J. Serb. Chem. Soc. 8 (11) 143 1433 (15) UDC 547.533+547.16+547.41.113: JSCS 488 541.1.34.:53.14 Orgnal scentfc paper Modelng of densty and calculatons of derved volumetrc propertes for n-hexane, toluene and dchloromethane at pressures.1 6 MPa and temperatures 88.15 413.15 K GORICA R. IVANIŠ #, ALEKSANDAR Ž. TASIĆ, IVONA R. RADOVIĆ #, BOJAN D. DJORDJEVIĆ #, SLOBODAN P. ŠERBANOVIĆ # and MIRJANA LJ. KIJEVČANIN* # Faculty of Technology and Metallurgy, Unversty of Belgrade, Karnegjeva 4, 111 Belgrade, Serba (Receved 18 March, revsed 3 June, accepted 1 July 15) Abstract: Denstes data of n-hexane, toluene and dchloromethane at temperatures 88.15 413.15 K and at pressures.1 6 MPa, determned n a prevous artcle, were ftted to the modfed Tat equaton of state. The ftted temperature pressure dependent densty data were used to calculate the followng derved propertes: the sothermal compressblty, the sobarc thermal expansvty, the dfference between specfc heat capacty at constant pressure and at constant volume and the nternal pressure, over the entre temperature and pressure ntervals specfed above. In order to assess the proposed modelng procedure, the obtaned values for the sothermal compressblty and the sobarc thermal expansvty were compared wth the correspondng lterature data. The average absolute percentage devatons for the sothermal compressblty were for n-hexane.1 3.64 %, for toluene.64.48 % and for dchloromethane 1.81 3. %, and for the sobarc thermal expansvty, they were for n-hexane 1.31 4.17 %, for toluene.71.45 % and for dchloromethane 1.16 1.61 %. By comparng the obtaned devatons values wth those found n the lterature, t could be concluded that the presented results agree well wth the lterature data. Keywords: hgh pressure; sobarc thermal expansvty; sothermal compressblty; n-hexane; toluene; dchloromethane. INTRODUCTION In a prevous artcle, 1 the expermental denstes of compressed lqud n- hexane, toluene and dchloromethane, determned usng a set-up based on a densty measurng cell AP DMA HP coupled wth a DMA 5 vbratng tube densmeter, were reported. The measurements were performed over the temperature * Correspondng author. E-mal: mrjana@tmf.bg.ac.rs # Serban Chemcal Socety member. do: 1.98/JSC153186I 143
144 IVANIŠ et al. and pressure ranges: 88.15 413.15 K and.1 6 MPa, respectvely. It was shown that the obtaned data are n good and acceptable agreement wth the correspondng lterature values. Ths paper, as well as the prevous one related to the densty measurements for the same pure compounds-solvents, s a part of the systematc nvestgaton of crucal thermodynamc propertes of mportant solvents, used n both ndustry and the laboratory. Correlaton of the volumetrc nformaton mentoned above allows the evaluaton of temperature and pressure effects on varous thermomechancal propertes, such as sothermal compressblty, κ T, and sobarc thermal expansvty, α p ; based on these propertes, the nternal pressure p nt and the dfference of the sobarc and sochorc heat capactes, C p C V, can also be determned. By couplng the mentoned sothermal and sobarc propertes κ T and α p, and usng the sentropc compressblty, κ S, whch s related to the thermodynamc speed of sound, some addtonal propertes can be calculated; n ths respect, the sobarc heat capacty, C p, and sochorc heat capacty, C V, could be mentoned. Varous methods, dealng wth the correlaton of measured denstes as a functon of temperature and pressure, have been establshed n the lterature. In the present work, a Tat-lke equaton was appled. The obtaned correlaton was used to compute the aforementoned parameters. CALCULATIONS The expermental densty data presented n a prevous paper 1 were ftted to the Tat-lke equaton: ρ ( T, p) = ref ρ ( T ) B( T) + p 1 CT ( )ln BT ( ) pref + where, T and p refer to temperature and pressure, respectvely, and C(T) and B(T) are parameters that were assumed to be temperature dependent and can be calculated from the second order polynomals: (1) B( T) = bt () CT ( ) = ct (3) ρ ref s the densty at the reference pressure, p ref, and t can be calculated from the second order polynomal: ρ ref ( T) = at (4) In Eqs. () (4), a, b and c are adjustable parameters.
DERIVED VOLUMETRIC PROPERTIES OF n-hexane, TOLUENE AND DICHLOROMETHANE 145 As stated prevously, a number of derved propertes exst that can be calculated from densty data. The sothermal compressblty, κ T, represents a measure of the senstvty of densty to a pressure change at constant temperature and t depends on the flud structure. It can be expressed as: 1 ρ κt = (5) ρ p T Couplng Eqs. (1) and (5) leads to the expresson for the calculaton of κ T : CT ( ) κ T = (6) B( T) + p ( BT ( ) + p) 1 CT ( )ln BT ( ) pref + The sobarc thermal expansvty, α p, s a measure of the relatve flud densty change n response to a temperature change at constant pressure: 1 1 p p 1 ρ α p = = ρ T ρ ρ T ρ When Eq. (1) s ncluded nto Eq. (7), an expresson for the calculaton of the sobarc thermal expansvty s obtaned: ref B ( T)( p p) B( T) + p CT ( ) + C ( T)ln ref ref ref ρ ( T ) ( BT ( ) + p)( BT ( ) + p ) BT ( ) + p (8) α p = ref ρ ( T ) BT ( ) + p 1 CT ( )ln ( ) ref + BT p where ρ ref (T), B (T) and C (T) stay for the dervatves of the parameters ρ ref, B(T) and C(T) of Eq. (1), respectvely: ρ T p (7) ref 1 =at (9) B = 1 bt (1) C = 1 ct (11) Another thermodynamc property that can be calculated from the aforementoned equatons s the rato of the sobarc thermal expansvty to the sothermal compressblty,.e., the thermal pressure coeffcent, γ: α γ (1) κ = p T The thermal pressure coeffcent s assocated wth the nternal pressure, p nt, whch represents the change of the nternal energy of a sample when t sothermally expands or
146 IVANIŠ et al. contracts, thus descrbng ntermolecular nteractons. It s easy to calculate the nternal pressure when the sobarc thermal expansvty and the sothermal compressblty are known: U p Tα p pnt = = T p= Tγ p= p (13) V T T ρ κt where, U denotes the nternal energy of a sample and V s ts volume. An mportant thermodynamc property that can be calculated from the quanttes defned above s the dfference between specfc heat capacty at constant pressure, c p, and specfc heat capacty at constant volume, C V : p T ρ Cp = CV + T (14) p ρ ρ T Lnkng Eqs. (5) and (7) wth Eq. (14) gves the expresson: α = p T Cp C V (15) ρκt The relaton gven by Eq. (15) s sgnfcant because t can be used to calculate the heat capacty at constant volume nstead of ts expermental determnaton. Heat capacty at constant pressure can be measured or calculated from speed of sound data and the above-defned coeffcents by combnng Eqs. (16) (18). Isentropc compressblty, κ S, s defned as: 1 ρ κs = (16) ρ p S and can be calculated as follows: 1 κs = ρu where, u s the speed of sound waves spreadng through the sample. Further, the specfc heat capacty at constant pressure can be obtaned from the equaton: ( κ ) (17) α pt C p = (18) ρ κ RESULTS AND DISCUSSION The crtera used to assess the qualty of fttng were the absolute average percentage devaton, AAD, the percentage maxmum devaton, MD, the average percentage devaton, Bas, and standard devaton, σ: exp cal 1 ρ ρ = N AAD (19) N exp 1 ρ T S
DERIVED VOLUMETRIC PROPERTIES OF n-hexane, TOLUENE AND DICHLOROMETHANE 147 exp cal max 1 ρ ρ = exp ρ exp cal 1 ρ ρ = N Bas N exp 1 ρ exp cal N ( ρ ρ ) σ = 1 N m MD, 1,N () (1) () where, ρ exp s the expermentally determned densty, ρ cal denotes the densty value obtaned from Eq. (1), N s the number of expermental data and m represents the number of parameters ncluded n the correlaton of the Tat-lke equaton. The reference pressure n Eq. (1) was chosen to be 1 MPa. Hence, the densty data of a sample at 1 MPa were frst used for the optmzaton of the a parameters. Then, all the densty data, presented for the studed compounds n a prevous work 1 were employed for the determnaton of the b and c parameters. All the parameters were obtaned applyng the Marquardt algorthm for the mnmzaton of the objectve functon,.e., the standard devaton. The obtaned parameters of Eqs. () (4) and the correspondng devatons are gven n Table I. TABLE I. Fttng parameters of Eqs. () (4) Parameter n-hexane Toluene Dchloromethane a / kg m -3 791.8 18.499 1599.9 a 1 / kg m -3 K -1.3571.55689.13 a / kg m -3 K -.1439 1 -.6343 1-3.8367 1 - b / MPa 349.513 54.19 654.433 b 1 / MPa K -1 1.46894 1.9693.8478 b / MPa K -.15855 1 -.1 1 -.3968 1 - c.1845.15696.9779 c 1 / K -1.58799 1-3.3937 1-3.11415 1 - c / K -.8778 1-6.54981 1-6.1599 1-5 AAD / %.3.8.19 MD / %.191.8.18 Bas / %..3.1 σ / kg m -3.9.9.33 From the presented devaton values, t could be concluded that the proposed Eqs.(1) (4) ft well the denstes of the selected substances.
148 IVANIŠ et al. The sothermal compressblty, the sobarc thermal expansvty, the dfference between the specfc heat capactes at constant pressure and constant volume and the nternal pressure for all three examned compounds, at temperatures 88.15 413.15 K and pressures.1 6 MPa, are presented n Tables S-I S-III of the Supplementary materal to ths paper. In order to verfy the accuracy of the densty data modelng and the procedure of calculatng the derved propertes lsted n the prevous secton, the obtaned values of the sothermal compressblty and sobarc thermal expansvty were compared to those found n the lterature. The crtera for the evaluaton of the agreement were the same as n a prevous paper: 1 the absolute average percentage devaton (AAD), the percentage maxmum devaton (MD) and the average percentage devaton (Bas), but here, nstead of densty, the correspondng values of the sothermal compressblty and the sobarc thermal were used. n-hexane Data on n-hexane densty were modeled and the calculated values of the sothermal compressblty, gven n Table S-I, were compared to those publshed by Dardon et al. 3 The comparson was performed at temperatures 93.15 373.15 K and at pressures.1 6 MPa; the obtaned devatons were: AAD = = 3.64 %, MD = 5.67 % and Bas = 3.64 % (Fg. 1a). Amng at assessng the accuracy of the presented densty modelng, the sentropc compressblty was compared. Values of the speed of sound were retreved from the lterature 3 and the sentropc compressblty was calculated usng Eq. (17); the obtaned values were compared to the lterature data 3 at temperatures 93.15 373.15 K and pressures 5 6 MPa. The correspondng devatons were AAD =.1 %, MD =.9 % and Bas =.1 %; these devatons represent one more proof that the modelng procedure was properly selected. The calculated sothermal compressblty data were compared to those gven by Rendzo et al. 4 at several temperatures n the range 313.15 363.15 K and at pressures.1 6 MPa. Values of the crtera were AAD =.63 %, MD = 4.73 % and Bas =.56 %. Rendzo et al. 4 also reported equatons for the calculaton of sobarc thermal expansvty. Hence, ther data wth those reported n Table S-I were compared at temperatures 88.15 393.15 K and at pressures.1 6 MPa. The obtaned devatons were AAD =.99 %, MD = 3.73 % and Bas =.68 %, ndcatng very good agreement (shown n Fg. 1b). Tronkoso et al. 5 gave the sothermal compressblty of n-hexane n the somewhat narrower ranges of pressure and temperature than dd the aforementoned authors, and hence the comparson s realzed at temperatures 93.15
DERIVED VOLUMETRIC PROPERTIES OF n-hexane, TOLUENE AND DICHLOROMETHANE 149 413.15 K and at pressures.1 4 MPa. The devatons were AAD =.1 %, MD = 3.31 % and Bas =.1 %. (a) (b) Fg. 1. Comparsons of the calculated derved propertes wth lterature data for n-hexane: a) the sothermal compressblty wth Dardon et al. 3 at () 93.15, () 33.15, () 313.15, () 33.15, () 33.15, () 343.15, () 353.15, () 363.15 and () 373.15 K; b) the sobarc thermal expansvty wth Rendzo et al. 4 at () 88.15, () 33.15, () 318.15, () 333.15, () 343.15, () 353.15, () 363.15, () 373.15 and () 393.15 K. The values of the sobarc thermal expansvty presented n ths work were compared to those reported by Zhou et al. 6 at temperatures 93.15 413.15 K and at pressures.1 4 MPa and the obtaned devaton crtera were: AAD = 4.17 %, MD = 6.8 % and Bas = 3.86 %. Carderne et al. 7 gave the sobarc thermal expansvtes at atmospherc pressure and at temperatures 93.15 333.5 K. The results of the comparson of those data and the present ones were AAD =.6 %, MD = 3.71 % and Bas = = 1.69 %. Consderng the values of the devatons that the cted authors reported n ther papers, t could be concluded that the agreement of the present results wth those found n the lterature was acceptable. Toluene The sothermal compressblty data of toluene, gven n Table S-II, were compared to those reported by Chorazewsk et al. 8 The comparson was performed at temperatures 33.15 373.15 K and pressures.1 6 MPa and the obtaned results were: AAD = 1.19 %, MD = 1.97 % and Bas = 1.18 % (Fg. a). Chorazewsk et al. 8 gave equatons for the determnaton of the sobarc thermal expansvtes of toluene and hence, the data presented n Table S-II were compared to these values over wde ranges of temperature and pressure, 88.15 413.15 K and.1 6 MPa, respectvely. The obtaned devatons were AAD = =.45 %, MD = 5.18 % and Bas =.9 % (Fg. a).
143 IVANIŠ et al. (a) (b) Fg.. Comparsons of the calculated derved propertes wth lterature data for toluene: a) the sothermal compressblty wth Chorazewsk et al. 8 at () 33.15, () 33.15 and () 373.15 K, and b) the sobarc thermal expansvty wth Chorazewsk et al. 8 at () 88.15, () 33.15, () 318.15, () 333.15, () 343.15, () 353.15, () 363.15, () 373.15, () 393.15 and () 413.15 K. The values of the sothermal compressblty obtaned n ths paper were also compared to those gven by Zéberg-Mkkelsen et al. 9 at temperatures 33.15 333.15 K and at pressures.1 3 MPa. The devatons were rather low: AAD = =.64 %, MD = 1.48 % and Bas =. %. In addton, Zéberg-Mkkelsen et al. 9 reported the sobarc thermal expansvtes of toluene at the same temperatures and pressures as they dd for the sothermal compressblty. For ths property, the values of the approprate crtera were AAD =.71 %, MD =.39 % and Bas =.17 %. Easteal et al. 1 presented the values for the sothermal compressblty and the sobarc thermal expansvty of toluene under the same condtons. The comparson was performed at temperatures 88.15 33.15 K and pressures.1 5 MPa and the attaned results were: for the sothermal compressblty AAD =.48 %, MD = 4.1 % and Bas =.38 % and for the sobarc thermal expansvty AAD = 1.9 %, MD =.63 % and Bas =.1 %. The values of the devaton crtera obtaned for the sothermal compressblty and the sobarc thermal expansvty of toluene were suffcently low for t to be concluded that the results calculated n ths work agree well wth the lterature data. Dchloromethane Gonçalves et al. 11 recently presented results for the sothermal compressblty and the sobarc thermal expansvty for dchloromethane at temperatures 3 4 K and pressures.1 1 MPa. The applcaton of Eqs. (1) (11) enabled the calculaton of the values for the mentoned propertes at the same temperatures and at pressures up to 6 MPa. A comparson of the calculated data wth
DERIVED VOLUMETRIC PROPERTIES OF n-hexane, TOLUENE AND DICHLOROMETHANE 1431 lterature results 11 resulted n the followng: for the sothermal compressblty AAD was 1.81 %, MD was 5.4 % and Bas was.38 % (Fg. 3a) and for the sobarc thermal expansvty AAD = 1.16 %, MD = 3.53 % and Bas =.19 % (Fg. 3b). (a) (b) Fg. 3. Comparsons of the calculated derved propertes wth lterature data for dchloromethane: a) the sothermal compressblty wth Ferrera et al. 11 at () 3, () 3, () 34, () 36, () 38 and () 4 K, and b) the sobarc thermal expansvty wth Ferrera et al. 11 at () 3, () 3, () 34, () 36, () 38 and () 4 K. Easteal et al. 1 gave the values of the sothermal compressblty and the sobarc thermal expansvty of dchloromethane at two temperatures, 88.15 and 98.15 K, and at pressures.1 5 MPa. The sothermal values presented n Table S-III agree wth those taken from lterature 1 wth: AAD of 3. %, MD of 3.56 % and Bas of 3. %. When comparng the sobarc thermal expansvtes from Table S-III, the followng devatons were obtaned: AAD = 1.61 %, MD = 3.33% and Bas = 1.43 %. Although the mentoned values of the crtera for the comparson of both propertes for dchloromethane were slghtly hgher than those for toluene, they could stll be consdered as good. CONCLUSIONS The densty data of n-hexane, toluene and dchloromethane, reported n a prevous paper, 1 were ftted to a Tat-lke equaton; the obtaned parameters, and the results of assessment of the modelng, by the comparson wth lterature nformaton, are gven n ths work. The calculated values of the sothermal compressblty, the sobarc thermal expansvty, the dfference between specfc heat capacty at constant pressure and at constant volume as well as the nternal pressure for each of three nvestgated compounds, at temperatures 88.15 413.15 K and at pressures.1 6 MPa, are also presented heren. In order to assess the proposed modelng procedure, the obtaned values of the sothermal compres-
143 IVANIŠ et al. sblty and the sobarc thermal expansvty were compared to correspondng lterature data, and the good agreement was acheved. SUPPLEMENTARY MATERIAL The calculated values of the sothermal compressblty, κ T, the sobarc thermal expansvty, α p, the dfference between the specfc heat capactes at constant pressure and constant volume, C p C V, and the nternal pressure, p nt, for n-hexane, toluene and dchloromethane, at temperatures 88.15 413.15 K and pressures.1 6 MPa, are avalable electroncally from http://www.shd.org.rs/jscs/, or from the correspondng author on request. Acknowledgements. The authors gratefully acknowledge the fnancal support receved from the Research Fund of Mnstry of Educaton, Scence and Technologcal Development (Project No. 1763) of the Republc of Serba, and the Faculty of Technology and Metallurgy, Unversty of Belgrade. ИЗВОД МОДЕЛОВАЊЕ ГУСТИНЕ И ПРОРАЧУН ИЗВЕДЕНИХ ВОЛУМЕТРИЈСКИХ СВОЈСТАВА НА ПРИТИСЦИМА ОД,1 ДО 6 MPa И НА ТЕМПЕРАТУРАМА ОД 88,15 ДО 413,15 K ГОРИЦА Р. ИВАНИШ, АЛЕКСАНДАР Ж. ТАСИЋ, ИВОНА Р. РАДОВИЋ, БОЈАН Д. ЂОРЂЕВИЋ, СЛОБОДАН П. ШЕРБАНОВИЋ и МИРЈАНА Љ. КИЈЕВЧАНИН Технолошко-металуршки факултет, Универзитет у Београду, Карнегијева 4, 111 Београд Густине n-хексана, толуена и дихлорметана на температурама 88,15 413,15 K и притисцима,1 6 MPa су моделоване прилагођеном Тејтовом једначином. У овом раду су наведене одговарајуће једначине за рачунање изотермске стишљивости, изобарске топлотне експанзивности, разлике између специфичног топлотног капацитета при сталном притиску и оног при сталној запремини и унутрашњег притиска у читавом поменутом опсегу притисака и температура, као и вредности наведених изведених величина. У циљу процене предложене методе моделовања поређене су добијене вредности коефицијента изотермске стишљивости и коефицијента изобарског топлотног ширења са онима из литературе. Просечно апсолутно процентуално одступање за изотермску стишљивост је било: за n-хексан,1 3,64 %, за толуен,64,48 % и за дихлорметан 1,81 3, %; a за изобарско топлотну експанзивност: за n-хексан 1,31 4,17 %, за толуен,71,45 % и за дихлорметан 1,16 1,61 %. Поређењем добијених одступања са оним која се могу наћи у литератури долази се до закључка да се представљени резултати добро слажу са литературним. (Примљено 18. марта, ревидирано 3. јуна, прихваћено 1. јула 15) REFERENCES 1. G. R. Ivanš, A. Ž. Tasć, I. R. Radovć, B. D. Djordjevć, S. P. Šerbanovć, M. Lj. Kjevčann, J. Serb. Chem. Soc. 8 (15) 173. J. Safarov, F. Mllero, R. Festel, A. Hentz, E. Hassel, Ocean Sc. 5 (9) 35 3. J. L. Dardon, B. Lagourette, J.-P. E. Groler, Int. J. Thermophys. 19 (1998) 145 4. S. L. Randzo, J.-P. E. Groler, J. R. Qunt, D. J. Eatough, E. A. Lews, L. D. Hansen, Int. J. Thermophys. 15 (1994) 415 5. J. Troncoso, D. Bessères, C. A. Cerderña, E. Carballo, L. Romaní, Flud Phase Equlb. 8 (3) 141 6. J. Zhou, R. Zhu, H. Xu, Y. Tan, J. Chem. Thermodyn. 4 (1) 149
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