J. Nig. Soc. Phys. Sci. 4 (2022) 711 Journal of the Nigerian Society of Physical Sciences Estimation of Critical and Thermophysical Properties of Saturated Cyclic Alkanes by Group Contributions C. Otobrise∗, G. A. Orotomah Department of Chemistry, Delta State University, P.M.B. 1, Abraka, Nigeria. Abstract Group Contribution Methods (GCMs) of Marrero & Gani (M & G), Constantinou & Gani (C & G) and Lydersen (LYD) were employed in the prediction of some critical and thermophysical properties(TPPs), namely; critical temperature (Tc ), critical pressure (Pc ), critical volume (Vc ), boiling temperature (Tb ) and melting temperature (Tm ), for various cycloalkanes. The predicted properties were compared with available experimental data. Experimental data for Tm were unfortunately very scanty in the open literature; no comparison was done to appraise any of the methods. For Tc, LYD, C & G and M & G gave average relative deviation (ARD) values of 0.02%, 1.57% and 14.64%, respectively. In the case of Vc, ARD values for LYD, C & G and M & G are 2.6 %, 23.97 %, -4.53 %, respectively. Predicted Pc values using the methods gave ARD of -31.55 %, -1.49 %, -277.98 %, respectively. C & G and M & G recorded ARD values of 0.63 % and 10.30 % for Tb, respectively. DOI:10.46481/jnsps.2022.711 Keywords: Group Contributions, Critical Properties, Thermophysical Properties, Cycloalkanes. Article History : Received: 16 March 2022 Received in revised form: 17 July 2022 Accepted for publication: 27 July 2022 Published: 20 August 2022 c© 2022 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license (https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Communicated by: Emmanuel Etim 1. Introduction Cycloalkanes have numerous applications. They are widely used in the pharmaceutical industry as organic solvent for the production of drugs. They are largely employed in the petroleum industry for the production of fuels. In modern chemical processes hundreds of thousands of components are used. Chemical processing units are designed on the basis of a set of physical and thermodynamic properties of compounds [1]. It is not feasible to measure them as the need arises due thermal instability in most cases, estimation/prediction meth- ods are generally employed in such situations. Different esti- ∗Corresponding author tel. no: +2348038930023 Email address: otobrisec@delsu.edu.ng ( C. Otobrise ) mation methods have been developed over the years to provide data that cannot be sourced readily in open literature [2]. Such models are validated by comparing available experimental data with predicted values [3]. Predictive methods can replace mea- surements/experiments if they provide sufficiently good estima- tions. More chemists now employ computational models to elu- cidate details of compounds observed in the laboratory [4]. The estimated properties cannot be as precise as experimental mea- surements, but for many purposes the quality of the estimated properties is sufficient [5]. Estimation methods depending on the required input data have been divided into Quantity-property-property-relationship (QPPR) or Quantity-structure-property-relationship (QSPR) [6]. QPPR methods are input data intensive. QSPR methods 1 C. Otobrise & G. A. Orotomah / J. Nig. Soc. Phys. Sci. 4 (2022) 711 2 need only knowledge of the chemical structure of a compound to predict the estimated property [6, 7]. GCMs are presented as empirical QSPR approaches. They have been found to be very suitable and easy to employ for the prediction of a large num- ber of pure components properties [8], as well as physicochem- ical properties of thousands of compounds [1]. A lot of GCMs have been designed already with varying predictive capabili- ties. They have been used to predict critical properties [9, 10], state equation parameters [11, 12], acentric factor [2, 9], activ- ity coefficients [13], normal boiling temperature [1, 14], liquid density [10, 15, 16], and flash temperatures [17]. The purpose of this work is to predict with good accuracy, some critical and TPPs of saturated cyclic hydrocarbons. Such data are signifi- cant in the design of process plants. 2. Numerical Methods The GCMs of LYD, C & G and M & G were employed in the determination of Tc, Pc, Vc, Tb and Tm, for C3 − C13 cycloalkanes. This range of cycloalkanes with or without side chains are 49 common and experimental data can be readily sourced for a good number of them. The rest are large, ther- mally labile compounds with paucity of experimental data to compare with the results of predictive models. 2.1. The Method of Lydersen The LYD estimates Tc, Pc, and Vc. The method is the proto- type for many new models. It employs structural combinations and the relations are shown in equations (1), (2), (3) below: Tc = Tb 0.567 + Σ∆T − (Σ∆T ) 2 (1) Pc = M (0.34 + Σ∆P) 2 (2) Vc = 40 + Σ∆V (3) M represents the molar mass of the compound, ∆T , ∆P, ∆V are the group contributions for different groups of atoms in the molecule. Tc and Tc are in kelvin (K), Pc is measured in at- mospheres (atm), and Vc is in Cubic centimeters per gram mole (cm3/gm). Contributions of atoms and molecules as stipulated by LYD are accessible [18]. 2.2. The Method of Constantinou & Gani This method makes use of first and second order level group contributions. The second level involves groups that per- mit a better description of proximity effects and differentiation among isomers. Property estimation in this model takes the form of equation (4) below: f (x) = Σi NiCi + WΣ j M j D j (4) Ci in equation (4) is the contribution of the first order group type-i which occurs Ni times and D j is the contribution of the second order group type- j with M j occurrence in a compound. For Tc, f (x) = exp(Tc/tc0). Hence, for Tc equation (4) becomes exp(Tc/tc0) = ( Σi Nitc1i + Σ j M jtc2 j ) (5) This can also be expressed as: Tc = tc0 ln ( Σi Nitc1i + Σ j M jtc2 j ) (6) tc0 is a constant (universal adjustable parameter) with the value 181.128 K, tc1i represents group contribution of the first order group of type-i, tc2 j represents group contribution of the second order group of type- j. For Pc f (x) = (Pc − Pci) −0.5 −Pc2. Hence, Pc takes the form (Pc − Pci) −0.5 − Pc2 = Σi Ni pc1i + Σ j M j pc2 j (7) Pc1 and Pc2 are universal constants (adjustable parameters) with values 1.3705 bar and 0.100220 82 bar−0.5, respectively, Pc1i represents group contribution of the first order of type-i, Pc2 j represents group contribution of the second order of type- j. For Vc, f (x) = Vc − Vc0. Therefore, we can write: Vc − Vc0 = Σi NiVc1i + Σ j M jVc2 j (8) Vc0 is a universal constant (adjustable parameter) with a value of -0.004350 m3/Kmol, Vc1i represents group contribution of the first order of type-i, Vc2 j represents group contribution of the second order of type- j. For Tm, f (x) = exp (Tm/tm0). Equation (4) will take the form of equation (9) for Tm exp (Tm/tm0) = Σi Nitc1i + Σ j M jtc2 j (9) This can be written as: Tm = tm0 ln ( Σi Nitm1i + Σ j M jtm2 j ) (10) tm0 is a constant (adjustable parameter) with value 102.425 k. tm1i represents group contribution of first order of type-i, tm2 j represents group contribution of the second order of type- j. For Tb, f (x) = exp (Tb/tb0). This can be expressed as: exp (Tb/tb0) = Σi Nitb1i + Σ j M jtb2 j (11) Tb = tb0 ln ( Σi Nitb1i + Σ j M jbm2 j ) (12) tb0 is a constant (adjustable parameter) with value 204.359 k. tb1i represents group contribution of first order of type-i, tb2 j represents group contribution of the second order of type- j. A table of the contributions for various atoms or groups as pro- posed by C & G can be readily sourced [2]. 2.3. The Method of Marrero & Gani The above method considers the molecular structure of a compound to be collection of three types of groups. The prop- erty estimation method takes the form of equation (13). f (x) = Σi NiCi + ωΣ j M j D j + zΣkOk Ek (13) 2 C. Otobrise & G. A. Orotomah / J. Nig. Soc. Phys. Sci. 4 (2022) 711 3 Table 1: Critical properties predicted by the GCMs Cycloalkanes Nc Tc/K Pc/bar Vc/cm3/gmol LYD C & G M & G LYD C & G M & G LYD C & G M & G Cyclopropane 3 397.04 379.91 320.46 66.14 55.75 611.42 173.5 162.92 176.79 Cyclobutane 4 463.25 459.93 386.98 64.48 49.85 346.54 218 217.82 233.07 Methylene cyclobutane 5 499.7 498.61 442.24 59.65 42.78 268 254.5 171.26 268.19 Spiropentane 5 511.81 480.44 388.3 60.97 58.35 331.72 249 217.79 249.08 Cyclopentane 5 514.52 510.83 438.58 59.89 43.85 223.94 262.5 273.58 289.35 Cyclohexane 6 553.59 558.22 480.74 54.71 37.74 157.34 307 331.85 345.63 Methyl cyclopentane 6 535.76 533.33 469.91 50.94 38.51 160.16 319 228.48 340.64 Cycloheptane 7 602.57 578.95 516.38 49.77 38.4 117.19 351.5 378.92 401.91 1,1-dimethyl cyclopentane 7 559.89 555.95 504.53 46.35 35.06 122.51 359 232.8 385.8 Cis-1,2 dimethyl cyclopentane 7 565.19 553.33 512.09 43.78 34.89 113.27 375.5 182.98 388.18 Trans-1,2 dimethyl cyclopentane 7 553.57 562.24 512.09 43.78 34.82 113.27 375.5 182.98 388.18 Trans-1,3 dimethyl 7 553.36 553.33 497.49 43.78 34.11 120.77 375.5 183.38 391.93 cyclopentane Ethyl cyclopentane 7 569.54 563.89 508.34 44.22 34.15 120.1 374 284.24 395.21 Methyl cyclohexane 7 571.35 575.78 507.13 46.64 33.48 118.96 363.5 286.74 396.92 Cyclooctane 8 642.25 603.14 547.26 45.36 30.69 91.12 396 441.73 458.19 1,1-dimethyl cyclohexane 8 598.11 593.9 536.92 42.51 30.69 94.67 403.5 291.06 442.08 Cis-1,2-dimethyl cyclohexane 8 601.16 599.02 543.51 40.35 30.5 88.49 420 241.24 444.46 Trans-1,2-dimethyl cyclohexane 8 592.19 599.02 543.51 40.35 30.5 88.49 420 241.24 444.46 Cis-1,3-dimethyl cyclohexane 8 586.97 591.78 530.82 40.35 29.92 93.52 420 241.64 448.21 Trans-1,3 dimethyl- cyclohexane 8 592.94 591.78 530.82 40.35 29.92 93.52 420 241.64 448.21 Cis-1,4-dimethyl cyclohexane 8 593.69 591.78 530.82 40.35 29.92 93.52 420 241.64 448.21 Trans-1,4-dimethyl cyclohexane 8 585.47 591.78 530.82 40.35 29.92 93.52 420 241.64 448.21 Ethyl cyclohexane 8 602.71 600.36 540.24 40.72 29.95 93.07 418.5 342.5 451.49 Propyl cyclopentane 8 0.00 590.03 540.24 38.82 30.52 93.07 429 340 451.49 Cis-octahydro-1h- 9 688.01 640.73 517.9 43.59 42.46 89.62 413.5 386.67 433.93 indene Trans-octahydro-1h- 9 677.04 640.73 517.9 43.59 42.46 89.62 413.5 386.67 433.93 indene Butyl cyclopentane 9 0.00 612.86 568.27 34.47 27.46 74.62 484 395.76 507.77 Propyl cyclohexane 9 624.61 622.01 568.27 35.98 26.98 74.62 473.5 398.26 507.77 Table 2: Predicted critical properties continued Cycloalkanes Nc Tc/K Pc/bar Vc/cm3/gmol LYD C & G M & G LYD C & G M & G LYD C & G M & G 1,1-bicyclopentyl 10 0.00 642.13 571.77 37.01 29.62 72.19 488 446.3 533.19 Cis- decahydronaphthalene 10 718.14 650.8 548.59 39.93 36.64 87.42 458 444.93 490.21 Trans- decahydronaphthalene 10 705.81 650.8 548.59 39.93 36.64 87.42 458 444.93 490.21 Butyl cyclohexane 10 646.31 641.34 593.26 32.15 24.96 61.47 482.5 454.02 564.05 1,4-diethyl cyclohexane 10 0.00 634.53 587.52 31.92 24.43 62.49 530 353.16 557.35 1,1-dimethylethyl cyclohexane 10 642.12 629.19 581.52 39.98 24.64 63.03 514.5 296.3 346 Decahydro-1- 11 0.00 667.78 567.33 32.74 25.76 60.29 544.5 406.2 584.48 methylnaphthalene Decahydro-2- methylnaphthalene 11 0.00 667.78 567.33 32.74 25.76 60.29 544.5 406.2 584.48 Ethyl octahydro-1h- indene 11 0.00 660.72 590.75 31.53 26.21 60.67 555 697.39 582.77 Decahydro dimethyl naphthalene 12 0.00 677.58 615.92 29.3 23.38 49.44 601 361.1 632.02 Ethyl decahydronaphthalene 12 0.00 682.97 613.54 29.49 23.4 51.17 599.5 461.97 639.05 1, ethyl decahydronaphthalene 12 0.00 682.97 613.54 29.49 23.4 51.17 599.5 461.97 639.05 2, ethyl decahydronaphthalene 12 0.00 682.97 613.54 29.49 23.4 51.17 599.5 461.96 639.05 Octahydro (1- methylethyl)-1h- indene 12 0.00 670.9 608.29 34.43 23.76 51.92 606 358.6 632.35 Decahydro (1- methylethyl) naphthalene 13 0.00 692 629.48 31.92 21.35 44.52 650.5 416.86 688.63 Decahydro-1-(1- methylethyl) naphthalene 13 0.00 692 629.48 31.92 21.35 44.52 650.5 416.86 688.63 Decahydro-1-propyl naphthalene 13 0.00 682.97 634.28 26.75 21.37 43.94 654.5 517.73 695.33 2 methyl-1,1- bicyclohexyl 13 0.00 696.99 633.58 28.56 21.37 43.72 633.5 517.73 697.04 1,1-methylene biscyclohexane 13 0.00 701.84 638.98 28.73 19.61 43.37 632 618.59 702.03 Heptyl cyclohexane 13 701.58 689.38 655.24 24.2 26.98 38.68 647.5 621.3 732.89 In equation (13), Ci is the contribution of the first order group of type-i that occurs Ni times, D j is the contribution of the second order group of type- j that occurs M j times, the Ek is the contri- bution of the third order group of type-k that has Ok occurrence in a compound. In the first level of estimation, the constants ω and z are assigned zero values because only first order groups are employed. In the second order level, the constants ω and z are assigned unit and zero values, respectively, because only 3 C. Otobrise & G. A. Orotomah / J. Nig. Soc. Phys. Sci. 4 (2022) 711 4 Table 3: Thermophysical properties predicted by the GCMs Cycloalkanes Nc Tm/K Tb/K C & G M & G C & G M & G Cyclopropane 3 145.79 -41.54 240.37 169.55 Cyclobutane 4 133.96 0.88 285.65 233.57 methylene cyclobutane 5 155.49 61.81 322.07 279.07 Spiropentane 5 184.01 4.59 291.06 228.28 Cyclopentane 5 170.92 33.78 320.75 283.23 Cyclohexane 6 200.95 60.67 356.78 323.81 methyl cyclopentane 6 168.87 54.36 343.67 308.2 Cycloheptane 7 218.76 83.4 391.93 358.11 1,1-dimethyl cyclopentane 7 198.67 130.45 363.88 334.1 cis-1,2 dimethyl cyclopentane 7 175.27 92.54 378.87 346.04 trans-1,2 dimethyl cyclopentane 7 175.27 92.54 378.87 346.04 trans-1,3 dimethyl cyclopentane 7 166.79 72.41 364.27 330.65 ethyl cyclopentane 7 185.63 67.43 376.04 344.97 methyl cyclohexane 7 199.43 78.01 376.16 344.81 Cyclooctane 8 204.96 103.09 408.47 387.83 1,1-dimethyl cyclohexane 8 222.33 145.01 393.53 366.97 cis-1,2-dimethyl cyclohexane 8 204.22 111.12 406.55 377.3 trans-1,2-dimethyl cyclohexane 8 204.22 111.12 406.55 377.3 cis-1,3-dimethyl cyclohexane 8 197.89 93.52 393.87 363.99 trans-1,3 dimethyl-cyclohexane 8 197.89 93.52 393.87 363.99 cis-1,4-dimethyl cyclohexane 8 197.89 93.52 393.87 363.99 trans-1,4-dimethyl cyclohexane 8 197.89 93.52 393.87 363.99 ethyl cyclohexane 8 212.12 89.21 404.08 376.37 propyl cyclopentane 8 190.05 89.21 403.98 376.37 cis-octahydro-1H-indene 9 254.65 87.62 408.76 352.06 trans-octahydro-1H-indene 9 254.65 87.62 408.76 352.06 butyl cyclopentane 9 212.66 108.19 428.55 403.89 propyl cyclohexane 9 223.42 108.19 428.65 403.89 Table 4: Predicted thermophysical properties continued Cycloalkanes Nc Tm/K Tb/K C & G M & G C & G M & G 1,1-bicyclopentyl 10 230.3 34.78 448.35 409.52 cis-decahydronaphthalene 10 269.01 106.79 432.89 382.54 trans-decahydronaphthalene 10 269.01 106.79 432.89 382.54 butyl cyclohexane 10 233.59 125 450.57 428.37 1,4-diethyl cyclohexane 10 222.2 113.11 442.47 418.87 1,1-dimethylethyl cyclohexane 10 210.17 155.69 433.76 411.37 decahydro-1-methylnaphthalene 11 246.85 55.22 461.11 424.02 decahydro-2-methylnaphthalene 11 246.85 55.22 461.11 424.02 ethyl octahydro-1H-indene 11 238.38 42.8 461.02 424.14 decahydro dimethyl naphthalene 12 245.88 93.21 468.97 447.27 ethyl decahydronaphthalene 12 255.02 68.22 479.95 446.6 1, ethyl decahydronaphthalene 12 255.02 68.22 479.95 446.6 2, ethyl decahydronaphthalene 12 255.02 68.22 479.95 446.6 octahydro (1-methylethyl)-1H-indene 12 237.33 50.41 472.88 437.85 decahydro (1-methylethyl) naphthalene 13 254.12 74.65 490.79 459.03 decahydro-1-(1-methylethyl) naphthalene 13 254.12 74.65 490.79 459.03 decahydro-1-propyl naphthalene 13 262.59 89.89 497.21 466.99 2 methyl-1,1-bicyclohexyl 13 262.59 99.02 497.21 466.9 1,1-methylene biscyclohexane 13 270.4 103.71 503.43 474.68 heptyl cyclohexane 13 259.15 166.29 505 488.9 4 C. Otobrise & G. A. Orotomah / J. Nig. Soc. Phys. Sci. 4 (2022) 711 5 Table 5: Experimental data for the various properties used in the comparison [19]. Cycloalkanes Nc Tc/K Pc/bar Vc/cm3/gmol Tb/K (M) g/mol Cyclopropane 3 397.91 55.77 162.8 240.37 42.081 Cyclobutane 4 459.93 49.85 210 285.66 56.107 methylene cyclobutane 5 NA NA NA NA 68.118 Spiropentane 5 499.74 52.13 236.5 312.19 68.118 Cyclopentane 5 511.76 45.02 258.3 322.4 70.134 Cyclohexane 6 553.54 40.75 307.9 353.87 84.161 methyl cyclopentane 6 532.79 37.85 318.9 344.96 84.162 Cycloheptane 7 604.32 38.4 359 391.94 98.188 1,1-dimethyl cyclopen- tane 7 547 34.45 360 361 98.188 cis-1,2 dimethyl cy- clopentane 7 565.15 34.45 370 372.68 98.188 trans-1,2 dimethyl cy- clopentane 7 553.15 34.45 360 365.02 98.188 trans-1,3 dimethyl cy- clopentane 7 553 34.45 360 364.88 98.188 ethyl cyclopentane 7 569.52 33.98 374.5 376.62 98.189 methyl cyclohexane 7 572.19 34.71 368 374.08 98.186 Cyclooctane 8 647.2 35.5 410 424.3 112.21 1,1-dimethyl cyclohex- ane 8 591.15 29.38 450 392.7 112.22 cis-1,2-dimethyl cyclo- hexane 8 606.15 29.38 460 402.94 112.21 trans-1,2-dimethyl cyclohexane 8 596.15 29.38 460 396.58 112.21 cis-1,3-dimethyl cyclo- hexane 8 591.15 29.38 450 393.24 112.21 trans-1,3 dimethyl- cyclohexane 8 598 29.38 460 397.61 112.21 cis-1,4-dimethyl cyclo- hexane 8 598.15 29.38 460 397.47 112.21 trans-1,4-dimethyl cyclohexane 8 590.15 29.38 450 392.51 112.21 ethyl cyclohexane 8 609.15 30.4 450 404.95 112.21 propyl cyclopentane 8 NA NA NA NA 112.21 cis-octahydro-1H- indene 9 NA NA NA NA 124.22 NA = Not available. the first and the second order groups are involved while the third level both ω and z are set to unity values. The left hand side of equation (13) is a simple function f (x) of the target property “x”. For Tc, f (x) = exp (TC/TC0). Hence, exp (TC/TC0) = ( Σi Nitc1i + Σ j M jTc2 j + ΣkOkTc3k ) (14) TC = TC0 ln ( Σi NiTc1i + Σ j M jTc2 j + ΣkOkTc3k ) . (15) Tc0 is a constant (adjustable parameter) with value 231.239 k. Tc1i represents group contribution of first order of type-i, Tc2 j represents group contribution of the second order of type- j, Tc3k represents group contribution of third order of type-k. For Pc, f (x) = (Pc − Pc1) −0.5 − Pc2. Therefore (Pc − Pc1) −0.5 −Pc2 = ( Σi Ni Pc1i + Σ j M j Pc2 j + ΣkOk Pc3k ) (16) Pc = 1/ ( Σi Ni Pc1i + Σ j M j Pc2 j + ΣkOk Pc3k + Pc2 ) +Pc1(17) Figure 1: Plot of experimental Tc versus Tcs obtained from the GCMs. Pc1 and Pc2 are both universal constants (adjustable parameters) with values 5.9827 bar and 0.108998 bar-0.5, respectively. Pc1i represents group contribution of first order of type-i, Pc2 j rep- resents group contribution of second order of type- j and Pc3k 5 C. Otobrise & G. A. Orotomah / J. Nig. Soc. Phys. Sci. 4 (2022) 711 6 Table 6: Experimental properties used in the comparison continued [19]. Cycloalkanes Nc Tc/K Pc/bar Vc/cm3/gmol Tb/K (M) g/mol trans-octahydro-1H- indene 9 NA NA NA NA 124.22 butyl cyclopentane 9 625.05 27.64 480.5 429.76 126.24 propyl cyclohexane 9 639.15 28.87 477 429.9 126.24 1,1-bicyclopentyl 10 NA NA NA NA 138.25 cis- decahydronaphthalene 10 702.25 32.42 480 468.97 138.25 trans- decahydronaphthalene 10 687.05 28.37 480 460.46 138.25 butyl cyclohexane 10 667 25.7 574 454.13 140.27 1,4-diethyl cyclohexane 10 NA NA NA NA 140.27 1,1-dimethylethyl cyclo- hexane 10 NA NA NA NA 152.28 decahydro-1- methylnaphthalene 11 NA NA NA NA 152.28 decahydro-2- methylnaphthalene 11 NA NA NA NA 152.28 ethyl octahydro-1H- indene 11 NA NA NA NA 152.28 decahydro dimethyl naphthalene 12 NA NA NA NA 166.3 ethyl decahydronaphtha- lene 12 NA NA NA NA 166.3 1, ethyl decahydronaph- thalene 12 NA NA NA NA 166.3 2, ethyl decahydronaph- thalene 12 NA NA NA NA 166.3 octahydro (1- methylethyl)-1H- indene decahydro (1-methylethyl) 12 NA NA NA NA 204.35 naphthalene decahydro- 1-(1-methylethyl) 13 NA NA NA NA 208.38 naphthalene 13 NA NA NA NA 208.38 decahydro-1-propyl naphthalene 13 NA NA NA NA 208.38 2 methyl-1,1- bicyclohexyl 13 NA NA NA NA 180.33 1,1-methylene biscyclo- hexane 13 NA NA NA NA 180.33 heptyl cyclohexane 13 NA NA NA NA 182.35 NA = Not available. Figure 2: Plot of experimental Pc versus Pcs obtained from the GCMs. Figure 3: Plot of experimental Vc versus Vcs obtained from the GCMs. 6 C. Otobrise & G. A. Orotomah / J. Nig. Soc. Phys. Sci. 4 (2022) 711 7 Table 7: Deviations of predicted properties from experimental data Cycloalkanes Nc Tc (% Dev.) Pc (% Dev.) Vc (% Dev.) Tb (% Dev.) LYD C & G M & G LYD C & G M & G LYD C & G M & G C & G M & G Cyclopropane 3 0.22 4.52 24.17 - 0.04 - -6.57 -0.07 -8.59 0.00 29.4 Cyclobutane 4 - 0.00 18.85 -18.59 0.00 -996 3 -3.81 -3.72 - 0.00 6 18.2 methylene cyclobutane 5 -0.72 - - -29.35 - 3 - 595. -1 - -10.99 - -3 Spiropentane 5 - 3.86 28.7 - - -7 -5.29 7.91 -5.32 6.77 26.8 Cyclopentane 5 -2.41 0.18 16.69 -16.96 11.93 2.6 -536 3 -1.63 -5.92 - 0.51 12.1 8 Cyclohexane 6 -0.54 -0.85 15.14 -33.03 7.39 -397.4 4 0.29 -7.78 -12.02 -0.82 5 8.50 methyl cyclopentane 6 -0.01 -0.1 13.38 -34.58 -1.74 286. - 3 1 -0.03 28.35 -6.82 12.25 0.37 10.6 Cycloheptane 7 0.56 0.29 4.2 17.03 -34.58 0.00 323. - 2 1 2.09 -5.55 - 0.00 8.63 6 1,1-dimethyl cyclopentane 7 - -1.64 8.42 -29.61 -1.77 5 205. - 1 0.28 35.33 -7.17 11.95 -0.88 7.45 cis-1,2 dimethyl 7 -2.36 2.09 10.36 -34.54 -1.28 255. - 7 6 -1.49 50.55 -4.91 -1.66 7.15 cyclopentane trans-1,2 dimethyl 7 -0.01 -1.64 8.02 -27.08 -1.07 -228.2 8 -4.31 49.17 -7.83 -3.79 5.2 trans-1,3 cyclopentane dimethyl 7 -0.08 -0.06 11.16 -27.08 0.99 1 - 228. 8 -4.31 49.06 -8.87 0.17 9.38 cyclopentane ethyl cyclopentane 7 0.06 0.00 0.99 12.03 -27.08 -0.5 - 1 250. 5 0.13 24.1 -5.53 0.15 8.4 methyl cyclohexane 7 0.15 -0.63 12.83 -30.14 3.54 253. - 8 4 1.22 22.08 -7.86 -0.56 7.83 Cyclooctane 8 0.77 6.88 18.26 -34.37 13.55 5 242. - 7 3.41 -7.74 - 3.73 8.6 1,1-dimethyl cyclohexane 8 - -0.47 10.1 -27.77 -4.46 156. - 2 6 10.33 35.32 11.75 1.76 -0.21 6.55 cis-1,2-dimethyl 8 0.18 0.82 1.18 11.53 -44.69 -3.81 222. - 8 2 8.70 47.56 3.38 -0.9 6.36 cyclohexane trans-1,2-dimethyl 8 0.66 -0.48 9.69 -37.34 -3.81 201. - 4 2 8.70 47.56 3.38 -2.51 4.86 cyclohexane cis-1,3-dimethyl 8 0.71 -0.11 11.37 -37.34 -1.84 -201 2 6.67 46.3 0.4 -0.16 7.44 cyclohexane trans-1,3 dimethyl- 8 0.85 1.04 12.66 -37.34 -1.84 -218 3 8.70 47.47 2.56 0.94 8.45 cyclohexane cis-1,4-dimethyl 8 0.75 1.06 12.68 -37.34 -1.84 -218.1 3 8.70 47.47 2.56 0.91 8.42 cyclohexane trans-1,4-dimethyl 8 0.79 -0.28 11.18 -37.34 -1.84 -218.1 3 6.67 46.3 0.4 -0.35 7.27 cyclohexane ethyl cyclohexane 8 1.06 1.44 12.76 -37.34 1.48 1 - 218. 3 7.00 23.89 -0.33 0.21 7.06 propyl cyclopentane 8 - - - -39.95 - 206. - 1 -1 - - - - cis-octahydro-1H-indene 9 - - - - - -6 - - - - - trans-octahydro-1H-indene 9 - - - - - - - - - - - butyl cyclopentane 9 - 1.95 9.99 - 0.65 - -0.73 17.64 -5.68 0.28 6.02 propyl cyclohexane 9 2.28 2.68 12.47 -24.71 6.55 -169 6 0.73 16.51 -6.45 0.29 6.05 1,1-bicyclopentyl 10 - - - -24.63 - 4 158. - -4 - - - - cis-decahydronaphthalene 10 - 7.33 28.01 - - -7 4.58 7.31 -2.13 7.69 18.4 trans-decahydronaphthalene 10 -2.26 5.28 25.24 -23.16 -13.02 -169 6 4.58 7.31 -2.13 5.99 16.9 3 butyl cyclohexane 10 2.73 3.10 3.85 12.43 -25.1 40.75 2.88 29.15 208. - 4 1 15.94 20.9 1.73 0.78 5.67 2 Figure 4: Plot of experimental Tb versus Tbs obtained from the GCMs. represents group contribution of third order of type-k. For Vc, f (x) = VC − Vc0. Consequently, VC − Vc0 = ( Σi NiVc1i + Σ j M jVc2 j + ΣkOkVc3k ) (18) VC = 1/ ( Σi NiVc1i + Σ j M jVc2 j + ΣkOkVc3k ) + Vc0 (19) Vc0 is a universal constant (adjustable parameter) with a value 7.95 cm3/mol, Vc1i represents group contribution of first order of type-i, Vc2 j represents group contribution of second order of type- j and Vc3k represents group contribution of third order of type-k. For Tm, f (x) = exp (Tm/Tm0). Therefore, exp (Tm/Tm0) = ( Σi NiTm1i + Σ j M jTm2 j + ΣkOkTm3k ) (20) Tm = Tm0 ln ( Σi NiTm1i + Σ j M jTm2 j + ΣkOkTm3k ) . (21) Tm0 is a constant (adjustable parameter) with value 147.450 k. Tm1i represents group contribution of first order of type-i, Tm2 j represents group contribution of the second order of type- j, Tm3k represents group contribution of third order of type-k. For Tb, f (x) = exp (Tb/Tb0). Hence, exp (Tb/Tb0) = ( Σi NiTb1i + Σ j M jTb2 j + ΣkOkTb3k ) (22) Tb = Tb0 ln ( Σi NiTb1i + Σ j M jTb2 j + ΣkOkTb3k ) . (23) Tb0 is a constant (adjustable parameter) with value 222.543 k. Tb1i represents group contribution of first order of type-i, Tb2 j represents group contribution of the second order of type- j, Tb3k represents group contribution of third order of type-k. A table of the contributions for various atoms or groups as proposed by M & G is readily available [1]. 3. Results and Discussion Three GCMs were employed in the prediction of the critical properties. TPPs were estimated by C & G and M & G. Tables 1 and 2 show the predicted Tc, Pc and Vc values. Tables 3 and 4 contain predicted Tb and Tm for the cycloalkanes. Experimental data for the various TPPs used in the comparison are presented on tables 5 and 6. LYD required molecular weight (M) and 7 C. Otobrise & G. A. Orotomah / J. Nig. Soc. Phys. Sci. 4 (2022) 711 8 T b as input parameters, they are also shown on tables 5 and 6 Where experimental Tb was not found in the literature for a par- ticular cycloalkane, Tb predicted by the method of C & G was utilized. Deviations of the predicted properties from available experimental data were calculated using the equations below: Deviation% = Exp (data) − Pred (data) Exp (data) × 100. Average Relative Deviation = ΣDeviation N Table 7 shows the deviations of predicted critical properties and TPPs from available experimental data. The experimental data were obtained from the Handbook of Chemical Compound Data for Process Safety [19]. LYD provided better results for Tc prediction with Average Relative Deviation of 0.02 %. M & G with ARD of 14.64 % under predicted Tc for all the compounds. C & G with ARD of 1.57 %, like Lydersen’s method, gave results comparable with experimental data. Figure 1 is a comparative plot of experimen- tal Tc and the Tcs obtained from the GCMs. As the number of carbon atoms increased, the predicted Pc decreased for the three methods. This trend is typical of organic compounds. LYD and M & G predicted Pc values that were similar. The latter with ARD of −1.49 % gave a better predic- tion of Pc. C & G with an ARD of −31.55 % under predicted Pc as shown on figure 2. C & G under predicted Vc for most of the compounds as can be seen on figure 3. As the number of carbon atoms increased the accuracy of the predicted Vc by LYD declined. LYD and M & G proved more efficient in this particular critical property as they gave values close to available experimental data with ARD of 2.6 % and −4.35 %, respectively. ARD for C & G was 23.97 %. The method of C & G with ARD of 0.63 % proved a more reliable method for the prediction of Tb. It was able to differen- tiate between structural isomers having different boiling points in most cases. 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