.03 3.55 3.45 8.1 8.0 218 0.39 0.61 1.93 5.2 7.6 7.5 8.0 2.9 4.9 13.2 16.3 85

190.6 98.9 111.6 38 2.60 5.50 0.160 0.415 8.90 8.93 3.40 3.42 7.9 7.9 193 0.38 0.62 2.2 5.4 7.8 7.0 7.9 2.7 4.9 13.3 16.2 87

209.5 92.2 119.7 34.7 2.66 5.66 0.158 0.420 9.03 9.18 3.37 3.50 8.0 8.2 230 0.38 0.60 1.80 5.1 7.4 7.5 8.2 2.8 4.9 15.6 16.7 85

289.7 119 165 44.6 2.67 5.73 0.158 0.421 9.17 9.23 3.44 3.52 8.2 8.2 251 0.38 0.60 1.65 5.0 7.2 7.6 8.2 2.9 4.9 18.5 17.0 85

Experimental data, refs 11 and 12.

Table 2. Experimental Comparisons of SPT Model with Aliphatic Hydrocarbons at Their Normal Boiling Pointsa property crit. temp. crit. vol. boiling temp. volume @ TB red. density comp. factor ηc ηB ΔSvap/k −ΔSp/k CED, δ2̃ CED, δ2 (J/cm3) α̃ = Tα 104κ (bar−1) 102κ̃ γ̃V = α̃ /κ̃ (CP − CV)/k σ (dyn/cm) 103σ̃ 105κ̃σ̃ a

Tc (K) Vc (cm3/mol) TB (K) VB (cm3/mol) ρR = Vc/VB −ln Zliq adjusted calcd exptl calcd exptl calcd exptl calcd exptl exptl calcd exptl exptl calcd exptl calcd exptl calcd exptl exptl exptl

CH3CHCH2

C3H8

(CH3)3CH

n-C4H10

C(CH3)4

365.6 188.4 225.5 69.1 2.73 5.6 0.153 0.418 9.8 9.1 4.2 3.5 8.8 8.1 239 0.46 0.61 1.9 5.2 7.6 8.8 8.1 4.1 4.9 16.9 15.2 79

369.8 200 230.8 75.8 2.63 5.5 0.158 0.416 9.8 9.0 4.3 3.5 8.8 7.9 223 0.46 0.62 2.6 5.0 7.6 9.2 8.0 4.2 4.9 15.8 14.8 74

407.8 258 261.8 97.9 2.63 5.4 0.157 0.412 9.8 8.8 4.4 3.4 8.8 7.8 196 0.46 0.63 2.3 5.0 8.1 9.2 7.8 4.7 4.9 14.4 14.1 71

425.1 255 272.2 96.6 2.64 5.5 0.156 0.413 9.9 8.9 4.4 3.4 8.9 7.9 209 0.49 0.63 2.1 5.0 7.9 9.8 8.0 4.7 4.9 14.9 13.9 69

433.7 305.8 282.7 120 2.55 5.3 0.160 0.408 9.7 8.6 4.4 3.3 8.7 7.6 170 0.49 0.65 2.5 4.8 8.6 10.1 7.6 4.9 4.9

Experimental data, refs 11 and 12. 9392

dx.doi.org/10.1021/jp504140z | J. Phys. Chem. B 2014, 118, 9386−9397

The Journal of Physical Chemistry B

Article

Table 3. Experimental Comparisons of SPT Model with Aliphatic and Aromatic Hydrocarbons at Their Normal Boiling Pointsa property crit. temp. crit. vol. boiling temp. volume @ TB red. density comp. factor ηc ηB ΔSvap/k −ΔSp/k CED, δ2̃ CED, δ2 (J/cm3) α̃ = Tα 104κ (bar−1) 102κ̃ γ̃V = α̃ /κ̃ (CP − CV)/k σ (dyn/cm) 103σ̃ 105κ̃σ̃ a

Tc (K) Vc (cm3/mol) TB (K) VB (cm3/mol) ρR = Vc/VB −ln Zliq adjusted calcd exptl calcd exptl calcd exptl calcd exptl exptl calcd exptl exptl calcd exptl calcd exptl calcd exptl exptl exptl

n-C6H14

C6H6

(CH2)6

n-C7H16

CH3C6H5

507.8 369.5 341.2 140.4 2.63 5.3 0.153 0.418 10.1 8.6 4.8 3.3 9.1 7.6 184 0.46 0.65 2.7 5.5 8.6 8.4 7.6 5.6 4.9 13.4 12.8 71

562 255.3 353.3 95.9 2.66 5.7 0.158 0.416 10.5 9.3 4.8 3.6 9.5 8.3 291 0.46 0.59 1.5 4.5 7.1 10.2 8.3 5.0 4.9 21.1 15.1 69

553.6 308.2 354 117 2.64 5.5 0.157 0.412 10.2 8.9 4.7 3.4 9.2 7.9 231 0.46 0.63 1.9 4.8 7.9 9.6 7.9 5.2 4.9 18.1 14.7 71

540.1 432 371.5 163.1 2.65 5.2 0.156 0.413 10.3 8.6 5.1 3.4 9.3 7.6 176 0.49 0.65 3.0 5.6 8.6 8.8 7.6 6.1 4.9 12.7 12.3 69

591.8 315.6 384 118.3 2.67 5.6 0.160 0.408 10.4 9.0 4.8 3.4 9.4 8.0 254 0.49 0.62 1.8 4.8 7.7 10.2 8.0 5.2 4.9 17.9 13.5 65

Experimental data, refs 11 and 12.

Table 4. Experimental Comparisons of SPT Model with Halogenated Hydrocarbons at Their Normal Boiling Pointsa property crit. temp. crit. vol. boiling temp. volume @ TB red. density comp. factor ηc ηB ΔSvap/k −ΔSp/k CED, δ2̃ CED, δ2 (J/cm3) α̃ = Tα 104κ (bar−1) 102κ̃ γ̃V = α̃ /κ̃ (CP − CV)/k σ (dyn/cm) 103σ̃ 105κ̃σ̃ a

Tc (K) Vc (cm3/mol) TB (K) VB (cm3/mol) ρR = Vc/VB −ln Zliq adjusted calcd exptl calcd exptl calcd exptl calcd exptl exptl calcd exptl exptl calcd exptl calcd exptl calcd exptl exptl exptl

CCl4

CCl3F

CCl2F2

CClF3

CF4

C2F6

C3 F 8

(CF2)4

556.4 276 349.3 103.8 2.66 5.6 0.158 0.419 10.3 9.1 4.7 3.5 9.3 8.1 260 0.49 0.61 1.67 4.7 7.6 10. 8.1 5.1 4.9 19.8 15.1 71

471 248 297 92.9 2.67 5.6 0.156 0.417 10.1 8.6 4.5 3.0 9.1 7.6 242 0.48 0.65 1.76 4.7 8.5 10. 7.6 4.7 4.9 17.9 14.9 70

385.1 214 243.5 81.3 2.63 5.5 0.158 0.415 10.0 9.0 4.5 3.5 9.0 8.0 224 0.49 0.62 1.95 4.9 7.8 10. 8.0 4.7 4.9 15.9 14.8 75

302 179.2 192 68.7 2.61 5.4 0.158 0.413 9.8 8.8 4.4 3.4 8.8 7.8 204 0.49 0.63 2.20 5.1 8.1 9.6 7.8 4.5 4.9 13.9 14.6 74

227.5 142 145 54.9 2.59 5.4 0.160 0.411 9.8 8.9 4.4 3.5 8.8 7.9 193 0.48 0.63 2.35 5.1 7.9 9.4 7.9 4.5 4.9 12.2 14.6 74

293 225 195 86 2.62 5.2 0.155 0.407 10.0 8.5 4.8 3.3 9.0 7.5 170 0.54 0.66 2.83 5.3 8.8 10. 7.5 5.1 4.9 11.0 13.2 70

345 299 236 116.6 2.57 5.1 0.157 0.403 10.1 8.3 5.0 3.2 9.1 7.3 153 0.58 0.68 2.90 4.9 9.4 12. 7.3 6.5 4.9 10.6 12.9 63

388.4 323 267 123.8 2.61 5.2 0.155 0.405 10.5 8.4 5.3 3.2 9.5 7.4 170 0.58 0.66 2.70 4.8 8.9 12. 7.4 6.9 4.9 11.9 13.3 64

Experimental data, refs 11 and 12. 9393

dx.doi.org/10.1021/jp504140z | J. Phys. Chem. B 2014, 118, 9386−9397

The Journal of Physical Chemistry B

Article

favor the condensed state. It might even be expected that the liquid state should appear whenever Uliq/kT ≈ 1, but it does not. Attractive forces must also overcome repulsive intermolecular forces, which in the context of a VDW model, are measured by the hard sphere contribution, HS(ρR). At the normal boiling point, HS(2.63) ≃ 7 and U(TB)/kTB = 1 + 7 ≈ 8 as observed for several Group I molecules (see Table 1). Any generic VDW model nicely separates the configurational contributions at the normal boiling point: kTB to overcome thermal forces and 7 kTB to overcome repulsive forces. In the NLR, Uliq ≃ ΔUvap = ΔEvap where ΔEvap is the change in the internal energy on vaporization. Equating Uliq to ΔUvap in the NLR usually underestimates Uliq by less than 2%. The thermal energy contribution to the internal energy cancels in forming ΔEvap, leaving only the configurational energy contribution. This thermal contribution is 3kT/2 for monatomic elements and 5kT/2 for diatomic elements. It is much more difficult to calculate the thermal energy contribution for polyatomic molecules. For this reason, very little experimental data on configurational energies spanning the range from triple to critical point are available. However, some data do exist and are illustrated in Figure 10.

the two yields the thermal pressure coefficient, which is accurately predicted. Also notice how well the experimental values of the thermal pressure coefficient and CED (γ̃V and δ2̃ ) agree with one another. As mentioned previously, any model that assumes a linear dependency of configurational energy on density will yield γ̃V = δ̃2 in the NLR. As will be discussed below, Group II liquids do not satisfy this condition. Group II liquid properties are shown in Tables 2−4; aliphatic and aromatic hydrocarbons are in Tables 2 and 3, and halogenated hydrocarbons are in Table 4. Like Group I liquids, the adjusted critical densities vary little from one liquid to another: 0.153 ≤ ηc ≤ 0.160

As can be seen in the tables, the calculated ΔSvap and ΔSP are too small by about 1−2 k, and the discrepancy increases with molecular size. It is clear from the experimental values of ΔSP that Group II liquids, which occupy molecular volumes larger than those of Group I, pay a larger entropic penalty in selfsolvation. With regards to entropy changes, the SPT model seems adequate for small Group I molecules but less so for the larger Group II molecules. Because the CED depends on ΔSvap, the calculated CED values are lower than the experimental values that average 9.2 ± 0.6. Note that the calculated volume fractions at the normal boiling point average 0.412 and do not statistically differ from those of Group I (0.415). As with the Group I liquids, the SPT model overestimates thermal expansions and compressibilities by about 50%. But unlike Group I, the calculated thermal pressure coefficients are not accurately predicted. The discrepancy generally increases with molecular size. Also note that unlike Group I liquids, experimentally γ̃V > δ̃2 with the inequality becoming stronger with increasing molecular size. The heat capacity difference, CP − CV, is an interesting property. Group I liquids have ΔC/k values of about 3 compared to Group II values that hover around 5. The SPT model has a flat minimum at ρR ≃ 2.5, where it has a value of 4.9. Consequently, all calculated values ΔC/k to two significant figures are 4.9. Surface tension is also an interesting property because there is no explicit thermodynamic relation to help guide in the selection of a scaling variable to render the surface tension dimensionless. However, it is known from the application of Cahn−Hilliard theory13 that the required thermodynamic quantity that needs to be integrated over the interfacial distance is the Helmholtz free-energy density that has units of pressure. Simple dimensional arguments, similar to what was done for the CED, suggest that the appropriate scaling variable for surface tension should be kTρ2/3. The results quoted in Tables 1−4 were calculated by dividing the experimental surface tension by the experimental value of kTρ2/3. The behavior of this dimensionless surface tension as a function of reduced density is illustrated in Figure 9. As can be seen from the tables, Group I and II liquids exhibit somewhat different values of this dimensionless surface tension at the same reduced density, with those of Group I about 10−20% larger than thsoe of Group II.

Figure 10. Liquid configurational energy for four Group I liquids, which also equals the dimensionless CED, δ̃2 . Group I types have their normal boiling points around δ̃2 = 8. According to any VDW model, the limiting value at the critical point should equal the internal pressure (see Supporting Information). Data are from refs 11 and 12.

B. CEDs and Solubility Parameters. Cohesive energy densities and solubility parameters have a long and respected history14 and are widely used in mixture models.15 The original motivation was to develop an intensive liquid state property that would measure “molecular cohesiveness”.14 Two liquids with similar molecular cohesions would be expected to be mutually soluble in one another. Internal pressure Pint was initially chosen as this measure, but in a VDW model, Pint and

IV. DISCUSSION A. Configurational and Thermal Energy. Qualitatively, the liquid state exists because attractive molecular interactions overcome thermal energy that varies as kT. Thermal energy favors the vapor state, whereas attractive molecular interactions 9394

dx.doi.org/10.1021/jp504140z | J. Phys. Chem. B 2014, 118, 9386−9397

The Journal of Physical Chemistry B

Article

Inspection of Tables 2−4 for Group II liquids reveals that traditional CEDs vary almost by a factor of 2 at their normal boiling points, with benzene at a high of 291 MPa and perfluoropropane at 153 MPa, whereas the dimensionless CED varies by less than 5% for the same 2 liquids: 9.5 for benzene to 9.1 for perfluoropropane. Clearly, there is a significant difference in how the two CEDs measure molecular cohesion. It remains for future studies to determine if the new CED is a better measure for mutual liquid solubility. C. Surface Tension. Because both compressibility and surface tension satisfy a CSP, their product also satisfies a CSP:

CED are equal to one another in the NLR. Because CEDs are more readily accessible experimentally, as well as for other technical reasons, CEDs have been used most often as a measure of molecular cohesion. Both properties have units of pressure, and neither satisfies a corresponding states principle. Dividing by the critical pressure renders them dimensionless, but they still do not obey a CSP. Any generic VDW model suggests that if either property is normalized by the pressure variable kTρ, the resulting dimensionless property becomes a function of reduced density only and does satisfy a CSP 2

δ ̃ ≡ δ 2/ρkT = Uliq /kT

(34)

κσ̃ ̃ = ρ1/3 κσ

This new dimensionless CED begs the question: does it offer a better measure of molecular cohesiveness and mutual solubility of one liquid in another? In Table 5 a dozen Group

This experimental product is tabulated in the tables. For Group I molecules, it varies from 80 to 90 and for Group II molecules from 65 to 80. It has long been recognized that the κσ product, which has the dimensions of length, is a fundamental length scale for liquids.16 Now it is understood that that this length scale varies as V1/3, which is a measure of the average distance between molecules. Thermodynamic work required to create free liquid surface involves an expenditure of free energy and governs the magnitude of the surface tension. In Figure 3 the favorable change (reduction) in chemical potential involved in the solvation step where the interactions are “turned on” is given by

Table 5. Comparison of Traditional CED with Dimensionless CED at 25 °Ca δ̃2=Uliq/RT@

substance bromine carbon disulfide benzene chloroform o-xylene toluene ethyl benzene carbon tetrachloride cyclohexane cyclopentane diethyl ether n-hexane

Uliq(TB) (kJ/mol)15

δ2 = Uliq/V @ 298 K (MPa)

TB

298 K

26.76 24.25 28.18 26.25 33.72 30.61 32.86 27.87

552 420 353 353 339 331 324 310

9.7 9.1 9.6 9.4 9.7 9.6 9.7 9.6

11.4 10.3 12.7 11.5 16.5 14.3 16.1 12.2

27.44 24.95 24.32 26.58

282 276 228 222

9.3 9.3 9.5 9.4

12.4 10.6 9.7 11.8

(35)

β[μ liq (T , ρliq ) − μig (T , ρliq )] = −(β ΔH vap + ΔSP /k) = −(ΔSvap + ΔSP)/k = ΔScond /k = ln Z liq = −ln B liq (36)

Notice that liquids both from Groups I and II with the more negative, and thus more favorable values of lnZliq also have the largest surface tensions. This is only a qualitative correlation, but it agrees with the intuitive idea that as free energy is lowered, more thermodynamic work is required to create a free liquid surface. D. Solvation Thermodynamics. Decomposing the entropy of vaporization into two contributions, as shown in Figure 3, illustrates the importance of the compressibility and insertion factors Zliq and Bliq on liquid state thermodynamics. In the NRL, ZliqBliq ≃ 1; therefore, knowledge of one yields the other, ln Bliq ≃ −ln Zliq. In transferring a molecule at a fixed position in an ideal gas to a fixed position in the liquid, the entropy change ΔSP/k is negative and equals −(ΔSvap/k + ln Zliq). This is the self-solvation entropy, and it measures the effects of repulsive and attractive interactions on the solvation process. Summing free energy around the thermodynamic cycle shows that the favorable free-energy change associated with this molecular transfer is ln Zliq = −ln Bliq (see discussion above on surface tension). With one well-known exception,2 the important role that these dimensionless factors play in solvation has not been fully appreciated. As mentioned above, the magnitude of the free-energy change on solvation correlates qualitatively with surface tension. The essential difference between Group I and II molecules is that the larger Group II molecules suffer a larger entropic penalty during solvation. Larger molecules require larger cavities, which are entropically unfavorable. This trend can be easily discerned in the tables.

a

The substances are arranged in descending order according to their traditional CEDs. Data source, ref 15.

II type molecular liquids are shown along with their traditional CEDs calculated at 25 °C that should be compared to the new dimensionless CED also evaluated at 25 °C. Bromine has the highest traditional CED that is 2.5 times larger than that of n-hexane. Although the available solubility information in the literature is primarily anecdotal, it appears that bromine is soluble in most organic solvents. On the basis of the observed large disparity in the traditional CEDs between bromine and the listed organic solvents, one might expect limited solubility of bromine in organic solvents. Below are some predictions based on CED differences: Best 3 solvents for bromine: Criterion: δ2 = Uliq/V carbon disulfide > benzene ≈ chloroform Criterion: δ2̃ = Uliq/RT chloroform > n-hexane > carbon tetrachloride Worst 3 solvents for bromine: Criterion: δ2 = Uliq/V n-hexane > diethyl ether > cyclopentane Criterion: δ2̃ = Uliq/RT o-xylene > ethyl benzene > toluene Clearly, the two CEDs disagree with each other on n-hexane; the traditional CED indicates that it is a poor solvent, and the dimensionless CED indicates that it is a good solvent. 9395

dx.doi.org/10.1021/jp504140z | J. Phys. Chem. B 2014, 118, 9386−9397

The Journal of Physical Chemistry B

Article

V. SUMMARY AND CONCLUSIONS Guided by the VDW model, all thermodynamic properties can be rendered dimensionless by a proper choice of scale parameters. The choice is not arbitrary, but is what any generic VDW model suggests to reduce the property to a function of reduced density only. Thus, comparing dimensionless thermodynamic properties at the same value of ρR yields the following corresponding states principle: liquids at the same reduced density ρR have comparable dimensionless thermodynamic properties. Properly expressing thermodynamic properties in dimensionless form helps to consolidate and harmonize liquid state properties. A summary of the multiplicative scale parameters and dimensionless variables are given in Table 6.

Isothermal compressibility may be the most universal property among Groups I and II. Among the 26 liquids in listed in Tables 1−4 at their normal boiling points, the average experimental value of this dimensionless compressibility is (5.1 ± 0.5) × 10−2, or about a 10% variation from the mean among these diverse liquids. A similar small variation is also observed in ln Zliq. The latter represents the favorable change in chemical potential in the self-solvation process. It remains to be seen whether the new dimensionless CED will be a better measure of molecular cohesion than the traditional CED. Certainly one attractive advantage of the new CED is that it indicates that Group II liquids will be near their normal boiling points when δ2̃ ≃ 9 and Group I liquids will boil near δ̃2 ≃ 8. As the tables indicate, the traditional CED does not possess this property. A closed-formed VDW equation for surface tension does not exist, but a scaling argument in the VDW spirit yields a dimensionless surface tension illustrated in Figure 9. Again, Group I and II behaviors split to some degree. The product of surface tension and compressibility has units of length, and this length scale is shown to vary as V1/3. By using scaled particle theory, an improved VDW equation of state is derived and is described in Supporting Information. Using critical density as the only adjustable parameter, the dimensionless thermodynamic properties are calculated at the normal boiling point and displayed in the tables. Excellent agreement is obtained for first-order properties of Group I liquids such as cohesive energy density and entropy of vaporization. Second-order properties such as compressibility and thermal expansion coefficient are predicted to be relatively constant, and they are, but quantitatively the model overestimates both properties. A second-order property that is accurately predicted is the thermal pressure coefficient for Group I liquids. Overall, the SPT model is less satisfactory for the larger Group II molecules, which is not too surprising. The SPT model is best suited for small and/or rigid molecules, not large flexible molecules such as n-heptane or larger alkanes. The SPT model predicts the difference in heat capacities ΔC/k to be relatively constant in the NLR; experimentally, it varies little in the NRL as predicted. It has values of 3 ± 0.3 for Group I and 5 ± 1 for Group II at the normal boiling point. The SPT model yields 4.9 for both Groups I and II. One of the contributions to the entropy of vaporization is the self-solvation entropy, ΔSp (see Figure 3). Experimentally, ΔSp is more punitive for the large molecules in Group II than for the smaller Group I molecules. It is the primary reason why Groups I and II differ in vaporization entropies. It is also the main reason why the SPT model does not work as well for Group II liquids because it underestimates the magnitude of ΔSp. Three new dimensionless critical properties have been identified. The first is the excess absolute fugacity, which is the product of the compressibility factor and the insertion factor, ZcBc. This product is unity for an ideal gas and also for a liquid in the NLR whose saturated vapor approximates an ideal gas. This product increases slowly with temperature along the saturation line and reaches a value of about 1.7 ± 0.1 at the critical point for nonpolar fluids. This is a preliminary estimate based on accurate empirical equations of state extended into the critical region for some simple fluids.1 A second dimensionless critical constant is the thermal pressure coefficient, γ̃V. Note that γ̃V = γ̃σ at the critical point, where γσ is the temperature coefficient of the vapor pressure curve

Table 6. Scaling Parameters Used to Render Thermodynamic Properties Dimensionless. Note That Neither the Critical Temperature nor Critical Pressure Is Involved in Scaling. property chem. potential, μ internal energy, E conf. energy, U entropy, S pressure, P density, ρ CED, δ2 thermal expansion coefficient, α isothermal compressibility, κ thermal pressure coefficient, γV = (∂P/∂T)V vapor pressure coefficient, γσ = (∂P/∂T)σ internal pressure Pint = (∂U/∂V)T heat capacity, CP or CV surface tension, σ

scale multiplier

notation

β = 1/kT β = 1/kT β = 1/kT 1/k βV = 1/kTρ 1/ρc βV = 1/kTρ T kTρ V/k = 1/kρ

μ̃ = βμ Ẽ = βE Ũ = βU S̃ = S/k P̃ = P/ρkT = Z ρR δ2̃ = δ2/kTρ α̃ = Tα κ̃ = kTρκ γ̃V = γV/kρ

V/k = 1/kρ

γ̃σ = γσ/kρ

βV = 1/kTρ 1/k βV2/3 = 1/kTρ2/3

P̃int = Pint/kTρ C̃ i = Ci/k σ̃ = σ/kTρ2/3

Depending on their vaporization entropies, liquids can be classified into three general groups. Group I includes small molecules such as monatomic and diatomic elements as well as methane (Table 1). Group II includes nonpolar and slightly polar organics that includes several common organic solvents such as carbon tetrachloride, benzene, heptane, and cyclohexane among others (Tables 2−4). Herein, the focus was on Group I and II liquids and not polar and hydrogen bonding liquids that comprise Group III. Normally in an EOS, density is considered the dependent variable with temperature and pressure as independent variables. Saturation pressures have little effect on liquid densities, and both pressure and temperature can be eliminated explicitly via the EOS because Zliq ≃ 0 in the NLR. Temperature effects are implicitly taken into account via scaling. For example, all energies are scaled by β = 1/kT. Thus, density becomes the only variable governing the thermodynamic properties of liquids in the NLR, and apparently all the way to the critical point. This is the new paradigm. Some thermodynamic properties of both Groups I and II follow the same “master curves”, such as isothermal compressibility illustrated in Figure 6 and the compressibility factor in Figure 4, while other properties split into two clear branches, as illustrated for the thermal expansion coefficient in Figure 7 and thermal pressure coefficient in Figure 8. 9396

dx.doi.org/10.1021/jp504140z | J. Phys. Chem. B 2014, 118, 9386−9397

The Journal of Physical Chemistry B

Article

(see Supporting Information). Experimental estimates of γ̃cσ are more certain than are those of γ̃V, and a few have been determined from extant data:12 water (1.77), hydrogen chloride (1.6), argon (1.6), nitrogen (1.7), methane (1.7), carbon tetrachloride (1.9), benzene (1.9), and n-heptane (1.8). Little variation in this new critical constant is seen in these diverse liquids. The third critical property is the internal pressure, which is a linear combination of other critical properties, P̃ cint = γ̃cV − Zc. According to a VDW model, the dimensionless CED, Uliq/kT, equals the internal pressure at the critical point. From Figure 10, it is seen that 1.8 < Uliq/kTc < 2.0, which is somewhat higher than estimated values of P̃cint. For hard spheres with attractive tails, the volume fraction of space η occupied is rigorously given by η = ηcρR. For the SPT model, ηc = 0.1287..., but this value yields calculated values of the vaporization entropy that are too small. Although ΔSvap differs for Group I, II, and III liquids, the ΔScond = k ln Zliq contribution to ΔSvap appears to be more universal, even including some Group III liquids such as methanol (see Figure 4). For this reason, ηc is “adjusted” to yield the correct value of ΔScond at the normal boiling point. This adjusted value of ηc is then used to predict all other thermodynamic properties and represents the only adjustable parameter in the SPT model. This adjusted value varies little between Group I and II liquids and most often lies in the range 0.154 ≤ ηc ≤ 0.16, but with a few exceptions noted in the tables. Values of ηc determined this way are shown in Tables 1−4. This value range for the critical density corresponds to what has been computed for a LennardJones fluid. Experimentally, the 18 Group II liquids in Tables 2−4 at their normal boiling points have ρR = 2.63 ± 0.07. The corresponding value of the calculated occupied volume fraction via the SPT model is ηB = 0.415 ± 0.10. Similar values are obtained for Group I liquids. A major conclusion is that occupied volume, or equivalently, free volume plays a dominant role in governing liquid thermodynamic properties for both Group I and II liquids.

■

(4) Widom, B. Potential-Distribution Theory and the Statistical Mechanics of Fluids. J. Phys. Chem. 1982, 86, 869−872. (5) 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. (6) Reiss, H.; Frish, H. L.; Lebowitz, J. L. Statistical Mechanics of Rigid Spheres. J. Chem. Phys. 1959, 31, 369−380. (7) Wertheim, M. S. Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres. Phys. Rev. Lett. 1963, 10, 321−323. (8) Johnson, J. K.; Zollweg, J. A.; Gubbins, K. E. The Lennard-Jones Equation Revisted. Mol. Phys. 1993, 78, 591−618. (9) Kolafa, J.; Nezbeda, I. The Lennard-Jones Fluid: An Accurate Analytic and Theoretically Based Equation of State. Fluid Phase Equilib. 1994, 100, 1−34. (10) Baltachev, G. S.; Nezbeda, I. Equation of State for LennardJones Fluid. High Temp. 2003, 41, 270−272. (11) 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 (12) Rowlinson, J. S., Swinton, F. L. Liquids and Liquid Mixtures, 3rd ed.; Butterworth Scientific: London, 1982; Chapter 2. (13) Cahn, J. W.; Hilliard, J. E. Free Energy of a Non-uniform System 1: Interfacial Free Energy. J. Chem. Phys. 1958, 28, 258−267. (14) Hildebrand, J. H.; Prausnitz, J. M.; Scott, R. L. Regular and Related Solutions: The Solubility of Gases, Liquids, and Solids; Van Norstrand Reinhold Co.: New York, 1970. (15) Barton, A. F. M. Handbook of Solubility Parameters and Other Cohesion Parameters; CRC Press: Boca Raton, FL, 1983. (16) Egelstaff, P. A.; Widom, B. Liquid Surface Tension Near the Triple Point. J. Chem. Phys. 1970, 53, 2667−2669.

ASSOCIATED CONTENT

S Supporting Information *

A complete description of the SPT model. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The author thanks Sean O’Keefe for his careful reading of the manuscript and helpful suggestions. Financial support from the William J. Murray, Jr. Endowed Chair in Engineering is gratefully acknowledged.

■

REFERENCES

(1) Sanchez, I. C.; Truskett, T. M.; in’t Veld, P. J. Configurational Properties and Corresponding States in Simple Fluids and Water. J. Phys. Chem. B 1999, 103, 5106−5116. (2) Ben-Naim, A. Solvation Thermodynamics; Plenum Press: New York, 1987. (3) Guillot, B.; Guissani, Y. A Computer Simulation of the Temperature Dependence of the Hydrophobic Hydration. J. Chem. Phys. 1993, 99, 8075−8094. 9397

dx.doi.org/10.1021/jp504140z | J. Phys. Chem. B 2014, 118, 9386−9397