Lennard-Jones fluid simulation website

8. TMMC: Vapor-phase Equation of State at subcritical temperatures - Linear-Force Shifted Potential

The vapor phase equation of state (ρ*,p*,lnz) was obtained by grand-canonical transition-matrix Monte Carlo and histogram re-weighting [8-12] over the reduced temperature range 0.65 to 0.90 at increments of 0.05. Mean values of the saturation pressure, density, potential energy per molecule, and activity (chemical potential- see below) for each phase are reported. Uncertainties of all properties will be added at a future date.

METHOD: Grand-canonical transition-matrix Monte Carlo [1] and histogram re-weighting [8-12]
V/σ3: 512
TRUNCATION: Linear Force Shifted at 2.5σ
Prob. of Disp. Move: 0.75
Prob, of Ins/Del. Move: 0.25
Biasing Function Update Frequency
1.0E6 trials
Simulation Length: 8.0E9 trials

Vapor Phase Properties
p/psat ln z* ρ*
ln z* ρ*
ln z* ρ*
ln z* ρ*
ln z* ρ*
ln z* ρ*
0.01 -9.17926E+00 1.03326E-04 -8.68131E+00 1.70091E-04 -8.25396E+00 2.60928E-04 -7.88206E+00 3.78879E-04 -7.55155E+00 5.27784E-04 -7.25153E+00 7.12896E-04
0.02 -8.48689E+00 2.06814E-04 -7.98927E+00 3.40563E-04 -7.56230E+00 5.22643E-04 -7.19085E+00 7.59242E-04 -6.86080E+00 1.05812E-03 -6.56135E+00 1.43004E-03
0.03 -8.08221E+00 3.10466E-04 -7.58492E+00 5.11419E-04 -7.15834E+00 7.85157E-04 -6.78734E+00 1.14111E-03 -6.45774E+00 1.59103E-03 -6.15885E+00 2.15142E-03
0.04 -7.79532E+00 4.14282E-04 -7.29836E+00 6.82664E-04 -6.87216E+00 1.04848E-03 -6.50161E+00 1.52449E-03 -6.17247E+00 2.12654E-03 -5.87412E+00 2.87705E-03
0.05 -7.57296E+00 5.18263E-04 -7.07633E+00 8.54300E-04 -6.65053E+00 1.31262E-03 -6.28042E+00 1.90940E-03 -5.95173E+00 2.66468E-03 -5.65393E+00 3.60697E-03
0.06 -7.39143E+00 6.22410E-04 -6.89513E+00 1.02633E-03 -6.46972E+00 1.57758E-03 -6.10005E+00 2.29585E-03 -5.77182E+00 3.20548E-03 -5.47455E+00 4.34125E-03
0.07 -7.23807E+00 7.26724E-04 -6.74211E+00 1.19876E-03 -6.31709E+00 1.84338E-03 -5.94786E+00 2.68385E-03 -5.62008E+00 3.74899E-03 -5.32334E+00 5.07998E-03
0.08 -7.10532E+00 8.31205E-04 -6.60970E+00 1.37159E-03 -6.18508E+00 2.11001E-03 -5.81629E+00 3.07342E-03 -5.48897E+00 4.29524E-03 -5.19274E+00 5.82325E-03
0.09 -6.98833E+00 9.35854E-04 -6.49305E+00 1.54483E-03 -6.06882E+00 2.37749E-03 -5.70047E+00 3.46457E-03 -5.37360E+00 4.84428E-03 -5.07791E+00 6.57115E-03
0.10 -6.88376E+00 1.04067E-03 -6.38882E+00 1.71847E-03 -5.96498E+00 2.64583E-03 -5.59708E+00 3.85731E-03 -5.27067E+00 5.39616E-03 -4.97549E+00 7.32379E-03
0.11 -6.78924E+00 1.14566E-03 -6.29464E+00 1.89252E-03 -5.87120E+00 2.91502E-03 -5.50373E+00 4.25167E-03 -5.17779E+00 5.95092E-03 -4.88313E+00 8.08123E-03
0.12 -6.70302E+00 1.25082E-03 -6.20876E+00 2.06698E-03 -5.78572E+00 3.18509E-03 -5.41869E+00 4.64765E-03 -5.09321E+00 6.50860E-03 -4.79908E+00 8.84357E-03
0.13 -6.62377E+00 1.35615E-03 -6.12985E+00 2.24186E-03 -5.70722E+00 3.45603E-03 -5.34062E+00 5.04529E-03 -5.01561E+00 7.06926E-03 -4.72200E+00 9.61088E-03
0.14 -6.55045E+00 1.46166E-03 -6.05688E+00 2.41715E-03 -5.63465E+00 3.72786E-03 -5.26849E+00 5.44459E-03 -4.94395E+00 7.63293E-03 -4.65087E+00 1.03832E-02
0.15 -6.48226E+00 1.56734E-03 -5.98903E+00 2.59286E-03 -5.56719E+00 4.00058E-03 -5.20147E+00 5.84559E-03 -4.87741E+00 8.19967E-03 -4.58485E+00 1.11607E-02
0.16 -6.41851E+00 1.67319E-03 -5.92563E+00 2.76900E-03 -5.50420E+00 4.27421E-03 -5.13891E+00 6.24830E-03 -4.81533E+00 8.76952E-03 -4.52330E+00 1.19435E-02
0.17 -6.35868E+00 1.77922E-03 -5.86615E+00 2.94556E-03 -5.44512E+00 4.54875E-03 -5.08027E+00 6.65275E-03 -4.75717E+00 9.34252E-03 -4.46566E+00 1.27315E-02
0.18 -6.30232E+00 1.88542E-03 -5.81013E+00 3.12256E-03 -5.38951E+00 4.82420E-03 -5.02510E+00 7.05896E-03 -4.70248E+00 9.91871E-03 -4.41150E+00 1.35249E-02
0.19 -6.24905E+00 1.99180E-03 -5.75721E+00 3.29998E-03 -5.33699E+00 5.10059E-03 -4.97302E+00 7.46695E-03 -4.65089E+00 1.04981E-02 -4.36044E+00 1.43238E-02
0.20 -6.19855E+00 2.09836E-03 -5.70706E+00 3.47784E-03 -5.28725E+00 5.37791E-03 -4.92371E+00 7.87677E-03 -4.60207E+00 1.10809E-02 -4.31216E+00 1.51284E-02
0.21 -6.15056E+00 2.20510E-03 -5.65942E+00 3.65613E-03 -5.24001E+00 5.65618E-03 -4.87692E+00 8.28842E-03 -4.55577E+00 1.16669E-02 -4.26639E+00 1.59386E-02
0.22 -6.10484E+00 2.31202E-03 -5.61405E+00 3.83487E-03 -5.19505E+00 5.93540E-03 -4.83240E+00 8.70193E-03 -4.51174E+00 1.22564E-02 -4.22290E+00 1.67547E-02
0.23 -6.06119E+00 2.41912E-03 -5.57075E+00 4.01405E-03 -5.15216E+00 6.21559E-03 -4.78994E+00 9.11735E-03 -4.46979E+00 1.28493E-02 -4.18149E+00 1.75768E-02
0.24 -6.01943E+00 2.52640E-03 -5.52934E+00 4.19368E-03 -5.11116E+00 6.49676E-03 -4.74939E+00 9.53468E-03 -4.42974E+00 1.34457E-02 -4.14197E+00 1.84049E-02
0.25 -5.97940E+00 2.63387E-03 -5.48967E+00 4.37375E-03 -5.07190E+00 6.77891E-03 -4.71058E+00 9.95396E-03 -4.39142E+00 1.40456E-02 -4.10420E+00 1.92393E-02
0.26 -5.94099E+00 2.74152E-03 -5.45161E+00 4.55428E-03 -5.03425E+00 7.06206E-03 -4.67337E+00 1.03752E-02 -4.35472E+00 1.46491E-02 -4.06804E+00 2.00800E-02
0.27 -5.90405E+00 2.84936E-03 -5.41502E+00 4.73527E-03 -4.99808E+00 7.34621E-03 -4.63764E+00 1.07985E-02 -4.31950E+00 1.52563E-02 -4.03337E+00 2.09271E-02
0.28 -5.86848E+00 2.95738E-03 -5.37981E+00 4.91672E-03 -4.96329E+00 7.63138E-03 -4.60330E+00 1.12238E-02 -4.28566E+00 1.58673E-02 -4.00008E+00 2.17809E-02
0.29 -5.83420E+00 3.06558E-03 -5.34588E+00 5.09864E-03 -4.92977E+00 7.91757E-03 -4.57023E+00 1.16511E-02 -4.25311E+00 1.64820E-02 -3.96807E+00 2.26415E-02
0.30 -5.80110E+00 3.17398E-03 -5.31314E+00 5.28102E-03 -4.89745E+00 8.20481E-03 -4.53836E+00 1.20806E-02 -4.22175E+00 1.71006E-02 -3.93726E+00 2.35090E-02
0.31 -5.76911E+00 3.28257E-03 -5.28151E+00 5.46388E-03 -4.86624E+00 8.49309E-03 -4.50760E+00 1.25121E-02 -4.19150E+00 1.77231E-02 -3.90758E+00 2.43835E-02
0.32 -5.73817E+00 3.39134E-03 -5.25093E+00 5.64721E-03 -4.83607E+00 8.78243E-03 -4.47789E+00 1.29459E-02 -4.16231E+00 1.83497E-02 -3.87894E+00 2.52652E-02
0.33 -5.70821E+00 3.50031E-03 -5.22132E+00 5.83102E-03 -4.80688E+00 9.07284E-03 -4.44916E+00 1.33818E-02 -4.13410E+00 1.89803E-02 -3.85129E+00 2.61543E-02
0.34 -5.67916E+00 3.60947E-03 -5.19263E+00 6.01531E-03 -4.77862E+00 9.36433E-03 -4.42136E+00 1.38199E-02 -4.10681E+00 1.96151E-02 -3.82457E+00 2.70509E-02
0.35 -5.65098E+00 3.71882E-03 -5.16481E+00 6.20009E-03 -4.75122E+00 9.65692E-03 -4.39442E+00 1.42602E-02 -4.08040E+00 2.02541E-02 -3.79873E+00 2.79551E-02
0.36 -5.62362E+00 3.82837E-03 -5.13781E+00 6.38537E-03 -4.72464E+00 9.95061E-03 -4.36831E+00 1.47029E-02 -4.05480E+00 2.08975E-02 -3.77371E+00 2.88673E-02
0.37 -5.59703E+00 3.93811E-03 -5.11158E+00 6.57114E-03 -4.69884E+00 1.02454E-02 -4.34297E+00 1.51478E-02 -4.02999E+00 2.15453E-02 -3.74947E+00 2.97874E-02
0.38 -5.57117E+00 4.04805E-03 -5.08609E+00 6.75741E-03 -4.67377E+00 1.05414E-02 -4.31837E+00 1.55951E-02 -4.00592E+00 2.21976E-02 -3.72597E+00 3.07158E-02
0.39 -5.54601E+00 4.15818E-03 -5.06128E+00 6.94418E-03 -4.64940E+00 1.08385E-02 -4.29446E+00 1.60448E-02 -3.98254E+00 2.28544E-02 -3.70317E+00 3.16525E-02
0.40 -5.52150E+00 4.26852E-03 -5.03714E+00 7.13147E-03 -4.62568E+00 1.11367E-02 -4.27122E+00 1.64968E-02 -3.95983E+00 2.35160E-02 -3.68105E+00 3.25979E-02
0.41 -5.49762E+00 4.37905E-03 -5.01362E+00 7.31927E-03 -4.60260E+00 1.14361E-02 -4.24861E+00 1.69513E-02 -3.93775E+00 2.41823E-02 -3.65955E+00 3.35522E-02
0.42 -5.47434E+00 4.48979E-03 -4.99070E+00 7.50758E-03 -4.58011E+00 1.17367E-02 -4.22659E+00 1.74083E-02 -3.91627E+00 2.48536E-02 -3.63867E+00 3.45154E-02
0.43 -5.45162E+00 4.60072E-03 -4.96835E+00 7.69642E-03 -4.55819E+00 1.20385E-02 -4.20515E+00 1.78679E-02 -3.89537E+00 2.55298E-02 -3.61836E+00 3.54880E-02
0.44 -5.42945E+00 4.71186E-03 -4.94655E+00 7.88579E-03 -4.53682E+00 1.23415E-02 -4.18426E+00 1.83300E-02 -3.87502E+00 2.62111E-02 -3.59860E+00 3.64701E-02
0.45 -5.40779E+00 4.82321E-03 -4.92526E+00 8.07568E-03 -4.51596E+00 1.26457E-02 -4.16389E+00 1.87947E-02 -3.85519E+00 2.68975E-02 -3.57937E+00 3.74621E-02
0.46 -5.38663E+00 4.93476E-03 -4.90446E+00 8.26612E-03 -4.49560E+00 1.29512E-02 -4.14401E+00 1.92620E-02 -3.83586E+00 2.75893E-02 -3.56065E+00 3.84641E-02
0.47 -5.36594E+00 5.04652E-03 -4.88414E+00 8.45709E-03 -4.47572E+00 1.32579E-02 -4.12462E+00 1.97320E-02 -3.81701E+00 2.82865E-02 -3.54241E+00 3.94765E-02
0.48 -5.34571E+00 5.15848E-03 -4.86427E+00 8.64861E-03 -4.45629E+00 1.35659E-02 -4.10568E+00 2.02048E-02 -3.79863E+00 2.89893E-02 -3.52463E+00 4.04996E-02
0.49 -5.32591E+00 5.27066E-03 -4.84484E+00 8.84068E-03 -4.43730E+00 1.38752E-02 -4.08718E+00 2.06804E-02 -3.78068E+00 2.96977E-02 -3.50730E+00 4.15336E-02
0.50 -5.30652E+00 5.38304E-03 -4.82583E+00 9.03331E-03 -4.41873E+00 1.41858E-02 -4.06910E+00 2.11587E-02 -3.76316E+00 3.04119E-02 -3.49040E+00 4.25790E-02
0.51 -5.28754E+00 5.49564E-03 -4.80722E+00 9.22650E-03 -4.40057E+00 1.44977E-02 -4.05143E+00 2.16400E-02 -3.74605E+00 3.11321E-02 -3.47391E+00 4.36361E-02
0.52 -5.26895E+00 5.60845E-03 -4.78900E+00 9.42025E-03 -4.38279E+00 1.48109E-02 -4.03415E+00 2.21242E-02 -3.72933E+00 3.18583E-02 -3.45782E+00 4.47052E-02
0.53 -5.25072E+00 5.72148E-03 -4.77115E+00 9.61458E-03 -4.36538E+00 1.51255E-02 -4.01724E+00 2.26113E-02 -3.71299E+00 3.25907E-02 -3.44211E+00 4.57867E-02
0.54 -5.23285E+00 5.83472E-03 -4.75365E+00 9.80948E-03 -4.34833E+00 1.54415E-02 -4.00070E+00 2.31015E-02 -3.69702E+00 3.33295E-02 -3.42677E+00 4.68811E-02
0.55 -5.21533E+00 5.94817E-03 -4.73650E+00 1.00050E-02 -4.33163E+00 1.57589E-02 -3.98450E+00 2.35948E-02 -3.68139E+00 3.40747E-02 -3.41178E+00 4.79886E-02
0.56 -5.19813E+00 6.06185E-03 -4.71968E+00 1.02010E-02 -4.31526E+00 1.60776E-02 -3.96864E+00 2.40913E-02 -3.66611E+00 3.48267E-02 -3.39714E+00 4.91098E-02
0.57 -5.18126E+00 6.17574E-03 -4.70319E+00 1.03977E-02 -4.29922E+00 1.63978E-02 -3.95310E+00 2.45909E-02 -3.65115E+00 3.55854E-02 -3.38283E+00 5.02452E-02
0.58 -5.16469E+00 6.28986E-03 -4.68700E+00 1.05950E-02 -4.28348E+00 1.67194E-02 -3.93788E+00 2.50938E-02 -3.63652E+00 3.63512E-02 -3.36884E+00 5.13951E-02
0.59 -5.14843E+00 6.40419E-03 -4.67111E+00 1.07928E-02 -4.26805E+00 1.70425E-02 -3.92296E+00 2.56001E-02 -3.62218E+00 3.71242E-02 -3.35517E+00 5.25601E-02
0.60 -5.13245E+00 6.51876E-03 -4.65551E+00 1.09913E-02 -4.25290E+00 1.73671E-02 -3.90833E+00 2.61097E-02 -3.60815E+00 3.79045E-02 -3.34179E+00 5.37407E-02
0.61 -5.11675E+00 6.63354E-03 -4.64019E+00 1.11904E-02 -4.23804E+00 1.76931E-02 -3.89399E+00 2.66228E-02 -3.59440E+00 3.86924E-02 -3.32871E+00 5.49375E-02
0.62 -5.10132E+00 6.74856E-03 -4.62514E+00 1.13901E-02 -4.22345E+00 1.80207E-02 -3.87992E+00 2.71395E-02 -3.58093E+00 3.94881E-02 -3.31592E+00 5.61510E-02
0.63 -5.08615E+00 6.86380E-03 -4.61035E+00 1.15904E-02 -4.20912E+00 1.83499E-02 -3.86612E+00 2.76597E-02 -3.56773E+00 4.02918E-02 -3.30339E+00 5.73819E-02
0.64 -5.07123E+00 6.97927E-03 -4.59582E+00 1.17914E-02 -4.19505E+00 1.86806E-02 -3.85258E+00 2.81836E-02 -3.55479E+00 4.11037E-02 -3.29114E+00 5.86308E-02
0.65 -5.05656E+00 7.09497E-03 -4.58153E+00 1.19930E-02 -4.18122E+00 1.90129E-02 -3.83929E+00 2.87113E-02 -3.54211E+00 4.19240E-02 -3.27915E+00 5.98984E-02
0.66 -5.04212E+00 7.21090E-03 -4.56749E+00 1.21953E-02 -4.16764E+00 1.93468E-02 -3.82624E+00 2.92428E-02 -3.52967E+00 4.27531E-02 -3.26740E+00 6.11855E-02
0.67 -5.02792E+00 7.32707E-03 -4.55367E+00 1.23982E-02 -4.15429E+00 1.96823E-02 -3.81342E+00 2.97782E-02 -3.51747E+00 4.35911E-02 -3.25591E+00 6.24929E-02
0.68 -5.01394E+00 7.44347E-03 -4.54008E+00 1.26017E-02 -4.14116E+00 2.00195E-02 -3.80084E+00 3.03176E-02 -3.50551E+00 4.44383E-02 -3.24466E+00 6.38214E-02
0.69 -5.00018E+00 7.56012E-03 -4.52670E+00 1.28060E-02 -4.12826E+00 2.03584E-02 -3.78847E+00 3.08611E-02 -3.49378E+00 4.52951E-02 -3.23363E+00 6.51719E-02
0.70 -4.98662E+00 7.67699E-03 -4.51354E+00 1.30109E-02 -4.11557E+00 2.06991E-02 -3.77633E+00 3.14088E-02 -3.48226E+00 4.61617E-02 -3.22284E+00 6.65454E-02
0.71 -4.97328E+00 7.79411E-03 -4.50058E+00 1.32164E-02 -4.10308E+00 2.10414E-02 -3.76439E+00 3.19607E-02 -3.47096E+00 4.70384E-02 -3.21227E+00 6.79431E-02
0.72 -4.96013E+00 7.91148E-03 -4.48783E+00 1.34227E-02 -4.09080E+00 2.13855E-02 -3.75266E+00 3.25171E-02 -3.45987E+00 4.79256E-02 -3.20192E+00 6.93660E-02
0.73 -4.94718E+00 8.02908E-03 -4.47526E+00 1.36297E-02 -4.07871E+00 2.17315E-02 -3.74113E+00 3.30778E-02 -3.44899E+00 4.88236E-02 -3.19178E+00 7.08155E-02
0.74 -4.93441E+00 8.14693E-03 -4.46289E+00 1.38373E-02 -4.06682E+00 2.20792E-02 -3.72980E+00 3.36432E-02 -3.43830E+00 4.97328E-02 -3.18184E+00 7.22928E-02
0.75 -4.92183E+00 8.26503E-03 -4.45071E+00 1.40457E-02 -4.05511E+00 2.24288E-02 -3.71865E+00 3.42132E-02 -3.42781E+00 5.06536E-02 -3.17211E+00 7.37995E-02
0.76 -4.90943E+00 8.38337E-03 -4.43870E+00 1.42548E-02 -4.04359E+00 2.27803E-02 -3.70769E+00 3.47880E-02 -3.41751E+00 5.15864E-02 -3.16258E+00 7.53372E-02
0.77 -4.89720E+00 8.50197E-03 -4.42687E+00 1.44646E-02 -4.03224E+00 2.31337E-02 -3.69691E+00 3.53678E-02 -3.40740E+00 5.25317E-02 -3.15324E+00 7.69077E-02
0.78 -4.88514E+00 8.62081E-03 -4.41520E+00 1.46751E-02 -4.02106E+00 2.34890E-02 -3.68631E+00 3.59525E-02 -3.39747E+00 5.34900E-02 -3.14409E+00 7.85129E-02
0.79 -4.87324E+00 8.73992E-03 -4.40371E+00 1.48864E-02 -4.01005E+00 2.38463E-02 -3.67587E+00 3.65425E-02 -3.38771E+00 5.44617E-02 -3.13513E+00 8.01550E-02
0.80 -4.86151E+00 8.85927E-03 -4.39238E+00 1.50984E-02 -3.99921E+00 2.42057E-02 -3.66561E+00 3.71377E-02 -3.37813E+00 5.54474E-02 -3.12636E+00 8.18364E-02
0.81 -4.84993E+00 8.97889E-03 -4.38120E+00 1.53111E-02 -3.98852E+00 2.45671E-02 -3.65551E+00 3.77383E-02 -3.36872E+00 5.64477E-02 -3.11776E+00 8.35596E-02
0.82 -4.83851E+00 9.09876E-03 -4.37018E+00 1.55247E-02 -3.97799E+00 2.49306E-02 -3.64557E+00 3.83446E-02 -3.35947E+00 5.74633E-02 -3.10934E+00 8.53278E-02
0.83 -4.82724E+00 9.21889E-03 -4.35931E+00 1.57390E-02 -3.96762E+00 2.52962E-02 -3.63578E+00 3.89566E-02 -3.35039E+00 5.84947E-02 -3.10110E+00 8.71442E-02
0.84 -4.81611E+00 9.33929E-03 -4.34859E+00 1.59541E-02 -3.95739E+00 2.56640E-02 -3.62615E+00 3.95744E-02 -3.34147E+00 5.95427E-02 -3.09303E+00 8.90126E-02
0.85 -4.80513E+00 9.45995E-03 -4.33801E+00 1.61699E-02 -3.94731E+00 2.60339E-02 -3.61667E+00 4.01984E-02 -3.33271E+00 6.06081E-02 -3.08513E+00 9.09374E-02
0.86 -4.79428E+00 9.58088E-03 -4.32758E+00 1.63866E-02 -3.93738E+00 2.64062E-02 -3.60734E+00 4.08286E-02 -3.32409E+00 6.16917E-02 -3.07740E+00 9.29236E-02
0.87 -4.78357E+00 9.70208E-03 -4.31728E+00 1.66041E-02 -3.92758E+00 2.67807E-02 -3.59815E+00 4.14652E-02 -3.31563E+00 6.27945E-02 -3.06983E+00 9.49770E-02
0.88 -4.77300E+00 9.82355E-03 -4.30711E+00 1.68224E-02 -3.91792E+00 2.71575E-02 -3.58910E+00 4.21084E-02 -3.30732E+00 6.39175E-02 -3.06243E+00 9.71047E-02
0.89 -4.76255E+00 9.94529E-03 -4.29707E+00 1.70415E-02 -3.90839E+00 2.75368E-02 -3.58018E+00 4.27585E-02 -3.29916E+00 6.50617E-02 -3.05519E+00 9.93154E-02
0.90 -4.75224E+00 1.00673E-02 -4.28717E+00 1.72614E-02 -3.89900E+00 2.79184E-02 -3.57141E+00 4.34156E-02 -3.29114E+00 6.62284E-02 -3.04811E+00 1.01620E-01
0.91 -4.74204E+00 1.01896E-02 -4.27739E+00 1.74822E-02 -3.88973E+00 2.83025E-02 -3.56276E+00 4.40800E-02 -3.28326E+00 6.74189E-02 -3.04120E+00 1.04037E-01
0.92 -4.73197E+00 1.03122E-02 -4.26773E+00 1.77039E-02 -3.88058E+00 2.86892E-02 -3.55425E+00 4.47519E-02 -3.27552E+00 6.86347E-02 -3.03445E+00 1.06592E-01
0.93 -4.72202E+00 1.04350E-02 -4.25819E+00 1.79264E-02 -3.87156E+00 2.90784E-02 -3.54586E+00 4.54317E-02 -3.26792E+00 6.98773E-02 -3.02787E+00 1.09349E-01
0.94 -4.71218E+00 1.05582E-02 -4.24877E+00 1.81498E-02 -3.86266E+00 2.94703E-02 -3.53760E+00 4.61194E-02 -3.26045E+00 7.11488E-02 -3.02146E+00 1.12469E-01
0.95 -4.70246E+00 1.06816E-02 -4.23947E+00 1.83742E-02 -3.85388E+00 2.98648E-02 -3.52946E+00 4.68156E-02 -3.25311E+00 7.24511E-02 -3.01518E+00 1.15211E-01
0.96 -4.69285E+00 1.08053E-02 -4.23028E+00 1.85993E-02 -3.84521E+00 3.02621E-02 -3.52145E+00 4.75204E-02 -3.24591E+00 7.37866E-02 -3.00910E+00 1.18580E-01
0.97 -4.68335E+00 1.09293E-02 -4.22120E+00 1.88254E-02 -3.83666E+00 3.06622E-02 -3.51355E+00 4.82342E-02 -3.23884E+00 7.51583E-02 -3.00318E+00 1.22175E-01
0.98 -4.67395E+00 1.10536E-02 -4.21223E+00 1.90524E-02 -3.82822E+00 3.10652E-02 -3.50577E+00 4.89573E-02 -3.23190E+00 7.65692E-02 -2.99745E+00 1.26147E-01
0.99 -4.66467E+00 1.11781E-02 -4.20336E+00 1.92804E-02 -3.81989E+00 3.14711E-02 -3.49810E+00 4.96902E-02 -3.22509E+00 7.80234E-02 -2.99189E+00 1.30394E-01
1.00 -4.65548E+00 1.13030E-02 -4.19460E+00 1.95093E-02 -3.81166E+00 3.18800E-02 -3.49055E+00 5.04332E-02 -3.21841E+00 7.95256E-02 -2.98651E+00 1.35122E-01


Uncertainties were obtained from 5 independent simulations and represent 67% confidence limits (one standard deviation). Liquid-vapor coexistence was determined by adjusting the activity such that the pressures of the liquid and vapor phases were equal. Following, identification of state points below the coexistence pressure was accomplished by adjusting the activity until the pressure of the vapor phase was equal to the desired fraction of psat[6]. Here, the pressure is not the conventional virial pressure [2, 3] but is the actual thermodynamic pressure, based on the fact that the absolute free energies can be obtained from the distributions determined from simulation [4]. Alternative methods, for example Gibbs-ensemble Monte Carlo and combination grand-canonical Monte Carlo and histogram re-weighting, can be used to determine liquid-vapor coexistence. A review of standard methods of phase equilibria simulations can be found in Ref. 5.

As introduced in Refs. 2 and 3, the activity, z, is defined as

z = Λ-3 exp(βμ)

where Λ is the de Broglie wavelength, β = 1/(kBT) (where kB is Boltzmann's constant), and μ is the chemical potential. It is sometimes more convenient to work with ln z in the simulations and in post-processing.

Phase-coexistence energies were obtained by determining the mean potential energy at a given value of N for an additional 8 billion MC trials. Combining this information with the particle number probability distribution, the mean potential energy of the coexisting phases can be calculated [6].

For the Lennard-Jones fluid, linear force shifted at 2.5σ, the critical properties are Tc*=0.937, ρc*=0.320, and pc*=0.0820 [7].

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[3] D. Frenkel and B. Smit, Understanding Molecular Simulation, 2nd ed. (Academic, San Diego, 2002).
[4] J. R. Errington and A. Z. Panagiotopoulos, J. Chem. Phys. 109, 1093 (1998).
[5] A. Z. Panagiotopoulos, J. Phys.: Condens. Matter 12, 25 (2000).
[6] J. R. Errington and V. K. Shen, J. Chem. Phys. 123, 164103 (2005).
[7] J. R. Errington P. G. Debenedetti, and S. Torquato, J. Chem. Phys. 118, 2256 (2003).
[8] V. K. Shen and D. W. Siderius, J. Chem. Phys., 140, 244106, (2014).
[9] V. K. Shen and J. R. Errington, J. Phys. Chem. B 108, 19595, (2004).
[10] V. K. Shen and J. R. Errington, J. Chem. Phys. 122, 064508, (2005).
[11] V. K. Shen, R. D. Mountain, and J. R. Errington, J. Phys. Chem. B 111, 6198, (2007).
[12] D. W. Siderius and V. K. Shen, J. Phys. Chem. 117, 5681, (2013).