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here

On the other hand, the

The heat capacity under the constant volume

The present study proposes an extended form of the van der Waals EOS to explain the

The extended van der Waals EOS for the Lennard-Jones system has only three parameters, two of which are the Lennard-Jones parameters. The remaining parameter is the effective volume of the hard sphere of the reference system. The

The present paper follows the statistical mechanical theory on the original van der Waals EOS [7], which is then extended to explain, at least qualitatively, the

Here, the potential energy of the system is written as

The thermodynamic quantities are written by the partition function as follows:

According to this formula, in the case of a perfect gas, the pressure is given as follows:

The interaction energy is approximated by the van der Waals approximation. The statistical mechanical theory on van der Waals EOS adopts the perturbation approximation. The total potential energy is divided into two parts:

where and are the potential energy of the reference system and that of the perturbed system. The quantity b is the inverse of temperature: b = 1/

This equation can be rewritten as follows:

The reference system is the hard sphere system, and its pair potential function

The pair correlation function

When is sufficiently small, the term in the configuration integral

The average is estimated by the above pair correlation function as follows:

Here

In this manner, the configuration integral can be written as follows:

In addition, is approximated as follows:

Using this configuration integral, one can write the pressure as follows:

The internal energy

Next, we introduce the molecular partition function of the attractive term

This molecular partition function is compared with that of the original EOS,

The contribution of the attractive interaction to pressure is as follows:

The total internal energy and pressure are as follows:

The Helmholtz free energy and entropy are calculated by the standard method from the partition function.

If we express the molecular interaction in the Lennard-Jones form, the parameters in the attractive part are given by the Lennard-Jones parameters. The Lennard-Jones potential is written as follows:

The constant e in Eq. (22) is equal to e

This value is used in Eq. (22).

The last parameter in the present model is the effective volume of the hard sphere

The obtained

Table 1. Lennard-Jones potential parameters for argon [9].

e/(kcal/mol) | (e/k)/K | s/m |
---|---|---|

0.2483 | 125 | 3.43E-10 |

Figure 1. The *pVT* relation calculated by MD simulation on fluid argon using the Lennard-Jones model. The ordinate shows pressure *p* given in Lennard-Jones units e/s^{3} (Table 1) and in atm [10]. The abscissa shows the number density *N*/*V* and the mass density r in units of g/cm^{3}. The constant temperature curves are shown at *T* = 10, 50, 100, 175, 250, and 500 K.

Figure 2. The *UVT* relation calculated by MD simulation on fluid argon using the Lennard-Jones model as a function of density. The ordinate shows the internal energy per molecule *U*/*N* in Lennard-Jones units e (Table 1) and in units of kJ/mol. The abscissa shows the number density *N*/*V* and the mass density r in units of g/cm^{3}. The constant temperature curves are shown at *T* = 10, 50, 100, 175, 250, and 500 K.

Figure 3. The *pVT* relation calculated by the extended van der Waals EOS. The ordinate is pressure *p* in Lennard-Jones units e/s^{3} (Table 1). The abscissa is the number density *N*/*V*. The constant temperature curves are shown at *T* = 10, 50, 100, 175, 225, 275, and 500 K.

Figure 4. The *UVT* relation calculated by the extended van der Waals EOS as a function of density. The ordinate shows the internal energy per molecule *U*/*N* in Lennard-Jones units e (Table 1). The abscissa shows the number density *N*/*V*. The constant temperature curves are shown at *T* = 10, 50, 100, 175, 225, 275, and 500 K.

The simulation results are usually fitted by EOS with several fitting parameters [11, 12]. The critical point is determined by such EOS [11, 12]. The critical point determined by the present EOS is compared with that shown in Table 2. The obtained result does not differ greatly from the MD simulation results even with only three parameters in the present extended EOS.

Figure 5. The *pVT* relation calculated by the extended van der Waals EOS compared with that obtained by MD simulation. The ordinate shows pressure *p* in Lennard-Jones units e/s^{3} (Table 1). The abscissa shows the number density *N*/*V*. The constant temperature curves are shown at *T* = 50, 200, and 500 K.

Figure 6. The *UVT* relation calculated by the extended van der Waals EOS compared with that obtained by MD simulation. The ordinate shows the internal energy per molecule in Lennard-Jones units e (Table 1). The abscissa shows the number density *N*/*V*. The constant temperature curves are shown at *T* = 50, 200, and 500 K.

T/(e/_{c}k) | p/(e/s_{c}^{3}) | r_{c}/s^{-3} | |
---|---|---|---|

Nicolas et al [11] | 1.35 | 0.142 | 0.35 |

Kolafa & Nezbeda [12] | 1.3396 | 0.1405 | 0.3108 |

Present Study | 1.8 | 0.213 | 0.33 |

T/K_{c} | p/atm_{c} | V/(cm_{c}^{3}/mol) | |
---|---|---|---|

Nicolas et al [11] | 169 | 60.0 | 69 |

Kolafa & Nezbeda [12] | 167 | 59.4 | 78 |

Present Study | 225 | 90.0 | 74 |

Table 3. The van der Waals coefficients for fluid argon [1].

a/(atm.L^{2}/mol^{2}) | b/(L/mol) |
---|---|

1.345 | 3.22E-02 |

As shown in Figure 7, the extended EOS pressure is different from the MD results at low temperatures. The effect of attractive interaction is not well incorporated in this case. This is due to the simplified cluster structure, which is determined by the short-range term in the molecular interaction. The original van der Waals EOS also yields poor results at low temperatures, as shown in Figure 7. This curve shows that the attractive force is too strong in this EOS. The van der Waals coefficients used in Figure 7 and Figure 8 are shown in Table 3 [1]. A comparison of internal energy values is shown in Figure 8 at low temperatures.

Figure 7. Pressures are compared for the extended van der Waals EOS, the original van der Waals EOS, and the MD results on the Lennard-Jones system as a model of argon at *T* = 10 K = 0.08 e/*k*. The abscissa shows the number density *N*/*V*.

Figure 8. Average potential energies per molecule *U _{e}*/

The value -6e shown in Figure 8 of the present study is obtained from the 12 nearest neighbors in the FCC lattice. With respect to the internal energy, the extended van der Waals EOS is better than the original van der Waals EOS, as shown in Figure 8. The shortcomings in the present study (as shown in Figure 7 and Figure 8) may be overcome by the inclusion of the long-range part of the attractive interaction.

The authors would like to thank the Research Center for Computing and Multimedia Studies, at Hosei University for use of computers.

[ 2] R. Yamamoto, H. Tanaka, K. Nakanishi, and X. C. Zeng,

[ 3] R. Yamamoto and K. Nakanishi,

[ 4] R. Yamamoto, O. Kitao, and K. Nakanishi,

[ 5] R. Yamamoto and K. Nakanishi,

[ 6] R. Yamamoto and K. Nakanishi,

[ 7] D. A. McQuarrie,

[ 8] M. P. Allen and D. J. Tildesley,

[ 9] http://software.fujitsu.com/jp/materials-explorer/

[10] The relation between the units of pressure is given as follows: 1 atm = 101,325 Pa.

[11] J. J. Nicolas, K. E. Gubbins, W. B. Streett, and D. J. Tildesley,

[12] J. Kolafa and I. Nezbeda,

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