ARTICLE pubs.acs.org/IECR
Development of Single Insertion Probability for Equation of State Applicable to Three Phases of Matter Ju Ho Lee,† Moon Sam Shin,‡ and Ki-Pung Yoo*,† † ‡
Department of Chemical and Biomolecular Engineering, Sogang University, Sinsu-Dong, Mapo-Gu, Seoul 121-742, Korea Department of Dermatological Health Management, Eulji University, 212 Yangji-dong, Sujeong-gu, Seongnam-si, Gyeonggi-do, 461-713, Republic of Korea ABSTRACT: We present an analytic equation of state (EOS) that describes the three phases of real substances and their transition in a single isotherm. To develop the repulsive contribution of the EOS for the solid-fluid transition, we reinterpret Alder’s correlated-cell model (Alder et al. Phys. Rev. Lett. 1963, 11 (6), 241) in terms of insertion probability and propose a simple mathematical function for insertion probability which approximately follows that of the correlated cell model. The resulting EOS was found to describe the solid-fluid phase transition of hard-sphere fluids qualitatively, avoiding the solid-fluid critical transition. To extend the model to the solid-vapor and solid-liquid phase transitions, we added the van der Waals attractive contribution to the EOS and tested the combined EOS against the equilibrium properties of eight substances ranging from simple gases to organic compounds. Calculation results show that for eight substances the combined model with four parameters closely reproduces the saturated vapor pressure, predicts the sublimation pressure reasonably, but underestimates the melting pressure, which results from the repulsive contribution of the EOS.
1. INTRODUCTION A volumetric equation of state (EOS) provides a well-defined mathematical relationship among the measurable properties of a substance. For the EOS to accurately describe the properties, the relationship should account for the existing forces such as repulsive and attractive forces as well as for their relative contribution affecting the phase of the matter. Since these forces exist in the solid, vapor, and liquid phases, a volumetric EOS should, in principle, apply to all three phases. However, its development has been mainly limited to the vapor-liquid transition since the transition is readily modeled by a single EOS.1 It is desirable, in the calculation of the phase equilibria, that the solid-fluid transition phase transition be also modeled by single EOS. However, the molecular simulation results for hard-sphere fluids have shown that the crystallization of hard-sphere fluids lead to the separate branch of the isotherm at the same density, an upper branch for the fluid phase and a lower branch for the solid phase.2 This distinguishes the solid-fluid transition from the vapor-liquid transition in three aspects: (1) no critical point exists in the solidfluid transition; (2) no attractive forces are involved in the phase transition; and (3) both branches show a monotonic rise as the density increases, indicating mechanical stability of the both phases. Thus, for the description of the solid-fluid phase transition, it has become customary to apply different EOS for each phase.3-7 For the reason given above, an approach based on a single EOS for all three phases has been mainly applied to the development of engineering-based empirical EOS,8-13 with incomplete theoretical rigorousness. The most notable EOS was proposed by Yokozeki12 who modified the repulsive term of the original van der Waals EOS to have two singularities, one for the upper limit of the fluid region and the other for that of the solid region. This approach yielded a discontinuous isotherm that eliminates the critical transition and implies a discontinuous r 2011 American Chemical Society
change of the phase transition. Although the developed unified equations of state have been applied, successfully to some extent, to the description of the three phases of real gases and other organic compounds, they demonstrate different behaviors in the shape of the isotherm in the solid-fluid phase transition—a main issue8,12,14-16 in model development. Besides, these models were found to violate physical constraints: that is, they predict a critical point in the solid-liquid phase transition,8 or a negative pressure12,14,17 occurs even without attractive forces. The cell theory18,19 is originally developed to model the liquid phase as an extension of the solid phase. This theory is also shown to be applicable to a description of the solid-fluid transition with single EOS, under the assumption19 that the change from one phase to the other can be followed by a continuous transition of a properly chosen variable. Following this approach, Mansoori and co-workers20,21 modified the theory and formulated an EOS capable of representing the vapor-liquid and the solid-liquid transitions simultaneously. However, their EOS poses the problems similar to those of the engineeringbased single EOS: the solid-liquid transition has a critical point and the description of the transition requires an attractive force. Since the single EOS based on the original cell theory violates the physical constraints, a different systematic approach is required for developing a consistent model. In this perspective, we note Alder’s correlated cell model22 for the melting transition of hard-disk fluids. Despite its relatively simple formulation, the model is considered to approximately account for the unique feature of the solid phase, the sliding of rows of particles past each other, and the reproduced single but Received: September 3, 2010 Accepted: January 14, 2011 Revised: December 27, 2010 Published: March 08, 2011 4166
dx.doi.org/10.1021/ie101853s | Ind. Eng. Chem. Res. 2011, 50, 4166–4176
Industrial & Engineering Chemistry Research
ARTICLE
sublimation pressure, and the heat of fusion of eight substances, including light gases and heavy compounds.
2. MODEL DERIVATION
Figure 1. Unit cell of the correlated cell model with (a) no overlap between hard disks, (b) overlap between hard disks in the same row, and (c) overlap between hard disks in the same row and the same column. Filled circles represent the hard disks, and dotted circles represent the excluded region inaccessible to the center of other hard disks.
continuous isotherm was found to be in reasonable agreement with the molecular simulation results, avoiding the critical transition. Thus, the correlated cell model gives some useful insight into the development of a single EOS for three phases. In this work, we develop a single unified EOS that shares the feature of the correlated cell model. The correlated cell model of Alder et al. is briefly reviewed, and the feature of the model is reinterpreted in terms of the insertion probability, a framework23-25 conventionally employed to explain the solid-fluid transition. By phenomenologically observing the insertion probability of the correlated cell model, we establish the condition required to model the solid-fluid transition in a single isotherm. Finally, we propose a simple mathematical form for the insertion probability that satisfies the established constraint and test the derived EOS against the measured melting pressure, the
2.1. Alder’s Correlated Cell Model. The hexagonal lattice unit cell of the original cell √ model is changed to a rectangular lattice of width d and height 3 d to allow the sliding of rows of particles past each other.22 Four hard disks of diameter σ are fixed to lattice positions, and a central disk is allowed to wander the region not occupied by the excluded volume of the fixed disks. Because the unit cell includes two particles, the packing fraction is defined as
π σ 2 η ¼ pffiffiffi 2 3 d
From the literature, the free √ area to the central disks is known to change form at d/σ = 2/ 3. However, we found that the form also changes at d/σ = 2, the condition for which the excluded regions of two particles in the same row of a unit cell begin to overlap. Figure 1 illustrates three representative unit cells of the correlated cell model. There is either no overlap at all between hard disks (Figure 1a), overlap only between hard disks in the same row (Figure 1b), or overlap between hard disks in the same row and column (Figure 1c). By introducing the dimensionless variable X = (d/σ)2, the corresponding mathematical formulation of the free area, Af, becomes
8 pffiffiffi > 3X - π > > > rffiffiffiffi > > > 1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi X < pffiffiffi Af 4X - X - 2 sin-1 3X ¼ 2 4 > σ2 rffiffiffiffi rffiffiffiffiffiffi > > > pffiffiffi 1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi X 1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 3X > -1 -1 > > 3X 4X - X - 2 sin - 12X - 9X þ 2 cos : 2 4 2 4 The generalized van der Waals theory26 allows us to relate the compressibility factor to the free area by PA D ln Af ¼ Z¼ ð3Þ kT D ln d2
PA0 PA A0 Z ¼ ¼ kT A X kT
ð5Þ
√ where A0 is the closed packed area defined by A0 = 3 σ 2. In Figure 2a,b, we plot the reduced pressure and its first derivative with respect to η. The isotherm shows a loop with a maximum at η = 0.680 and a minimum at η = 0.687, similar to
for d=σ > 2 pffiffiffi for 2 > d=σ > 2= 3
ð2Þ
pffiffiffi for 2= 3 > d=σ > 1
where P is the pressure, √ R is the gas constant, T is the temperature, and A = 3 d2 is the area of the unit cell. Accordingly, Z has a different mathematical form depending on the ratio of d to σ,
8 pffiffiffi 2 > > ð 3X=Af Þσ > > > ffi > < pffiffi3ffiX - 1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4X - X 2 =Af σ 2 Z¼ 2 > > > pffiffiffi 1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi 1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 > > > 3X - 4X - X - 12X - 9X =Af σ 2 : 2 2 The reduced pressure can be obtained from the relation
ð1Þ
for d=σ > 2 pffiffiffi for 2 > d=σ > 2= 3
ð4Þ
pffiffiffi for 2= 3 > d=σ > 1
that of vapor-liquid equilibria (VLE) in the presence of an extreme point. However, we note that the present loop is fundamentally different from that of VLE in the shape of pressure maximum. In VLE, at the pressure maximum, the isotherm is convex upward and its first derivative of pressure becomes zero. In contrast, as revealed by Figure 2b, the first derivative of the reduced pressure becomes discontinuous at the pressure maximum: as η approaches from the left side of η = 0.68, the 4167
dx.doi.org/10.1021/ie101853s |Ind. Eng. Chem. Res. 2011, 50, 4166–4176
Industrial & Engineering Chemistry Research
Figure 2. (a) Reduced pressure isotherm of the correlated cell model from eqs 4 and 5 and (b) its first derivative, as a function of the packing fraction.
derivative is positive finite but diverges to negative infinity as η slightly exceeds this value. Except the unstable region existing between extremes in the pressure isotherm, the single isotherm of the correlated cell model approximately follows the approach of employing different EOS for the solid and fluid phases which requires positive finite value of the first derivative of pressure over the entire density region. Since the pressure isotherm of the correlated cell model exhibits a cusp without attractive force, the critical solid-fluid transition is absent in the correlated cell model. In the next section, we reinterpret Alder’s model within the framework of the insertion probability. 2.2. Insertion Probability of Alder’s Model. The insertion probability, p*, is defined as the probability that a randomly selected molecule can be inserted into a system without overlapping other molecules. It is related to the EOS by the osmotic equation23,27 Z Fn 1 ZðFn , βÞ ¼ 1 - ln p þ ln pðψ, βÞ dψ ð6Þ Fn 0 where Z is the compressibility factor, ψ is the dummy integration variable that corresponds to the number density Fn, β = 1/k/T, and k is the Boltzmann constant. Equation 6 is found to be applicable to various fluids such as lattice and off-lattice systems. In off-lattice fluids, it is rewritten as23 Z 1 j Zðj, βÞ ¼ 1 - ln p þ ln pðω, βÞ dω ð7Þ j 0 where j is the volume fraction defined as b/V, b is the covolume of hard-sphere fluids, V is the molar volume, and ω is the dummy integration variable that corresponds to j. If an insertion probability satisfying the boundary conditions p*(0,β) = 1 and p*(1,β) = 0 is given as a mathematical function, we can derive a
ARTICLE
Figure 3. (a) Insertion probability of the correlated cell model and (b) its first derivative with respect to the packing fraction, η.
volumetric EOS. Conversely, the inverse form of eq 7 allows us to derive p* from Z differentiable with respect to j: Z j Zðω, βÞ - 1 ð8Þ dω ln pðj, βÞ ¼ 1 - Zðj, βÞ ω 0 A linear relation between j and η transforms eq 8 to Z η Zðζ, βÞ - 1 ln p ðη, βÞ ¼ 1 - Zðη, βÞ dζ ζ 0
ð9Þ
where ζ is the dummy integration variable that corresponds to η. If Z has a simple mathematical formulation, we can derive p* from eq 9. However, the following relation, derived in the Appendix, directly relates p* to the free area, Af: Af η DAf η DA ln p ðη, βÞ ¼ ln ð10Þ þ Af Dη A Dη A In Figure 3a,b, we plot p* calculated by substituting eq 2 into eq 10 and the first derivative of p*, respectively. As the density is increased, the calculated insertion probability monotonically decreases, passes through a minimum and a maximum, and then decreases to zero. The increase in p* is approximately consistent with the molecular simulation results23,25 that the insertion probability of hard-sphere or hard-disk fluid should increase during the fluid-to-solid transition. The behavior of the calculated insertion probability is reasonably similar to that of the reduced pressure shown in Figure 2a,b: as η approaches 0.680 from the left, the first derivative of the insertion probability approaches a finite negative value, but becomes infinitely positive when η approaches from the right side, as demonstrated in Figure 3. We apply these features to obtain a mathematical formulation for a single insertion probability. 4168
dx.doi.org/10.1021/ie101853s |Ind. Eng. Chem. Res. 2011, 50, 4166–4176
Industrial & Engineering Chemistry Research
ARTICLE
chain-connectivity effect, pr ðjÞ ¼ ðpðjÞÞr 0 1r j rj ¼ @m þ ð1 - mÞ1 - A ð1 - jÞr exp a 1-j ð13Þ The compressibility factor of r-mer molecules is obtained by substituting eq 13 into eq 7 and the reduced pressure is related to the compressibility factor by Pb/k/T = Zj/r. Thus, the reduced pressure for j e a becomes Pb j j2 a ð1 - mÞj ¼ ð1 - rÞ þ ln 1 kT r a 1-j 1-m ð14Þ And for j g a, it becomes Pb j j2 2am ¼ ð1 - rÞ þ ln m kT r 1-j 1-m að2m - 1Þ ð1 - mÞj þ ln 1 þ - 2ð1 - mÞ 1-m a Figure 4. (a) Insertion probability of monomeric fluid (r = 1) calculated by eq 13 and (b) the reduced pressure isotherm calculated by eqs 14 and 15, with a = 0.5 and m = 0.1.
2.3. New Insertion Probability Applicable to the SolidFluid Phase Transition. A mathematical formulation of the
insertion probability can be developed to account for all of the above features. However, describing the jump in Figure 3b would require a very complex mathematical form. Since our goal is to develop as simple and concise a form as possible, we approximate the behavior of the insertion probability of the correlated cell model as follows: as the density approaches the minimum in p* from the right, its first derivative with respect to density has a finite, rather than infinite, positive value. Thus, we write simply j ð11Þ pðj, βÞ ¼ pðjÞ ¼ m þ ð1 - mÞ1 - a where m is the minimum value of p* constrained between 0 e m e 1 and a is the density at the minimum, i.e., the upper limit of the fluid phase. The effect of β is ignored since eq 11 only deals with the repulsive contribution. The proposed function reproduces the above behavior and satisfies one of the boundary conditions, p*(0) = 1. However, it fails to satisfy the other boundary condition, p*(1) = 0. To remedy this problem, we multiply eq 11 by the insertion probability27 of van der Waals EOS, which yields zero at the closed-packed limit, 0 1 j j A @ p ðjÞ ¼ m þ ð1 - mÞ1 - ð1 - jÞ exp a 1-j ð12Þ The insertion probability for r-mer molecules, pr*, can be derived if we follow the Flory approach neglecting the
ð15Þ
Although our EOS has a different form depending on j, both equations have the same value at j = a. In Figure 4a,b, we plot the insertion probability of eq 13 and the reduced pressure given by eqs 14 and 15 for a = 0.5, m = 0.1, and r = 1. The reproduced insertion probability and pressure shows behavior similar to those of Alder’s model: they show a cusp at the pressure maximum while keeping P > 0. The derived EOS also satisfies ideal gas limits since Z obtained from eq 7 approaches 1 as j goes to 0. The reduced chemical potential becomes 0 1 μ j ¼ ln j - r ln@m þ ð1 - mÞ1 - A kT a - r lnð1 - jÞ þ
rj 1-j
ð16Þ
2.4. Single Equation of State for Solid-Fluid Phases. The proposed insertion probability contains two parameters, a and m, for r-mer fluids. These two parameters have a diverse combination, and we have to find the combination that yields the solidfluid transition. In the region j < a, for all combination of a and m, the reduced pressure of eq 14 monotonically increases with increasing density because the insertion probability of eq 13 monotonically decreases. However, for j > a, the reduced pressure of eq 15 shows a different behavior. As an illustration, Figure 5 shows plots of reduced pressure by eqs 14 and 15 for m = 0.3, 0.2, and 0.1 with a fixed to 0.5 for monomer fluid. The reduced isotherms for m = 0.3 and 0.2 show monotonically increasing behavior while the isotherm for m = 0.1 yields a loop exhibiting a cusp, which represents the solid-fluid transition. For an isotherm to have such a loop, the first derivative of the reduced pressure as the density approaches j = a from the right should be negative, DðPb=kTÞ 1 1 1-m < 0 ð17Þ ¼ þ -1 Dj r ð1 - aÞ2 m jfaþ0 4169
dx.doi.org/10.1021/ie101853s |Ind. Eng. Chem. Res. 2011, 50, 4166–4176
Industrial & Engineering Chemistry Research
ARTICLE
Pb j j2 j ¼ ð1 - rÞ þ - a ln1 - kT r a 1-j
ð20Þ
Although this EOS poses a discontinuity, the resulting Helmholtz energy has finite value except j = a, yielding finite chemical potential.28 The chemical potential becomes μ j rj ð21Þ ¼ ln j - r ln1 - - r lnð1 - jÞ þ kT a 1-j
Figure 5. Reduced pressure isotherms calculated by eqs 14 and 15 for m = 0.3 (dashed-dotted line), 0.2 (continuous line), and 0.1 (dashed line) For all isotherms, a and r are fixed at 0.5 and 1, respectively.
In our development of the model, we use the simplified EOS, eqs 20 and 21, since this form always exhibits the solid-fluid transition and requires fewer parameters than those of eqs 14-16. We aim to show that the EOS can describe the three phases of matter if an attractive contribution is considered. As attempted by Longuet-Higgins and Widom,29 we derive a unified EOS by incorporating a simple van der Waals attractive term into eq 20: Pb j j2 j ε 2 ¼ ð1 - rÞ þ j - a ln1 - ð22Þ kT r a kT 1-j where ε is the van der Waals attractive energy. The chemical potential then becomes μ j rj 2rε ¼ ln j - r ln1 - - r lnð1 - jÞ þ j kT a 1 - j kT
Figure 6. Reduced pressure isotherm of the combined EOS calculated by eq 22 for a = 0.7 and ε/k/T = 5.5. The dashed line on the left side indicates the vapor-liquid phase transition, and the dashed line on the right side indicates the solid-liquid phase transition.
This condition is rewritten as m
