Simple estimations of the speed of sound in ionic liquids, with and without any physical property data available

Fluid Phase Equilibria 503 (2020) 112291

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Simple estimations of the speed of sound in ionic liquids, with and without any physical property data available Reza Haghbakhsh, Simin Keshtkari, Sona Raeissi* School of Chemical and Petroleum Engineering, Shiraz University, Shiraz, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 2 May 2019 Received in revised form 14 July 2019 Accepted 22 August 2019 Available online 23 August 2019

Ionic Liquids (ILs) are designer solvents with very unique properties, resulting in the exponential growth of publications in the field. Speed of sound can be considered as one of the important thermodynamic properties of compounds, since many other thermophysical properties can be determined using the speed of sound, including density, isentropic compressibility, isothermal compressibility, thermal conductivity, heat capacity, Joule-Thomson coefficient, and bulk modulus. Since ILs are designer solvents, much of their properties are unknown, hence, knowledge of their speeds of sound can be quite valuable. Two new straightforward models, with totally different approaches and input parameters, are proposed to estimate the speed of sound in ILs: an atomic contribution model, which only considers the atoms as building blocks to create the molecule and estimate its speed of sound; and a novel correlation. The atomic contribution model is the first which requires knowledge of only the chemical formula of the IL, making it needless of, not only any physical properties, but also the molecular structure which group contribution methods do require. This is considerable progress, as it will cover the majority of future ILs, which have not even been synthesized, and it does not have the ambiguities and difficulties of conventional group contribution (GC) methods for such complex structures. The further notable progress is its easy incorporation into computer programs, which is a serious setback with GC models. However, while being very straightforward and easy-to-use, it is more global than literature models. In addition to the atomic contribution method, a novel empirical correlation is proposed, with a new perspective. Both proposed models are quite reliable, while being very simple, and general. © 2019 Published by Elsevier B.V.

Keywords: Ionic liquid Speed of sound Group contribution Correlation Physical property

1. Introduction In addition to being an independent physical property of concern to certain applications, the speed of sound has been used historically as a useful property that can be used to calculate other thermophysical properties of the substance under study [1]. This makes it a valuable basic property. The thermophysical properties that can be determined using speed of sound, such as density, isentropic and isothermal compressibilities, thermal conductivity, heat capacity, Joule-Thomson coefficient, and bulk modulus, are widely used as controlling parameters in the optimization and design of industrial processes. For example, the design of a reactor for any endothermic or exothermic reaction involving an ionic liquid in the chemical or petroleum industries requires calorific

* Corresponding author. School of Chemical and Petroleum Engineering, Shiraz University, 71348-51154, Mollasadra Ave., Shiraz, Iran. E-mail address: [email protected] (S. Raeissi). https://doi.org/10.1016/j.fluid.2019.112291 0378-3812/© 2019 Published by Elsevier B.V.

information on the IL, i.e., its heat capacities at various temperatures. As another example, information of the inversion curve (the curve which shows where the JouleeThomson coefficient becomes zero) is extremely essential to the design of throttling processes [2e6]. Since the speed of sound can be determined experimentally with high precision over wide temperature and pressure ranges [1], it has also been used as a reliable source to determine virial coefficients, van der Waals constants, Lennard-Jones potential parameters, and equation of state constants, especially when the critical properties are not accessible [7,8]. However, until recently, only limited experimental data were available in literature on the speed of sound in ionic liquids, therefore, they have not yet been used extensively to derive other ionic liquid thermophysical properties [9]. By taking into account the huge number of Ionic Liquids (ILs) and their industrial potentials, and also by considering the functionalities of thermophysical properties on the speed of sound, it seems necessary to produce a large experimental database of speed

2

R. Haghbakhsh et al. / Fluid Phase Equilibria 503 (2020) 112291

of sound for this class of compounds. However, since experimental data points are not available at all of the desired operating conditions for all of the ILs of interest, the development of accurate and applicable predictive methods to estimate the speed of sound in ILs is very useful [1]. Only a few theoretical and empirical studies are available on the estimation of speeds of sound in ionic liquids. Gardas and Coutinho were the first researchers who developed a model by obtaining optimized values for the constants of Auerbach's Equation for ionic liquids [4]. Auerbach's Equation is a well-known empirical formula that relates the speed of sound in liquids to their surface tension and density [10]. Gardas and Coutinho's model is specific to 14 imidazolium-based ILs (optimized to 133 data points), giving an overall relative deviation of 1.96%. Therefore, it is not a general relation for ILs. Additionally, it requires experimental data on density and surface tension, the latter often being unavailable [1]. Because of this, Gardas and Coutinho also developed correlations to compute the surface tensions and densities of ILs in the cases that experimental data were lacking. However, this increases the required computations and complexity and decreases the accuracy [9]. Their approach was continued by Singh and Singh who extended the model for three additional imidazolium-based ILs [11]. Sattari et al. [9] used the method of least squared support vector machine (LSSVM) to develop a nonlinear group contribution model. Their database contained a total of 446 data points from 41 different ILs, contsisting of 29 cations and 11 anions. The novelty of their work was in determining the most effective functional groups using the forward feature selection method. Eight parameters, consisting of temperature and substructural-related groups, were identified as the most effective variables, and considered as the LSSVM model parameters. The method had an overall AARD% of 0.36%. The model's ability to predict the speed of sound of new ILs was also verified by an AARD% value of 0.87% on a test dataset [9]. However, the most important problems of this model, similar to all “black-box” computer models, are that they are not reproducible and they are not easy to use by all. Wu et al. [1] estimated the speeds of sound of 96 ILs over a wide temperature range, made of 51 cations and 23 anions, using a second-order corresponding states group contribution method. First and second order structural groups were defined to provide the basic and the supplementary information on the various IL molecular structures. In order to consider the decreasing effect of temperature on the speed of sound, a corresponding states method was applied where the critical temperature was calculated by Valderrama's group contribution method [12]. An overall relative deviation of 2.34% was reported, while the maximum error was lower than 20%. Although this is a more general model, it does have its shortcomings. It is not a global model, considering its limited groups which cannot cover the uninvestigated ILs. Furthermore, in the case of ILs with their complicated structures, it is not always easy to find the proper groups. In this study, we have investigated the feasibility of proposing a simple atomic contribution model to determine the speed of sound in ILs, as a very straightforward function of only the chemical formula. The aim of the study was to achieve simplicity and generality, while at the same time, keeping the accuracy at an acceptable level. Furthermore, for cases where accuracy is the first priority and experimental data are available, a new empirical correlation is proposed to estimate the speed of sound. 2. Model development 2.1. Atomic contribution model The basic technique in the group contribution approach is that

the molecular structure is decomposed into a number of functional groups. Each of these functional groups contributes to the value of the physical property under study [9]. Obviously, various methods can be chosen to decompose the molecule to get the desired groups. When groups constructed of a number of atoms are the basic functional groups, the model can suffer from generality because substances can often be found whose structures are not covered by the available groups of the GC model. Although a general shortcoming, this issue is more problematic for ILs because of their particularly complicated structures. Furthermore, even if a structure has the potential to be completely covered by the available groups, it happens at times that the correct decomposition and assignment of groups is very difficult because of certain ambiguities involved in the proposed groups. Even in cases where all of the necessary groups are available in the list of contributions and there are no uncertainties in the proper selection of groups, the process is still not straightforward and requires a time-consuming procedure of hand-picking the groups by going through the (often long) list of groups. Because of the complexities of structure decomposition, such models cannot be easily coupled to engineering software. In order to overcome the above-mentioned problems of group contribution models, the contribution segments can be reduced to single atoms. An atomic contribution model is the simplest of all “group” contribution models. Decomposition is not affected by the complexity of the molecular structure of the compound. In fact, it is not even necessary to know the structure since the chemical formula alone suffices for calculations. Therefore, difficult timeconsuming decompositions and the risks of incorrect group selection are alleviated. Atomic models can be easily incorporated into computer codes and engineering software without any complicated algorithms. With this idea in mind, we have proposed a new atomic contribution model using genetic programming [13,14]. A very simple and straightforward mathematical equation is suggested, which takes into account the temperature effect on the speed of sound, without the need for the critical temperature as was the case for Wu et al.‘s CSGC model [1]. This is a particular advantage because the critical temperatures of ionic liquids cannot be experimentally measured and can only be predicted approximately. The proposed atomic contribution equation is given as Equation (1):

U¼

A þBT MW

(1)

where U is the speed of sound in m/s, T is the temperature in degrees Kelvin, and MW is the molecular weight of the desired IL. The constants of the proposed model, A and B, are calculated by atomic contribution following Equations (2) and (3):

A¼

n X

ni  DAi

(2)

ni  DBi

(3)

i¼1

B¼

n X i¼1

where DAi and DBi are the contributions of each atom to the speed of sound. The values of DAi and DBi were optimized using genetic algorithm, based on the objective function presented in Equation (4):

  ndp  calc: X  uexp:  u Obj:Fun: ¼     uexp:

(4)

i¼1

where ni is the number of occurrence of atoms of type i in the

R. Haghbakhsh et al. / Fluid Phase Equilibria 503 (2020) 112291

structure, ndp is the number of data points, and ucalc. and uexp. are the calculated and experimental speed of sound values, respectively. In this way, the diversity of atoms in the IL structure is taken into account. To obtain the weights corresponding to each atom by optimization, a large experimental database is required. Experimental

3

speed of sound data from a total of 140 different ILs, consisting of 3790 data points, were gathered from open literature. Table 1 presents the investigated ILs, and the literature ranges of temperature and speed of sound for each. The number of data, chemical formula, and molecular weight of each IL are also presented in this table.

Table 1 The ionic liquids investigated in this study and the ranges of temperatures and speeds of sound. Mw (g/ mol)

Chemical formula

T (K)

u (m/s)

ndpa ref.

Optimization set 1 O1 propylammonium acetate

119.16

C5H13NO2

[15]

O2

2-hydroxy-N-methylethanammonium formate

121.14

C4H11NO3

241

[16]

3

O3

diethylammonium acetate

133.19

C6H15NO2

1417.5 e1517.1 1738.6 e1854.7 1599e1608

11

2

3

[17]

4

O4

N-methyl-2-hydroxyethylammonium acetate

135.16

C5H13NO3

241

[16]

5

O5

triethylammonium acetate

161.24

C8H19NO2

3

[17]

6

O6

N-methyl-2-hydroxyethylammonium butanoate

163.21

C7H17NO3

241

[16]

7

O7

1-ethyl-3-methylimidazolium acetate

170.21

C8H14N2O2

13

[18]

8

O8

N-methyl-2-oxopyrrolidinium propionate

173.21

C8H15NO3

11

[19]

9

O9

1-butyl-3-methylimidazolium chloride

174.67

C8H15ClN2

5

[20]

10

O10

2-hydroxy-N-methylethanammonium pentanoate

177.24

C8H19NO3

241

[16]

11

O11

N-methyl-2-oxopyrrolidinium butanoate

187.24

C9H17NO3

11

[19]

12

O12

2-hydroxy-N,N,N-trimethylethanammonium (S)-2-hydroxypropanoate

193.24

C8H19NO4

7

[21]

13

O13

N,N-dimethylbenzylammonium acetate

195.26

C11H17NO2

6

[22]

14

O14

1-butyl-3-methylimidazolium thiocyanate

197.30

C9H15N3S

5

[23]

15

O15

1-ethyl-3-methylimidazolium tetrafluoroborate

197.97

6

[24,25]

16

O16

triethylammonium dihydrogen phosphate

199.19

C6H18NO4P

4

[26]

17

O17

N-methyl-2-oxopyrrolidinium pentanoate

201.26

C10H19NO3

11

[19]

18

O18

1-methyl-3-propylimidazolium bromide

205.10

C7H13BrN2

19

O19

1-butyl-3-methylimidazolium dicyanamide

205.26

C10H15N5

20 21

O20 O21

1-ethyl-3-methylimidazolium methanesulfonate 1,3-dimethylimidazolium methylsulfate

206.26 208.24

C7H14N2O3S C6H12N2O4S

22

O22

1-butyl-1-methylpyrrolidinium dicyanamide

208.30

C11H20N4

23

O23

1-propylpyridinium tetrafluoroborate

208.99

C8H12BF4N

24

O24

benzyldimethylammonium propionate

209.28

C12H19NO2

25

O25

N-methyl-2-oxopyrrolidinium hexanoate

215.29

C11H21NO3

26

O26

1-butyl-3-methylimidazolium bromide

219.12

C8H15BrN2

27

O27

1,3-dimethylpyridinium methyl sulfate

219.26

C8H13NO4S

28

O28

1-ethyl-3-methylimidazolium methylsulfate

222.26

C7H14N2O4

29

O29

1-butylpyridinium tetrafluoroborate

223.02

C9H14BF4N

30

O30

1-hexyl-3-methylimidazolium thiocyanate

225.35

C11H19N3S

31

O31

1-butyl-3-methylimidazolium tetrafluoroborate

226.02

C8H15BF4N2

293.15 e343.15 278.15 e338.15 298.15 e308.15 278.15 e338.15 298.15 e308.15 278.15 e338.15 283.15 e343.15 293.15 e343.15 298.15 e328.15 278.15 e338.15 293.15 e343.15 293.15 e323.15 293.15 e318.15 293.15 e313.15 278.15 e328.15 293.15 e313.15 293.15 e343.15 288.15 e308.15 293.15 e343.15 303e333 283.15 e343.15 293.15 e343.15 278.15 e338.15 293.15 e318.15 293.15 e343.15 298.15 e308.15 293.15 e343.15 288.15 e343.15 278.15 e338.15 288.15 e303.15 283.15 e343.15

#

IL IL codeb

1691.3 e1848.3 1798e1840 1489.3 e1683.5 1628e1778 1282.6 e1460.3 1804e1868 1425.6 e1616.6 1259.1 e1435.6 1831.1 e1935.6 1344.7 e1438.5 1732.1 e778.8 1550.9 e1677.0 1730e1794 1255.0 e1427.7 1648.3 e1699.3 1629e1750

5

[27]

14

[18,28]

1708e1796 1708e1851

7 13

[29] [30]

1700e1823

14

[28,31]

1549.5 e1691.9 1321.9 e1416.9 1260.2 e1431.4 1642.1 e1664.4 1715.6 e1834.9 1652e1781

25

[32]

6

[22]

11

[19]

2

[33]

11

[34]

12

[35]

25

[36]

4

[37]

19

[38e40]

1510.8 e1649.8 1654.3 e1697.4 1462.1 e1604.5

(continued on next page)

4

R. Haghbakhsh et al. / Fluid Phase Equilibria 503 (2020) 112291

Table 1 (continued ) #

IL IL codeb

Mw (g/ mol)

Chemical formula

T (K)

u (m/s)

ndpa ref.

32

O32

N,N-diethyl-N-methylethanammonium methyl sulfate

227.32

C8H21NO4S

[41]

O33

1-methyl-3-octylimidazolium chloride

230.78

C12H23ClN2

18

[42e44]

34

O34

1-pentyl-3-methylimidazolium bromide

233.15

7

[45]

35

O35

2-ethyl-1-methylpyridinium methylsulfate

233.28

C9H15NO3S

11

[34]

36

O36

1-hexyl-3-methylimidazolium dicyanamide

233.31

C12H19N5

1761.5 e1853.5 1510.2 e1885.4 1550.2 e1678.6 1747.0 e1869.2 1578e1700

8

33

14

[28,46]

37

O37

1-butyl-3-methylimidazolium hydrogen sulfate

236.29

C8H16N2O4S

8

[47]

38

O38

1-butyl-3-methylpyridinium tetrafluoroborate

237.05

C10H16BF4N

37

[48e51]

39

O39

1-ethyl-1-methylpyrrolidinium ethyl sulfate

239.33

C9H21NO4S

8

[41]

40

O40

1-butyl-2,3-dimethylimidazolium tetrafluoroborate

240.05

C9H17BF4N2

7

[45]

41

O41

3-hexyl-1-methyl-1H-imidazolium bromide

247.18

C10H19BrN2

7

[52]

42

O42

1-butyl-3-methylimidazolium methylsulfate

250.32

C9H18N2O4S

14

[42]

43

O43

benzyldimethylammonium hexanoate

251.36

C15H25NO2

6

[22]

44

O44

1-butyl-1-methylpyrrolidinium methylsulfate

253.36

C10H23NO4S

10

[41]

45

O45

1-methyl-3-octyl-1H-imidazolium thiocyanate

235.41

C13H23N3S

4

[37]

46

O46

1-hexyl-3-methylimidazolium tetrafluoroborate

254.08

C10H19BF4N2

19

[49,50,52]

47

O47

1-butyl-3-methylimidazolium iodide

266.12

C8H15IN2

3

[53]

48

O48

1-butyl-3-methylimidazolium hexafluorophosphate

284.18

C8H15F6N2P

45

49

O49

1-butyl-3-methylimidazolium trifluoromethanesulfonate

288.29

C9H15F3N2O3S

50

O50

1-butyl-1-methylpyrrolidinium trifluoromethanesulfonate

291.33

C10H20F3NO3S

51

O51

N-octyl-3-methylpyridinium tetrafluoroborate

293.15

C14H24BF4N

52

O52

1-butyl-3-methylpyridinium trifluoromethanesulfonate

299.31

C11H16F3NO3S

53

O53

3-butyl-1-ethyl-1H-imidazolium trifluoromethanesulfonate (1:1)

302.31

C10H17F3N2O3S

54

O54

1-hexyl-3-methylimidazolium hexafluorophosphate

312.24

C10H19F6N2P

55

O55

1-hexyl-3-methylimidazolium trifluoromethanesulfonate

316.34

C11H19F3N2O3S

56

O56

tributylmethylphosphonium methylsulfate

328.45

C14H33O4PS

57

O57

1-butyl-3-methylimidazolium octylsulfate

348.50

C16H32N2O4S

58

O58

tributyloctylphosphonium chloride

350.99

C20H44ClP

59

O59

2-isopropoxy-2-oxoethan-1-ammonium dodecyl sulfate

383.54

C17H37NO6S

60

O60

tributylethylphosphonium diethylphosphate

384.47

C18H42O4P2

61

O61

1-ethylpyridinium bis[(trifluoromethyl)sulfonyl]imide

388.31

C9H10F6N2O4S2

62

O62

triisobutylmethylphosphonium p-toluenesulfonate

388.54

C20H37O3PS

63

O63

2-isobutoxy-2-oxoethan-1-ammonium dodecyl sulfate

397.57

C18H39NO6S

64

O64

1-ethyl-2-methylpyridinium bis((trifluoromethyl)sulfonyl)amide

402.33

C10H12F6N2O4S2

65

O65

1-methyl-1-propylpyrrolidinium bis[(trifluoromethyl)sulfonyl]imide

408.38

C10H18F6N2O4S2

66

O66

(S)-1-isobutoxy-1-oxopropan-2-ammonium dodecyl sulfate

411.6

C19H41NO6S

67

O67

1-butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide

419.36

C10H15F6N3O4S2

68

O68

1-butyl-1-methylpyrrolidinium bis[(trifluoromethyl)sulfonyl]imide

422.41

C11H20F6N2O4S2

308.15 e343.15 278.15 e343.15 288.15 e318.15 293.15 e343.15 293.15 e343.15 298.15 e333.15 278.15 e328.15 308.15 e343.15 288.15 e318.15 288.15 e318.15 278.15 e343.15 293.15 e318.15 298.15 e343.15 288.15 e303.15 288.15 e318.15 293.15 e308.15 283.15 e343.15 293.15 e343.15 293.15 e343.15 278.15 e328.15 288.15 e308.15 278.15 e338.15 278.15 e343.15 303.15 e343.15 293.15 e343.15 278.15 e343.15 293.15 e343.15 303.15 e343.15 293.15 e343.15 288.15 e338.15 298.15 e333.15 303.15 e343.15 278.15 e338.15 29315 e343.15 288.15 e343.15 298.15 e323.15 278.15 e343.15

1601.5 e1689.8 1511.0 e1633.9 1665.7 e1750.2 1593.9 e1668.6 1517.8 e1639.8 1552.1 e1711.3 1309.5 e1400.1 1625.5 e1741.6 1601.3 e1644.3 1473.3 e1548.8 1473.3 e1508.1 1329.4 e1479.8 1294e1403

28

[38e40,54 e58] [28,46,49,50]

1359e1473

14

[28,47]

1433.6 e1571.9 1389.6 e1436.1 1286.1 e1420.6 1318.4 e1490.7 1270e1358

21

[59]

3

[28]

7

[60]

27

[52,54,55,61]

9

[46]

11

[62]

50

[42,63]

11

[62]

9

[64]

11

[62]

21

[65]

8

[47]

9

[64]

25

[66]

11

[67]

12

[64]

6

[68]

25

[46,69]

1420.3 e1564.0 1557.2 e1347.1 1401.0 e1627.9 1245.8 e1373.4 1285.6 e1439.9 1183.8 e1289.6 1508.3 e1628.8 1248.7 e1372.1 1212.8 e1345.1 1176.2 e1281.9 1246.6 e1447.6 1171.3 e1227.8 1173e1316

R. Haghbakhsh et al. / Fluid Phase Equilibria 503 (2020) 112291

5

Table 1 (continued ) #

IL IL codeb

Mw (g/ mol)

Chemical formula

T (K)

u (m/s)

ndpa ref.

69

O69

(S)-2-(isopropoxycarbonyl)pyrrolidin-1-ium dodecyl sulfate

423.61

C20H41NO6S

[64]

O70

(S)-1-isopropoxy-3-methyl-1-oxobutan-2-ammonium dodecyl sulfate

425.62

C20H43NO6S

12

[64]

71

O71

430.39

C12H16F6N2O4S2

11

[46]

72

O72

1-butyl-3-methylpyridinium 1,1,1-trifluoro-N-[(trifluoromethyl)sulfonyl] methanesulfonamide (S)-2-(isobutoxycarbonyl)pyrrolidinium dodecyl sulfate

1284.0 e1455.5 1254.9 e1438.4 1152e1260

12

70

437.63

C21H43NO6S

12

[64]

73

O73

1-hexyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide

447.42

C12H19F6N3O4S2

28

[40,46,70]

74

O74

1-ethyl-3,5-dimethyl-2-pentylpyridinium bis(trifluoromethylsulfonyl)imide

486.49

C16H24F6N2O4S2

10

[71]

75

O75

(S)-1,5-diisopropoxy-1,5-dioxopentan-2-ammonium dodecyl sulfate

497.69

C23H47NO8S

76

O76

500.52

C17H26F6N2O4S2

77

O77

1-hexyl-2-ethyl-3,5-dimethylpyridinium 1,1,1-trifluoro-N-[(trifluoromethyl) sulfonyl]methanesulfonamide 1-butyl-3,5-dimethyl-2-pentylpyridinium bis(trifluoromethylsulfonyl)imide

514.55

C18H28F6N2O4S2

78

O78

(S)-1,5-diisobutoxy-1,5-dioxopentan-2-ammonium dodecyl sulfate

525.74

C25H51NO8S

79

O79

1-hexyl-3,5-dimethyl-2-pentylpyridinium bis(trifluoromethylsulfonyl)imide

542.60

C20H32F6N2O4S2

80

O80

tetradecyl(trihexyl)phosphonium dicyanamide

549.90

C34H68N3P

81

O81

648.85

C27H54F6N2O4S2

82

O82

methyltrioctylammonium 1,1,1-trifluoro-N-[(trifluoromethyl)sulfonyl] methanesulfonamide 1-butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide

419.37

C10H15F6N3O4S2

83 84

O83 O84

1-pentyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide 1-octyl-3- methylimidazolium bis(trifluoromethylsulfonyl)imide

433.39 475.47

C11H17F6N3O4S2 C14H23F6N3O4S2

85

O85

1-butyl-3-methylimidazolium methyl sulfate

250.32

C9H18N2O4S

86

O86

1-ethyl-3-methylimidazolium ethyl sulfate

236.29

C8H16N2O4S

87

O87

1,3-dimethylimidazolium methyl sulfate

208.24

C6H12N2O4S

88

O88

L-Glutamic

525.75

C25H51NO8S

89

O89

L-Valine

439.66

C21H45NO6S

90

O90

1-propyl-3-methylpyridinium bis(trifluoromethylsulfonyl)imide

416.37

C11H14F6N2O4S2

91 92 93 94

O91 O92 O93 O94

N-Ethyl-N-(2-hydroxyethyl)-N,N-dimethylammonium butanesulfonate N-Ethyl-N,N- dimethylbutylammonium Ethylsulfate N-Ethyl-N-(2-hydroxyethyl)-N,N- dimethylammonium 1-hexyl-3-methylimidazolium bromide

240.34 241.35 243.32 247.18

C10H25NO4S C9H23NO4S C8H21NO5S C10H19BrN2

95

O95

1-propyl-2-methylpyridinium bis(trifluoromethylsulfonyl)imide

416.37

C11H14F6N2O4S2

96

O96

N-methyl- 2-hydroxyethylammonium butyrate

163.22

C7H17NO3

97

O97

1-propyl-3-methylimidazolium bromide

205.10

C7H13BrN2

98

O98

2-hydroxyethylammonium pentanoate

163.22

C8H19NO3

99

O99

1-butylpyridinium triflate

285.29

C10H14F3NO3S

237.05

C10H16BF4N

383.55

C17H37NO6S

102 O102 triethylammonium sulfat

199.27

C6H17NO4S

103 O103 2-hydroxy diethylammonium pentanoate

207.27

C9H21NO4

288.15 e343.15 288.15 e343.15 293.15 e343.15 288.15 e343.15 283.15 e343.15 298.15 e343.15 288.15 e343.15 298.15 e343.15 298.15 e343.15 323.15 e343.15 298.15 e343.15 278.15 e343.15 298.15 e313.15 283.15 e323.27 293.25 293.15 e323.28 278.15 e313.15 293.15 e343.15 288.15 e343.15 323.15 e343.15 288.15 e343.15 293.15 e343.15 298.15 298.15 298.15 288.15 e318.15 278.15 e338.15 278.15 e338.15 288.15 e308.15 278.16 e338.15 298.16 e338.15 288.15 e338.15 303.15 e343.15 298.15 e313.15 278.15 e338.15

Validation set 104 V1 propylammonium formate

105.14

C4H11NO2

105 V2

N-methyl-2-oxopyrrolidinium formate

145.16

C6H11NO3

106 V3

2-hydroxy-N-methylethanammonium propionate

149.19

C6H15NO3

acid diisobutylester lauryl sulphate

isobutylester lauryl sulphate

100 O100 1-butyl-2-methylpyridinium tetrafluoroborate 101 O101

L-glycine

isopropylester lauryl sulphate

293.15 e343.15 293.15 e343.15 278.15 e338.15

1286.2 e1460.2 1129.6 e1262.0 1198e1309 1233.6 e1439.7 1192e1303

12

[64]

10

[71]

1179e1293

10

[71]

1270.5 e1330.6 1176e1291

5

[64]

10

[71]

1390e1599

14

[69]

1212e1260

4

[72]

1264.55 13 e1172.92 1227 1 1175.2e1232 10 1617.92 e1708.3 1566.4 e1703.9 1708e1838

[40,68,73,74] [40] [40,74]

35

[42,55,75,76]

16

[77e79]

16

[30,55]

5

[64]

12

[64]

11

[46]

1694 1817.9 1714 1517.79 e1639.85 1199.35 e1332.01 1489.31 e1683.51 1699.3 e1648.33 1468.15 e1684.75 1331.57 e1416.11 1552.01 e1671.42 1245.8 e1373.4 1859e1874

1 1 1 7

[80] [80] [80] [52]

25

[66]

241

[16]

5

[27]

241

[16]

17

[36]

21

[81]

9

[64]

4

[82]

1464.17 e1684.36

240

[83]

1452.0 e1553.1 1320.5 e1496.0 1570.6 e1752.4

11

[15]

11

[19]

241

[16]

1270.5 e1330.6 1249.5 e1434.4 1154e1261

(continued on next page)

6

R. Haghbakhsh et al. / Fluid Phase Equilibria 503 (2020) 112291

Table 1 (continued ) #

IL IL codeb

Mw (g/ mol)

Chemical formula C7H13NO3

107 V4

N-methyl-2-oxopyrrolidinium acetate

159.18

108 V5

1-butyl-3-methylimidazolium acetate

198.26

109 V6

1-methylpyridinium methylsulfate

205.23

110 V7

1-butyl-3-methylpyridinium dicyanamide

216.28

111 V8

1-ethyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide

391.31

112 V9

1-ethyl-3-methylpyridinium bis((trifluoromethyl)sulfonyl)amide

402.00

113 V10

2-methyl-1-propylpyridinium bis((trifluoromethyl)sulfonyl)amide

416.36

114 V11

1-methyl-3-propylimidazolium bis[(trifluoromethyl)sulfonyl]imide

405.34

115 V12

1-methyl-3-octylimidazolium tetrafluoroborate

282.13

116 V13

1,2-diethylpyridinium ethylsulfate

261.34

117 V14

1-octyl-3-methylimidazolium hexafluorophosphate

340.29

118 V15

1-butyl-3-methylimidazolium ibuprofenate

344.49

119 V16

1-butylpyridinium trifluoromethanesulfonate

285.28

120 V17

1-ethyl-3-methylimidazolium trifluoromethanesulfonate

260.23

121 V18

1-methylpyridinium methyl sulfate

205.24

122 V19

1,2-diethylpyridinium ethyl sulfate

261.34

123 V20

1-butyl-4-methylpyridinium tetrafluoroborate

237.05

124 V21

L-Alanine

397.58

125 V22

L-Glutamic

126 V23

L-Valine

127 V24

L-Alanine

128 V25

L-glycine

129 V26

trimethylammonium acetate

119.16

130 V27

3-butyl-1-ethylimidazolium trifluoromethanesulfonate

302.32

131 V28

1-butyl-1-ethylpyrrolidinium ethylsulfate

281.42

132 V29

triethylmethylammonium methylsulfate

227.33

133 V30

tetramethylammonium hydroxide

91.15

134 V31

tetraethylammonium hydroxide

147.26

135 V32

tetrapropylammonium hydroxide

203.37

136 V33

tetrabutylammonium hydroxide

259.48

137 V34

2-hydroxyethylammonium acetate

121.14

138 V35

2-hydroxyethylammonium oleate

343.55

139 V36

1-ethyl-3-methylimidazolium l-(þ)-lactate

200.24

140 V37

1-ethylpyridinium ethyl sulfate

233.29

a b

isopropylester lauryl sulphate acid diisopropylester lauryl sulphate

isopropylester lauryl sulphate

497.69 425.63

isobutylester lauryl sulphate

411.6

isobutylester lauryl sulphate

397.58

ndp: the number of data. Oi: Optimization data set, Vi: Validation data set.

T (K)

293.15 e343.15 283.15 C10H18N2O2 e343.15 293.15 C7H11NO4S e343.15 278.15 C12H16N4 e328.15 C8H11F6N3O4S2 287.05 e343.15 C10H12F6N2O4S2 293.15 e343.15 C11H14F6N2O4S2 278.15 e338.15 C9H13F6N3O4S2 293.15 e343.15 283.15 C12H23BF4N2 e343.15 293.15 C11H19NO4S e343.15 278.15 C12H23F6N2P e343.15 288.15 C21H32N2O2 e318.15 C10H14F3NO3S 298.15 e338.15 C7H11F3N2O3S 278.15 e338.15 288.15 C9H15NO4S e343.15 288.15 C11H19NO4S e343.15 278.15 C10H16BF4N e328.15 C18H39NO6S 288.15 e343.15 C23H47NO8S 288.15 e343.15 C20H43NO6S 288.15 e343.15 C19H41NO6S 288.15 e343.15 C18H39NO6S 303.15 e343.15 298.15 C5H13NO2 e313.15 C10H17N2F3O3S 278.15 e338.15 328.15 C12H27NO4S e343.15 308.15 C8H21NO4S e343.15 298.15 C4H13NO e313.15 298.15 C8H21NO e313.15 298.15 C12H29NO e313.15 298.15 C16H37NO e313.15 288.15 C5H13NO3 e323.14 278.14 C20H41NO3 e343.15 278.15 C9H16N2O3 e343.15 298.15 C9H15NO4S e343.15

u (m/s)

ndpa ref.

1311.5 e1491.5 1534e1700

11

[19]

13

[18]

1772.8 11 e1888.6 1684.0 21 e1803.4 1146e1265.8 18

[34]

1164.2 e1270.6 1199.3 e1332.0 1137e1243

[84] [46,85]

11

[86]

25

[66]

11

[87]

14

[38,88,89]

11

[34]

20

[54,55,90]

4

[91]

17

[36]

7

[92]

13

[18]

14

[28,34]

25

[48,49]

12

[64]

12

[64]

12

[64]

12

[64]

9

[64]

4

[93]

1286.13 7 e1420.57 1564.9e1602 4

[60]

1361.1 e1522.5 1632.7 e1759.3 1294.6 e1481.6 1510.5 e1647.0 1331.6 e1416.1 1348.5 e1482.23 1722.8 e1901.1 1632.7 e1774.5 1524.7e1650 1266.3 e1432.3 1233.6 e1439.7 1254.9 e1438.4 1246.6 e1447.6 1248.7 e1372.1 1490e1544

[41]

1761.5 e1853.5 1781e1814

8

[41]

4

[94]

1781e1814

4

[94]

1769e1801

4

[94]

1767e1798

4

[94]

1735.99 e1817.23 1380.12 e1601.51 1614.9 e1799.1 1608e1711

15

[95]

239

[96]

67

[97,99]

10

[98]

Total

3790

R. Haghbakhsh et al. / Fluid Phase Equilibria 503 (2020) 112291 Table 2 The values of AARD% of the two proposed models for the different datasets considered in this study. Data set

Number of data AARD% Atomic contribution model Proposed correlation

Optimization 2854 Validation 936 Overall 3790

3.00 4.33 3.33

[99] developed a very simple correlation that required only one viscosity data point. The approach was based on the empirical observance that the sensitivity of viscosity to changes in temperature depends primarily on the value of the viscosity, and not on the substance itself. With this concept, the resulting equation was:

h0:2661 ¼ h0:2661 þ k

0.65 0.67 0.66

An additional advantage of this study is the organization and compilation (Table 1) of a more comprehensive and up-to-date database of ionic liquid speed of sound data than was available when earlier studies were conducted. In the next step, among all of the available ionic liquids, a number of them were randomly set aside to be used for validation, and the remaining ionic liquids were used for optimization. The optimization dataset was used to obtain the model's contribution for each atom. The validation dataset, consisting of those ionic liquids that were not used in optimization, was then used to test the capability of the model by calculating results for “new” ionic liquids. About 75% of all of the data (2854 data from 103 ILs) were allocated to the optimization set, and the remaining 25% (936 data of the 37 remaining ILs) were considered within the validation set. The corresponding optimization and validation ILs are distinguished in Table 2. Table 3 presents the optimized atom contributions for the parameters A and B in the proposed model. From the formulation of Equation (1), it can be concluded that the value of A is the main parameter which determines the speed of sound in each IL, and the value of B determines the slope of the speed of sound with temperature. Larger atoms, such as I, Br, Cl, P, and S, have large DAi values, which can affect the values of the estimated speeds of sound significantly, even when their number of occurrence in the molecule is small. 2.2. Correlation The correlation of this study takes on a different perspective compared to conventional correlations where a physical property is determined as a function of two or more other physical properties. The inspiration of the proposed relation was taken from the correlation of Lewis and Squires [99] for estimating the viscosities of commonly-used substances. While the conventional methods of viscosity correlations were based on Arrhenius-type functions of viscosity where the logarithm of viscosity is a linear function of reciprocal absolute temperature, requiring two viscosity data points to obtain the constants of the correlation, Lewis and Squires

Table 3 Optimized values of atomic contributions to calculate the parameters A and B of the proposed atomic contribution model Equations (2) and (3). Atom

DAi

DB i

C H B F N P O S Cl Br I

15450.041 520.455 182721.376 15970.979 73665.262 210596.678 44485.777 104359.119 194261.269 188736.426 141865.214

0.037871 0.013983 1.589838 0.004395 0.632333 1.389037 0.390676 0.000065 1.885907 0.915839 0.473567

7

T  Tk 233

(5)

where h is the desired viscosity at temperature T, and hk is the viscosity of the compound at an arbitrary reference temperature of Tk. In this way, apart from one single viscosity value of the desired compound as input data, no other physical properties are necessary for the calculation of the desired viscosity. Taking inspiration from the methodology and formulation of the viscosity correlation of Lewis and Squires [99], and using the same optimization and validation datasets as presented in Table 1, we propose a new correlation as follows,

u0:1352432 ¼ u0:1352432 þ k

T  Tk 9533:114

(6)

where u and uk are the calculated and reference values of speed of sound, respectively, in m/s. T and Tk are the desired and reference temperatures, respectively, in Kelvin or degrees Centigrade. The reference point (k) in this equation can be chosen at any arbitrary temperature with the restriction that the IL remains in the liquid state. If the reference temperature is chosen closer to the desired temperature, the results are expected to be more reliable. In this way, one single equation with fixed constants works for all ionic liquids. 3. Results and discussion 3.1. Atomic contribution model As explained earlier, to estimate the speed of sound at various temperatures, the atomic contribution model requires only the chemical formula and the corresponding molecular weight of each IL. The speeds of sound of all of the ILs presented in Table 1 were estimated using the atom contribution values given in Table 3. The results were rather accurate for the varying types of ILs. In order to have a quantitative comparison of errors with respect to the experimental data, values of absolute average relative deviation percent (AARD%) were calculated according to Equation (7),

   ucalc: uexp:   uexp:   i¼1 

ndp P

AARD% ¼

ndp

 100

(7)

in which ndp is the number of data points, and ucalc. and uexp. are the calculated and experimental speed of sound values, respectively. Table 2 presents, separately, the values of AARD% for the optimization and validation datasets, as well as the overall dataset consisting of all of the ionic liquids. It is observed in this table that the results for both the optimization and validation datasets are in good agreement with the corresponding experimental values. The overall AARD% value of 3.33% confirms the reliability of this simple, general, and straightforward atomic contribution model. In order to present a more detailed investigation of errors, the values of AARD% for each IL is presented separately in Table 4. The proposed atomic contribution model is the first of the atomic category for the speed of sound in ILs. For comparison with group contribution methods, the results of one of the best GC models of literature, namely the method of Wu et al. [1], are

8

R. Haghbakhsh et al. / Fluid Phase Equilibria 503 (2020) 112291

Table 4 Comparison of the AARD% values of the atomic contribution model and the proposed correlation with the literature GC model of Wu et al. [1] for the investigated ILs. #

IL IL code2

Optimization 1 O1 propylammonium acetate 2 O2 2-hydroxy-N-methylethanaminium formate 3 O3 diethylammonium acetate 4 O4 N-methyl-2-hydroxyethylammonium acetate 5 O5 triethylammonium acetate 6 O6 N-methyl-2-hydroxyethylammonium butanoate 7 O7 1-ethyl-3-methylimidazolium acetate 8 O8 N-methyl-2-oxopyrolidinium propionate 9 O9 1-butyl-3-methylimidazolium chloride 10 O10 2-hydroxy-N-methylethanaminium pentanoate 11 O11 N-methyl-2-oxopyrolidinium butanoate 12 O12 2-hydroxy-N,N,N-trimethylethanaminium (S)-2-hydroxypropanoate 13 O13 N,N-dimethylbenzylammonium acetate 14 O14 1-butyl-3-methylimidazolium thiocyanate 15 O15 1-ethyl-3-methylimidazolium tetrafluoroborate 16 O16 triethylammonium dihydrogen phosphate 17 O17 N-methyl-2-oxopyrrolidinium pentanoate 18 O18 1-methyl-3-propylimidazolium bromide 19 O19 1-butyl-3-methylimidazolium dicyanamide 20 O20 1-ethyl-3-methylimidazolium methanesulfonate 21 O21 1,3-dimethylimidazolium methylsulfate 22 O22 1-butyl-1-methylpyrrolidinium dicyanamide 23 O23 1-propylpyridinium tetrafluoroborate 24 O24 benzyldimethylammonium propionate 25 O25 N-methyl-2-oxopyrrolidinium hexanoate 26 O26 1-butyl-3-methylimidazolium bromide 27 O27 1,3-dimethylpyridinium methyl sulfate 28 O28 1-ethyl-3-methylimidazolium methylsulfate 29 O29 1-butylpyridinium tetrafluoroborate 30 O30 1-hexyl-3-methylimidazolium thiocyanate 31 O31 1-butyl-3-methylimidazolium tetrafluoroborate 32 O32 N,N-diethyl-N-methylethanaminium methyl sulfate 33 O33 1-methyl-3-octylimidazolium chloride 34 O34 1-pentyl-3-methylimidazolium bromide 35 O35 2-ethyl-1-methylpyridinium methylsulfate 36 O36 1-hexyl-3-methylimidazolium dicyanamide 37 O37 1-butyl-3-methylimidazolium hydrogen sulfate 38 O38 1-butyl-3-methylpyridinium tetrafluoroborate 39 O39 1-ethyl-1-methylpyrolidinium ethyl sulfate 40 O40 1-butyl-2,3-dimethylimidazolium tetrafluoroborate 41 O41 3-hexyl-1-methyl-1H-imidazolium bromide 42 O42 1-butyl-3-methylimidazolium methylsulfate 43 O43 benzyldimethylammonium hexanoate 44 O44 1-butyl-1-methylpyrolidinium methylsulfate 45 O45 1-methyl-3-octyl-1H-imidazolium thiocyanate 46 O46 1-hexyl-3-methylimidazolium tetrafluoroborate 47 O47 1-butyl-3-methylimidazolium iodide 48 O48 1-butyl-3-methylimidazolium hexafluorophosphate 49 O49 1-butyl-3-methylimidazolium trifluoromethanesulfonate 50 O50 1-butyl-1-methylpyrolidinium trifluoromethanesulfonate 51 O51 N-octyl-3-methylpyridinium tetrafluoroborate 52 O52 1-butyl-3-methylpyridinium trifluoromethanesulfonate 53 O53 3-butyl-1-ethyl-1H-imidazolium trifluoromethanesulfonate (1:1) 54 O54 1-hexyl-3-methylimidazolium hexafluorophosphate 55 O55 1-hexyl-3-methylimidazolium trifluoromethanesulfonate 56 O56 tributylmethylphosphonium methylsulfate 57 O57 1-butyl-3-methylimidazolium octylsulfate 58 O58 tributyloctylphosphonium chloride 59 O59 2-isopropoxy-2-oxoethan-1-aminium dodecyl sulfate 60 O60 tributylethylphosphonium diethylphosphate 61 O61 1-ethylpyridinium bis[(trifluoromethyl)sulfonyl]imide 62 O62 triisobutylmethylphosphonium p-toluenesulfonate 63 O63 2-isobutoxy-2-oxoethan-1-aminium dodecyl sulfate 64 O64 1-ethyl-2-methylpyridinium bis((trifluoromethyl)sulfonyl)amide 65 O65 1-methyl-1-propylpyrrolidinium bis[(trifluoromethyl)sulfonyl]imide 66 O66 (S)-1-isobutoxy-1-oxopropan-2-aminium dodecyl sulfate 67 O67 1-butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide 68 O68 1-butyl-1-methylpyrolidinium bis[(trifluoromethyl)sulfonyl]imide 69 O69 (S)-2-(isopropoxycarbonyl)pyrrolidin-1-ium dodecyl sulfate 70 O70 (S)-1-isopropoxy-3-methyl-1-oxobutan-2-aminium dodecyl sulfate 71 O71 1-butyl-3-methylpyridinium 1,1,1-trifluoro-N-[(trifluoromethyl)sulfonyl] methanesulfonamide

Proposed correlation

Atomic contribution model

Wu et al.‘s CSGC model [1]

1.5 1.7 0.8 1.1 0.1 0.2 1.3 1.1 1.2 0.1 1.2 0.4 0.6 0.5 0.9 0.2 1.1 0.4 1.4 0.7 1.6 1.5 1.1 0.7 1.0 0.4 1.6 1.5 1.1 0.2 1.0 1.2 2.8 0.4 1.6 1.3 1.0 0.8 1.2 0.5 0.4 1.2 0.6 1.2 0.2 0.4 0.3 0.6 0.9 0.9 0.3 0.3 0.7 0.5 0.7 0.3 0.2 2.3 0.9 0.4 0.5 0.4 0.7 0.5 0.6 1.4 0.2 0.5 0.4 0.8 0.5

18.5 0.6 4.7 3.2 13.2 1.8 1.6 12.7 2.6 2.2 12 20.6 9.0 13.1 7.8 6.9 10.3 0.5 4.6 2.7 0.1 3.6 0.6 9.5 8.2 1.5 5.6 0.6 0.6 12.3 4.3 9.4 3.6 1.4 9.7 0.1 0.3 1.3 5.9 3.5 1.4 0.8 8.5 5.9 11.2 1.5 0.7 0.6 5.1 1.8 2.2 2.9 4.8 2.3 4.7 6.7 1.3 2.3 1.2 0.1 2.5 2.1 1.1 0.6 2.0 2.4 1.6 1.8 3.9 2.4 2.8

e 3.22 3.31 1.98 4.94 e 3.78 e e 3.00 e e e e 5.38 e e e 2.96 e e 0.76 1.99 e e e 1.99 e 1.60 e 2.52 e 3.67 e 0.73 1.74 e 0.93 4.05 e e e e 2.31 e 0.12 e 1.49 2.37 1.86 e 1.11 e 2.30 1.10 2.23 5.06 6.04 e e e e e 1.62 e e 5.34 2.74 e e e

R. Haghbakhsh et al. / Fluid Phase Equilibria 503 (2020) 112291

9

Table 4 (continued ) #

IL IL code2

Proposed correlation

Atomic contribution model

Wu et al.‘s CSGC model [1]

72 73 74 75 76

O72 O73 O74 O75 O76

0.5 0.3 0.1 1.5 0.1

4.5 2.9 1.6 14.6 0.9

e 7.83 e e e

77 78 79 80 81

O77 O78 O79 O80 O81

0.1 0.3 0.1 0.1 0.4

0.9 15.2 1.5 2.5 8.8

e e e 8.23 e

0.3 0.2 0.2 1.1 1.1 1.3 0.3 0.5 0.5 0.0 0.0 0.0 0.4 0.5 0.2 0.4 0.3 1.0 1.1 0.9 1.2 0.3

1.33 0.5 3.6 0.8 0.9 0.1 15.2 3.2 2.9 2.8 9.6 6.3 1.4 0.1 2.0 0.5 2.6 4.9 4.6 1.2 5.4 6.7

6.20 5.91 6.89 1.05 3.41 3.77 1.22 5.84 5.06 1.55 5.45 0.62 2.60 1.89 2.10 2.62 1.27 1.47 0.82 3.06 11.11 1.66

1.5 0.9 0.5 1.0 0.8 1.8 1.0 0.4 0.6 0.5 0.5 0.4 1.3 0.3 0.8 1.0 0.7 1.5 1.1 1.8 0.3 1.5 0.8 1.4 0.7 0.4 0.7 0.4 1.2 0.7 0.7 0.7 0.7 0.9 0.3 1.0 1.6

22.3 17.5 1.7 13.7 2.2 5.8 2.7 1.0 2.7 0.1 1.7 0.9 7.4 2.9 9.2 4.9 4.8 5.9 7.6 2.1 2.3 14.6 2.4 2.4 1.1 15.7 4.8 4.2 9.4 1.4 11.8 15.0 15.1 1.9 4.4 8.4 1.8

e e 0.81 e 1.53 e 0.64 6.10 4.25 e 7.24 2.38 e 5.2 e e 4.63 1.98 1.56 0.63 6.11 3.39 5.52 5.24 3.23 13.22 1.67 3.13 0.36 6.25 1.78 3.88 9.87 3.20 2.35 2.90 3.08

82 O82 83 O83 84 O84 85 O85 86 O86 87 O87 88 O88 89 O89 90 O90 91 O91 92 O92 93 O93 94 O94 95 O95 96 O96 97 O97 98 O98 99 O99 100 O100 101 O101 102 O102 103 O103 Validation 104 V1 105 V2 106 V3 107 V4 108 V5 109 V6 110 V7 111 V8 112 V9 113 V10 114 V11 115 V12 116 V13 117 V14 118 V15 119 V16 120 V17 121 V18 122 V19 123 V20 124 V21 125 V22 126 V23 127 V24 128 V25 129 V26 130 V27 131 V28 132 V29 133 V30 134 V31 135 V32 136 V33 137 V34 138 V35 139 V36 140 V37

(S)-2-(isobutoxycarbonyl)pyrrolidinium dodecyl sulfate 1-hexyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide 1-ethyl-3,5-dimethyl-2-pentylpyridinium bis(trifluoromethylsulfonyl)imide (S)-1,5-diisopropoxy-1,5-dioxopentan-2-aminium dodecyl sulfate 1-hexyl-2-ethyl-3,5-dimethylpyridinium 1,1,1-trifluoro-N-[(trifluoromethyl)sulfonyl] methanesulfonamide 1-butyl-3,5-dimethyl-2-pentylpyridinium bis(trifluoromethylsulfonyl)imide (S)-1,5-diisobutoxy-1,5-dioxopentan-2-aminium dodecyl sulfate 1-hexyl-3,5-dimethyl-2-pentylpyridinium bis(trifluoromethylsulfonyl)imide tetradecyl(trihexyl)phosphonium dicyanamide methyltrioctylammonium 1,1,1-trifluoro-N-[(trifluoromethyl)sulfonyl] methanesulfonamide 1-butyl-3-mthylimidazolium bis(trifluoromethylsulfonyl)imide 1-pentyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide 1-octyl-3- methylimidazolium bis(trifluoromethylsulfonyl)imide 1-butyl-3-methylimidazolium methyl sulfate 1-ethyl-3-methylimidazolium ethyl sulfate 1,3-dimethylimidazolium methyl sulfate L-glutamic acid diisobutylester lauryl sulphate L-valine isobutylester lauryl sulphate 1-propyl-3-methylpyridinium bis(trifluoromethylsulfonyl)imide N-ethyl-N-(2-hydroxyethyl)-N,N-dimethylammonium butanesulfonate N-ethyl-N,N- dimethylbutylammonium ethylsulfate N-ethyl-N-(2-hydroxyethyl)-N,N- dimethylammonium 1-hexyl-3-methylimidazolium bromide 1-propyl-2-methylpyridinium bis(trifluoromethylsulfonyl)imide N-methyl- 2-hydroxyethylammonium butyrate 1-propyl-3-methylimidazolium bromide 2-hydroxyethylammonium pentanoate 1-butylpyridinium triflate 1-butyl-2-methylpyridinium tetrafluoroborate L-glycine isopropylester lauryl sulphate triethylammonium sulfat 2-hydroxy diethylammonium pentanoate set propylammonium formate N-methyl-2-oxopyrrolidinium formate 2-hydroxy-N-methylethanaminium propionate N-methyl-2-oxopyrrolidinium acetate 1-butyl-3-methylimidazolium acetate 1-methylpyridinium methylsulfate 1-butyl-3-methylpyridinium dicyanamide 1-ethyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide 1-ethyl-3-methylpyridinium bis((trifluoromethyl)sulfonyl)amide 2-methyl-1-propylpyridinium bis((trifluoromethyl)sulfonyl)amide 1-methyl-3-propylimidazolium bis[(trifluoromethyl)sulfonyl]imide 1-methyl-3-octylimidazolium tetrafluoroborate 1,2-diethylpyridinium ethylsulfate 1-octyl-3-methylimidazolium hexafluorophosphate 1-butyl-3-methylimidazolium ibuprofenate 1-butylpyridinium trifluoromethanesulfonate 1-ethyl-3-methylimidazolium trifluoromethanesulfonate 1-methylpyridinium methyl sulfate 1,2-diethylpyridinium ethyl sulfate 1-butyl-4-methylpyridinium tetrafluoroborate L-alanine isopropylester lauryl sulphate L glutamic acid diisopropylester lauryl sulphate L-valine isopropylester lauryl sulphate L-alanine isobutylester lauryl sulphate L-glycine isobutylester lauryl sulphate trimethylammonium acetate 3-butyl-1-ethylimidazolium trifluoromethanesulfonate 1-butyl-1-ethylpyrolidinium ethylsulfate triethylmethylammonium methylsulfate tetramethylammonium hydroxide tetraethylammonium hydroxide tetrapropylammonium hydroxide tetrabutylammonium hydroxide 2-hydroxyethylammonium acetate 2-hydroxyethylammonium oleate 1-ethyl-3-methylimidazolium l-(þ)-lactate 1-ethylpyridinium ethyl sulfate

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as can be seen in Fig. 1. This is consistent with the nature of a best fit approach for the contribution of the atoms using a large data bank, i.e., some temperature slopes will be under-predicted and some will be over-predicted. In order to investigate the effects of different cations and anions on the proposed atomic contribution method, Figs. S4eS6 have been presented in the Supplementary material section. In these figures, by considering the cations or anions as fixed, the effect of changing anions and cations are shown, respectively.

3.2. Correlation

Fig. 1. Comparison of the behavior of speed of sound with respect to temperature by the proposed atomic contribution model for some of the investigated ILs.

contrasted to those of the proposed model. These results are also presented in Table 4. The model of Wu et al. is a structural GC model, which is not straightforward in comparison to an atomic model, and the decomposition of the IL structure can be difficult and time-consuming. Sometimes, there can even be more than one way to decompose the structure. Furthermore, we were confronted with the issue that certain “more recent” ionic liquids could not be estimated with Wu et al.‘s model because the structure could not be fully covered by the list of available groups (the present study contains a larger databank, including 44 ionic liquids which were not investigated in Wu et al.‘s study). Furthermore, because of the dependency of Wu et al.‘s model on critical temperatures, which they had suggested to be calculate by Valderama's GC model [12], a further number of ionic liquids had to be set aside from the comparison because in some cases, Valderrama's model additionally lacked the necessary groups to cover the new ILs. Table 4 indicates the number of ILs which were not possible to estimate with Wu et al.‘s method. The results of this table indicate that, not only is the proposed atomic contribution model capable to calculate the speed of sound in all of the investigated ILs, the errors are also smaller for almost half of those ILs which could be calculated with Wu et al.‘s model. Figs. S1 and S2 of the Supplementary section show the distribution of relative deviations for the whole range of investigated speeds of sound for the optimization and validation datasets, respectively. Fig. S3 shows the distribution of the percentage of data with different relative deviation percent values for the overall dataset of this study. The normal and reliable statistical behavior shown in Figs. S1eS3 further confirm the accuracy of the proposed atomic contribution model. Fig. 1 is a qualitative investigation of the behavior of the atomic contribution model with respect to experimental data, as a function of temperature. Because of the extensive number of investigated ILs and the close values of speeds of sound, we have shown only some of the ILs to avoid the overlapping of data, which would render the figure ineffective. The experimental data have almost linear decreasing trends with increasing temperature. Such trends are generally well-followed by the atomic contribution model. However, there is a slight increase in errors as temperature increases (Table S2 of the Supplementary section), although there is no particular positive or negative trend for this. While the estimated slope at higher temperatures is greater than the experimental data for triisobutylmethylphosphonium p-toluenesulfonate (062), it is smaller for 1butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide (O67),

Apart from the atomic contribution model, the proposed empirical correlation also needs to be verified. One of the important issues, when using the suggested correlation, is the reference point. As mentioned previously, the reference point can be any arbitrary point, as long as the IL remains in the liquid phase. It is expected that the estimated value of speed of sound will be more accurate if the selected reference temperature is closer to the desired temperature. Because of this issue, in order to avoid an unfair evaluation, and also to have a standard and unique procedure of estimation for all of the investigated ILs, we have tried wherever possible, to take the same reference temperature for every IL. With this idea in mind, the temperature of 298.15 K was taken as the reference temperature whenever experimental data was available at this temperature. This covered the majority of the investigated ILs. However, in some cases, this temperature was outside of the investigated temperature range. In such cases, the minimum available temperature was taken as the reference temperature. Table S1 of the Supplementary section presents the reference temperatures and the corresponding experimental values of speed of sound for all of the investigated ILs. It is important to note that there is absolutely no need for the user to adopt the reference points listed in Table S1. Any desired temperature at which the IL is in the liquid phase can be taken as the reference temperature. With the reference points of Table S1, the speeds of sound in all of the investigated ILs were calculated. Table 2 presents the values of AARD% of the proposed correlation for the different datasets. As can be seen, the errors are always lower than one percent, indicating that the proposed correlation is quite reliable for both the optimization and validation datasets. In order to compare the errors of each IL separately, Table 4 gives the individual AARD% values. A comparison of the different ILs with each other indicates that the ionic liquid with the maximum error by the proposed correlation has an AARD% of 2.8%. This shows a high level of reliability. Therefore, with only one reference point and with no other physical property data, this correlation can reliably estimate the speeds of sound of all ionic liquids in a very easy and simple manner. One must note that there are no component-specific constants in the equation, and one global equation is being used for all of the ILs. Statistical graphs are also presented (Figs. S7eS9) for the proposed correlation. Fig. S7 indicates the relative deviation for both of the datasets. Based on Table 2, the errors of the optimization and validation datasets are almost equal for the proposed correlation. In addition to the quantitative presentation of errors above, Fig. 2 shows the temperature trends of the proposed correlation and compares them to the experimental values. The behavior of the 1-alkyl-3-methylimidazolium family with four different commonly-used anions, consisting of tetrafluoroborate [BF4], hexafluorophosphate [PF6], bis[(trifluoromethyl)sulfonyl]imide [Tf2N], and trifluoromethanesulfonate [TFO] are shown. It is obvious from this figure that the proposed correlation gives very good agreement with the experimental trends of decreasing speed of sound with increasing temperatures.

R. Haghbakhsh et al. / Fluid Phase Equilibria 503 (2020) 112291

11

Fig. 2. The speed of sound behavior of the proposed correlation with respect to temperature and comparison with experimental data for some common ILs. Filled markers and dashed lines represent the experimental data and proposed model, respectively.

4. Conclusions There are very few studies in literature which present models to estimate the speed of sound in ILs. Each method has its own shortcomings, such as complexity, irreproducibility, difficulty of use, dependence on other physical properties, and/or not being global for vast numbers and types of ionic liquids. In this study, two new estimation models have been proposed for the speeds of sound in different types of ILs, in an attempt to alleviate some of the above shortcomings. An atomic contribution model has been proposed which does not require any physical properties as input parameters. It is a very straightforward, easy-to-use, general, and reliable method. The chemical formula and the molecular weight of the desired IL are the only information required to easily estimate the speed of sound. In addition to this model, a new correlation with a new perspective has been proposed. This global correlation is very simple, general, accurate, and does not require any other physical properties to calculate the speed of sound other than one single speed of sound data point. Both of the models have been developed by the most comprehensive and up-to-date databank of IL speed of sound data, available in open literature. A total of 3790 data points from 140 different types of ILs were collected. This large database was divided randomly into two datasets for optimization (75%) and

validation (25%). The estimated speeds of sound had AARD% values of 3.33% and 0.66% for the atomic contribution model and the correlation, respectively. Such error values show the good agreement of both models with the corresponding experimental data. In addition to the reliable quantitative errors of both models, they both showed good agreement with the linearly-decreasing temperature trends of the different ILs, as shown by experiments. Acknowledgments The authors are grateful to Shiraz University for supporting this research. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.fluid.2019.112291. References [1] K. Wu, O. Chen, C. He, Speed of sound of ionic liquids: database, estimation, and its application for thermal conductivity prediction, AIChE J. 60 (2014) 1120e1131. [2] A.J. Queimada, J.A.P. Coutinho, I.M. Marrucho, J.L. Daridon, Correspondingstates modeling of the speed of sound of long-chain hydrocarbons, Int. J. Thermophys. 27 (2006) 1095e1109. [3] C.W. Lin, J.P.M. Trusler, The speed of sound and derived thermodynamic

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