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THERMAL PROPERTIES OF SC The isochoric heat capacity, Cv, can be calculated from thermodynamic relations as: CO2 BY MC SIMULATIONS 407 h›kVl i2 ›T P b2P CP2Cv1⁄42T ›kVl 1⁄4TkVlk ð9Þ ›P T Finally, the Joule–Thomson coefficient, mJT, and speed of sound, u, can be expressed as: ›T 1 ›kVl mJT1⁄4 ›P 1⁄4C T ›T 2kVl hPP 1⁄4 kVl 1⁄2TbP 2 1 CP ð10Þ u 1⁄42Cv›kVl N 1⁄4Cv N k ð11Þ CP kVl2 MNA CP MNA kVl ›P T where M is the molecular weight. Expressions for other thermodynamic properties, such as the adiabatic expansivity or the Gru ̈neisen parameter can be obtained from combinations of the relations in this section. POTENTIAL MODEL The model of Mo ̈ ller and Fischer for a 2CLJQ fluid [2] was used to describe CO2. A molecule is composed of two identical Lennard–Jones sites a distance l apart plus a point quadrupole moment Q placed in the geometric center of the molecule. The full potential, u2CLJQ, is written as: u2CLJQðrij; vi; vj; l; Q2Þ 1⁄4 u2CLJðrij; vi; vj; lÞ þuQðrij;vi;vj;Q2Þ ð12Þ 2 where u2CLJðrij; vi; vj; lÞ 1⁄4 41 r a1⁄41 b1⁄41 and uQðrij;vi;vj;Q Þ1⁄444p10jrijj51⁄2125ðci þcjÞ "# X2 X2 s 12 2 s 6 r 23Q2 22 215c2ic2j þ2ðc25cicjÞ2 ð14Þ ab ab ð13Þ FIGURE 1 consists of two LJ centers of diameter s separated by a distance l. Dispersion – repulsion interactions between the sites are calculated for all distances rab. (b) Point quadrupoles are located at the center of each 2CLJ molecule. Quadrupole – quadrupole interactions between two molecules depend on the orientation of the molecules (u1, u2, f) and the distance rij between the two molecules. Eq. (14), where rij is the center-to-center distance between two molecules, i and j (see Fig. 1b). The vectors vi and vj represent the orientation of molecules i and j; ui and uj are the angles between the axis of the molecule and the center-to-center connection line, and fij is the difference in azimuthal angles of molecules i and j. In this equation, 10 is the permittivity of free space. We used potential parameters suggested by Mo ̈ller and Fischer [2]: 1=kB 1⁄4 125:317 K; s 1⁄4 3:0354 A ; L 1⁄4 l=s 1⁄4 0:699 and Q*2 1⁄4 Q2=ð4p10s5Þ 1⁄4 3:0255: SIMULATION DETAILS The fluctuation method based on Monte Carlo simulations in the isothermal – isobaric ensemble was used. Simulations containing 500, 864 and 1320 molecules, initially placed on a face-centered cubic lattice, were performed at low and high temperature, and it was verified that the results were independent on the size of the system. Therefore, the results in this work correspond to a total of N 1⁄4 500 molecules. Periodic boundary conditions and minimum image conventions were applied. Simulations were orga- nized in cycles. Each cycle consisted of N attempts to displace or rotate a randomly chosen molecule and an attempt of volume change. A variable spherical cut-off radius rc, equal to half the box length was used (e.g. at 330.39 K and P 1⁄4 2:5 MPa; rc 1⁄4 15:5s 1⁄4 4:7 nm; and at 330.39 K and P 1⁄4 72:3 MPa; rc 1⁄4 5:4s 1⁄4 1:6 nm), and long-range corrections were recalculated after each volume Representation of the 2CLJQ model. (a) Each molecule In Eq. (13), rab is one of the four Lennard – Jones site-to-site distances (see Fig. 1a) where a counts the two sites of molecule i, and b counts those of molecule j. The Lennard – Jones parameters s and 1 represent the size of each site and the well depth of the potential energy, respectively. The contribution to the potential energy due to the quadrupole – quadrupole interactions is given byPDF Image | Thermal Properties of Supercritical Carbon Dioxide
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