Lennard-Jones equation of state

The equation of state of the Lennard-Jones model.

Johnson, Zollweg and Gubbins

Johnson, Zollweg and Gubbins proposed an equation of state based on 33 parameters within a modified Benedict, Webb and Rubin equation of state, which accurately reproduces the vapour-liquid equilibrium curve.

Kolafa and Nezbeda

The Kolafa and Nezbeda equation of state provides us with the Helmholtz energy function: (Eq. 30):

\[A=A_{\mathrm{HS}} + \exp (-\gamma \rho^2) \rho T \Delta B_{2,{\mathrm{hBH}}} + \sum_{ij} C_{ij} T^{i/2} \rho^j\]

the compressibility factor (Eq. 31)

\[z \equiv \frac{P}{\rho T}= z_{\mathrm{HS}} + \rho(1-2\gamma\rho^2) \exp (-\gamma \rho^2) \Delta B_{2,{\mathrm{hBH}}} + \sum_{ij} jC_{ij} T^{i/2-1} \rho^j\]

and the internal energy (Eq. 32)

\[U= {3(z_{\rm HS}-1)\over d_{\rm hBH}}\, {\partial d_{\rm hBH}\over \partial (1/T)} + \rho \exp(-\gamma\rho^2)\,{\partial \Delta B_{\rm2,hBH}\over\partial (1/T)} - \sum_{ij} \left({i\over2}-1\right) C_{ij}\, T^{i/2} \rho^j \]

On the following page is the FORTRAN code for the Kolafa and Nezbeda equation of state.

Ree

The Ree equation of state is an extension of the earlier work of Hansen in the high temperature region.

Boltachev and Baidakov

Boltachev and Baidakov have paid particular attention to including data from the metastable region .

Melting line

The solid and liquid densities along the melting line are given by the following equations

van der Hoef

van der Hoef (Ref. Eqs. 25 and 26):

\[\rho_{\mathrm {solid}} = \beta^{-1/4} \left[ 0.92302 - 0.09218 \beta + 0.62381 \beta^2 -0.82672 \beta^3 + 0.49124 \beta^4 -0.10847 \beta^5\right]\]

and

\[\rho_{\mathrm {liquid}} = \beta^{-1/4} \left[ 0.91070 - 0.25124 \beta + 0.85861 \beta^2 -1.08918 \beta^3 + 0.63932 \beta^4 -0.14433 \beta^5\right]\]

Mastny and de Pablo

Mastny and de Pablo (Ref Eqs. 20 and 21):

\[\rho_{\mathrm {solid}} = \beta^{-1/4} \left[ 0.908629 - 0.041510 \beta + 0.514632 \beta^2 -0.708590\beta^3 + 0.428351 \beta^4 -0.095229 \beta^5\right]\]

and

\[\rho_{\mathrm {liquid}} = \beta^{-1/4} \left[ 0.90735 - 0.27120 \beta + 0.91784 \beta^2 -1.16270\beta^3 + 0.68012 \beta^4 -0.15284 \beta^5\right]\]

References

Related reading

• J. J. Nicolas, K. E. Gubbins, W. B. Streett and D. J. Tildesley "Equation of state for the Lennard-Jones fluid", Molecular Physics 37 pp. 1429-1454 (1979)
• Karel Aim, Jirí Kolafa, Ivo Nezbeda and Horst L. Vörtler "The Lennard-Jones fluid revisited: new thermodynamic data and new equation of state", Fluid Phase Equilibria 83 pp. 15-22 (1993)
• Hertanto Adidharma and Maciej Radosz "The LJ-Solid Equation of State Extended to Thermal Properties, Chain Molecules, and Mixtures", Industrial and Engineering Chemistry Research 43 pp. 6890 - 6897 (2004)
• David M. Eike, Joan F. Brennecke, and Edward J. Maginn "Toward a robust and general molecular simulation method for computing solid-liquid coexistence", Journal of Chemical Physics 122 014115 (2005)
• Sergey A. Khrapak and Gregor E. Morfill "Accurate freezing and melting equations for the Lennard-Jones system", Journal of Chemical Physics 134 094108 (2011)