Pressure dependence of the solubility of light fullerenes in n-nonane

J. Chem. Thermodynamics 112 (2017) 259–266

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Pressure dependence of the solubility of light fullerenes in n-nonane Konstantin N. Semenov a,⇑, Nikolay A. Charykov c, Enriqueta R. López b, Josefa Fernández b, Vladimir V. Sharoyko a, Igor V. Murin a a

Institute of Chemistry, Saint-Petersburg State University, Saint-Petersburg 198504, Universitetskii pr. 26, Russia Nafomat Group, Applied Physics Department, Santiago de Compostela University, Santiago de Compostela E-15782, Spain c Saint-Petersburg State Technological Institute (Technical University), Saint-Petersburg 190013, Moskovskii pr. 26, Russia b

a r t i c l e

i n f o

Article history: Received 12 July 2016 Received in revised form 25 April 2017 Accepted 11 May 2017 Available online 13 May 2017 Keywords: Light fullerenes C60 C70 Solubility High pressure n-Nonane

a b s t r a c t Solubility of light fullerenes (C60 and C70) in n-nonane was investigated in the ranges of pressure form 0.1 MPa up to 100 MPa and temperature from 298.3 K to 353.3 K. Under isothermal conditions, the solubility, expressed as weight fraction of the fullerene in the solution, increases monotonously with increasing pressure. At ambient pressure, we have found that the temperature dependence of the solubility of C60 in n-nonane is non-monotonic: the solubility diagram consists of two branches corresponding to crystallization of different solid phases and one invariant point corresponding to simultaneous saturation of both phases. At 0.1 MPa, the solubility diagram of the binary system C70 – n-nonane in the analysed temperature range consists of only one branch corresponding to crystallization of nonsolvated C70. Ó 2017 Published by Elsevier Ltd.

1. Introduction In recent years one of the most developing areas of modern chemistry is physical chemistry of nanostructures, in particular of carbon nanoclusters (fullerenes and their derivatives) [1–3]. These compounds present unique properties in the context of electronic structure, physical and chemical properties [1]. Phase equilibria research of systems containing fullerenes is extremely important for the development of extraction and crystallization isolation of fullerenes from the fullerene mixture and fullerene black, for elaboration of chromatographic and prechromatographic methods of the fullerenes separation, for investigation of chemical reactions in systems containing fullerenes, for preparation of biologically active phases based on fullerenes and for optimization of the light fullerenes applications as nanomodifiers, among others [4–9]. The great relevance of this research topic in the fullerene-containing systems can be easily illustrated by the large amount of experimental data on solubility of individual light fullerenes (C60 and C70) in various organic and inorganic solvents as well as in solvent mixtures under different T, P conditions [4–20]. In this regard, several reviews on phase equilibria of fullerenecontaining systems as well as on physicochemical properties of fullerene solutions were published [4–7]. In addition, extraction equilibria in several systems (C60 – C70 – o-xylene – butylamine ⇑ Corresponding author. E-mail address: [email protected] (K.N. Semenov). http://dx.doi.org/10.1016/j.jct.2017.05.017 0021-9614/Ó 2017 Published by Elsevier Ltd.

– H2O, C60 – C70 – o-xylene – monoethanolamine, C60 – C70 – o-xylene – dymethylformamide – H2O, C60 – C70 – toluene – dymethylformamide – H2O, C60 – C70 – a – pinene – ethanol – H2O, C60 – C70 – o-xylene – ethanol – H2O, C60 – C70 – 1,2,4trichlorobenzene – ethanol – H2O) were studied [7,9]. These systems can be effectively used for purification of light fullerenes and for separation of industrial fullerene mixtures. Another set of scientific papers is devoted to investigation of sorption equilibria in systems containing fullerenes. In particular, authors of Refs. [21–23] investigated the adsorption properties of the Norit-Azo carbon and multi-walled carbon nanotubes (MWCNT) in relation to light fullerenes. The present paper is devoted to the investigation of individual light fullerenes solubility in n-nonane in the temperature range from 298.3 K to 353.3 K and pressures up to 100 MPa, as well as to the thermodynamic description of the obtained experimental data. Analysis of the literature reveals that the experimental data concerning the P-T-x diagrams of binary fullerene-solvent systems are scarce due to the considerable difficulty of such experimental investigations. Up to this time, only few systems were studied: C60-1-hexanol and C70-1-hexanol (in the ranges of pressure 0.1 MPa–100 MPa and temperature 298.15 K–363.15 K) [24], C60 – toluene [25] (in the range of temperatures from 278.2 K to 308.2 K and pressures up to 340 MPa), C60 – n-hexane [26] (at 298.15 K in the range of pressure up to 400 MPa), C60 – toluene and C60 – water (in the temperature range 313 K–371 K and in the range of pressure 0.1 MPa–103.1 MPa) [27,28]. It is noteworthy

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that no studies on the solubility at atmospheric pressure of the individual light fullerenes in n-nonane were performed. 2. Experimental 2.1. Materials We have used samples of C60 fullerene (99.9% wt) and C70 fullerene (99.5% wt) purchased from ILIP, St. Petersburg, with controllable principal admixtures C70 in C60 and C60 in C70 of (0.1 and 0.5)% wt, respectively. The n-nonane sample was anhydrous (>99% wt) and purchased from Sigma Aldrich. The samples were used without further purification. The characteristics of the samples are indicated in Table 1. 2.2. Solubility measurements techniques at atmospheric pressure The temperature dependence of the light fullerenes (C60 or C70) solubility in n-nonane in the temperature range 293.3 K–353.3 K was carried out by the method of isothermal saturation in ampoules. The saturation time was equal to 8 h. Temperature was measured with an uncertainty of 0.1 K (k = 2). For the saturation of the fullerene solutions, a thermostatic shaker (LAUDA ET 20) was used at a shaking frequency x
80 Hz. The fullerenes concentrations (after the dilution and cooling of saturated solutions) was determined using a double-beam spectrophotometer (Specord M40, Karl Zeiss, Germany) at characteristic wavelengths of (335 and 472) nm corresponding to the maximum absorbance. The accuracy of wavelength was 0.5 nm, the photometric accuracy (DD) was 0.005, and the thick of the absorption layer was 1 cm. The experimental method was previously used to study the C60 (or C70) solubility in 1-hexanol [24]. The relative expanded uncertainty of the solubility values was 10%. Relative air humidity was (40–50)%. For the determination of the solvent content in solid crystal solutes, the following experimental method was used. The solid phase deposited from n-nonane solution was filtered on a Schott filter (porosity factor 10), rinsed quickly with ethanol, and then dried for (10–15) min at 293 K. Then, the solid phase was weighted, repeatedly washed with ethanol in a Soxhlet apparatus at 351 K and 0.101 MPa, dried for 1 h under vacuum (13.3 Pa) at 473 K, and weighed again. The weight change corresponded to the n-nonane content in the initial crystal solutes. The estimated uncertainty the solid solvate concentration in the mixture is 5% (k = 2). 2.3. Solubility measurement technique at high pressures High pressure phase equilibria measurements have been performed in a cylindrical stainless steel variable-volume view cell. Both the experimental device and procedure have been described in detail previously [29]. The cell supports working pressures and temperatures up to 100 MPa and 423 K, respectively. Two sapphire windows are located in the front and in the lateral wall of the cell. The second one permits lighting inside the cell whereas in the first one it is located an endoscope which allows us observe the sample under study. The pressure is measured by means of a pressure transducer (Kulite, model HEM375) with a typical uncertainty less Table 1 Provenance and mass fraction purity of the samples studied in this work.

a

Name

Supplier

Mole fraction puritya

Analysis method

n-Nonane C60 C70

Aldrich ILIP ILIP

>0.99 0.999 0.995

Gas chromatography Liquid chromatography Liquid chromatography

The purity analysis was performed by supplier.

than ± 0.03 MPa. The temperature is measured with a Pt100 probe with an uncertainty of ±0.02 K. Initially the cell was charged with a known amount of solution of the light fullerene (C60 or C70) in nnonane precisely measured with a Sartorius MC210P balance. The light fullerene concentration in this initial solution was determined using the spectrophotometric method. After that a weighted sample of fullerene powders was added to the cell. Under isothermal conditions, the mixture of known composition was compressed to achieve a single phase under continuous stirring. For a fixed temperature, several trials have been performed, being the lowest value associated to the experimental equilibrium pressure. After that, a new temperature is setup. When all the selected temperatures are investigated, a new portion of fullerene (C60 or C70) was added. The overall uncertainty of the fullerene weight fraction is 10% (k = 2). For the equilibrium pressure the uncertainty is 0.1 MPa (k = 2). 3. Results and discussion 3.1. Experimental values for solubility of C60 or C70 in n-nonane Table 2 contains values to illustrate the temperature dependence of solubility of the individual fullerenes (C60 or C70) in Table 2 Solubility of individual light fullerenes (C60, C70) in n-nonane at 0.1 MPa. w is the mass fraction of fullerene in the saturated solution in weight percentage, T – temperature. T/K

w (C60)/%

Solid phase

w (C70)/%

Solid phase

298.3

0.0044 0.0043 0.0044

C60n-C9H20

0.0030 0.0028 0.0029

C70

303.3

0.0045 0.0046 0.0044

C60n-C9H20

0.0032 0.0027 0.0033

C70

308.3

0.0045 0.0043 0.0045

C60n-C9H20

0.0036 0.0034 0.0037

C70

313.3

0.0046 0.0046 0.0043

C60n-C9H20

0.0049 0.0052 0.0049

C70

318.3

0.0047 0.0046 0.0048

C60n-C9H20

0.0071 0.0070 0.0068

C70

322.3

0.0082 0.0081 0.0080

C60n-C9H20

323.3

0.0139 0.0140 0.0142

C60n-C9H20 + C60

0.0090 0.0089 0.0091

C70

328.3

0.0149 0.0146 0.0145

C60

0.0105 0.0108 0.0104

C70

333.3

0.0155 0.0151 0.0152

C60

0.0108 0.0110 0.0108

C70

338.3

0.0158 0.0161 0.0162

C60

0.0120 0.0121 0.0119

C70

343.3

0.0165 0.0174 0.0176

C60

0.0137 0.0136 0.0138

C70

348.3

0.0207 0.0198 0.0196

C60

0.0151 0.0151 0.0149

C70

353.3

0.0225 0.0222 0.0223

C60

0.0162 0.0160 0.0158

C70

Expanded uncertainties (k = 2) are U(T) = ±0.1 K, Ur(p) = 0.5% and Ur(w) = 10%.

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Fig. 1. Temperature dependence of the fullerene solubility in n-nonane at 0.1 MPa, expressed as weight fraction of the fullerene in the solution, w: C60 (s) and C70 (d). Points represent the mean values of those reported in Table 2 for each temperature. Dashed line corresponds to crystallization of C60n-C9H20, solid lines correspond to crystallization of non-solvated fullerenes (C60 and C70). Dashed circle is a non-variant point, corresponding to simultaneous saturation of both solid phases.

n-nonane at 0.1 MPa, expressed as weight fractions of fullerene in the saturated solution. The mean solubility values versus temperature curve of C70 in n-nonane at 0,1 MPa (Fig. 1) has a sigmoid shape, which reflects a solvate-free nature of the dissolution. The solubility diagram consists of monovariant line of the crystallization of individual C70. Such types of dependences are rather often take place in the fullerene-containing binary systems (C70 – Cn-1H2n-1COOH (n = 6, 7, 8), C70 – 1,2dimethylbenzene, C70 – styrene, etc.) [4,7–9]. The solubility diagram of the C60 – n-nonane binary system can be characterized by non-monotonic temperature dependence of solubility and consists of two monovariant lines: the low-temperature branch of the solubility diagram corresponds to crystallization of the monosolvated fullerene C60n-C9H20, and the high-temperature branch corresponds to crystallization of non-solvated C60 (Fig. 1). Additionally, the C60 – n-nonane diagram consists of an invariant point corresponding to three phase equilibrium: solid C60 – solid C60n-C9H20 – saturated binary solution. Significant changes in the slopes of the crystallization branches often take place in binary systems, containing light fullerenes (for example in the case of C60-o-xylene system [7]. Experimental p-T-w data in the binary systems fullerene C60 – n-nonane (Fig. 2a) and fullerene C70 – n-nonane (Fig. 2b) in the range of pressures (0.1–100) MPa and in the range of temperatures (298.3–353.3) K are presented in Table 3 and in Fig. 2. In the case of C70 – n-nonane system in the whole range of temperatures and pressures the only one tentative solid phase was in the equilibrium with saturated liquid solution (non-solvated C70) and no solid phase transitions were detected. In the case of C60 – n-nonane system (Fig. 2a) we propose that the temperature range 298.3 K  T  320.8 K corresponds to the solubility of C60n-C9H20 and the temperature range 320.8 K  T  353.3 K corresponds to non-solvated C60. Also, we understand that the temperature of the C60n-C9H20 dissociation process (Tdiss) according to the reac(l) tion C60n-C9H20 (s) ! C(s) 60 + n-C9H20 is dependent on pressure, but according to our experimental data (solubility along isobars) we can conclude that the pressure dependence of Tdiss (in the pressure range 0.1 MPa  P  100 MPa) is weak and corresponds to the temperature range 318 K  Tdiss  323 K at all the investigated pressures.

Fig. 2. P-T-w values for binary systems C60 – n-nonane (a) and C70 – n-nonane (b) within the pressure range (0.1 – 100) MPa. (s) 298 K, (h) 303 K, (4) 308 K, (r) 313 K, (e) 318 K, (d) 323 K, (j) 328 K, (N) 333 K, (.) 338 K, (r) 343, (X) 348 K, (+) 353 K. Dashed area corresponds to crystallization of C60n-C9H20.

3.2. Thermodynamic description of the solid-liquid equilibrium in the individual light fullerene – n-nonane binary systems It is known that two-phase ða  bÞ equilibrium in the multicomponent system can be described by the following system of differential equations in vector – matrix form (Eqs. (1)-(3)) using Gibbs energy potential:

^ ðaÞ d~ ð~ X ðbÞ  ~ X ðaÞ ÞG X ðaÞ ¼ ½SðaÞ  SðbÞ þ ð~ X ðbÞ  ~ X ðaÞ ÞrSðaÞ dT  ½V ðaÞ  V ðbÞ þ ð~ X ðbÞ  ~ X ðaÞ ÞrV ðaÞ dP;

ð1Þ

^ ðbÞ d~ ð~ X ðaÞ  ~ X ðbÞ ÞG X ðbÞ ¼ ½SðbÞ  SðaÞ þ ð~ X ðaÞ  ~ X ðbÞ ÞrSðbÞ dT X ðaÞ  ~ X ðbÞ ÞrV ðbÞ dP;  ½V ðbÞ  V ðaÞ þ ð~

ð2Þ

^ ðbÞ d~ ^ ðaÞ d~ X ðaÞ  rSðaÞ dT þ rV ðaÞ dP ¼ G X ðbÞ  rSðbÞ dT þ rV ðbÞ dP; G ð3Þ where V

ðsÞ

and S

ðsÞ

are the molar volumes and entropies of phases

(s = a or b); rV ðsÞ and rSðsÞ are the gradients of these last properties with the concentration, ~ X ðsÞ is a vector, characterizing the state of X ðsÞ is the figurative point of the phase s in concentration space, d~ ðsÞ ~ a vector which characterize the displacement of X according with ^ ðsÞ is an operator, displacement of the two-phase equilibrium; G ðsÞ

corresponding to the matrix of the second derivatives Gij by Eq. (4):

given

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Table 3 Experimental (PTw) values in the binary systems C60 – n-nonane and C70 – n-nonane; w is the mass percentage of the fullerene in the saturated solution, T – temperature, P – pressure. In the case of the C70 – n-nonane system, the equilibrium with saturated liquid solution the tentative solid phase is non-solvated C70. For the C60 – n-nonane system we estimate that the temperature range 298.3 K  T  318.3 K corresponds to the solubility of C60n-C9H20, the range 323.3 K  T  353.3 K corresponds to nonsolvated C60. The equilibrium solid phase in the temperature range 318.3 K  T  323.3 K is not identified (italic type); the authors propose that this temperature region corresponds to an incongruent dissolution of solid crystalline solvate. P/MPa

w/%

C60 – n-nonane 298.3 K 29.6 0.0052 55.5 0.0062 64.0 0.0065 92.5 0.0076 313.3 K 21.3 29.4 66.9

0.0062 0.0065 0.0076

328.3 K 15.8 28.3 46.6 76.9

0.0161 0.0170 0.0182 0.0195

343.3 K 8.5 15.7 35.9 85.5

0.0182 0.0195 0.0225 0.0252

ðsÞ

ðlÞ

ðlÞ

ðlÞ

ðX 1  X 1 ÞG11 dX 1 ¼ Sðs!lÞ dT  V ðs!lÞ dP;

ð5Þ

where Sðs!lÞ , V ðs!lÞ are the changes of S or V function in the process of formation of one mole of phase l from an infinitely big mass of the phase s at constant T and P given by Eqs. (6), (7).

"

S

ðs!lÞ

ðlÞ

¼ S S

ðsÞ

þ

ðsÞ ðX 1



ðlÞ X1 Þ



@S @X 1

ðlÞ # ;

ð6Þ

"  ðlÞ # @V ðsÞ ðlÞ : V ðs!lÞ ¼ V ðlÞ  V ðsÞ þ ðX 1  X 1 Þ @X 1

ð7Þ

P/MPa

w/%

P/MPa

w/%

303.3 K 20.1 40.1 54.0 84.3

0.0052 0.0062 0.0065 0.0076

308.3 K 12.0 35.01 42.8 78.6

0.0052 0.0062 0.0065 0.0076

318.3 K 16.6 21.4 31.8 64.9 92.9

0.0062 0.0065 0.0076 0.0100 0.0115

323.3 K 27.9 40.2 69.3 95.8

Similar equations may be written for the equilibrium of the fullerene crystalline solvates (in our case C60n-C9H20) and saturated liquid fullerene solution. On the basis of Eq. (5) we can easily derive some qualitative analogues of some well-known thermodynamic laws:

0.0161 0.0170 0.0182 0.0195

1. When X 1 ¼ const we can obtain Eq. (8)

333.3 K 18.0 29.8 48.7 93.0

ðlÞ



@P @T

 ðlÞ

¼

Sðs!lÞ V ðs!lÞ

<0

ð8Þ

0.0170 0.0182 0.0195 0.0225

338.3 K 17.5 31.6 60.6 97.2

0.0182 0.0195 0.0225 0.0252

348.3 K 15.6 38.3 85.6

0.0225 0.0252 0.0284

353.3 K 16.8 43.2 94.6

0.0252 0.0284 0.0301

C70 – n-nonane 298.3 K 94.2 0.0042

303.3 K 61.4

0.0042

308.3 K 23.3 99.7

0.0042 0.0064

313.3 K 33.5 61.3 96.4

0.0064 0.0075 0.0096

318.3 K 11.0 51.4 94.6

0.0075 0.0096 0.0114

323.3 K 15.8 50.9 80.5 98.0

0.0096 0.0114 0.0130 0.0145

328.3 K 22.2 55.2 73.9 99.6

0.0114 0.0130 0.0145 0.0164

333.3 K 29.3 49.5 72.5 98.4

0.0130 0.0145 0.0164 0.0182

338.3 K 10.3 30.1 54.3 84.0 99.8

0.0130 0.0145 0.0164 0.0182 0.0195

343.3 K 11.2 38.3 59.6 89.1

0.0145 0.0164 0.0182 0.0201

348.3 K 23.9 48.2 78.8 98.0

0.0164 0.0182 0.0201 0.0215

353.3 K 3.0 26.8 51.9 74.9 98.5

where is the molar entropy of dissolution of the solid fullerene with formation of very diluted solution (in the extreme case, an infinitely diluted solution). In the case of equilibrium fullerene crystalline solvate (C60n-C9H20)-saturated solution we can easily obtain Eq. (10.2):

0.0164 0.0182 0.0201 0.0215 0.0233

Sðs!lÞ
S2 þ S1  S2
DSsol cr ;

Expanded uncertainties (k = 2) are U(T) = ±0.02 K, U(P) = 0.1 MPa and Ur(w) = 10%.

ðsÞ Gij

¼

@ 2 GðsÞ

ðsÞ ðsÞ @xi @xj ðsÞ

! ;

ð4Þ

T;P;xk–j;n

is the molar Gibbs energy potential of the phase s [30]. ^ ðsÞ According to the phase stability criterion, the matrix of the G operators is positively definite, as well as the minors of its main diagonal [30]. Let us consider the two-phase equilibrium of saturated fullerene solution (phase l) – non-solvated light fullerenes (C60 or C70) (phase s). Bottom index (1) will be referred to n-nonane, and index (2) to individual light fullerene (C60 or C70). In the case of description of the two-phase solid (s) – liquid (l) equilibrium in binary systems we can rewrite Eqs. (1)-(3) in the scalar form (see Eq. (5)): where G

X1

The sign of the ð@P=@TÞX ðlÞ derivative may be easily determined 1

from the obtained experimental data (see Tables 2, 3). Obviously we can rewrite Eq. (6) in another form and to obtain Eq. (9): ðlÞ ðlÞ

ðlÞ ðlÞ

ðsÞ

ðsÞ

ðlÞ

ðlÞ

ðlÞ

Sðs!lÞ ¼ ½X 1 S1 þ X 2 S2  S2 þ ðX 1  X 1 ÞðS1  S2 Þ;

ð9Þ

ðaÞ

where Si is the partial molar entropy of the ith component in the phase a. The partial molar entropy of individual fullerene in the solid phase (or of the fullerene crystalline solvates) is equal to its average molar entropy. Taking into account the assumption of simðlÞ

ðlÞ

ðsÞ

ilar order of the S1 ; S2 ; S1 functions and due to the fact that the fullerenes solutions in n-nonane are very diluted (X(l) 2 takes values ðlÞ

ðsÞ

in the range 105–106 so X 1
1; and X 1 ¼ 0) we can obtain from Eq. (9) the Eq. (10.1) ðlÞ

ðsÞ

Sðs!lÞ
S2  S2
DSsol 2 ;

ð10:1Þ

DSsol 2

ðlÞ

ðlÞ

ðsÞ

ð10:2Þ

ðlÞ

where: S1 is the partial molar entropy of the liquid solvent n-C9H20. The volume change in the process of formation of one mole of phase (s) from a infinitely big mass of the phase (l) at constant T and P, V(sl) can be expressed by Eq. (11), which can be obtained from Eq. (7) in a similar way as Eq. (9): ðlÞ

ðlÞ

ðlÞ

ðlÞ

ðsÞ

ðsÞ

ðlÞ

ðlÞ


1,

ðsÞ X1

ðlÞ

V ðs!lÞ ¼ ½X 1 V 1 þ X 2 V 2  V 2 þ ðX 1  X 1 ÞðV 1  V 2 Þ Taking again into account that

ðlÞ X1

ð11Þ

¼ 0 and ðlÞ

together with the assumption of similar order of the V 1 ; functions we can obtain Eqs. (12.1) and (12.2): ðlÞ

ðsÞ

V ðs!lÞ
V 2  V 2
DV sol 2

ðlÞ X2 ðlÞ V2 ;


0 ðsÞ

V1

ð12:1Þ

where DV sol 2 is the molar volume change in the dissolution process of light solid fullerene with formation of infinitely diluted solution. In the case of equilibrium fullerene crystalline monosolvate - saturated solution we can obtain Eq. (12.2): ðlÞ

ðlÞ

ðsÞ

V ðs!lÞ
V 2 þ V 1  V 2
DV sol cr

ð12:2Þ

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K.N. Semenov et al. / J. Chem. Thermodynamics 112 (2017) 259–266 ðlÞ

where V 1 is the partial molar volume of the liquid solvent n-C9H20. Also we can unify the similar effects (in the cases of individual fullerene-saturated solution and fullerene crystalline monosolvate – saturated solution equilibria) and denote DV sol and DV sol 2 cr as sol sol DV sol , and DSsol 2 and DScr as DS . Thus, we can rewrite Eq. (8) and to obtain Eq. (13):   @P DSsol ¼ <0 ð13Þ @T XðlÞ DV sol 1

It can be seen that Eqs. (8) and (13) are analogues of the Clausius–Clapeyron relation. These equations describe the P-T dependences in the two-phase systems in the following cases: (i) in one-component systems; (ii) in azeotropic n-component systems, and (iii) in the n-component systems containing a liquid phase with constant composition [31]. We can conclude that in the studied fullerene-n-nonane systems the contrary change of pressure and temperature takes place.

4. The analogue of the third Gibbs-Konovalov law for our systems also cannot be formulated, because the partial derivatives (@X(1 s) (s) (l) /oX(l) 1 )T and (@X1 /oX1 )p are null, i.e., they cannot be positive as requested in this law. 5. Additionally we can derive semi-quantitative equations (Eq. (21)) for determining the differential molar volume and entropy effects of the phase transition (see Eq. (5)): ðlÞ

G11 ¼ ð@ 2 G=@X21 ÞT;P ¼ ð@ l1 =@X1 ÞT;P  ð@ l2 =@X1 ÞT;P ; ðlÞ

ðlÞ

ðlÞ

ð@P=@X 1 ÞT ¼ ½G11 =DV sol > ð<Þ0; if DV sol > ð<Þ0 ðlÞ

ðlÞ

ð@T=@X 1 ÞP ¼ ½G11 =DSsol > ð<Þ0; if DSsol < ð>Þ0; ðlÞ

ðlÞ

ð@X 2 =@PÞT ¼ DV sol =G11 > ð<Þ0; if DV sol < ð>Þ0 ðlÞ

ðlÞ

ð@X 2 =@TÞP ¼ DSsol =G11 > ð<Þ0; if DSsol > ð<Þ0;

ðlÞ cðlÞ 1
const
1; c2
const–1;

where c is the activity coefficient of ith component in the liquid solution.Thus, we can rewrite the Eq. (21) and to obtain Eq. (23):

thus DV sol < 0

ð18Þ

ðlÞ

thus DSsol > 0

ð19Þ

ð@X 2 =@TÞP > 0;

Also we can formulate the analogue of the first GibbsKonovalov law for our systems: At constant temperature, the solubility of both light fullerenes in n-nonane increases when the pressure increases then the change of molar volume in the process of the light fullerenes dissolution with formation of diluted solution is negative. At constant pressure, the solubility of both light fullerenes in n-nonane increases when the temperature increases then the molar entropy change in the process of the light fullerene dissolution with formation of diluted solution is positive. 3. Besides, we can analyse the feasibility of the second GibbsKonovalov law in the case where of the pressure temperature curve of the (l-s) equilibrium has an extremum (analogue of azeotrope [32,33]). Eq. (5) leads to the following result along the equilibrium curves:

dT P ¼ 0; dPT ¼ 0 when

ðsÞ 1

¼

ðlÞ X1

ð20Þ

However in the case of the present systems the equality of comðsÞ

positions cannot be possible because X 1 ¼ 0 (one-component sysðlÞ X1

ðlÞ

ðlÞ

> 0 (binary system). Thus we can conclude the tem) and 1 > following: Temperature (at constant pressure) or pressure (at constant temperature) of the two-phase equilibrium cannot pass through an extremum, so in the considered systems the analogous of azeotropes are not possible.

ð23Þ

According to Eqs. (16), (17), (23) we can obtain Eqs. (24), (25): ðlÞ

ð@RT ln X 2 =@TÞP
DSsol

ðlÞ

ð@X 2 =@PÞT > 0;

ðlÞ

G11 ¼ RT=ðX 1 X 2 Þ
RT=X 2

ð15Þ

From Tables 2, 3 it can be concluded for both fullerene-nnonane systems the following signs for the partial derivatives (Eqs. (18), (19)):

ð22Þ

(l) i

ð@ ln X 2 =@PÞT RT
DV sol ;

ð17Þ

ð21Þ

where l is the chemical potential of ith component in the liquid phase. In the studied case, we can easily make the assumption that the fullerene solutions in n-nonane are infinitely diluted, so we can postulate Eq. (22):

ð14Þ

ð16Þ

ðlÞ

(l) i

ðlÞ

2. In the case of isothermal changes or isobaric changes we can obtain Eqs. (14)–(17):

ðlÞ

ð24Þ

ðlÞ

ð25Þ

3.2.1. Thermodynamic calculations for the individual light fullerene – n-nonane binary systems under isothermal conditions For determination of the differential molar volume and entropy effects of the phase transition we have assumed the folðlÞ

lowing approximation: that the values of ðln X 2 ÞRT at a constant ðlÞ ðln X 2 ÞT RT,

follow a quadratic function of pressure temperature T, in the temperature range (298.3 K  T  320.8 K) for monosolvated C60, in temperature range (320.8 K  T  353.3 K) for nonsolvated C60 and in the temperature range 298.3 K  T  353.3 K for C70, being the coefficients temperature-dependent according to Eq. (26): ðlÞ

ðln X 2 ÞT RT ¼ AðTÞ þ BðTÞP þ CðTÞP2 ;

ð26Þ

being X2 the mole fraction of C60 or C70 in the saturated solution. Thus, DV sol ðP; TÞ can be approximated as Eq. (27): ðlÞ

ð@ ln X 2 =@PÞT RT
DV sol
BðTÞ  2CðTÞP;

ð27Þ

ðlÞ

Fig. 3 shows the result of the ðln X 2 ÞT RT ¼ f ðPÞ function approximation using Eq. (26) for the C60-n-nonane (at 318.3 K) and C70-nnonane (at 353.3 K) binary systems. According to Eq. (28) we can calculate the change of the isothermal compressibility in the process of the fullerene dissolution (Dbsol T ) with formation of infinitely diluted solution according to Eq. (28):

Dbsol T

! !   1 @ DV sol @ ln DV sol ¼ ¼ @P @P DV sol T T

ð28Þ

It is necessary to point out that the average molar volume (V) and isothermal compressibility (bT ) are positively defined in the region of thermodynamic stability of equilibrium phases, but the

D V sol and Dbsol T functions can have the arbitrary signs. In our case is nonlinear and in the first approximation is the function Dbsol T weakly dependent on temperature (see Table 4 and Eq. (29)):

Dbsol T


2CðTÞ ; ½BðTÞ þ 2CðTÞP

ð29Þ

D(P/MPa)

1.64 + (P-20)0.033 1.12  (P-20)0.016 1.15  (P-20)0.0012

E(P/MPa)

K.N. Semenov et al. / J. Chem. Thermodynamics 112 (2017) 259–266

970  (P-20)20.3 -800 + (P-20)10.6 747 + (P-20)0.67

264

Fig. 5 shows the result of the temperature dependence of ðlÞ

ðRT ln X 2 =@TÞP for the binary systems C60-n-nonane (at 20 MPa) and C70-n-nonane (at 80 MPa) using Eq. (31). Table 4 shows the obtained D(P) and E(P) functions. The change of the isobaric heat capacity in the process of the fullerene dissolution (DC sol P ) with formation of infinitely diluted solution is given by Eq. (32):

DC sol P ¼

@ DSsol @T

! T

ð32Þ

P

Using Eqs. (31) and (32) we obtain Eq. (33):

DC sol P ¼ 2EðPÞT;

ð33Þ

From the E(P) functions given in Table 4, it can be concluded that the branch of crystallization of C60n-C9H20 in the range of pressures 0:1 MPa 6 P 6 100 MPa and the branch of crystallization of C60 in the range of pressures 0:1 MPa 6 P 6 90 MPa correspond sol becomes negative to positive values of DC sol P , although the DC P on the branch of crystallization of C60 at P > 90 MPa. In the case

of the C70 –n-nonane system the function DSsol changes the sign

F(P/MPa) C(T/K)

0.045  (T-298.3)0.0071 0.025  (T-323.3)0.0030 0.0073  (T-313.3)0.0012 20.4 + (T-298.3)1.01 10.5 + (T-323.3)0.30 21.4-(T-313.3)0.20

B(T/K)

ð31Þ

A(T/K)

ðlÞ

ð@ðRT ln X 2 Þ=@TÞP
DSsol
DðPÞ þ 2EðPÞT;

298:3K 6 T 6 320:8K 320:8K 6 T 6 353:3K 298:3K 6 T 6 353:3K

Thus, the DSsol ðP; TÞ dependence can be approximated by Eq. (31):

T/K

where X2 is the mole fraction of C60 or C70 in saturated solution.

C60n-C9H20 C60 C70

ð30Þ

Solid phase

ðlÞ

ðln X 2 ÞP RT ¼ FðPÞ þ DðPÞT þ EðPÞT 2 ;

Table 4 Functions A(T), B(T), C(T), F(P), D(P), E(P) of the Eqs. (26) and (30) for the light fullerenes dissolution in n-nonane.

ðlÞ

C60 or C70 in saturated solution, X 2 , in the temperature range (298:3K 6 T 6 320:8K) corresponding to crystallization of C60nC9H20, (320:8K 6 T 6 353:3K) to the crystallization of C60 and (313:3K 6 T 6 353:3K) to the crystallization of C70 were fitted using the Eq. (30):

29300  (T-298.3)45 28300  (T-323.3)43 30400  (T-313.3)17

3.2.2. Thermodynamic calculations in the individual light fullerene – n-nonane binary systems under isobaric conditions The isobaric data are presented in Fig. 4 versus temperature. The data presented in Fig. 4 (temperature dependences of solubility at various pressures) were obtained by interpolation according to obtain pTw experimental values. Second order polynomial temperature dependences of solubility expressed as mole fraction of

181000 + (P-20)4500 112000  (P-20)16000 -150700  (P-20)78

ðlÞ

Fig. 3. Isothermal ðln X 60;70 ÞT RT values as a function of pressure. Experimental ðlÞ values of ðln X 60;70 ÞT for the binary system C60 – n-nonane at 318.3 K (s) and for C70 – n-nonane at 353.3 K (d). Lines represent the results of Eq. (26) with the parameter values indicated in Table 2.

265

K.N. Semenov et al. / J. Chem. Thermodynamics 112 (2017) 259–266 2 sol ð@ DSsol =rPÞT ¼ ð@ DV sol =@TÞP ¼ @ 2 Dlsol 0 =@P@T ¼ @ Dl0 =@T@P;

ð34Þ where: ðlÞ ðsÞ Dlsol 0 ¼ l0  l0 ;

ð37Þ

is the change of standard chemical potential in the process of dissolution of fullerenes (C60 or C70) or of the solvated fullerene C60nC9H20 in n-nonane with formation of infinitely diluted liquid solutions. In terms of our approximation (see Table 4) we can easily determine the mixed derivatives according to the following equations:

ð@ DSsol =@PÞT ¼ ðdD=dPÞ þ 2ðdE=dPÞT;

ð38Þ

ð@ DV sol =@TÞP ¼ ðdB=dTÞ þ 2ðdC=dTÞP

ð39Þ sol

Thus, we can calculate the mixed derivatives ð@ DS =@PÞT as function of T and ð@ DV sol =@TÞP as function P. The borders of the ð@ DSsol =@PÞT and ð@ DV sol =@TÞP derivatives changes are presented in the Fig. 6 for the processes of dissolution of the individual light fullerenes (C60 and C70) as well as for solvated fullerene C60nC9H20. One can see a satisfactory agreement between these functions taking into account that successive derivatives lead to high uncertainties. The lower discrepancies in the system C70 – nC9H20 in comparison with those of C60 – n-C9H20 are connected with the fact that the values of mixed derivatives ð@ DSsol =@PÞT and ð@ DV sol =@TÞP for the first system are considerably lower than for the second system, but the relative discrepancies in both systems are similar.

Fig. 4. P-T-w values for binary systems C60 – n-nonane (a) and C70 – n-nonane (b). (s) 20 MPa, (h) 40 MPa, (4) 60 MPa, (r) 80 MPa, (e) 100 MPa.

3.3. Equation of solid-liquid equilibrium in the fullerene-n-nonane binary systems Also, we can formulate the approximated differential equation of the (s)-(l) equilibrium (see Eq. (5)) by Eq. (40): ðlÞ

dRT ln X 2 ¼ DV sol dP þ DSsol dT ¼ ðBðTÞ þ 2CðTÞÞdP þ ðDðPÞ þ 2EðPÞTÞdT;

ð40Þ

and also formulate the alternative integral equations of the (s)-(l) equilibrium (see Eqs. (26), (30)):

ðlÞ

Fig. 5. Isobaric ðln X 60;70 ÞP RT values as a function of temperature. Experimental values for the binary systems C60 – n-nonane (s) at 20 MPa and C70 – n-nonane (d) at 80 MPa. Lines represent the results of Eq. (30) with the parameter values indicated in Table 2.

in a narrow temperature region T ¼ 323  5K, at lower temperatures DSsol >0, and at higher temperatures DSsol < 0 (in the pressure range 0:1 MPa 6 P 6 100 MPa), whereas DC sol P < 0 in all cases. 3.2.3. Verification of the calculated thermodynamic parameters It is possible to check the goodness of the approach of both sol

sol

DS ðT; PÞ and DV ðT; PÞ functions by the parity of mixed derivatives (see Eq. (34)).

Fig. 6. Borders of the (dDSsol/dP)T and (dDVsol/dT)P derivatives, changes in the process of fullerenes (C60, C70) and solvated fullerene (C60n-C9H20) dissolution. I) (dDSC60C9H20/dP)T against temperature; II) (dDVC60C9H20/dT)P against pressure; III) (dDSC60/dP)T; IV) (dDVC60/dT)P; V) (dDSC70/dP)T; VI) (dDVC70/dT)P.

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K.N. Semenov et al. / J. Chem. Thermodynamics 112 (2017) 259–266

Additionally, we have considered inexpedient to perform our calculations under simultaneous variation of T and P parameters of the DSsol and DV sol functions, and to carry out the calculation of the polynomial parameters (BðT; PÞ;CðT; PÞ, etc) directly from the experimental data due to the variable fitting parameters have very low stability if the number of such parameters is relatively big and higher stability in the case of constrained number of parameters, but in the latter case the goodness of the fitting is low. 4. Conclusions The solubility of light fullerenes (C60 and C70) in n-nonane was investigated within the pressure range of (0.1–100) MPa and temperature range from 298.3 K to 353.3 K. Along isotherms, solubility increases monotonously with increasing pressure. Thermodynamic calculations in the individual light fullerene (C60, C70) – n-nonane binary systems under isobaric and isothermal conditions were performed. At atmospheric pressure, we have observed that in the case of C60 – n-nonane binary system the temperature dependence of solubility is non-monotonic in contrast with the C70 – n-nonane system. This fact is connected with de-solvatation of the C60nC9H20 solvate (the solubility diagram consists of an invariant point corresponding to phase transition of the monosolvated C60). Acknowledgements The work was supported by the interuniversity exchange program between Saint-Petersburg State University (Russia) and University of Santiago de Compostela (Spain) and by Grant of President of Russian Federation for supporting of young scientists MK4657.2015.3. Part of this research was performed by using the equipment of the Resource Center ‘GeoModel’ and Center for Chemical Analysis and Materials Research of Research park of St. Petersburg State University. J.F. and E.R.L. acknowledge the financial support of Spanish Ministry of Economy and Competitiveness and of UE FEDER (ENE2014-55489-C2-1-R) and of Xunta de Galicia (AGRUP2015/11 and GRC ED431C 2016/001). References [1] L.N. Sidorov, M.A. Yurovskaya, Fullerenes, Ekzamen, Moscow, 2005. [2] F. Cataldo, T. da Ros, Carbon Materials: Chemistry and Physics: Medicinal Chemistry and Pharmacological Potential of Fullerenes and Carbon Nanotubes, Springer, 2008. [3] A.V. Eletskii, V.Yu. Zitserman, G.A. Kobzev, High Temp. 53 (2015) 130–150.

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JCT 16-566