Article pubs.acs.org/JPCC
Isomorphism in Fluid Phase Diagrams: Kulinskii Transformations Related to the Acentric Factor Qi Wei and Dudley R. Herschbach* Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843, United States Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, United States ABSTRACT: For a wide class of molecular fluids, the temperature−density phase diagrams exhibit two prominent generic properties: a nearly linear locus, termed the Zeno line, along which the compressibility factor, Z = P/ρRT = 1 (same as an ideal gas), and the widely arching border of the vapor−liquid coexistence region, termed the binodal curve, with gas and liquid branches meeting at the critical point. The Zeno and binodal loci have been known for more than a century, yet only during the past two decades were striking empirical correlations between them recognized. Recently, Kulinskii introduced a remarkably simple projective transformation, wherein the linearity of the Zeno line and its relation to the binodal curve are geometrical consequences of an approximate isomorphism of the fluid with a venerable theoretical model, the lattice gas (equivalent to the Ising spin model). Here we show the Kulinskii transformation is significantly improved in accuracy and scope by using as input, in place of the lattice gas, the original van der Waals equation or simulation results for the Lennard-Jones potential. Moreover, the key parameters in these transformations can be expressed in terms of the acentric factor, introduced by Pitzer to extend corresponding states.
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INTRODUCTION An intriguing correlation was recognized 20 years ago between two features prevalent in the temperature-density phase diagrams of a wide variety of fluids.1,2 These features, known for more than a century, are both manifest in approximately straight lines. One line, mentioned in myriad textbooks, has the quaint name “line of rectilinear diameters.” This line (RDL) is defined, in the two-phase region below but not too near to the critical point (CP), by < ρ >/ρC = 1 + AD(TC − T )/TC
(1)
where = (ρL + ρV)/2 is the average of the densities at temperature T on the liquid and vapor branches of the binodal curve. For typical fluids, the constant AD ≈ 1 ± 0.25. The other line, now termed the “Zeno line” (ZL), is much less familiar; we know of only two textbooks that discuss it.3,4 The ZL is defined as the contour in the T−ρ plane along which the compressibility factor, Z = P/ρRT = 1, the same as for an ideal gas. As illustrated in Figure 1, the Z = 1 contour extends from the Boyle point of the dilute gas (where ρ → 0 and the second virial coefficient vanishes) down to near the triple point in the dense liquid region yet remains nearly linear all the way. The ideal ZL is conveniently specified by its intercepts on the temperature and density axes T /TZ + ρ /ρZ = 1
Figure 1. Phase diagram for CO2, with several Z contours, constructed from experimental data from NIST.5 The Zeno line (Z = 1) contour is extrapolated down to the density axis to display that it becomes tangent to the extrapolated liquid branch of the binodal curve. Also shown is the line of rectilinear diameters in the vapor−liquid coexistence region and its extrapolations (short dashed lines).
(2)
where TZ is the Boyle point temperature and ρZ can be determined2 from the ratio of the Boyle volume and third virial coefficient at that temperature. The strong correlation observed between the RDL and ZL lines was intriguing because it implies that a remarkable kinship exists among the Boyle point, which pertains to supercritical © XXXX American Chemical Society
Special Issue: Ron Naaman Festschrift Received: April 3, 2013
A
dx.doi.org/10.1021/jp403307g | J. Phys. Chem. C XXXX, XXX, XXX−XXX
The Journal of Physical Chemistry C
Article
As defined in Figure 2, the LG variables (x, t) are dimensionless analogs for density and temperature; the corresponding fluid variables (ρ, T) are scaled to the CP values (ρC, TC). Table 1
dilute gas, and both the subcritical two-phase region and the liquid domain.1,2,6 In further studies of the ZL, Apfelbaum and coworkers7,8 discovered other striking correlations by boldly extrapolating to T → 0 (as illustrated in Figure 1). From empirical scrutiny9 and computer simulations,10 they confirmed that the extrapolated ZL becomes tangent to the liquid branch of the binodal curve that borders the two-phase region. Apfelbaum et al. also noted that the extrapolated density intercept for the RDL was close to ρZ/2, so they suggested the RDL could be regarded as approximately a “Zeno meridian” line: T/TZ + ρ/2ρZ = 1. These clues led Kulinskii to propose a simple geometrical interpretation derived from the topological isomorphism of the phase diagrams for the lattice gas model and real fluids.11−13 The isomorphism had long been invoked in heuristic fashion.14 Kulinskii took it much further by applying a projective mapping that provided natural explanations for the linearity of the ZL and RDL, their correlation, and the asymptotic congruence with the liquid branch. Here we show the Kulinskii transformation is significantly improved in accuracy and scope by using as input, in place of the lattice gas, the original van der Waals equation, simulation results for the Lennard-Jones potential, or empirical data for a reference fluid (e.g., Ar). The only parameters required to implement the various transformations are the ratios of the ZL intercepts to the CP quantities: TZ/TC and ρZ/ρC. We provide approximate formulas for those ratios, expressed solely as functions of the acentric factor, ω, introduced by Pitzer to extend corresponding states.15 Extensive tables of ω derived from experimental data are available.16 The link to the acentric factor seems likely to further enhance the unifying role of the Kulinskii mapping in wider applications to thermodynamics.13 Projective Transformations. Figure 2 depicts key aspects of the projective mapping between the lattice gas-phase
Table 1. Input for Lattice Gas Transformationa lattice gas
fluid diagram
point
x
t
ρ/ρC
T/TC
consequences
1 2 3 4 5 6
0 1/2 1 0 1/2 0
0 1 0 ∞ 0 ∞
0 1 ρZ/ρC 0 ρD/ρC 0
0 1 0 TZ/TC 0 TD/TC
c = 0, l = 0 a = d + 2e + 2 = 2k ρZ/ρC = a/(d + 1); h = 0 TZ/TC = k/e; b = 0 ρD/ρC = a/(d + 2); h = 0 TD/TC = k/e; b = 0
a
Points mapped in the transformation are specified in Figure 2.
exhibits how the eight transformation coefficients (a, b...) are determined by mapping between the LG and fluid diagrams four points (1−4 in Figure 2) located on the binodal curve and ZL. That yields ρ /ρC =
(a /2)t ax ; T /TC = 1 + dx + et 1 + dx + et
(4)
where the nonzero coefficients 1 a ; d= − 1; a= 1 − 1/(ρZ /ρC ) − 1/(TZ/TC) ρZ /ρC a e= 2TZ/TC (5) depend solely on the ratios, ρZ/ρC and TZ/TC, of the ZL and CP parameters. In eq 6 of ref 11, Kulinskii adopted a simpler form of the transformation: ρ T x αt ρ /ρC = Z · ; T /TC = B · ρC 1 + αt TC 1 + αt (6) where α = TC/(TB − TC). This results from an approximation involving the extrapolated density intercept, ρD, of the rectilinear diameter line (point 5 in Figure 2 and Table 1). The ratio of that intercept to the ZL density intercept (also extrapolated) is given by ρD /ρZ = (d + 1)/(d + 2)
The added approximation sets this ratio to 1/2, thereby specifying d = 0. Then, eqs 4 reduce to eqs 6, with e = α (as seen from Table 1, via points 2 and 4). The ρD/ρZ ratio is exactly 1/2 for the van der Waals equation. That is a fair approximation for many actual fluids; deviations of ρD/ρZ from 1/2 are typically small (often
