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SAT-TMMC: Liquid-Vapor coexistence properties - TraPPE Carbon Dioxide - 12Å LFS

Liquid-vapor coexistence properties of Carbon Dioxide, modeled by the TraPPE Force Field [1], obtained by grand-canonical transition-matrix Monte Carlo and histogram re-weighting. Mean values of the saturation pressure, density, and activity (chemical potential- see below) for each phase are reported.

METHOD Grand-canonical transition-matrix Monte Carlo [2] and histogram re-weighting
Fluid Carbon Dioxide
Model TraPPE [1]
V 27000 Å3
Lennard-Jones 12 Å + Linear Force Shift
Electrostatics 12 Å + Ewald Summation
Prob. of Disp. Move 0.3
Prob. of Rot. Move 0.2
Prob. of Ins/Del Move 0.5
Biasing Function Update Frequency 1.0E6 trial moves
Simulation Length 1.0E9 trial moves


T (K)

ρvap (mol/L)


ρliq (mol/L)


psat (bar)




230 8.561E-01 2.852E-04 2.424E+01 6.074E-03 1.383E+01 8.010E-03 -7.882E+00 2.303E-04
235 1.028E+00 5.896E-04 2.372E+01 4.841E-03 1.654E+01 7.744E-03 -7.741E+00 1.875E-04
240 1.229E+00 9.578E-04 2.318E+01 4.076E-03 1.963E+01 8.486E-03 -7.610E+00 2.079E-04
245 1.466E+00 2.150E-03 2.261E+01 4.470E-03 2.313E+01 2.137E-02 -7.486E+00 1.407E-04
250 1.744E+00 1.194E-03 2.200E+01 3.482E-03 2.707E+01 1.847E-02 -7.370E+00 2.524E-04
255 2.074E+00 1.530E-03 2.135E+01 2.026E-03 3.149E+01 5.843E-03 -7.260E+00 1.945E-04
260 2.472E+00 3.582E-03 2.063E+01 6.449E-03 3.643E+01 1.463E-02 -7.157E+00 1.923E-04
265 2.962E+00 3.386E-03 1.981E+01 5.863E-03 4.191E+01 1.460E-02 -7.060E+00 1.258E-04
270 3.597E+00 2.645E-03 1.886E+01 1.822E-03 4.803E+01 1.443E-02 -6.968E+00 8.604E-05
275 4.499E+00 4.120E-03 1.764E+01 1.000E-02 5.486E+01 3.804E-02 -6.822E+00 1.167E-04
280 5.745E+00 1.451E-02 1.610E+01 2.762E-02 6.247E+01 9.978E-03 -6.800E+00 1.508E-04



Uncertainties were obtained from four independent simulations and represent 95% confidence limits based on a standard t statistic. Liquid-vapor coexistence was determined by adjusting the activity such that the pressures of the liquid and vapor phases were equal. Here, the pressure is not the conventional virial pressure [3,4] but is the actual thermodynamic pressure, based on the fact that the absolute free energies can be obtained from the distributions determined from simulation [5]. 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. 6.

As introduced in Refs. 3 and 4, 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. The reported activity has units of Å-3.


  1. J. A. Potoff and J. I. Siepmann, AIChE J., 47, 1676–1682 (2001).
  2. J. R. Errington, J. Chem. Phys. 118, 9915 (2003).
  3. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, New York, 1989).
  4. D. Frenkel and B. Smit, Understanding Molecular Simulation, 2nd ed. (Academic, San Diego, 2002)., pp.37-38.
  5. J. R. Errington and A. Z. Panagiotopoulos, J. Chem. Phys., 109, 1093 (1998).
  6. A. Z. Panagiotopoulos, J. Phys.: Condens. Matter, 12, R25-R52, (2000).
Created September 20, 2013, Updated August 2, 2018