Theretical study f third virial cefficient with Kihara ptential Jurnal: Manuscript ID cjp-017-0705.r Manuscript Type: Article Date Submitted by the Authr: 6-Dec-017 Cmplete List f Authrs: Smuncu E.; Giresun Üniversity Mamedv Bahtiyar; University Askerv I.M.; Giresun Universitesi Keywrd: Is the invited manuscript fr cnsideratin in a Special Issue? : Virial cefficient Third virial cefficient Kihara ptential virial equatin f state thermdynamic 33rd Internatinal Physics Cnference f Turkish Physical Sciety
Page 1 f 5 Theretical study f third virial cefficient with Kihara ptential E. Smuncu 1 B. A. Mamedv I. M. Askerv 1 1 Department f Physics Faculty f Arts and Sciences Giresun University Giresun Turkey. Department f Physics Faculty f Arts and Sciences Gazismanpasa University Tkat Turkey. Crrespnding authr: E. Smuncu (email: elf_smnc@htmail.cm) Abstract: In this paper a new frmula has been presented fr accurate calculatin f the third virial cefficient with Kihara ptential. The btained frmula allws us t the determinatin f thermdynamic prperties f imperfect gases. The validity f the frmula has been tested by applicatin t sme gasesc H 6 CH C( CH 3) and n CH 4. The btained results have been cmpared with the ther studies. These cmparisns shw that the frmula develped in this study are in gd agreement with the data available in the literature. Keywrds: Virial cefficient; Third virial cefficient; Kihara ptential 1. Intrductin The determinatin f the equatins f state are an imprtant issue t define thermdynamic prperties (heat capacity internal energy sund velcity etc.) f imperfect gases [1-3]. Many equatins f state have been prpsed t accurate evaluatin f the thermdynamic prperties f imperfect gases [3 4]. One f equatin f state is the virial equatin f state. The imprtance f the virial equatin f state lies in its direct relatin t intermlecular interactin [56]. The Lennard-Jnes ( n n) ptential that it is described the interactin between simple spherical mlecules nly has been extensively studied by bth experimental and theretical methds [7]. Kihara ptential is defined the interactin between tw mre cmplex mlecules and fund wide applicatins because f its well definitin f thermdynamic prperties [7 8]. In spite f many studies the evaluatins f the third virial cefficient fr varius types f the intermlecular interactin are still ne f the main crucial prblems in physics and biphysical chemistry [9-11]. In this paper we have presented an efficient methd fr the third virial cefficient with Kihara ptential. Fr sme f gases example applicatins are given t demnstrate the effectiveness f the present frmula. The results f calculatin fr the third virial cefficient with Kihara ptential are in gd agreement in the literature.. Definitin and expansins relatins fr the third virial cefficient with Kihara ptential The virial equatin f state expresses the deviatin frm ideal behavir f real gases and can be defined in the general series frm
Page f 5 PV n n = Z = 1 + B (T) + B3 (T) +... (1) nrt V V which is expansin in pwers f the number f mlecules per unit vlume n V B (T) B (T)... 3 are called the secnd virial cefficient the third virial cefficient and s n [1]. The virial cefficients depend nly n temperature and ptential energy f interactin between mlecules. The third virial cefficient in terms f intermlecular ptential ur ( ij ) are given in the fllwing frms N 3( ) A f ( r1 ) f ( r ) f ( r3) dr1dr B T = 3 () ( ij) ( ( ij) B ) 1 f r = exp u r kt (3) where f ( rij ) is Mayer functins ( ij) u r is the intermlecular ptential N A is Avgadr s cnstant kb is the Bltzmann cnstant and T is temperature. [1]. In this paper fr evaluatin f the third virial cefficient we have used the Kihara ptential fr mlecules with spherical cres in the fllwing frm: r< a 1 6 u( r) = σ a σ a 4ε r a r a r a (4) where a is the radius f spherical mlecular cre ε is the depth f the ptential well σ is the cllisin diameter and r is the distance between the particles [4 1]. In this sectin fr evaluatin f third virial cefficient with Kihara ptential the three bdy systems are illustrated in Fig. 1. Figure 1. The interactin f three atms Frm the Fig. 1 the relatins between f parametersr 1 r andr 3 can be btained frm as r r1 = r3 = r1 + r r1r csϕ (5) dr = 4πrdr (6)
Page 3 f 5 dr = πr dr sinϕdϕ 1 1 1 The substituting Eq. (5)-(7) int the Eq. () we have btained the fllwing frmula B3 T π 1 1 1 1 1 3 r f r r f r f r r r r η dη dr dr. (8) 0 0 1 1 N ( ) 8 A = ( ) ( ) ( + ) We have btained the fllwing frm fr third virial cefficient (7) 4 1 6 4 1 6 6 ( r ) ( ) ( ) ( ) ( ) 1 a r 1 a r a r a T T B3 T = e 1 e 1 6 + 0 0 ( 1 a ) ( r ) ( ) 1 r r 1 r η a r 1 r r 1 r η a 1 1 6 4 + + T 1 e 3 B3 T = B3 T b0 b0 = πn A σ 3 T = kt B ε where ( ) ( ) a σ a= a 1 r1 r dη dr1dr r ij σ a= r and ij (9) 3. Numerical results and discussin In this paper we have presented an efficient frmula fr third virial cefficient using Kihara ptential. The established frmula is general and free f any restrictin n its applicatins t gases. The Mathematica 7.0 internatinal mathematical sftware has been used t calculate the frmula btained in this paper. The integrals in Eq. (9) have been evaluated by using numerical methds. The btained results are cmpared with theretical data and shw a gd agreement in the literature data [14]. The results f calculatins f the third virial cefficient are presented in Tables 1- fr 0.50 T 100. Table 1. The results f calculatin fr the third virial cefficient 3( ) diameter σ and the radius f spherical mlecular cre a T CH 6 CH ( ) C CH n CH 0.50-8.57673-5.49-3.85708-1.0765 0.80 0.553937 0.59589 0.590318 0.515087 1.00 0.56631 0.55 0.48778 0.38978.00 0.338168 0.3441 0.3178 0.309801 3.00 0.37838 0.39057 0.3339 0.343681 4.00 0.33481 0.341493 0.346876 0.365788 5.00 0.340174 0.349509 0.356364 0.378317 6.00 0.34854 0.353753 0.361471 0.38515 10.00 0.339408 0.3533 0.36614 0.389901 100.00 0.17305 0.3679 0.49875 0.88048 3 4 B T by the cllisin
Page 4 f 5 Table. The results f calculatin fr the third virial cefficient by the parameter a T 0.00 0.05 0.1 0.5 0.50-40.1638-31.9503-5.576-4.78344-1.5460 0.80-0.849145-0.39466-0.07073 0.593839 0.538579 1.00 0.49769 0.5187 0.56953 0.508616 0.411836.00 0.437073 0.417365 0.399518 0.31698 0.3108 3.00 0.35309 0.345039 0.3393 0.39788 0.340075 4.00 0.366 0.35697 0.3567 0.343447 0.360938 5.00 0.31508 0.3176 0.3058 0.3505 0.37907 6.00 0.30765 0.3148 0.317037 0.35664 0.379477 10.00 0.86068 0.94367 0.30369 0.356765 0.383438 100.00 0.14535 0.154097 0.16589 0.41677 0.79047 0.75 The results f calculatin frm the Eq. (9) are pltted in Figure. Figure. Third virial cefficient fr Kihara ptential 06 C H 6 C 3 H 8 C(CH 3 ) 4 04 n-c 5 H 1 B 3 (T ) 0 00 0 0 40 60 80 100 T -0 It is understd frm the suitability f the graphics that the results are in gd agreement with literature data [14].
Page 5 f 5 The parameters f Kihara ptential fr gasesch 6 presented in Table 3 [4 15]. CH ( ) C CH and n CH are 3 4 Mlecules a a [4] [15] Table 3. The parameters f Kihara ptential σ ( A ) [4] σ ( A ) [15] B ( ) ε k K [4] B ( ) ε k K CH 6 0.359 0.166 3.504 4.000 496.69 414.8 CH 0.470 0.168 4.611 4.519 501.89 493.7 ( ) 3 4 C CH 0.551 5.76 557.75 n CH 0.818 5.09 837.8 [15] In cnclusin the btained frmula ffers a direct and precise calculatin advantage f the third virial cefficient fr 0.50 T 100 temperature range. Acknwledgements This wrk has been supprted by the Scientific and Technlgical cuncil f Turkey (TUBITAK) Science Fellwships and Grant Prgrammes Department (BIDEB). References 1. I. Nezbeda W.R. Smith. Fluid Phase Equilib. 16 183 (004).. C. Vega C. McBride C. Menduiña. Phys. Chem. Chem. Phys. 4 3000 (004). 3. D.A. McQuarine. Statistical Mechanics. Harper & Rw New Yrk (1973). 4. J. M. Prausnitz R. N. Lichtenthaler E. G. de Azeved. Mlecular Thermdynamic f Fluid-Phase Equilibria. Third Editin Prentice Hall PTR New Jersey (1999). 5. T. Kihara. Rev. Md. Phys. 5 831 (1953). 6. J.E. Mayer M.G. Mayer. Statistical Mechanics. Wiley New Yrk (1948). 7. G. J. Chen C. Y. Sun. Chem. Eng. Sci. 56 7045 (001). 8. W. Witschel. Int. J. Themphys. 11 1075 (1990). 9. M. Deszczynski S.E. Harding D.J. Winzr. Biyphys. Chem. 10 106 (006). 10. C. Haas J. Drenth W.W. Wilsn. J. Phys. Chem. 103 808 (1999). 11. A. Maghari M.H.K. Jafari. IEEE Cnf. Pub. 994 (009). 1. J. O. Hirschfelder C. F. Curtiss R. B. Bird. Mlecular Thery f Gases and Liquids. Jnh Wiley &Sns (1954).. I.G. Kaplan. Intermlecular Interactins: Physical Picture Cmputatinal Methds and Mdel Ptentials. Jnh Wiley & Sns (006). 14. A. E. Sherwd J. M. Prausnitz. J. Chem. Phys. 41 4 (1964). 15. G. Mradi E. Khsravani. Fluid Phase Equilibria. 338 179 (0).