Article pubs.acs.org/jced
A New Scaling Principles−Quantitative Structure Property Relationship Model (SP-QSPR) for Predicting the Physicochemical Properties of Substances at the Saturation Line Vitaly Zhelezny,† Vitaliy Sechenyh,*,‡ and Anastasia Nikulina† †
Institute of Power Engineering and Ecology, Odessa National Academy of Food Technologies, 1/3 Dvorynskaya Street, Odessa, 65082 Ukraine ‡ Microgravity Research Center, CP-165/62, Université Libre de Bruxelles, 50, av. F. D. Roosevelt, Brussels, B-1050 Belgium ABSTRACT: In recent years, there has been rapid development in new methods for calculating the physicochemical properties of substances based on scaling principles (SP) and quantitative structure−property relationships (QSPR). This paper presents an extended SP-QSPR model that can be used to predict the refractive index, surface tension, density, and viscosity of liquids on the saturation line. The temperature dependence of the molar refractivity is also considered. In this paper, we propose a new additive increment scheme for alkanes and its halogenated derivatives which can be used for calculations of orthochor, scaling values of parachor, molar refractivity, critical molar volume and the molar volume of a liquid subcooled to the absolute zero (T = 0 K). The correlation between structural-additive properties and critical molar volume is also discussed.
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INTRODUCTION To date, science and industry have not yet resolved the problem of how to guarantee reliable data about the physicochemical properties of substances. Industry uses a large number of heat-transfer media working fluids, intermediate and final products of chemical synthesis. Therefore, information about the properties of fluids cannot be gathered using only experimental methods. On the other hand, sometimes it is not possible or very difficult to calculate the properties of polymolecular substances and solutions to the required accuracy using methods based on statistical physics and thermodynamics. For precisely these reasons, the question of further developing models to predict the physicochemical properties of substances using minimal amounts of available experimental data remains pertinent. A considerable number of different phenomenological approaches1−4 have so far been suggested to calculate the physicochemical properties of substances. It should also be mentioned that computational structure−property models were proposed by authors.4−17 However, experience of using structure−property models to calculate the physicochemical properties of substances has shown that such methods have certain weaknesses. A succession of authors1,3,5,9 have emphasized that the macrophysical properties of a substance y and the values representing the relationship between these properties y′ are only indirectly related to its composition and molecular structure. Factors such as mass, volume, shape, polarizability, dielectric permeability and the dipole moment of a molecule can have a significant influence on a substance’s properties. Therefore, in some cases © XXXX American Chemical Society
the aforementioned values y and y′ will not follow the additivity rule. It is obvious that the changes to the properties of a molecule that are brought about by the introduction of some sort of new element (for example, a new atom or group of atoms) do not only depend on the characteristics of the element that is introduced. Moreover, its physicochemical properties can depend on two or more of the above factors simultaneously, which complicates the development of calculation methods that are based on the structure−property model. Therefore, such models usually use to calculate the thermal and caloric properties of certain homologous series of substances. In order to improve the quality of predictions some authors have recommended that substances should be classified according to the values of Stiel’s polar factor,1 intermolecular interaction, etc. Therefore, introducing various fitting coefficients, which can take into account changes in the behavior of intermolecular interaction when certain atoms are substituted or added within a molecule of the substance into structure− property models helps to improve the quality of calculations regarding the properties of substances.10 Moreover, proposed methods for calculating the physicochemical properties of substances usually do not consider the temperature dependence of structural-additive properties such as the parachor, molar refractivity and the molar speed of sound.3 Hence, it is not possible using mentioned structure− Received: October 23, 2013 Accepted: January 10, 2014
A
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Table 1. Values for the Structural-Additive Properties and Complexes of Certain Substances V0
Vc
Vnb
molecular formula
Or
[P]c
Ac
cm3/mol
cm3/mol
cm3/mol
C2H6 (ethane) C3H8 (propane) C4H10 (butane) C6H14 (hexane) C7H16 (heptane) C9H20 (nonadecane) C10H22 (decane) C12H26 (dodecane) C13H28 (tridecane) C14H30 (tetradecane) C15H32 (pentadecane) C16H34 (hexadecane) C17H36 (heptadecane) C20H42 (eicosane) CCl3F (trichlorofluoromethane) CCl2F2 (dichlorodifluoromethane) CClF3 (chlorotrifluoromethane) CF4 (tetrafluoromethane) C2Cl3F3 (1,1,2-trichloro-1,2,2-trifluoroethane) C2Cl2F4 (1,2-dichloro-1,1,2,2-tetrafluoroethane) C2ClF5 (chloropentafluoroethane) C2F6 (hexafluoroehtane) CH2F2 (difluoromethane) C2ClH3F2 (1-chloro-1,1-difluoroethane) C2H3F3 (1,1,1-trifluoroethane) C2H2F4 (1,1,1,2-tetrafluoroethane) CH(CH3)3 (2-methylpropane) CF3CH2CHF2 (1,1,1,3,3-pentafuolropropane) C2HF5 (pentafluoroethane) CHF3 (trifluoromethane) CHClF2 (chlorodifluoromethane) C3F8 (octafluoropropane) C2H4F2 (1,1-difluoroethane) (CH3)2O (dimethylether) CHF2CF2CH2F (1,1,2,2,3-pentafluoropropane) CF3CHFCHF2(1,1,1,2,3,3-hexafluoropropane) CF3CH2CF3 (1,1,1,3,3,3-hexafluoropropane) CHClFCF3 (1-chloro-1,2,2,2-tetrafluoroethane)
45.9 62.0 78.9 113.1 131.3 162.2 183.3 212.7 230.7 245.7 261.5 280.8 302.0 347.5 78.6 67.9 56.6 46.3 105.9 95.0 82.0 72.1 37.5 68.7 60.1 63.5 80.2 83.5 69.8 40.4 51.5 92.6 55.0 55.8 84.4 88.7 88.8 77.5
116.4 157.3 194.4 276.8 322.0 411.6 458.9 547.8 591.5 637.6 685.0 737.4 790.4 943.6 196.7 167.8 136.2 102.5 246.7 220.9 188.0 160.5 94.5 173.9 145.4 149.7 197.0 199.0 156.0 100.1 131.6 215.2 136.0 132.5 197.2 203.7 208.5 184.6
383.6 524.1 662.4 955.3 1105.3 1420.0 1583.0 1889.3 2039.9 2198.6 2361.7 2542.1 2724.4 3250.7 649.2 561.5 467.8 367.9 858.9 768.3 667.1 579.2 308.5 584.7 525.6 518.7 669.9 686.0 527.1 344.2 434.3 764.0 469.4 426.3 696.0 693.0 708.0 640.0
38.65 52.41 66.09 94.08 108.20 137.19 150.72 179.81 194.64 209.82 224.58 240.22 253.50 298.58 63.87 55.79 47.26 37.64 84.58 76.47 68.39 59.00 29.62 57.31 48.85 48.86 66.70 69.55 53.43 32.95 42.40 75.94 45.33 45.03 67.30 69.43 70.84 62.91
145.6 201.8 255.1 369.6 431.9 552.5 609.7 718.7 768.1 826.6 885.0 943.5 1001.9 1177.2 248.0 214.0 179.2 140.6 334.6 294.7 251.9 225.0 122.6 224.3 195.0 199.3 259.1 259.3 209.2 133.0 165.0 299.4 179.5 166.3 256.0 270.0 275.8 243.7
55.26 75.87 96.61 140.48 163.03 210.46 235.42 287.4 314.2 341.08 368.56 396.3 424.51 511.07 92.80 81.26 68.62 54.86 124.16 112.52 99.81 85.94 42.87 84.21 72.01 74.07 97.82 98.15 79.25 48.40 61.33 116.59 65.28 62.63 96.66 102.48 105.17 92.56
boiling and condensation lines for a wide range of temperatures.
property models to predict properties such as viscosity, surface tension, density, refractive index, etc. to the necessary degree of accuracy for a wide range of parameters, particularly in the wide vicinity of the critical point. Despite the observations, structure−property models have a great deal of potential for predicting the physicochemical properties of substances. This goal can be achieved by introducing few-constant equations based on scaling principles into the calculation methodology.10,12−15,18−20 This approach is promising for several reasons. First, it will extend the range of parameters for which it is possible to use structure−property models. Second, it becomes possible to use sets of the universal amplitudes. These sets include thermodynamically consistent critical parameters, exponents, and amplitudes for a large number of thermophysical properties of fluids.10,12−15,18,19 Using the example of alkanes and their halogenated derivatives, we present the development of QSPR models designed for the calculation of critical parameters, viscosity, surface tension, refractive index and density differences on the
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COMPUTATIONAL METHODS
Structural-additive properties and complexes such as the critical molar volume, molar refractivity, parachor, and orthochor are widely used to predict the thermophysical properties of fluids. In the present study we analyze these structural-additive properties in more detail, and also consider the correlation between them and the critical molar volume. Parachor. One of the most frequently used structuraladditive properties is parachor, [P]. In 1924, Sugden6 proposed calculating this value according to the formula [P] = M 4 B =
Mσ 0.25 , ρ′ − ρ″
(1)
where B is the coefficient, depending on the individual properties of the fluids; σ is the coefficient for surface tension, B
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Table 2. Relationships between Structural-Additive Properties and Complexes Ac
[P]c
V0
Vc
Vnb
8.467·Or AAD = 3.78
2.467·Or AAD = 4.09 0.2914·Ac AAD = 1.87
0.815·Or AAD = 2.42 0.09625·Ac AAD = 3.00 0.3304·[P]c AAD = 3.04
3.212·Or AAD = 3.1 0.3795·Ac AAD = 1.79 1.302·[P]c AAD = 2.37 3.942·V0 AAD = 2.09
1.269·Or AAD = 6.6 0.1499·Ac AAD = 3.31 0.5145·[P]c AAD = 4.23 1.557·V0 AAD = 5.69 0.3951·Vc AAD = 4.66
Or Or Ac
0.1181·Ac AAD = 3.83 0.4053·[P]c AAD = 4.18 1.227·V0 AAD = 2.40 0.3113·Vc AAD=3.12 0.788·Vnb AAD = 6.77
[P]c V0 Vc Vnb
3.432·[P]c AAD = 1.86 10.39·V0 AAD = 2.90 2.635·Vc AAD = 1.78 6.671·Vnb AAD = 3.43
3.027·V0 AAD = 2.92 0.768·Vc AAD = 2.33 1.944·Vnb AAD = 4.39
N·m−1; ρ′ and ρ″ represent liquid and vapor density on the saturation line, kg·m−3. A large number of experimental and theoretical studies have been dedicated to examination the parachor values of various fluids. Lennard-Jones and Sugden1,6 considered that parachor is linked to a molecule’s volume. Several authors21 have attempted to provide a theoretical justification for the parachor concept. Moreover, there are known correlations between parachor and critical parameters, parachor and Lennard-Jones potential parameters21 and also with critical amplitudes of various thermodynamic functions.12 Thus, parachor plays a very important role in the formation of new methods of computational analysis of the thermophysical properties of fluids. At the same time, questions about the temperature dependence of parachor and the value of the exponents in eq 1 have not been sufficiently studied.10 Some authors10,12 introduced the idea of Ising parachor as a critical amplitude set, which could be calculated using the formula [P]c =
[π ] =
f (n) =
f (n) =
(4)
where F(t) and f(t) are universal crossover functions for nonassociated fluids. Values for crossover functions in the range from the triple point to the critical point can be calculated using the following equations:10,20 F(t ) = 1 − 1.2278
t2 t3 + 1.3282 , ln t ln t
A V
(8)
n2 − 1 n2 + 2
(9)
Thanks to its theoretical validity, the Lorentz−Lorenz formula has virtually replaced other formulas for calculating molar refractivity that were developed empirically. However, various authors9,22−24 have noted the existence of temperature dependence for function. Experiments carried out by Bottcher25 show that function f(n) should include certain individual properties of a substance−its polarizability α and molecular radius k:
(3)
σ = σ0t μf (t )
(7)
The coefficient A, specific to a given substance, is known as a molar refractivity. A fair number of attempts have been made to find types of function f(n) for which there is weak dependence between temperature and pressure.9 Both Lorenz in 1880 and Lorentz somewhat earlier concluded that the function f(n) should look as follows:9
where σ0 and ρ0 are critical amplitudes which depend on the properties of the thermodynamic system being considered; β and μ are individual critical indices for different substances for density and surface tension, which authors of the papers10,15,20 propose to determine as fitting coefficients in following extended scaling equations: δρ = ρ′ − ρ″ = ρ0 t
[P] = t β[f (t ) − F(t )] [P]c
where [P] is the parachor value for a given temperature. Ising parachor values for certain alkanes and their halogenated derivatives are presented in Table 1. As was demonstrated by Zhelezny et al.,10 the parachor of different fluids is functionally related to critical molar volume (see Table 2). Molar Refractivity. Theoretical studies of the relation between density ρ and refractive index n and analysis of experimental data both demonstrate linear dependence between an arbitrary function f(n) and the molar volume of a liquid V:
(2)
βF(t )
2.531·Vnb AAD = 4.81
temperature range. This correlation makes it possible to take the temperature dependence of parachor into account:
Mσ0β / μ ρ0
0.2537·Vc AAD = 2.14 0.6423·Vnb AAD = 5.95
α k
(
(n2 − 1) 2n2 + 1 + 2 3 (n2 − 1)
(5)
f (n) =
t 1.5 t2 t3 f (t ) = 1 − 0.03534 − 0.31656 + 0.34246 , ln t ln t ln t
n
2
) (10)
From the above it follows that currently no universal function f(n) exists that does not depend on the individual properties of a substance. To solve this problem, we would like propose function f(n) which may be considered in the following form:
(6) 10,15
Zhelezny et al. and Knotts et al., propose a correlation for calculations of reduced parachor in order to improve the quality of predictions of the surface tension of liquids over a wide
f (n) = n B1 / B1n C
(11)
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where B1 and B1n are coefficients (critical amplitudes) of extended scaling equations for a refractive index n′ and density ρ′ of a liquid on the saturation line, as reported by Zhelezny:20 ln
ln
ρ′ = B1θ βF1(θ) ρc
(12)
n′ = B1nθ βF1(θ) nc
(13)
It should be emphasized that the molar refractivity in this equation is only related to the critical parameters nc and Vc and the amplitudes B1 and B1n. One of the purposes of the present study is the attempt to generalize the temperature dependence of molar refractivity. We propose to consider molar refractivity values at different temperatures from values for the molar refractivity of different substances in the reduced form: A − Ac [A] = Ac (17)
where nc and ρc are the refractive index and density of the substance at the critical point; θ = ln(Tc/T) is the reduced temperature; Tc is the critical temperature; β is the exponent, the value of which is equal to the value for the critical index used in scaling theory26,27 (β = 0.3245); F1(θ) is the universal crossover function for normal liquids:20 0.4 F( 1 θ ) = 1 − 1.113θ /ln θ ,
The temperature dependence of function [A] = f(θ) was investigated using refractive index and density data for various alkanes and their halogenated derivates at a range of reduced temperatures 0 ≤ θ ≤ 0.6 (see Figure 2). Generalized
(14)
It should be emphasized that for the case in which data on critical temperature is available to define amplitudes a limited amount of experimental data for the density and refractive index on the saturation line is needed. The analysis carried out in this paper for the temperature dependence of the refractive index close to the critical point showed that it is essential to clarify the crossover function F1(θ). From this study, a new crossover function for the refractive index is proposed which has been fitted to the following equation (Figure 1): F1n(θ ) = 1 + 0.743θ 0.255/ln θ ,
(15) Figure 2. Temperature dependence of the reduced molar refractivity [A].
temperature dependence of molar refractivity was fitted with the next simple power function: [A] = 1.273·θ 0.688
Using this correlation it is possible to predict the value of the molar refractivity, which is functionally dependent on other structural-additive properties (see Table 2). Table 1 presents values of the molar refractivity calculated using eq 16 for certain alkanes and their halogenated derivatives. The undertaken studies show that the molar refractivity of different fluids is in a perfect relationship with the critical molar volume and other structural-additive properties (see Table 2). Orthochor. Another structural-additive property is also used to predict the viscosity of liquidsthe orthochor. The orthochor concept was first introduced by Litvinov30 when the dependence of the fluidity of a liquid 1/η on its molar volume V was analyzed. In this paper, we propose identifying the orthochor value for the liquid by fitting the dependence η−1(V) to zero fluidity of a liquid. The orthochor is a value which represents the minimal limiting molar volume of the liquid which can be achieved by cooling it. The orthochor can be calculated by summation of the structural increments values corresponding to certain elements of a molecule.11,30 It was proposed previously11 to fit data for the viscosity of nonassociated liquids using the following equation: 1 = dV − e ηf (19)
Figure 1. Temperature dependence of the crossover functions for density F(τ) and refractive index Fn(τ).
For the range of reduced temperatures 0.03 ≤ θ ≤ 0.6, the experimental data for the refractive index22−24,28,29 for the halogenated alkanes CH2F2 (difluoromethane), CH2F2CF3 (pentafluoroethane), CF 3 CH 3 (1,1,1-trifluoroethane), CHClFCF3 (1-chloro-1,2,2,2-tetrafluoroethane), CHF2CF2CH 2 F (1,1,2,2,3-pentafluoropropane), CF 3 CH 2 CHF 2 (1,1,1,3,3-pentafluoropropane), CF3CHFCHF2 (1,1,1,2,3,3hexafluoropropane), CF3CH2CF3 (1,1,1,3,3,3-hexafluoropropane) and C7H16 (n-heptane) deviate from the values calculated using eq 13 and eq 15 within a range of 0.1 % to 0.15 %. From eq 8 and eq 11 the following relation for molar refractivity can be obtained: Ac = VcncB1 / B1n
(18)
(16) D
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where η is the coefficient of dynamic viscosity of a liquid on the saturation line, mPa·s; d, e and f are coefficients which depend on the individual properties of a substance. One of the main advantages of eq 19 in comparison with the similar correlations which include two adjustable parameters (for example correlation of Litvinov30) is the presence of the additional coefficient f, which makes it possible to describe experimental data on viscosity of the long-chain heavy hydrocarbons (including, for example, industrial oils). Equation 19 makes it possible to describe experimental data for the viscosity of long-chain heavy hydrocarbons, alkanes, and its halogenated derivatives within a reduced temperature interval of 0.08 ≤ 1 − T/Tc ≤ 0.55 with an absolute deviation average within 2 % to 3 %.11 If Or = e/d, eq 19 can be reformulated as 1 = a(V − Or)b η
modern methods for predicting critical molar volume are highly accurate. For example, Phillipov2 indicates that the average absolute deviation when calculating the critical molar volume of 67 alkanes and 60 cyclic hydrocarbons is no more than 0.5 % to 0.6 % and correlates well with the experimental uncertainty when determining this thermodynamic property.
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RESULTS AND DISCUSSION The SP-QSPR Model for Predicting the Physicochemical Properties of Substances. In recent years, many authors2,12−15,18−20 have recommended to include scaling principles (SP) into a model based on quantitative structure− property relations (QSPR). The basic concept of scaling principles is using universal ratios between critical exponents and critical amplitudes. Using universal crossover functions to calculate effective exponents, it is possible to achieve a good prediction quality for the thermophysical and electrophysical properties of nonassociated liquids for a wide range of temperatures.10,20 In correlations based on the scaling principles, critical amplitudes, whose values depend on the individual properties of the object being studied, are used as coefficients, as are universal critical exponents. Values for critical parameters and critical amplitudes can be determined from a limited amount of experimental data on the properties of substances.10,19,20,22 As a consequence, scaling principles can be combined with structure−property models to calculate the physicochemical properties of substances on the saturation line.10,12,19,20 Analysis showed2,10,20 that scaling equations 3, 4, 12, and 13 adequately describe experimental data for a wide range of state parameters. Therefore, we propose to use the principle of two-parameter universality together with the assertion made by Zhelezny et al.10 and Zhelezny20 about the universality of crossover functions for critical exponents. An important advantage of the proposed SP-QSPR model for calculating the thermophysical properties of substances is that it establishes perfect relationships between structural-additive properties and critical molar volume (see Table 2). Using this approach to prediction, if two values of the amplitudes are available, others values can be calculated from known universal ratios (which are constant for different substances). Several of the universal sets of the amplitudes are presented below:19
(20)
where a and b are coefficients which depend on the individual properties of a substance. As was previously demonstrated,11 eq 20 can be successfully applied for predicting the dynamic viscosity of the binary and ternary mixtures of alkanes and its halogenated derivatives based on the limited amount of the initial information about the viscosity and density of the components of the mixtures. Table 1 presents orthochor values for certain alkanes and its halogenated derivatives. As has been reported,11 the orthochor of different fluids is in a perfect relationship with the critical molar volume (see Table 2). Table 1 also presents the data about molar volume at the critical and normal boiling points for certain alkanes and its halogenated derivatives. Table 1 gives values for other structural-additive properties which were calculated using the data about their thermophysical properties taken from different sources.10,11,22−24,28,29,31 Molar Volume of a Liquid Subcooled to the Absolute Zero (T = 0 K). The methods for predicting the properties of a fluids developed by Phillipov2 and Zhelezny20 propose using a hypothetical value which corresponds to the density of the subcooled condensed phase at absolute zero T = 0 K. It follows from eq 3 that this value can be considered to be the critical amplitude ρ0. Our study demonstrates that quantity V0, which is equal to the ratio of the molecular mass M to the critical amplitude for a difference of orthobaric densities ρ0, can be considered as a structural-additive property. The molar volume V0 is linked to other structural-additive properties (see Table 2). This circumstance makes it possible to use the quantity V0 to develop methods for predicting properties such as viscosity, surface tension, and the capillary constant. Table 1 also presents values of V0 for certain alkanes and its halogenated derivatives. As was demonstrated by Zhelezny,20 the value of V0 for different fluids is functionally related to their critical molar volume (see Table 2). Critical Molar Volume. As various studies have noted,1,2,19 structural-additive properties and complexes (above all critical molar volume) play a very important role in the development of methods for predicting the thermophysical properties of fluids. The critical molar volume is a fundamental parameter which is linked to the volume of a molecule of a substance and its characteristic length parameter.2 Phillipov2 claims that critical volume is determined above all by the composition of a molecule rather than its structure. There are a large number of prediction methods for calculating molar volume at the critical and normal boiling points.1−4,14 We are able to conclude that
[P]c =
S0 =
Mσ0β / μ 2B0 ρc
2/3 σ0 ⎛ M ⎞ ⎜⎜ ⎟⎟ kBTc ⎝ NAρc ⎠
(21)
(22)
⎛ σ a2 ⎞ zσ = ⎜ 0 c ⎟Y c−μ ⎝ kBTc ⎠
(23)
z B = Y c−βZc1/2B0
(24)
where ac = [kBTc/Pc]1/3, Zc = (PcM)/(ρcNAkBTc), Yc = [γcTc/ Pc] − 1, γc = (∂[P(T,ρc)]T→Tc)/∂T; Pc, Tc, ρc are the pressure, temperature, and density at the critical point; NA is the Avogadro constant, mol−1; kB is the Boltzmann constant, kJ/ (kg·K) E
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Table 3. Values of the Structural Increments Vc, Vnb, V0, [P]c, Ac, and Or for Alkanes CnH2n+2 and Halogenated Hydrocarbons CnHiCljFk group contribution
V0
≥3
−CH −CCl −CF −CH −CCl −CF −(CH2)n−
5.9 18.2 9.51 6.13 18.4 9.9 10.8
