Properties of real fluids in critical region: third virial coefficient

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1 Indian J hys (February 2014) 88(2): DOI /s ORIGINAL AER roperties of real fluids in critical region: third virial coefficient R Khordad*, B Mirhosseini and M M Mirhosseini Department of hysics, College of Sciences, Yasouj University, Yasouj, Iran Received: 29 July 2013 / Accepted: 19 September 2013 / ublished online: 5 October 2013 Abstract: In present work, a perturbed virial expansion truncated at second order (third virial coefficient) is applied to calculate fluid critical points. We have used square-well (SW) potential model with constant width of the attractive well to obtain critical properties of several real fluids. Then, we have considered the width of attractive well as a function of temperature and studied its effect on critical properties of real fluids. Finally, we have studied the effect of different compressibility factors on critical properties of real fluids. Results reveal that critical temperatures and pressures calculated by using variable (temperature-dependent) width of the attractive well are in better agreement with experimental data. Selection of suitable compressibility factor has an important influence on our results. In general, we have theoretically improved critical points of some real fluids by using SW model with variable width of the attractive well and suitable compressibility factor. Keywords: ACS Nos.: Square-well potential; Third virial coefficient; Critical points Jk; fd 1. Introduction With development of nuclear and molecular theory a molecular interpretation of thermodynamics has been obtained. A relation between macroscopic and microscopic properties can be established thermodynamically [1 6]. Knowledge of microscopic forces between constituent particles of a system provides physical behavior of a macroscopic system by using statistical mechanics [1 8]. In this regard, parameters such as temperature (T), volume (V), pressure (), energy (E), entropy (S) and virial coefficients are obtained using the properties related to moving constitutive particles and interaction between those (including quantum phenomena) [9, 10]. In thermodynamics, interrelationship among macroscopic properties is studied. Using several thermodynamic properties like temperature, pressure and volume, one can describe a given thermodynamical system. Relation between thermodynamic properties of a system is called equation of state (EOS). We know that the simplest system in thermodynamics is ideal gas [2, 11]. An ideal gas represents the behavior of real gases fairly *Corresponding author, khordad@mail.yu.ac.ir well for high temperatures and low pressures [11]. The ideal gas EOS is very simple, but its range is limited. It is desirable to have (EOS) that represents V T behavior of substances accurately over a large region with no limitation [12, 13]. One of the most interesting EOS is virial EOS. Virial EOS is introduced to represent the behavior of real gases. The word virial stems from Latin word Vis (pleura vires) which means force and it refers to interaction forces between molecules [12]. Virial expansion is another set of EOS for imperfect gasses which has more parameters than Van-der-Waals equation. Therefore, virial expansion is more accurate to calculate the thrmodynamical properties of imperfect gases. The virial expansion is originally proposed by Thiesen and developed by Kamerling-Onnes [14, 15]. Thiesen have calculated the second and third virial coefficients by using a power series. As we know, constitutive particles of a fluid easily move within it. So, we can use classical statistical mechanics which contains particles correlation. The various interactions between atoms and molecules can be calculated by using statistical mechanics [16, 17]. In study of fluids, two problems are important, particle s shape and interaction potential between particles [18]. It is to be noted that interplay between intermolecular forces and external potential plays an important role in a fluid Ó 2013 IACS

2 186 R Khordad et al. structure and its physical properties. Many scientists [7, 9] have tried to predict intermolecular potentials in fluids by using the laws of physics. They have also tried to formulate the intermolecular potential models by simple mathematical expressions. In practice, one requires some means to accurately model the intermolecular potential using a simple empirical expression that can be rapidly calculated. Hitherto, many intermolecular potential models have been created to study real fluids [9, 10]. Basically, interaction between particles in the real fluids is very complex. Therefore, various potential models have been presented to describe the interaction between particles in a real fluid. Examples of these models are the hard-sphere (HS), Sutherland, Lenard- Jones and square-well (SW) models [16, 17]. These models have advantages and shortcomings and they are only employed for specific categories of fluids. For engineering purposes, use of virial expansion is practical only where convergence is very rapid. This means that two or three terms suffice for reasonably close approximation to the values of series. The virial expansions realized for gasses and vapours at low to moderate pressures [2, 18]. It is worth mentioning that usefulness of virial expansion has been a little broader. The virial expansion can be employed to predict critical properties of real fluids [19, 20]. So far, many efforts have been made in studying the critical points by using different potential models and the virial expansion [20 23]. This procedure has been first explored a long time ago by Barker and Monaghan [24]. Also, recent calculations for Kihara fluids [25, 26] and water [27] have shown accuracy of this simple procedure. Recently, virial expansion has also been applied for obtaining critical properties of long-chain polymers [28, 29]. Nezbeda et al. [8] have calculated fluid critical points of several fluids by using the second virial coefficient. They have used Lennard-Jones and SW potentials and perturbed virial expansion to locate fluid critical points. Recently, we have applied Sutherland potential and SW models to calculate critical points of several real fluids [16, 17]. 2. Theory 2.1. Second and third virial coefficients Square-well potential model provides a better understanding of the effects of molecular interactions in real fluids. It is to be noted that SW model is an extension of hard sphere potential and it includes an attractive term. It is worth mentioning that SW is yet simple enough to use analytically. This model can provide a useful reference system in statistical mechanical perturbation theories for molecular fluids [9]. The mathematical form of SW potential model is given by U SW ðþ¼ r 8 < 1 r\r e r\r\kr; : 0 r [ kr ð1þ where k is range (width) of the attractive well, is usually taken to be between 1.5 and 2.0. Also, r means collision radius between molecules and r is particle separation. roperties of square-well fluid have also been studied extensively by computer simulations [17, 30]. This potential model represents hard spheres of diameter r surrounded by an attractive core of strength e which extends to separations kr. Using this potential, one can obtain second virial coefficient as B 2 ðtþ ¼ b 0 1 k 3 1 x : ð2þ where x ¼ e be 1 ð3þ and quantity b 0 is HS second virial coefficient 2=3pr 3. Also, b ¼ 1=k B T where k B is the Boltzman constant. Accurate data for third virial coefficient are far less common than for second virial coefficient. By using Eq. (1), we calculate third virial coefficient as below B 3 ðtþ ¼ 1 4 b 0 5 k 6 18k 4 þ 32k 3 15 x þ 2k 6 þ 34k 4 32k 3 18k 2 þ 16 x 2 ð4þ 6k 6 18k 4 þ 18k 2 6 x 3 Note that SW potential is particularly useful to describe EOS behavior of gases made up of complex molecules [31]. The virial coefficient depends only on temperature and on particular gas under consideration, but it is independent of density or pressure [32]. 3. Critical points 3.1. rediction of critical points with different compressibility factors The word critical point is often used to denote specifically vapor liquid critical point of a material. Vapor liquid critical point describes the condition above which distinct liquid and gas phases do not exist [2, 33]. In thermodynamics, one often assumes equations of state to be polynomials of a variable. The virial expansion can be written in closed form as [9, 10] qk B T ¼ 1 þ X1 i¼1 B iþ1 q i ð5þ

3 roperties of real fluids in critical region 187 where p is pressure, T absolute temperature, and q N=V is the number density of the gas. Virial coefficients B i appear as coefficients in virial expansion of pressure of a many-particle system in powers of density, providing systematic corrections to ideal gas law. They are characteristic of interaction potential between the particles and in general depend on temperature. The second virial coefficient B 2 depends only on pair interaction between the particles, third term (B 3 ) depends on 2-and non-additive 3-body interactions and so on. An EOS, where all virial coefficients would vanish beyond the second one would not be good enough to predict a liquid vapor phase transition. To calculate the critical properties of a fluid, we use perturbed virial expansion which has been mentioned by Nezbeda et al. [20]. erturbed virial expansion can be obtained from expanding compressibility factor of both fluid interest and a suitable compressibility factor in powers of density, qk B T ¼ qk B T þ X DB i q i1 ; ref i 2 ð6þ where DB i are residual virial coefficients, DB i ¼ B i B ref ;i. Dimensionless terms B ref ;i are known analytically for i ¼ 1; 2; 3[10, 32]. The obvious choice of compressibility factor in this case is HS fluid of diameter r, with the few first virial coefficients being B HS;2 ¼ b 0 ; B HS;3 ¼ 0:625b 2 0 ; B HS;4 ¼ 0:2906b 3 0 To calculate the fluid critical points, we use three different compressibility factors instead of the first term of Eq. (5). In following, three compressibility factors are given by [34 36]. (ercus Yevick theory) u Y # ¼ qk BT u Y c ¼ ref;1 Reiss-Frisch-Lebowitz (1959) ¼ and u cs ¼ qk BT qk BT 1 þ 2g þ 3g2 ð1 gþ 2 ; ð7þ ¼ 1 þ g þ g2 ref;2 ð1 gþ 3 ; ð8þ ¼ 1 þ g þ g2 g 3 ref;3 ð1 gþ 3 : ð9þ We intend to show the influence of compressibility factors on critical points of real fluids. Critical point results from solution of the set of two equations: o oq ¼ 0 cr ð10þ o 2 oq 2 ¼ 0 cr where subscript cr denote the critical point. This relation can be used to evaluate two parameters for an EOS in terms of the critical properties. The critical point is defined by critical temperature T c, critical pressure c and critical density q c If T c and c are obtainable, we can estimate critical viscosity by using the following relation [33]: l c ¼ 7:70M 1 2ð c Þ 2 3 ðtc Þ 1 6 ; ð11þ where M is molecular weight (relative molar mass). Reduced property is commonly expressed as a fraction of the critical property: T r ¼ T=T c ; r ¼ = c ; and l r ¼ l=l c 3.2. Temperature-dependent width of attractive well We know that for SW potential, derivative of second virial coefficient with respect to temperature is always positive. Therefore, this potential cannot show any maximum value which has been observed experimentally [17]. This difference in behavior is mainly due to existence of a hardcore diameter in such potential model. Hitherto, many efforts have been made to correct this problem. For example, recently, some authors obtained second virial coefficient (for SW and Sutherland potentials) by considering molecular diameter as a function of temperature [17, 37]. Even though considerable attentions have been made to study SW potential under various conditions, temperature-dependent width of the attractive well has not been investigated so far. We may note that when temperature increases, real molecules are expected to penetrate more into each other and therefore width of the attractive well is expected to become variable. We can include the above-mentioned character to improve the potential model without losing its simple functional form by supposing that the width of the attractive well is temperature-dependent. Table 1 Values of coefficients of Eq. (11) for kðtþ Substance A B (K) C (K 2 ) D (K 3 ) Ar , : CH , : Kr , : Xe N CO Benzen ,

4 188 R Khordad et al. Width of the attractive well has been fitted with respect to temperature for several substances such as Ar, CH 4, Kr, Xe, N 2,CO 2, water and benzene. We have selected these substances due to existence of experimental data over spread range of temperatures. Using temperature-dependent width of attractive well, the critical points for abovementioned substances have been calculated. We have considered width of the attractive well as a function of temperature and fitted it into a polynomial using experimental data. We have found that width of the attractive well is quite well fitted into the following polynomial. kðtþ ¼ A þ B 1 þ C 1 2 þd 1 3 : ð12þ T T T The coefficients A, B, C and D for various substances are given in Table 1. It is worth mentioning that the best fit for water has been obtained as follows. kðtþ ¼ 1:007 þ ffiffiffi T p 4. Results and discussion þ 4:5 1 T : T ð13þ We have reported the obtained results for critical points. First, we have used Eqs. (6) (8) to determine the critical points with constant width of attractive well. Table 2 shows the dimensionless critical points by using the SW potential with k = 1.5. The dimensionless critical points are defined as, Tc ¼ k BT c ; q c e ¼ q cr 3 ; c ¼ cr 3 : ð14þ e In Table 2, we have applied different compressibility factors [Eqs. (6) (8)] as reference fluids in Eq. (5). The obtained results have been compared with simulation data [38]. It is seen from the table that the obtained results using k = 1.5 and compressibility factor 2 [Eq. (7)] are in better agreement with simulation data. Table 3 displays the dimensionless critical points by using SW potential with k = 3 for all reference fluids given by Eqs. (6) (8). Table 2 Dimensionless critical points using square-well potential with k = 1.5, perturbed virial expansion and different compressibility factors T c q c c [1] [2] [3] Simulation Table 3 Dimensionless critical points using square-well potential with k = 3, perturbed virial expansion and different compressibility factors T c q c c [1] [2] [3] Simulation data Table 4 Critical properties of several real fluids estimated by the perturbed virial expansion [using the compressibility factor 1] Substance T c ðkþ q c ðmol=cm 3 Þ c ðbarþ Ar 15 [144.1] 108 [108] [36.09] (150.9) (138) (48.98) CH [182.0] 077 [077] [32.43] (19) (101) (45.99) Kr [171.5] 121 [121] [47.94] (209.4) (069) (55.02) Xe [249.3] 092 [092] [52.88] (289.7) (056) (58.40) N [119.6] 121 [121] [33.50] (126.2) (085) (34.00) CO [356.1] 094 [094] [76.72] (304.2) (112) (73.83) Benzene [779.1] 038 [038] [67.82] (562.2) (039) (48.98) Water [58] 095 [095] [127.47] (647.1) (178) (220.55) Table 5 Critical properties of several real fluids estimated by the perturbed virial expansion [using the compressibility factor 2] Substance T c ðkþ q c ðmol=cm 3 Þ c ðbarþ Ar [142.9] 104 [104] [35.34] (150.9) (138) (48.98) CH [18] 074 [074] [31.74] (19) (101) (45.99) Kr [170.1] 116 [116] [47.02] (209.4) (069) (55.02) Xe [247.4] 088 [088] 53 [51.96] (289.7) (056) (58.40) N [118.6] 116 [116] [32.29] (126.2) (085) (34.00) CO [353.4] 090 [090] [75.40] (304.2) (112) (73.83) Benzene 55 [773.5] 036 [036] [67.17] (562.2) (039) (48.98) Water [576.3] 091 [091] [125.14] (647.1) (178) (220.55)

5 roperties of real fluids in critical region 189 Table 6 Critical properties of several real fluids estimated by the perturbed virial expansion [using the compressibility factor 3] Substance T c ðkþ q c ðmol=cm 3 Þ c ðbarþ Ar [143.3] 105 [105] [35.83] (150.9) (138) (48.98) CH [181.1] 075 [075] [32.09] (19) (101) (45.99) Kr [17] 117 [117] [47.33] (209.4) (069) (55.02) Xe [248.1] 089 [089] 53 [52.25] (289.7) (056) (58.40) N [119.0] 117 [117] [32.98] (126.2) (085) (34.00) CO [354.3] 091 [091] [75.94] (304.2) (112) (73.83) Benzene [774.8] 037 [037] [66.81] (562.2) (039) (48.98) Water [577.9] 092 [092] [126.57] (647.1) (178) (220.55) From results in this Table 3, we have concluded that compressibility factor 1, from Eq. (6) gives better results. We can say that reference fluid has an important effect to obtain dimensionless critical points. In following, we intend to apply the variable (temperature-dependent) width of the attractive well. In Table 4, we have applied the compressibility factor 1 [34] and calculated the critical points for several real substances by using temperature-dependent width of attractive well. The obtained results have been compared with experimental data [2, 17, 33] and constant width of the attractive well k = 1.5. Results in Tables 5 and 6 are same as those given in Table 4, but for compressibility factor 2 [35] and compressibility factor 3 [36], respectively. Comparing the Tables 4, 5, and 6, one finds that: (i) Critical temperatures obtained by using temperaturedependent width of the attractive well and compressibility factor 1 [Eq. (6)] have been improved (except CH 4 ). They are in fairly agreement with experimental data [2, 17, and 33]. (ii) It is interesting that use of temperature-dependent width of attractive well has no influence on the critical density for all reference fluids. (iii) The critical pressures obtained with temperaturedependent width of the attractive well have been improved by using the compressibility factor 2. They are in good agreement with experimental data except for benzene and CO 2. (a) / c (b) / c (c) / c Fig. 1 Variation reduced pressure with reduced temperature for (a) Xe, (b) N 2 and (c) CO 2 Figures 1(a) 1(c) show the variation of reduced pressure as a function of reduced temperature for several real fluids such as Xe, N 2 and CO 2, respectively for using compressibility factor 1 [Eq. (6)].

6 190 R Khordad et al. (a) (d) (b) (e) (c) (f) Fig. 2 Variation reduced viscosity with reduced temperature for (a) Xe, (b) N 2, (c) CO 2 (d) Ar, (e) Kr and (f) CH 4 It is clear that reduced pressure increases by increasing the reduced temperature. It is seen from the figures that our theoretical results [constant k; kðtþ] are close to each other and with experimental data at low temperatures [2, 17]. This point clearly shows that at low temperature, width of the attractive well is not sensitive to temperature. When temperature increases, the difference between our theoretical results increases [constant k; kðtþ]. Results obtained by using temperature-dependent width of attractive well are closer to experimental data at moderate and high temperatures. From

7 roperties of real fluids in critical region 191 obtained results in present work, we have found that our consideration about temperature-dependent width of the attractive well is fairly correct especially at moderate and high temperatures. Better results can be obtained by using higher orders in perturbed virial expansion in addition to temperature-dependent width of the attractive well. Using obtained critical points, we have calculated another critical property like critical viscosity. Figs. 2(a) 2(f) show the variation of reduced viscosity as a function of reduced temperature for several real fluids such as Xe, N 2,CO 2,Ar,Kr and CH 4, respectively for compressibility factor 1 [Eq. (6)]. As we observe, reduced viscosity increases when reduced temperature increases. It is to be noted that reduced viscosity has been obtained by using SW potential with constant and variable width of the attractive well. Obtained results have been compared with experimental data [33]. It is observed from these figures that the results obtained by using variable width of the attractive well are in better agreement with experimental data. 5. Conclusions In regard to the results obtained in present work, it is deduced that: Critical temperatures and pressures obtained with temperature-dependent width of the attractive well have been improved with respect to experimental data. Best values for critical temperatures and pressures can be obtained by using the temperature-dependent width of the attractive well and by choosing the sufficient compressibility factor. We have also calculated reduced viscosity of several real fluids by using SW potential and the temperature-dependent width of the attractive well. It is deduced that reduced viscosity calculated by using variable width of the attractive well in addition to compressibility factor 1 [Eq. (6)] have been improved with respect to available data. References [1] F Reif Fundamental of statistical and thermal physics (New York: McGraw- Hill) (1965) [2] J M Smith, H C Van Ness and M M Abbott Introduction to Chemical Engineering Thermodynamics (New York: McGraw- Hill) (2005) [3] M R Gupta et al. Indian J. hys (2012) [4] X Y Zhao and X W Lei Indian J. hys (2012) [5] B Dutta, R De and J Chowdhury Indian J. hys (2013) [6] K Fakhar, A H Kara, R Morris and T Hayat Indian J. hys (2013) [7] T Dorlas Statistical Mechanics Fundamentals a Model Solution (London: Institute of physics) (1999) [8] K Morita Indian J. hys (2011) [9] J Hansen and I R McDonald Theory of Simple Liquids (London: Academic press) (1986) [10] D A McQuarri Statistical Mechanics (London: Harper and Row) (1976) [11] Y A Gengel and M A Boles Thermodynamics: An Engineering Approach (New York: McGraw- Hill) (2006) [12] N N Ajitanand Indian J. hys (2010) [13] E Fermi Thermodynamics (New York: Dover publication) (1936) [14] M Thiesen Annalen der hysik (1885) [15] H K Onnes Commun. hys. Lab.Univ. Leiden 71 3 (1901) [16] R Khordad and B Mirhosseini hys. Chem. Indian J (2012) [17] R Khordad and B Mirhosseini Int. J. Mod. hys. B (2012) [18] R Sadus Molecular Simulation of Fluids, Theory Algorithms and Object Orientation (Amsterdam: Elsevier) (1999) [19] B Tomberli, S Goldman and C G Gray Fluid hase Equilib (2001) [20] I Nezbeda Fluid hase Equilib (1993) [21] J Kolafa and I Nezbeda Fluid hase Equilib (1994) [22] B E Turner, C L Costello and C Jurs J. Chem. Inf. Comput. Sci (1998) [23] A R Katritzky, U Maran, V S Lobanov and M Karelson J. Chem. Inf. Comput. Sci (2000) [24] J A Barker and J J Monaghan J. Chem. hys (1962) [25] J Janecek and T Boublik Mol. hys (2000) [26] J Janecek and T Boublik Mol. hys (2000) [27] G Kusalik, F Liden and I M Svishchev J. Chem. hys (1995) [28] C Vega and L G MacDowell Mol. hys (2000) [29] L Lue, D G Friend and J R Elliott Jr Mol. hys (2000) [30] V Ya Antonchenko, N V Gloskovskaya and V V IIyin J. Mol. Liq (2006) [31] T Kihara Rev. Mod. hys (1953) [32] J O Hirschfelder, C F Courtiss and R B Bird Molecular Theory of Gases and Liquid (New York: Wiley) (1954) [33] Y Demirel Nonequilibrium Thermodynamics 2nd Ed. (Amsterdam: Elsevier Science & Technology Books) (2007) [34] A Mulero, C Galan and F Cuadros hys. Chem. Chem. hys (2001) [35] Attard J. Chem. hys (1989) [36] N F Carnahan and K E Starling J. Chem. hys (1970) [37] G A arsafar, M Khanpour and A A Mohammadi Chem. hys (2006) [38] G Orkoulas and A Z anagiotopoulos J. Chem. hys (1999)