Prediction of Radon-222 Phase Behavior by Monte Carlo Simulation

Article pubs.acs.org/jced

Prediction of Radon-222 Phase Behavior by Monte Carlo Simulation Jason R. Mick, Mohammad Soroush Barhaghi, and Jeffrey J. Potoff* Department of Chemical Engineering and Materials Science, Wayne State University, Detroit, Michigan 48202, United States S Supporting Information *

ABSTRACT: Histogram-reweighting Monte Carlo simulations in the grand canonical ensemble are used to determine saturated liquid and vapor densities, vapor pressures, heats of vaporization, and compressibility factors for radon-222 from the normal boiling point to the critical point. An optimized intermolecular potential is developed by fitting parameters to reproduce experimental vapor pressures and the critical temperature. Vapor pressures are reproduced by simulation to within 2.2% of experiment, while the critical temperature and normal boiling point are reproduced exactly within the combined uncertainty of simulation and experimental data. The predictions of simulation are used to evaluate the reliability of reported experimental liquid densities, and those predicted by equations of state. While the predictions of simulation are in agreement with equations of state for the liquid density, significant differences are observed between the predictions of simulation and experiment. These results suggest that previously reported experimental liquid densities for radon-222 may be erroneous.

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radon, including crystal properties,59 solution behavior,27,60−64 and absorption into porous solids.65−68 The Lennard-Jones parameters used in these simulations were derived from correlations of the dispersion energy and the atomic polarizability with the number of electrons,69−73 or corresponding states theory.74 To date, however, these potentials have not been validated against experimental phase coexistence data. This work is, therefore, motivated by the limited experimental data and intermolecular potentials for 222Rn. Grand canonical histogram-reweighting Monte Carlo simulations are used to predict a variety of physical properties for 222 Rn, which include saturated liquid and vapor densities, vapor pressures, heats of vaporization and compressibility factors as a function of temperature. New Lennard-Jones parameters are optimized for 222Rn by fitting to reproduce experimental vapor pressures and the critical temperature. The predictions of simulation are used to evaluate the reliability of experimental data, especially the saturated liquid density, where large uncertainties have been reported, and to assess the accuracy of existing Lennard-Jones parameters for the prediction of 222 Rn vapor−liquid equilibria.

INTRODUCTION Radon’s scarcity presents a major challenge to its study.1,2 The concentration of radon in the atmosphere is approximately 6 × 10−11 ppb,3 which is a billion times lower than that of xenon. The densest naturally occurring member of the noble gas family of atomic gases, radon is continuously emanated from certain sediments that are distributed around the globe. Radon emanations are short-lived due to their isotopic instability; radon-222 (222Rn), the most stable isotope, has a half-life of 3.8235 days.4 Following the discovery of radon in 1900, methods were developed to refine approximately one cubic millimeter of 222 Rn gas from radium ores.5−7 Over the next 20 years, the normal boiling point,5,6 vapor pressure,6 vapor density at standard conditions,7 heat of vaporization,8 and critical temperature and pressure6 were determined. The critical density has not been measured experimentally, and has instead been calculated using empirical correlations.9,10 At least two attempts were made to determine the density of liquid 222 Rn;6,11,12 however, these estimates had large self-reported uncertainties. In the absence of new experimental data, there have been ongoing attempts to use corresponding state theory and equations of state to predict the physical properties of pure radon.9,10,13−19 However, there is significant disagreement between the various efforts. Liquid densities predicted by equations of state vary by up 10%9,10,13 and are 20−30% smaller than the experimental data.6,12 Similarly, reported heats of vaporization vary by up to 10%.8,13,16 While 222Rn has little commercial value, there is significant interest in its detection and remediation.20−22 Research has been performed to assess such properties of radon-222 as solution behavior,23−35 diffusion characteristics,36−38 transport rates,39−42 and adsorption20,43−49 and absorption44,45,50−58 behavior. Computer simulations have also been used to study © XXXX American Chemical Society

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FORCE FIELD Interactions between 222Rn atoms were described by a LennardJones potential:75 ⎡⎛ ⎞12 ⎛ ⎞6 ⎤ σij σij U (rij) = 4εij⎢⎢⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ ⎥⎥ r ⎝ rij ⎠ ⎦ ⎣⎝ ij ⎠

(1)

Received: November 24, 2015 Accepted: February 25, 2016

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DOI: 10.1021/acs.jced.5b01002 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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where rij, εij, and σij are the separation, well depth, and collision diameter, respectively, for the pair of interaction sites i and j. Optimized Lennard-Jones parameters for 222Rn are presented in Table 1. Six additional parametrizations found in the literature are included for comparison.69−74 Table 1. Lennard-Jones Parameters for

Table 2. Run Conditions for Grand Canonical HistogramReweighting Monte Carlo Simulations Used to Determine the Phase Behavior of 222Rn

222

Rn

model

εi/K

σi/Å

this work Saxena and Srivastava69 Miller70 Gopal73 Chakraborti71 Slyusar et al.72 van Loef74

292.0 484 290 283 280 292.8 300

4.145 4.48 4.35 4.36 4.15 4.212 4.17

run

T (K)

μ-β−1 ln Λ3 (K)

⟨N⟩

state

1 2 3 4 5 6 7 8 9 10

310 340 375 350 320 290 270 240 220 210

−2650 −2650 −2650 −2515 −2410 −2320 −2270 −2185 −2150 −2130

3.54 8.94 59.38 133.22 147.43 158.92 165.69 176.24 182.13 185.08

vapor vapor bridge liquid liquid liquid liquid liquid liquid liquid

ρliq − ρvap 2

The Lennard-Jones potential was chosen to model interactions between radon atoms based on previous calculations by our group,76 and others,77 which show that the Lennard-Jones potential provides a reasonable approximation of interactions between noble gas atoms. While it may be possible to improve the accuracy of the force field slightly by using a Mie potential instead of the 12−6 Lennard-Jones potential, the limited amount of available experimental data, and the uncertainty of those data, do not justify the introduction of additional parameters in the potential energy function.

(3)

and to the density scaling law for the critical temperatures: ρliq − ρvap = B(T − TC)β

(4)

where β = 0.325. A and B were constants fit to the saturated vapor and liquid densities. To avoid errors due to finite-size effects, these equations were applied to the coexistence densities on the range 255 ≤ T ≤ 340 K. The critical pressures PC and the normal boiling points TNBP were calculated by fitting vapor pressure data to the Clausius−Clapeyron equation: 83

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ΔH v +C RT

ln P = −

COMPUTATIONAL METHODS The phase behavior and physical properties of 222Rn were calculated with histogram-reweighting Monte Carlo simulations in the grand canonical ensemble. 78 Calculations were performed using the development version of GOMC.79 All simulations were run for 42 million Monte Carlo steps (MCS), where the first 4 million MCS were used for equilibration of the system. Packmol80 was used to generate the initial atomic coordinates, while psfgen was used to create the connectivity (*.psf) file.81 All simulations were performed using a fixed box size of L = 25 Å. Lennard-Jones interactions were truncated at 10 Å and analytical tail corrections were applied. The ratio of moves was 70% particle insertion/deletion and 30% particle displacement. To generate the phase diagram, 10 simulations were performed; one bridging the gas and liquid phases near the critical temperature, two in the gas phase, and seven in the liquid phase. The specific run conditions and average number of atoms in each simulation are listed in Table 2. Histogram data were collected in the form of a list, updated every 200 MCS, containing the number of particles and energy of the system. Statistical uncertainties were determined from five independent sets of simulations, each started with different random number seeds. The heat of vaporization (ΔHv) was determined via the difference in energies and molar volumes in each phase:82 ΔH v = (UV − UL) + P(VV − VL)

= ρC + A(T − TC)

(5)

where P is the vapor pressure, ΔH is the heat of vaporization, R is the gas constant, and C is a constant. v

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RESULTS AND DISCUSSION Given the limited availability of experimental liquid densities, Lennard-Jones parameters for 222Rn were fit to reproduce experimental vapor pressures and the critical temperature. A preliminary search was performed near the critical point to determine the parameter space that produced reasonable agreement for TC and the vapor pressure. On the basis of these results, phase diagrams were calculated for a total of 187 parameter sets, spaced on 0.01 Å increments along σi to give 11 values on the range σi = (4.10 to 4.20) Å and spaced on 0.5 K increments along εi to give 17 values on the range εi = (288.0 to 296.0) K. A scoring function was used to identify the optimal parameter set n

1 Scorej = 0.5*Err(TC) + 0.5* (0.91 ∑ Err(PV(Ti )) n i=0 n−1

+ 0.09 ∑ Err(PV(Ti + 1)) − Err(PV(Ti ))) i=0

(6)

where n is the number of experimental data points and

(2)

Err(X ) =

Where P is the saturation pressure and UV and UL; and VV and VL are the energy per mole; and molar volumes of the gas and liquid phases, respectively. The critical temperature (TC) and density (ρC) for each model was calculated by fitting the saturated liquid and vapor densities to the law of rectilinear diameters:

(Xsim − Xexpt) Xexpt

100 (7)

The relative error in the critical temperature was 50% of Scorej. The other 50% of Scorej was determined from the summation of the average absolute errors in the vapor pressure with respect to experiment, and by a summation of the difference between B



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Figure 1. Normalized error in TC (top left) and vapor pressure (lower left), and total value of the scoring function (right). Red represents the closest agreement with experimental data, while blue is the worst.

consecutive relative errors. The smaller consecutive difference factor is based upon the differential form of the exponential growth function and distinguishes between models with similar absolute error. In Figure 1, the performance of the potential parameters is visualized as heat maps, with errors normalized to the maximum error found in the parameter space. The heat maps in the upper left and lower left illustrate the performance of various parameter sets for the prediction of critical temperature and vapor pressure, respectively. The larger rightmost heat map depicts the final total score produced by eq 6. The heat maps show that the combination of vapor pressure and critical temperature is sufficient to define the optimal potential parameters uniquely. From these data, the optimal potential was determined to be σi = 4.145 Å and εi = 292.0 K. The uncertainty in the optimized parameters is approximately 1.0 K for epsilon and 0.03 Å for sigma. In Figure 2, the vapor−liquid coexistence curve, vapor pressure, and heat of vaporization predicted by the optimized potential are presented, with experimental data for comparison. Selected vapor−liquid coexistence data are presented in Table 3, while critical properties are listed in Table 4. The optimized model reproduced experimental vapor pressures6 to within 2.2% for temperatures between the normal boiling point and 0.9Tc. While the uncertainty in the experimental vapor pressures was not given in the original work, we have estimated to be 2% based on relative differences between the raw data and the best-fit curve. The model predicted a critical temperature of 377.6 K, which is in exact agreement with the experimental value, within the combined uncertainty of simulation and the experimental data.6 These results are expected, since the vapor pressure and critical temperature were used as target data for the optimization process. Small deviations from experiment were observed for the critical pressure. Simulation predicts Pc = 65.0 bar, while Ramsay and Gray determined Pc = 63.2 bar from their vapor pressure data.6 Refitting Ramsay and Gray’s smoothed data to eq 5 produced Pc = 64.5 bar, which suggests that most of the observed error between simulation and experiment was due to the process used to extrapolate the vapor pressure to the critical point. The optimized potential predicted a normal boiling point of 210.5 K, which is in good agreement with prior values determined from experimental vapor pressures of 208.11 to 211.12 K.5,6 At the normal boiling

Figure 2. (a) Vapor−liquid coexistence curve, (b) heat of vaporization, (c) P−ρ coexistence curve and isotherm at standard temperature and pressure, and (d) Clausius−Clapeyron plot. Experimental vapor pressures (black line),6 heat of vaporization (black star),6,8 and vapor density (black star).7 Simulation data (red diamonds); data at the normal boiling point (filled diamonds).

point, simulations predicted ΔHV = 16.36 kJ·mol−1, which was within 1.3% of the experimental value of ΔHV = 16.57 kJ· mol−1.8 Similarly, the saturated vapor density predicted by simulation ρgas = 10.021 ± 0.001 kg·m−33 is in excellent agreement with the experimental value of ρgas = 10.0 ± 0.2 kg· m−3.7 Published liquid densities for 222Rn show considerable scatter.7,11 Gray and Ramsay determined the liquid density near the normal boiling point by measuring the volume of the condensate. This work produced an average density of 5800 kg· m−3, with a standard deviation of 1100 kg·m−3.7 A second experimental measurement by Rutherford and Soddy produced a liquid density of 6000 ± 1000 kg·m−3.11,12 Part of the large uncertainty in the liquid density may be attributed to use of very small sample volumes of 0.00025 mm3 and 0.000120 mm3 for the experiments performed by Gray7 and Rutherford,11,12 respectively. Extrapolation of experimental saturated liquid densities at the normal boiling point for lighter noble gases to 222 Rn predicted a liquid density of 4350 kg·m−3,13 while Monte C



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Table 3. Selected Phase Coexistence Data for 222Rn Predicted by Monte Carlo Simulation. Number in Parentheses Denotes the Statistical Uncertainty in the Last Digit

Herreman proposed an alternative approach, and instead applied the principle of corresponding states to develop an empirical correlation for the viscosity9,19

ρvap

ρliq

P

ΔHv

K

kg·m−3

kg·m−3

bar

kJ·mol−1

Z

360 350 340 330 320 310 300 290 280 270 260 250 240 230 220 210

644.0(9) 522.0(9) 421.0(9) 340.9(9) 276.9(6) 224.4(4) 180.9(3) 144.7(2) 114.4(2) 89.3(1) 68.6(1) 51.7(1) 38.1(1) 27.3(1) 19.0(1) 12.7(1)

2690(5) 2914(2) 3086(1) 3226(2) 3351(3) 3464(3) 3570(3) 3668(2) 3763(2) 3853(2) 3940(1) 4024(2) 4105(2) 4184(2) 4260(1) 4332(4)

51.00(9) 43.37(7) 36.56(5) 30.56(4) 25.30(3) 20.71(2) 16.74(2) 13.35(1) 10.47(1) 8.07(1) 6.09(1) 4.50(1) 3.23(1) 2.25(1) 1.52(1) 0.99(1)

7.57(3) 8.83(2) 9.91(1) 10.81(1) 11.58(1) 12.25(1) 12.84(1) 13.37(1) 13.86(1) 14.31(1) 14.72(1) 15.09(1) 15.45(1) 15.78(1) 16.09(2) 16.38(4)

0.588(3) 0.634(3) 0.682(3) 0.725(2) 0.762(2) 0.795(2) 0.824(2) 0.849(2) 0.873(2) 0.894(1) 0.912(1) 0.929(1) 0.944(2) 0.958(4) 0.972(7) 0.990(9)

ηR =

ρC =

(9)

where T Tc

(

ρ

)

− 0.3201

(12)

0.3201 RT 0.667 8PC C

+ 0.00270

(13)

Using the critical parameters of Tc = 377.65 K and Pc = 63.31 bar as input, eq 13 predicted ρc = 1613 kg·m−3. At the normal boiling point, Herreman predicted ρliq = 4330 kg·m−3, which is within 1% of the predictions of simulation. Using the predictions of simulation for the critical properties, it is possible to resolve discrepancies in past equation of state calculations for the saturated liquid density. In their empirical correlations, both Grosse and Herreman used a critical pressure that was 3% lower than the value predicted by simulation. In addition, the difference in critical densities was approximately 5%. In Figure 3, the deviation between the two equations of state and simulation is presented. The original parametrization by Grosse underpredicts the saturated liquid density by approximately 7%, while the Herreman equation of state predicts saturated liquid densities that are within 1% simulation. Using the critical density and critical temperature predicted by simulation in both the Guggenheim correlation (applied by Grosse), and the Herreman equation of state, produced relative differences in the saturated liquid densities of less than 1.2% versus simulation for 211 ≤ T ≤ 350 K (Figure 3). This confirms that observed disagreements between past applications of equations of state were due largely to the use of different critical properties. Six existing Lennard-Jones potentials for 222Rn were also evaluated. The relative deviation from experiment for the vapor pressure is plotted as a function of temperature in Figure 4. The predicted critical points constants are listed in Table 4. The model of Saxena and Srivastava69 predicted a critical temperature that was almost twice the experimental value, hence these data have been excluded from Figure 4. The models of Miller70

and

τ=1−

0.5348 ρc

Since the critical density was unknown, it was determined from the following empirical correlation

(8)

ρvap = ρC (1 + 0.75τ − 1.75 3 τ )

(11)

where ηR is the reduced viscosity and Tr = T/Tc. Knowing ηR, liquid densities were calculated via

Carlo simulations using the optimized potential predicted a liquid density of 4329 ± 4 kg·m−3 at 210.5 K. Saturated liquid densities for 222Rn have also been predicted from equations of state. Grosse10 used the Guggenheim relationship84 to predict the saturated liquid and vapor densities of 222Rn. This expression is given as ρliq = ρC (1 + 0.75τ + 1.75 3 τ )

1.694 Tr

ln ηR = −0.709 +

T

(10) −3

The critical density of 1528 kg·m was calculated from the known experimental critical temperature (377.5 K) and pressure (63.23 bar), and critical compressibility factor Zc = 0.292. Zc was assumed to be the same as that of argon, krypton, and xenon. At the normal boiling point, Grosse predicted ρliq = 4070 kg·m−3 which is approximately 30% lower than the value reported by Gray and Ramsey, but within 5% of the value prediction by Monte Carlo simulations.

Table 4. Critical Parameters and Physical Properties for 222Rn Predicted from Simulation. Number in Parentheses Denotes the Statistical Uncertainty in the Last Digita property TC/K ρC/kg·m−3 PC/bar ZC TNBP/K ΔHV/kJ·mol−1 ρliq,/kg·m−3 ρvap/kg·m−3 a

experiment 377.7

6

63.26 211.12,6 208.11,5 19885,86 16.59,8 16.7816 5800,6 600012 10.0(2)7,87

this work

Saxena et al.69

Miller70

Gopal73

Chakraborti71

Slyusar et al.72

van Loef74

377.6(3) 1629(1) 65.0(3) 0.2810(1) 210.5(2) 16.36(4) 4329(4) 10.021(1)

622.4(1) 1287(1) 83.7(1) 0.2789(3) 338.7(1) 27.60(5) 3498(5) N/A

373.5(2) 1411(1) 55.2(1) 0.2795(6) 212.6(1) 16.22(2) 3744(2) 10.037(1)

364.0(1) 1405(1) 53.2(1) 0.2779(6) 208.3(2) 15.85(3) 3716(3) 10.0330(5)

361.2(1) 1633(1) 61.4(1) 0.2781(3) 202.7(3) 15.73(6) 4306(6) 10.012(5)

378.3(2) 1556(2) 62.2(2) 0.2820(9) 212.2(1) 16.40(2) 4128(5) 10.0261(2)

388.2(2) 1596(1) 65.7(1) 0.2845(9) 216.1(1) 16.80(2) 4259(7) 10.029(1)

ΔHv and ρliq were evaluated by simulation at the normal boiling point, while ρvap was evaluated at standard temperature and pressure. D



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corresponding state theory, and extrapolations of liquid densities from lighter noble gases.9,10,13 These data suggest that reported experimental values for the density of liquid Rn222 may be erroneous. The critical density predicted from simulation was used in the equations of state proposed by Grosse10 and Herreman.9,19 When using the same critical density and pressure, both methods were found to predict saturated liquid densities within 2% each other, and within 1.2% of simulation. This resolves past discrepancies of up to 10% in the saturated liquid densities predicted by these two equations of state. Additional calculations performed for six force fields found in the literature shows that the optimized potential offers significant improvements with respect to the prediction of vapor−liquid equilibria and critical properties of 222Rn.

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Figure 3. Relative error in the predictions of the Guggenheim relation for the saturated liquid density of 222Rn relative to the predictions of simulation: original (blue diamonds),10 and reevaluated with ρc and Pc predicted by this work (red diamonds). Predictions of the equation of state of Herreman: original (black squares),9 and reevaluated with ρc and Pc predicted by this work (green squares).

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.5b01002. Tabulated experimental vapor pressure data from the work of Gray and Ramsay, and additional predictions from various Lennard-Jones parameters for 222Rn (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: jpotoff@wayne.edu. Tel: 313 577 9357. Funding

Funding from the National Science Foundation OCI-1148168 and computational time from Grid Computing at Wayne State University is gratefully acknowledged. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Porstendörfer, J. Properties and behaviour of radon and thoron and their decay products in the air. J. Aerosol Sci. 1994, 25, 219−263. (2) Cohen, B. L. Radon: characteristics, natural occurrence, technological enhancement, and health effects. Prog. Nucl. Energy 1979, 4, 1−24. (3) Stanley, R. E.; Moghissi, A. A.; Center, N. E. R. Noble Gases; U.S. Environmental Protection Agency: 1975. (4) Fry, C.; Thoennessen, M. Discovery of the astatine, radon, francium, and radium isotopes. At. Data Nucl. Data Tables 2013, 99, 497−519. (5) Rutherford, E. The boiling point of the radium emanation. Nature 1909, 79, 457−458. (6) Gray, R. W.; Ramsay, W. CXXII.-Some physical properties of radium emanation. J. Chem. Soc., Trans. 1909, 95, 1073−1085. (7) Gray, R. W.; Ramsay, W. The density of niton (″radium emanation″) and the disintegration theory. Proc. R. Soc. London, Ser. A 1911, 84, 536−550. (8) Rudorf, G. Ü ber einige numerische Konstanten der Radiumemanation und deren Beziehung zu denen der Edelgase. Z. Elktrochem. Angew. P 1909, 15, 748−749. (9) Herreman, W. Calculation of the density of liquid radon. Cryogenics 1980, 20, 133−134. (10) Grosse, A. V. Some physical and chemical properties of element 118 (Eka-Em) and element 86 (Em). J. Inorg. Nucl. Chem. 1965, 27, 509−519. (11) Rutherford, E.; Soddy, F. LIX. Condensation of the radioactive emanations. Philosophical Magazine Series 6 1903, 5, 561−576. (12) Rutherford, E. LXVII. Condensation of the radium emanation. Philosophical Magazine Series 6 1909, 17, 723−729.

Figure 4. Relative deviation in the predicted vapor pressure with respect to the experiment.6 This work (red circles); Miller (brown stars);70 Chakraborti (green diamonds);71 Slyusar et al. (purple triangles);72 Gopal (orange downward-facing triangles);73 van Loef (blue squares).74

and Slyusar et al.72 each predicted TC to within 10 K of experiment. These models produced saturated liquid densities at the normal boiling point from ρliq = 3744 kg·m−3 (Miller)72 to ρliq = 4128 kg·m−3 (Slyusar et al.).72 The model from van Loef74 predicted ρliq = 4259 kg·m−3, which is within 1% of the value predicted by the optimized force field. In terms of the vapor pressure, only the predictions of models from Slyusar and Gopal were within 10% experimental data. Overall, the models of Slyusar et al. and Miller showed the least combined error of the existing force fields from the literature.

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CONCLUSIONS In this work, an optimized intermolecular potential for 222Rn was developed by fitting parameters to reproduce experimental vapor pressures and the critical temperature. Vapor pressures were reproduced by simulation to within 2.2% of experiment, while the critical temperature and normal boiling point and vapor density at STP were reproduced exactly. Liquid densities predicted by simulation were approximately 25% lower than experimental values, while being in close agreement with E



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(13) Pearson, T. G.; Robinson, P. L. 161. The volatile hydrides. Part I. Periodicity as a means of correcting and supplementing determined physical properties. Part II. The parachor, the molar volume at absolute zero, and the electronic structure and properties of compounds. J. Chem. Soc. 1934, 736−743. (14) Rankine, A. O. VII. On the relation between viscosity and atomic weight for the inert gases; with its application to the case of radium emanation. Philosophical Magazine Series 6 1911, 21, 45−53. (15) Kelley, K. K. Contributions to the Data on Theoretical Metallurgy. I, Entropies of Inorganic Substances. U.S. G.P.O.: Washington [D.C.], 1932. (16) Kelley, K. K. Contributions to the Data on Theoretical Metallurgy. Iii, The Free Energies of Vaporization and Vapor Pressures of Inorganic Substances; U.S. G.P.O.: Washington [D.C.], 1935. (17) Herz, W. Berechnung der Nullpunktsdichte einiger Elemente. Z. Anorg. Allg. Chem. 1919, 105, 171−174. (18) Slyusar, V. D.; Rudenko, N. S.; Tretyakov, V. M. Experimental study of elementary substance viscosity along the saturation line and under pressure, II argon, krypton, xenon. Ukranian J. Phys. 1972, 17, 190−194. (19) Herreman, W. A principle of corresponding states for the viscosity of non-quantum mono- and diatomic condensed gases. Cryogenics 1977, 17, 415−418. (20) Cohen, B. L.; Cohen, E. S. Theory and practice of radon monitoring with charcoal adsorption. Health Phys. 1983, 45, 501−508. (21) Jönsson, G. Radon gas  where from and what to do? Radiat. Meas. 1995, 25, 537−546. (22) Sanden, M. Capturing radon on tobacco smoke. Ph.D. Thesis, Chalmers University of Technology, 2011. (23) Nussbaum, E.; Harsh, J. B. Radon solubility in fatty acids and triglycerides. J. Phys. Chem. 1958, 62, 81−84. (24) Battino, R., 3-Radon-222. In Krypton, Xenon & Radon; Clever, H. L., Ed.; Pergamon: Amsterdam, 1979; pp 227−338. (25) Stein, L. Ionic radon solutions. Science 1970, 168, 362−364. (26) Gevantman, L. H. Solubility of Selected Gases in Water. In CRC Handbook of Chemistry and Physics, 95th ed.; Haynes, W. M., Ed.; CRC Press: New York, 2014; pp 5−150. (27) Dias, A.; Pamies, J.; Vega, L.; Coutinho, J.; Marrucho, I. Modelling the solubility of gases in saturated and substituted perfluoroalkanes. Polish J. Chem. 2006, 80, 143−152. (28) Kal’muk Bryk, S. D.; Makitra, R. G.; Pal’chikova, E. Y. Solubility and distribution of radon in organic solvents. Russ. J. Gen. Chem. 2007, 77, 503−508. (29) Gerrard, W., 4−Radon-219. In Krypton, Xenon & Radon; Clever, H. L., Ed.; Pergamon: Amsterdam, The Netherlands, 1979; pp 339− 342. (30) Gerrard, W., 1−Radon-220. In Krypton, Xenon & Radon; Clever, H. L., Ed.; Pergamon: Amsterdam, The Netherlands, 1979; pp 343− 345. (31) Key, R. M.; Brewer, R. L.; Stockwell, J. H.; Guinasso, N. L., Jr; Schink, D. R. Some improved techniques for measuring radon and radium in marine sediments and in seawater. Mar. Chem. 1979, 7, 251−264. (32) Smethie, W. M., Jr; Schell, W. R. Vacuum extraction of natural levels of radon from seawater. Mar. Chem. 1980, 8, 251−260. (33) Pressyanov, D.; Mitev, K.; Dimitrova, I.; Georgiev, S. Solubility of krypton, xenon and radon in polycarbonates. Application for measurement of their radioactive isotopes. Nucl. Instrum. Methods Phys. Res., Sect. A 2011, 629, 323−328. (34) Nikitin, B. A. Radon hydrate. Zeitschrift für anorganische und allgemeine Chemie (1950) 1936, 227, 81−93. (35) Juarez, R.; Zavala-Oseguera, C.; Jimenez-Halla, J. O. C.; Bickelhaupt, F. M.; Merino, G. Radon hydrides: structure and bonding. Phys. Chem. Chem. Phys. 2011, 13, 2222−2227. (36) Hirst, W.; Harrison, G. E. The diffusion of radon gas mixtures. Proc. R. Soc. London, Ser. A 1939, 169, 573−586. (37) Neufeld, P. D.; Aziz, R. A. Test of three new corresponding states potentials for Ne, Ar, Kr, and Xe with application to thermaldiffusion. J. Chem. Phys. 1973, 59, 2234−2243.

(38) Jacucci, G.; McDonald, I. R. Structure and diffusion in mixtures of rare-gas liquids. Phys. A 1975, 80, 607−625. (39) Saxena, S. C. Generalised relations for the three elementary transport coefficients of inert gases. Physica 1956, 22, 1242−1246. (40) Key, R. M.; Guinasso, N. L., Jr; Schink, D. R. Emanation of radon-222 from marine sediments. Mar. Chem. 1979, 7, 221−250. (41) Kestin, J.; Knierim, K.; Mason, E. A.; Najafi, B.; Ro, S. T.; Waldman, M. Equilibrium and transport properties of the noble gases and their mixtures at low density. J. Phys. Chem. Ref. Data 1984, 13, 229−303. (42) Ballentine, C. J.; Burnard, P. G. Production, release and transport of noble gases in the continental crust. Rev. Mineral. Geochem. 2002, 47, 481−538. (43) Lind, S. C.; Livingston, R. Adsorption of radon by glass. J. Phys. Chem. 1936, 40, 419−420. (44) Burtt, B. P.; Kurbatov, J. D. Mixed adsorption of radon and argon on silica gel. J. Am. Chem. Soc. 1948, 70, 2278−2282. (45) Ackley, R. D. Removal of radon-220 from HTGR fuel reprocessing and refabrication off-gas streams by adsorption (based on a literature survey); ORNL-TM-4883 United States; Oak Ridge National Laboratory: 1975. (46) Scarpitta, S. C. A theoretical model for 222Rn adsorption on activated charcoal carnisters in humid air based on polaynyis’s potential theory. Health Phys. 1995, 68, 332−339. (47) Eichler, B.; Zimmermann, H. P.; Gaggeler, H. W. Adsorption of radon on ice surfaces. J. Phys. Chem. A 2000, 104, 3126−3131. (48) Eichler, R.; Schadel, M. Adsorption of radon on metal surfaces: A model study for chemical investigations of elements 112 and 114. J. Phys. Chem. B 2002, 106, 5413−5420. (49) Brünner, S. Study of radon adsorption on activated carbon for a purification system in XENON1T. Ph.D. Thesis, Karl-FranzensUniversität Graz, 2013. (50) Mortazavi, S. M. J.; Mehdizadeh, S.; Zehtabian, M.; Sina, S. Development of an economical radon-resistant construction technique that is applicable in national radon-reduction programmes. Int. J. Low Radiat. 2009, 6, 113−118. (51) Paschalides, J. S.; Marinakis, G. S.; Petropoulos, N. P. Passive, integrated measurement of radon using 5A synthetic zeolite and blue silica gel. Appl. Radiat. Isot. 2010, 68, 155−163. (52) Dyer, A.; Las, T.; Zubair, M. The use of natural zeolites for radioactive waste treatment: studies on leaching from zeolite/cement composites. J. Radioanal. Nucl. Chem. 2000, 243, 839−841. (53) Hedström, H.; Foreman, M.; Ekberg, C.; Ramebäck, H. Radon capture with silver exchanged zeolites. Radiochim. Acta 2012, 100, 395. (54) Trofimov, A.; Kazankin, Y. N. Clathrate compounds of pi-cresol with inert gases. II. compounds of pi-cresol with krypton and radon. Radiokhimiya 1966, 8, 720−723. (55) Stein, L. The chemistry of radon. Radiochim. Acta 1983, 32, 163−171. (56) Jacobson, D. R.; Khan, N. S.; Colle, R.; Fitzgerald, R.; LaureanoPerez, L.; Bai, Y. B.; Dmochowski, I. J. Measurement of radon and xenon binding to a cryptophane molecular host. Proc. Natl. Acad. Sci. U. S. A. 2011, 108, 10969−10973. (57) Laureano-Perez, L.; Collé, R.; Jacobson, D. R.; Fitzgerald, R.; Khan, N. S.; Dmochowski, I. J. A novel application for 222Rn emanation standards: Radon−cryptophane host chemistry. Appl. Radiat. Isot. 2012, 70, 1997−2001. (58) Lee, E. P. F.; Wright, T. G. Interaction energy of the radonwater (Rn-H2O) complex. J. Phys. Chem. A 1999, 103, 7843−7847. (59) Magomedov, M. N. The calculation of the parameters of the Mie-Lennard-Jones potential. High Temp. 2006, 44, 513−529. (60) McLure, I. A.; Ramos, J. E.; del Rio, F. Accurate effective potentials and virial coefficients in real fluids. 1. Pure noble gases and their mixtures. J. Phys. Chem. B 1999, 103, 7019−7030. (61) Ramos, J. E. Viscosity and thermal conductivity from nonconformal potentials. I Theory and application to noble gases. Mol. Phys. 2005, 103, 2323−2334. F



Article

(62) del Río, F.; Guzmán, O.; Eloy Ramos, J.; Ibarra-Tandi, B. Effective intermolecular potentials in theoretical thermodynamics of pure substances and solutions. Fluid Phase Equilib. 2007, 259, 9−22. (63) del Rio, F.; Diaz-Herrera, E.; Guzman, O.; Moreno-Razo, J. A.; Ramos, J. E. Analytical equation of state with three-body forces: Application to noble gases. J. Chem. Phys. 2013, 139, 139. (64) Lazaridis, T.; Paulaitis, M. E. Simulation studies of the hydration entropy of simple, hydrophobic solutes. J. Phys. Chem. 1994, 98, 635− 642. (65) Guillot, B.; Guissani, Y. The solubility of rare gases in fused silica: A numerical evaluation. J. Chem. Phys. 1996, 105, 255−270. (66) Parkes, M. V.; Staiger, C. L.; Perry, J. J., IV; Allendorf, M. D.; Greathouse, J. A. Screening metal-organic frameworks for selective noble gas adsorption in air: effect of pore size and framework topology. Phys. Chem. Chem. Phys. 2013, 15, 9093−9106. (67) Van Heest, T.; Teich-McGoldrick, S. L.; Greathouse, J. A.; Allendorf, M. D.; Sholl, D. S. Identification of metal-organic framework materials for adsorption separation of rare gases: Applicability of ideal adsorbed solution theory (IAST) and effects of inaccessible framework regions. J. Phys. Chem. C 2012, 116, 13183− 13195. (68) Chen, L.; Reiss, P. S.; Chong, S. Y.; Holden, D.; Jelfs, K. E.; Hasell, T.; Little, M. A.; Kewley, A.; Briggs, M. E.; Stephenson, A.; Thomas, K. M.; Armstrong, J. A.; Bell, J.; Busto, J.; Noel, R.; Liu, J.; Strachan, D. M.; Thallapally, P. K.; Cooper, A. I. Separation of rare gases and chiral molecules by selective binding in porous organic cages. Nat. Mater. 2014, 13, 954−960. (69) Srivastava, B. N.; Saxena, S. C. Generalised relations for the thermal diffusion factor of inert gas mixtures with one invariable constituent. Physica 1956, 22, 253−262. (70) Miller, G. A. The Intermolecular Force Constants of Radon. J. Phys. Chem. 1960, 64, 163−165. (71) Chakraborti, P. K. Intermolecular Potential of Radon. J. Chem. Phys. 1966, 44, 3137−3139. (72) Slyusar, V. P.; Tret’yakov, V. M.; Rudenko, N. S. Thermal conductivity of krypton and xenon at constant density and pressures up to 2700 atm. Law of corresponding states. Sov. J. Low. Temp. Phys. 1978, 4, 363−368. (73) Gopal, R. On intermolecular Lennard-Jones potential energy parameters for radon. J. Indian Chem. Soc. 1962, 39, 791−792. (74) van Loef, J. J. On the thermophysical properties of liquid radon222. Physica B+C 1981, 103, 362−364. (75) Lennard-Jones, J. E. The equation of state of gases and critical phenomena. Physica 1937, 4, 941−956. (76) Mick, J.; Soroush Barhaghi, M.; Jackman, B.; Rushaidat, K.; Schwiebert, L.; Potoff, J. J. Optimized Mie potentials for phase equilibria: Application to noble gases and their mixtures with nalkanes. J. Chem. Phys. 2015, 143, 143. (77) Vrabec, J.; Stoll, J.; Hasse, H. A set of molecular models for symmetric quadrupolar fluids. J. Phys. Chem. B 2001, 105, 12126− 12133. (78) Potoff, J. J.; Errington, J. R.; Panagiotopoulos, A. Z. Molecular simulation of phase equilibria for mixtures of polar and non-polar components. Mol. Phys. 1999, 97, 1073−1083. (79) GOMC. http://gomc.eng.wayne.edu (accessed Nov 24, 2015). (80) Martinez, L.; Andrade, R.; Birgin, E. G.; Martinez, J. M. PACKMOL: A package for building initial configurations for molecular dynamics simulations. J. Comput. Chem. 2009, 30, 2157− 2164. (81) Humphrey, W.; Dalke, A.; Schulten, K. VMD: Visual molecular dynamics. J. Mol. Graphics 1996, 14, 33−38. (82) Martin, M. G.; Biddy, M. J. Monte Carlo molecular simulation predictions for the heat of vaporization of acetone and butyramide. Fluid Phase Equilib. 2005, 236, 53−57. (83) Potoff, J. J.; Panagiotopoulos, A. Z. Critical point and phase behavior of the pure fluid and a Lennard-Jones mixture. J. Chem. Phys. 1998, 109, 10914−10920. (84) Guggenheim, E. A. The Principle of Corresponding States. J. Chem. Phys. 1945, 13, 253−261.

(85) Takahashi, N. Studies on the Physicochemical Properties of Astatine and Radon by Radiogas chromatography. Ph.D. Thesis, Osaka University, January, 1983. (86) Takahashi, N.; Otozai, K. Estimation of boiling point of radon by radiogas chromatography. International Journal of Radiation Applications and Instrumentation. Part A. Applied Radiation and Isotopes 1986, 37, 191−194. (87) Ramsay, S. W.; Whytlaw-Gray, R. La Densité de l’émanation du Radium. C. R. Hebd. Seances Acad Sci. 1910, 151, 126−128.

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Prediction of Radon-222 Phase Behavior by Monte Carlo Simulation

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