**4. Thermodynamic functions in the ideal gas states**

The absolute entropy, changes of the enthalpy and Gibbs energy in three aggregate states are calculated on the basis of smoothed heat capacity values. The experimental data on the heat capacities of the substances under study in the temperature intervals from (6/8 to 372) K were fitted by polynomial (11). Extending the heat capacities to *T* 0 were performed by Debye equation:

$$\mathbf{C\_{S,m}} = \mathbf{n} D(\Theta \mid T) \tag{18}$$

where *D* is the Debye function, *n* = 3, and denote Debye characteristic temperature. Testing the *C*s,m values at*T* 0 was performed by fitting the heat capacities in small temperature interval below (10 to 12) K by equation:

$$\text{C}\_{\text{S,m}} / \text{T} = \alpha \text{T}^2 + \text{y}\_{\text{M}} \tag{19}$$

where and are coefficients. If = 0, it can be accepted that *C*s,m 0 and extrapolation of the heat capacity to *T* 0 can be carried out by equation (18) or by Debye cube's law,

Fig. 12. Molar entropies of fusion <sup>0</sup> *fusSm* for some bicyclic perfluorocarbons and appropriate hydrocarbons: *cis-* [1] and *trans-* [2] perfluorobicyclo(4,3,0)nonanes; *cis-* [3] and *trans*- [4] bicyclo (4,3,0) nonanes; *cis-* [5] and *trans-* [6] perfluorobicyclo(4,4,0)decanes; *cis-* [7] and

Analysis of the entropies of fusion for *cis-* and *trans-*isomers of perfluorobicyclo (4,3,0)nonanes and -(4,4,0) decanes and appropriate values of hydrogen - containing analogous (Table 5, Fig. 12) permits to interpret the influence of the structure and chemical nature on molecular mobility and thermodynamic properties of the solids. Comparesing the entropies of fusion shows that the mobility of the molecules increases in going from hydrocarbons to appropriate perfluorocarbons and from the *cis*-isomers to the *trans*-ones. The larger molecular mobility of the perfluorocarbons can be explained by more weak intermolecular interactions for these compounds compared with the hydrocarbons. The greater ability of the molecules of *cis*-isomer to reorient in the solid state seems to be due to the steric factors. The nature of the solid-to-solid transitions in *cis-* [7] and *trans*- [8] bicyclodecanes and *cis-* [5] and *trans-* [6] perfluorobicyclodecanes (Fig. 12) were discussed in

The absolute entropy, changes of the enthalpy and Gibbs energy in three aggregate states are calculated on the basis of smoothed heat capacity values. The experimental data on the heat capacities of the substances under study in the temperature intervals from (6/8 to 372) K were fitted by polynomial (11). Extending the heat capacities to *T* 0 were performed by

where *D* is the Debye function, *n* = 3, and denote Debye characteristic temperature.

*s,m*

Testing the *C*s,m values at*T* 0 was performed by fitting the heat capacities in small

 

of the heat capacity to *T* 0 can be carried out by equation (18) or by Debye cube's law,

*<sup>С</sup>*s,m *nD T* ( /) (18)

<sup>2</sup> *C TT* / , (19)

= 0, it can be accepted that *C*s,m 0 and extrapolation

*trans-* [8] bicyclo(4,4,0)decanes

Debye equation:

where and 

the order-disorder concept in reference (Kolesov, 1995).

temperature interval below (10 to 12) K by equation:

are coefficients. If

**4. Thermodynamic functions in the ideal gas states** 

*s,m* <sup>3</sup> *C AT* . But, in the case 0, appropriate extending of the heat capacity could be conducted by (18), provided the parameters *n* and are adjusted thus to allow one to obtain zero *Cs,m* values at *T* = 0 K. The ( / )( / ) *C C TV T* , , *p m sm p p T s* difference evaluated at *T* = 298.15 K was smaller than uncertainties of *Cs m*, for the substances under study and was not taken into account. The smoothed values of *Cp,m* and thermodynamic functions *<sup>0</sup>S T <sup>m</sup>*( ) , *0 0* { ( ) (0)} *HT H m m* , and *0 0* { ( ) (0)} *GT H m m* for the condensed states were calculated by numerical integrating the *Cp,m f* ( ) *T* functions obtained by equations (18) and (11) and adding the enthalpies and entropies of the solid-to-solid transition and fusion. The errors of thermodynamic functions were estimated by the law of random errors accumulation using the uncertainties of the heat capacity measurements. The ideal gas absolute entropy, *<sup>0</sup>S T <sup>m</sup>*( ) , the changes of the enthalpy and the free Gibbs energy at 298.15 K were calculated using the appropriate functions in the liquid state, enthalpies and entropy of vaporization and the entropy of the ideal gas compression, *S R pT* ln{ ( ) /(101.325) } <sup>k</sup> <sup>P</sup> <sup>a</sup> calculated from the vapor pressure data.

### **4.1 Theoretical calculations of the thermodynamic functions**

The ideal gas absolute entropy and heat capacity were calculated by statistical thermodynamics, additive principle (Poling et al., 2001; Domalski et al., 1993; Sabbe et al., 2008), and empirical difference method of group equations (Cohen & Benson, 1993).

The statistical thermodynamic method was used with quantum mechanical [QM] calculation on the basis of the density functional theory [DFT]. The QM calculation was performed on the level B3LYP/6-31G(d,p) using the Gaussian 98 and 03 software packages (Frisch et al., 2003). As a result, the following constants can be calculated: the moments of inertia of the entire molecule, the moments of inertia for internal rotors, the normal vibrational frequencies, and the barrier to internal rotation. The potential functions of internal rotation were determined by scanning the torsion angles from (0 to 360)o at 10o increments and allowing all other structural parameters to be optimized at the same level with the subsequent frequency calculation. The calculated potential energies were fitted to the cosine-based Fourier series:

$$V(\varphi) = V\_{\text{O}} + 0.5 \Sigma\_{\text{I}} V\_{\text{II}} (1 - \cos n\varphi) \, , \tag{20}$$

where *V* )( denotes potential energy function, is torsional angle.

The ideal gas entropies and heat capacities in dependence on the temperature were calculated by standard statistical thermodynamics formulae using the rigid-rotor harmonic oscillator [RRHO] approximation. To account for the internal rotation processes, the torsional frequencies were omitted in the calculation of thermodynamic function. A contribution of the internal rotation for each rotor was calculated by direct summation over the energy levels obtained by diagonalization of the one-dimensional Hamiltonian matrix associated with potential function from equation (20). The RRHO approximation, is known, results in overestimated entropy values for flexible molecules due to coupling the internal rotations. One-dimensional hindered rotor correction has been applied by (Vansteenkiste et al., 2003; Van Speybroeck et al., 2000) assuming decoupled internal rotations.

The method of group equation is suitable for calculation of some additive properties of a compound, namely *S T <sup>m</sup>*( ) , *C T*( ) *p,m* , *<sup>o</sup> <sup>f</sup> Hm* values on the basis of reliable appropriate

Thermodynamics of the Phase Equilibriums of Some Organic Compounds 625

*N* are the number of states of disordered and ordered phases. Thus, the residual entropy of CFCl2CHFCl and CF2ClCHCl2 is )0( *<sup>o</sup> Sm* <sup>=</sup>*R*·ln(3) = 9.1 JK-1mol-1. The value )0( *<sup>o</sup> Sm* = 10.1 JK-1mol-1 of CF2ClCFCl2, obtained in (Higgins & Lielmers, 1965), is in good agreement with the *R*·ln(3) value. After taking into account residual entropies of these freons, the calorimetric and theoretical values of absolute entropies agree within errors limits from (0.1 to 1.3) %. At the same time, testing the low-temperature*Cp*,*m* values of CFCl2CHFCl and CF2ClCHCl2 by equation (19) showed an absence of the residual entropies for these freons. Their heat capacities in the temperature interval from (5 to 8) K obeyed Debye cub's law with experimental error of *Cp*,*m* value, 2 %. A disagreement of two methods evaluation of the residual entropy can not be explained on the basis of available physico-chemical data of

123a

385.7 386.9 364.2 372.5 378.6 352.6 368.1 399.1 367.1 305.2 306.0 320.1

*cis*- *trans*- *cis*- *trans*- *cis*- *cis*- *trans*- EA

610.83 606.41 377.32*<sup>a</sup>* 365.30*<sup>a</sup>* 633.33 378.81 373.81 391.4 413.4 400.0

±0.3

e x p t

Table 7. The absolute entropies at *T =* 298.15 K, calculated by statistical thermodynamic, ( ) *<sup>o</sup> S stat <sup>m</sup>* , and determined from the calorimetric data on the heat capacity, entropy of

vaporization, and the entropy of the ideal gas compression, ( )( ) *<sup>o</sup> S g <sup>m</sup>*

vaporization, and the entropy of the ideal gas compression, ( )( ) *<sup>o</sup> S g <sup>m</sup>*

trimethyladamahtane [1,3,5-TMA] and 1-ethyladamahtane [1-EA]

Compounds C9F16 C9H16 C10F18 C10H18 1,3-

374.20 365.39 630.7

Table 8. The absolute entropies at *T =* 298.15 K, calculated by statistical thermodynamics, ( ) *<sup>o</sup> S stat <sup>m</sup>* , and determined from the calorimetric data on the heat capacity, entropy of

perfluorobicyclo(4,3,0)nonanes [*cis-* and *trans*- C9F16], *cis-* and *trans-* bicyclo(4,3,0)nonane, [*cis-* and *trans-* C9H16], *cis-*perfluorobicyclo(4,4,0)decane, [*cis-*C10F18], *cis-* and *trans*bicyclo(4,4,0)decanes [*cis-* and *trans*- C10H18]; 1,3-dimethyladamahtane [1,3-DMA], 1,3,5-

R-243 R-253fa

352.4 365.6 394.7 365.3 304.4 307.8 321.0

e x p t

378.86 374.43 392.9

e x p t

<sup>+</sup> (0) *<sup>o</sup> Sm* , were (0) *<sup>o</sup> Sm a*=10.1 JK-1mol-1

DMA

±1.0

1,1- DClE

1,2- DClE

for some freons

1,3,5- TMA

415.2 ±1.0

for *cis-* and *trans*-

1-

400.9 ±1.0

1,1,1- TClE

R-112 R-113 R-114 R-122 R-122a R-123 R-

366.2 375.4*<sup>b</sup>*

363.8 363.1 372.2*<sup>b</sup>*

*a,b* Absolute entropies were calculated as ( )( ) *<sup>o</sup> S g <sup>m</sup>*

602.8 ±3.0

1

these freons.

Comp ounds

*<sup>g</sup>* )( *<sup>o</sup> mS* )( e x pt , JK-1mol-1

*stat*)( *<sup>o</sup> mS* JK-1mol-1 382.3 378.9 382.0*<sup>a</sup>*

and (0) *<sup>o</sup> Sm b*=9.1 JK-1mol-1.

and cloroethanes

( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t

JK-1mol-1

( ) *<sup>o</sup> S stat <sup>m</sup>* , JK-1mol-1

,

605.3 ±3.0

*<sup>a</sup>*Values were taken from (Frenkel et al., 1994).

properties of the structurally similar compounds. For example, determination of *S T <sup>m</sup>*( ) value of ferrocenylmethanol using absolute entropies of the components of hypothetical reaction:

$$\text{Fe(C\_5H\_5)(C\_5H\_4\cdot CO\cdot H)} = \text{[Fe(C\_5H\_5)\_2]} + \text{[C\_5H\_5\cdot CO\cdot H]} - \text{[C\_5H\_6]}\tag{21}$$

The calculation of the absolute entropy by additive methods requires taking into account corrections for the symmetry and the optical isomerism of the molecule. Otherwise, the principle of group additivity can be broken if these parameters alter when changing the structure of the molecules in the series of compounds. Therefore, an additive calculation of the absolute entropy, 0*S*<sup>m</sup> , is conducted by using so-called intrinsic entropy, <sup>0</sup> m,int *<sup>S</sup>* , which allows to exclude an influence of the rotary components, depending on the symmetry and optical activity of the molecule:

$$S\_{\rm m,int}^{0} = S\_{\rm m}^{0} + R \cdot \ln(\sigma\_{\rm tot} \,/\, n) \tag{22}$$

where tot *R n* ln( / ) is the correction for the symmetry and the chirality of the molecule, tot and *n* denote a total number of the symmetry and the number of the optical isomers. The total symmetry of the molecule is calculated as tot ex in , where ex is the number of symmetry of the external rotation of the molecule, as a whole, and in denotes the number of the internal rotation equal to the product (*P*) of the order of independent axes in the rotating group (l) raised to the power of the number of these axes (k), <sup>k</sup> <sup>l</sup> in *P* .

Thus, an additive calculation of the absolute entropy by group equation method requires of a missing intrinsic entropy <sup>0</sup> intm, *<sup>S</sup>* determination and following recalculation it to the 0*S*<sup>m</sup> value by formula (20).

Comparing between the values of absolute entropy determined on the basis of calorimetric measurements, ( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t , group equation method, and statistical thermodynamic calculations, *stat*)( *<sup>o</sup> Sm* , are applied for prediction of the missing data, verification of their reliability, and mutual congruence in the series of the same type compounds, or homologous. Below, a critical analysis of the ideal gas entropies is performed for some series of functional organic compounds.

**Freons and cloroalkanes.** Table 7 lists the ideal gas absolute entropies determined on the basis of experimental data and those ones obtained by theoretical calculation for some freons [R] and cloroethanes [ClE]. The ( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t and *stat*)( *<sup>o</sup> Sm* values for halogenethanes and freons, including *g*)( *<sup>o</sup> Sm* values with marks (*a* and *b*), agree within errors limits: from (0.1 to 1.3) %. These compounds have rather compact and symmetric molecules and exist as an equilibrium mixture of *trans-* and *gauche-* conformers in three aggregate states and as single more stable conformers in the low-temperature crystal phase. There are disagreements between ( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t and ( ) *<sup>o</sup> S stat <sup>m</sup>* values in a group of three freons CF2ClCFCl2 [R-113], CFCl2CHFCl [R-122a], and CF2ClCHCl2 [R-122]. A characteristic feature of these compounds is availability of residual entropies caused by orientational or conformational disorders. The mixtures of the *trans-* and two *gauche-* conformers of these freons were probably frozen at low helium temperatures. The entropy change caused by disorder of this type can be evaluated by the formula ) 1 / 2 ln( *NNRS* , where <sup>2</sup> *N* and

properties of the structurally similar compounds. For example, determination of *S T <sup>m</sup>*( ) value of ferrocenylmethanol using absolute entropies of the components of

Fe(C5H5)(C5H4-CO-H) = [Fe(C5H5)2] + [C6H5-CO-H] – [C6H6] (21)

The calculation of the absolute entropy by additive methods requires taking into account corrections for the symmetry and the optical isomerism of the molecule. Otherwise, the principle of group additivity can be broken if these parameters alter when changing the structure of the molecules in the series of compounds. Therefore, an additive calculation of

allows to exclude an influence of the rotary components, depending on the symmetry and

m,int <sup>m</sup> tot *S SR n* ln( / )

the number of the internal rotation equal to the product (*P*) of the order of independent axes in the rotating group (l) raised to the power of the number of these axes (k), <sup>k</sup>

Thus, an additive calculation of the absolute entropy by group equation method requires of

intm, *<sup>S</sup>* determination and following recalculation it to the 0*S*<sup>m</sup> value by formula (20). Comparing between the values of absolute entropy determined on the basis of calorimetric

calculations, *stat*)( *<sup>o</sup> Sm* , are applied for prediction of the missing data, verification of their reliability, and mutual congruence in the series of the same type compounds, or homologous. Below, a critical analysis of the ideal gas entropies is performed for some series

**Freons and cloroalkanes.** Table 7 lists the ideal gas absolute entropies determined on the basis of experimental data and those ones obtained by theoretical calculation for some

> e x p t

and freons, including *g*)( *<sup>o</sup> Sm* values with marks (*a* and *b*), agree within errors limits: from (0.1 to 1.3) %. These compounds have rather compact and symmetric molecules and exist as an equilibrium mixture of *trans-* and *gauche-* conformers in three aggregate states and as single more stable conformers in the low-temperature crystal phase. There are

CF2ClCFCl2 [R-113], CFCl2CHFCl [R-122a], and CF2ClCHCl2 [R-122]. A characteristic feature of these compounds is availability of residual entropies caused by orientational or conformational disorders. The mixtures of the *trans-* and two *gauche-* conformers of these freons were probably frozen at low helium temperatures. The entropy change caused by

and *n* denote a total number of the symmetry and the number of the optical isomers.

is the correction for the symmetry and the chirality of the molecule,

 

, group equation method, and statistical thermodynamic

m,int *<sup>S</sup>* , which

(22)

<sup>l</sup> in 

*P* .

ex is the

denotes

*N* and

, where

and *stat*)( *<sup>o</sup> Sm* values for halogenethanes

1 / 2 ln( *NNRS* , where <sup>2</sup>

and ( ) *<sup>o</sup> S stat <sup>m</sup>* values in a group of three freons

the absolute entropy, 0*S*<sup>m</sup> , is conducted by using so-called intrinsic entropy, <sup>0</sup>

0 0

number of symmetry of the external rotation of the molecule, as a whole, and in

The total symmetry of the molecule is calculated as tot ex in

hypothetical reaction:

optical activity of the molecule:

a missing intrinsic entropy <sup>0</sup>

measurements, ( )( ) *<sup>o</sup> S g <sup>m</sup>*

of functional organic compounds.

disagreements between ( )( ) *<sup>o</sup> S g <sup>m</sup>*

e x p t

freons [R] and cloroethanes [ClE]. The ( )( ) *<sup>o</sup> S g <sup>m</sup>*

e x p t

disorder of this type can be evaluated by the formula )

where tot *R n* ln( / ) 

tot 

1 *N* are the number of states of disordered and ordered phases. Thus, the residual entropy of CFCl2CHFCl and CF2ClCHCl2 is )0( *<sup>o</sup> Sm* <sup>=</sup>*R*·ln(3) = 9.1 JK-1mol-1. The value )0( *<sup>o</sup> Sm* = 10.1 JK-1mol-1 of CF2ClCFCl2, obtained in (Higgins & Lielmers, 1965), is in good agreement with the *R*·ln(3) value. After taking into account residual entropies of these freons, the calorimetric and theoretical values of absolute entropies agree within errors limits from (0.1 to 1.3) %. At the same time, testing the low-temperature*Cp*,*m* values of CFCl2CHFCl and CF2ClCHCl2 by equation (19) showed an absence of the residual entropies for these freons. Their heat capacities in the temperature interval from (5 to 8) K obeyed Debye cub's law with experimental error of *Cp*,*m* value, 2 %. A disagreement of two methods evaluation of the residual entropy can not be explained on the basis of available physico-chemical data of these freons.


*a,b* Absolute entropies were calculated as ( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t <sup>+</sup> (0) *<sup>o</sup> Sm* , were (0) *<sup>o</sup> Sm a*=10.1 JK-1mol-1 and (0) *<sup>o</sup> Sm b*=9.1 JK-1mol-1.

Table 7. The absolute entropies at *T =* 298.15 K, calculated by statistical thermodynamic, ( ) *<sup>o</sup> S stat <sup>m</sup>* , and determined from the calorimetric data on the heat capacity, entropy of vaporization, and the entropy of the ideal gas compression, ( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t for some freons and cloroethanes


*<sup>a</sup>*Values were taken from (Frenkel et al., 1994).

Table 8. The absolute entropies at *T =* 298.15 K, calculated by statistical thermodynamics, ( ) *<sup>o</sup> S stat <sup>m</sup>* , and determined from the calorimetric data on the heat capacity, entropy of

vaporization, and the entropy of the ideal gas compression, ( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t for *cis-* and *trans*perfluorobicyclo(4,3,0)nonanes [*cis-* and *trans*- C9F16], *cis-* and *trans-* bicyclo(4,3,0)nonane, [*cis-* and *trans-* C9H16], *cis-*perfluorobicyclo(4,4,0)decane, [*cis-*C10F18], *cis-* and *trans*bicyclo(4,4,0)decanes [*cis-* and *trans*- C10H18]; 1,3-dimethyladamahtane [1,3-DMA], 1,3,5 trimethyladamahtane [1,3,5-TMA] and 1-ethyladamahtane [1-EA]

Thermodynamics of the Phase Equilibriums of Some Organic Compounds 627

Fig. 13. Absolute entropies (JK-1mol-1) as functions of molecular weights ( ) *M* at *T* = 298.15 K for **(a)** series of alkylferrocenes in condensed (F, *n*-PF, and *n-*BF) and ideal gas states ( F', EF', *n*-PF', and *n-*BF') where straight lines correspond to equations: ( , ) *<sup>o</sup> S cond AlF <sup>m</sup>* <sup>=</sup> 3.250· *<sup>M</sup>* - 377.1 and ( , ) *<sup>o</sup> Sm <sup>g</sup> AlF* = 3.840· *<sup>M</sup>* - 361.8; and **(b)** series of acylferrocenes in the condensed (F, *FF*, AF, and POF) and ideal gas states (F', FF', AF', and POF') where the straight lines correspond to equations: ( , ) *<sup>o</sup> S cond AcF <sup>m</sup>* = 1.392· *<sup>M</sup>* - 50.68 and ( , ) *<sup>o</sup> Sm <sup>g</sup> AcF* <sup>=</sup>

of the entropy of vapor compression calculation over the liquid phase because there are no vapor pressure data over the crystals. The verification of reliability of the other ( )( ) *<sup>o</sup> S g <sup>m</sup>*

values was analyzed on the basis of the additivity principle for extensive properties in

BF, and FF are 5-10 times larger than the errors of these values. To find the reasons for these discrepancies, the linear correlations between the absolute entropies and molecular weights of the compounds were analyzed in homologous series of alkyl-, ( ) *<sup>o</sup> S AlF <sup>m</sup>* , and acyl-,

The root-mean-square deviation [RMS] of the calculated ( , ) *<sup>o</sup> S cond AlF <sup>m</sup>* values from

by the value of 10.5 % from the linear dependence of the other homologues. These deviations are several times larger than the uncertainty of ( ) *<sup>o</sup> S cond <sup>m</sup>* values and can be explained by errors in calorimetric heat capacities, obtained in (Karyakin, et al., 2003). After excluding of this value, the coefficient *R2* of linear correlation increased from 0.9147 to

from the smoothed straight line (Fig. 13) by only (3.9 and 3.2) %, respectively, nevertheless the

large. After replacing the absolute entropy of *n*-BF, obtained from experimental data, on the

e x p t

( )( ) *<sup>o</sup> Sm <sup>g</sup> calc* value, calculated by the empirical method of group equations, the

e x p t

*s* = 29 JK- <sup>1</sup>mol-1. Besides, the entropy of ethylferrocene deviates

entropies for *n*-propylferrocene and *n*-butylferrocene deviate

*<sup>s</sup>* value decreased to 4 JK-1mol-1. Thus, the value ( ) *<sup>o</sup> S cond <sup>m</sup>* for EF is

values from initial ones

As is seen from Table 9, the discrepancies between ( )( ) *<sup>o</sup> S g <sup>m</sup>*

e x p t

and ( ) *<sup>o</sup> Sm <sup>g</sup>* values of EF, *n-*

*s* = 29 JK-1mol-1 is fairly

*s* deviation

2.664· *M* - 144.6

homologous series of compounds.

experimental ones was

0.9992, and the

( ) *<sup>o</sup> S AcF <sup>m</sup>* , derivatives of ferrocene (Fig. 13).

unreliable. The ( )( ) *<sup>o</sup> S g <sup>m</sup>*

e x p t

RMS deviation of calculated ( )( ) *<sup>o</sup> S g <sup>m</sup>*

**Cyclic hydrocarbons and perfluorocarbons**. The absolute entropies obtained by the third thermodynamic law from the experimental data and statistical thermodynamics for some cyclic and bicyclic hydrocarbons and perfluorocarbons are listed in Table 8. The values of the absolute entropies determined by independent methods agree within errors limits from (0.1 to 1) % that proves their mutual conformity and also the reliability of the all experimental data used for computing the value of ( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t , namely the heat capacities, enthalpies of vaporization, and saturation vapor pressures.

**Ferrocene derivatives [FD].** The experimental data on thermodynamics of the phase equilibriums for FD are very scarce. Therefore, critical analysis of the available data on thermodynamic properties and using them for science prognosis of failing properties are the urgent problems. A verification of reliability and mutual congruence of the properties of some derivatives of ferrocene were carried out on the basis of absolute entropies, which were computed by the third thermodynamic law from the experimental data and by statistical thermodynamics and group equation method. Comprising the *<sup>o</sup> Sm* and *Cp,m* values determined by both theoretical methods showed that they agree within errors limits, 1.2 %. The values of absolute entropies in homologous series of alkyl- and acyl-ferrocenes obtained in this work and available in the literature are presented in Table 9.


*<sup>a</sup>* ( ) *<sup>o</sup> S cond <sup>m</sup>* and ( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t denote the entropies determined based on experimental data; *<sup>b</sup>* ( )( ) *<sup>o</sup> Sm <sup>g</sup> calc* are values calculated by DFT and group equation methods; *c* ( )( ) *<sup>o</sup> Sm <sup>g</sup> recom* denotes recommended values; d (Emel'yanenko et al., 2010); e (Karyakin, et al., 2003).

Table 9. The absolute entropies (JK-1mol-1) in condensed and ideal gas states of ferrocene, [F], and some derivatives of ferrocene: ethylferrocene [EF], *n-*propylferrocene [*n*-PF], *iso*butylferrocene [*i-*BF], *n-*butylferrocene [*n-*BF], formylferrocene [FF], acetylferrocene [AF], propionylferrocene [POF], *iso-*butyrylferrocene [*i-*BTF] at *T* = 298.15 K

The ( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t and ( )( ) *<sup>o</sup> Sm <sup>g</sup> calc* values for ferrocene, *n-*propyl- and *iso-*butylferrocene agree to within their errors that proves the reliability of these values and all the thermodynamic data used for their calculation. The mutual consistency of the absolute entropies of *iso-*butyrylferrocene and *iso-*butylferrocene in the liquid state was revealed using the Benson additive method. The difference between the entropies of these compounds, *S =* 2 JK-1mol-1, is close to the entropy increment *S* = 1.4 JK-1mol-1, which fit for replacing the СН2 by CO group in the passage from *iso-*butyl- to *iso-*butyrylferrocene. The difference between the ( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t and ( )( ) *<sup>o</sup> Sm <sup>g</sup> calc* values for crystal propionylferrocene, which exceeds measurement errors, can be explained by uncertainties

**Cyclic hydrocarbons and perfluorocarbons**. The absolute entropies obtained by the third thermodynamic law from the experimental data and statistical thermodynamics for some cyclic and bicyclic hydrocarbons and perfluorocarbons are listed in Table 8. The values of the absolute entropies determined by independent methods agree within errors limits from (0.1 to 1) % that proves their mutual conformity and also the reliability of the all

**Ferrocene derivatives [FD].** The experimental data on thermodynamics of the phase equilibriums for FD are very scarce. Therefore, critical analysis of the available data on thermodynamic properties and using them for science prognosis of failing properties are the urgent problems. A verification of reliability and mutual congruence of the properties of some derivatives of ferrocene were carried out on the basis of absolute entropies, which were computed by the third thermodynamic law from the experimental data and by statistical thermodynamics and group equation method. Comprising the *<sup>o</sup> Sm* and *Cp,m* values determined by both theoretical methods showed that they agree within errors limits, 1.2 %. The values of absolute entropies in homologous series of alkyl- and acyl-ferrocenes obtained in this work and available in the literature are presented in Table 9.

> e x p t

F (cr) 211.85 ± 1.1*<sup>d</sup>* 361.0 ± 3.6 362.6 ± 7 362 ± 3.1 EF(liq) (354.1 ± 1.1)*<sup>e</sup>* (525.3 ± 2.9)*<sup>e</sup>* 453 ± 7*<sup>d</sup>* 453 ± 7 *n-*PF (liq) 356.9 ± 0.9 494.3 ± 5.0 489 ± 7 492 ± 6 *i-*BF (liq) 377.6 ± 1.9 517.3 ± 5.2 513 ± 7 515 ± 6 *n-*BF (liq) 398.1 ± 1.2*<sup>e</sup>* (586.3 ± 3.4)*<sup>e</sup>* 527 ± 7 527 ± 7 FF (cr) 241.3 ± 0.7*<sup>e</sup>* (398.1 ± 2.8)*<sup>e</sup>* 435 ± 7*<sup>d</sup>* 435 ± 7 AF (cr) 264.6 ± 0.8 478.3 ± 2.9 - 478 ± 3 POF (cr) 291.0 ± 1.2 502.7 ± 5.1 513 ± 7 507 ± 6 *i-*BTF (liq) 379.6 ± 1.2 - 541 ± 7 541 ± 7

denote the entropies determined based on experimental data;

and ( )( ) *<sup>o</sup> Sm <sup>g</sup> calc* values for ferrocene, *n-*propyl- and *iso-*butylferrocene

*<sup>b</sup>* ( )( ) *<sup>o</sup> Sm <sup>g</sup> calc* are values calculated by DFT and group equation methods; *c* ( )( ) *<sup>o</sup> Sm <sup>g</sup> recom* denotes

Table 9. The absolute entropies (JK-1mol-1) in condensed and ideal gas states of ferrocene, [F], and some derivatives of ferrocene: ethylferrocene [EF], *n-*propylferrocene [*n*-PF], *iso*butylferrocene [*i-*BF], *n-*butylferrocene [*n-*BF], formylferrocene [FF], acetylferrocene [AF],

agree to within their errors that proves the reliability of these values and all the thermodynamic data used for their calculation. The mutual consistency of the absolute entropies of *iso-*butyrylferrocene and *iso-*butylferrocene in the liquid state was revealed using the Benson additive method. The difference between the entropies of these compounds, *S =* 2 JK-1mol-1, is close to the entropy increment *S* = 1.4 JK-1mol-1, which fit for replacing the СН2 by CO group in the passage from *iso-*butyl- to *iso-*butyrylferrocene.

> e x p t

propionylferrocene, which exceeds measurement errors, can be explained by uncertainties

recommended values; d (Emel'yanenko et al., 2010); e (Karyakin, et al., 2003).

propionylferrocene [POF], *iso-*butyrylferrocene [*i-*BTF] at *T* = 298.15 K

e x p t , namely the heat capacities,

*<sup>a</sup>* ( )( ) *<sup>o</sup> Sm <sup>g</sup> calc <sup>b</sup>* ( )( ) *<sup>o</sup> Sm <sup>g</sup> recom <sup>c</sup>*

and ( )( ) *<sup>o</sup> Sm <sup>g</sup> calc* values for crystal

experimental data used for computing the value of ( )( ) *<sup>o</sup> S g <sup>m</sup>*

enthalpies of vaporization, and saturation vapor pressures.

Compound ( ) *<sup>o</sup> S cond <sup>m</sup><sup>a</sup>* ( )( ) *<sup>o</sup> S g <sup>m</sup>*

*<sup>a</sup>* ( ) *<sup>o</sup> S cond <sup>m</sup>* and ( )( ) *<sup>o</sup> S g <sup>m</sup>*

The ( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t

e x p t

The difference between the ( )( ) *<sup>o</sup> S g <sup>m</sup>*

Fig. 13. Absolute entropies (JK-1mol-1) as functions of molecular weights ( ) *M* at *T* = 298.15 K for **(a)** series of alkylferrocenes in condensed (F, *n*-PF, and *n-*BF) and ideal gas states ( F', EF', *n*-PF', and *n-*BF') where straight lines correspond to equations: ( , ) *<sup>o</sup> S cond AlF <sup>m</sup>* <sup>=</sup> 3.250· *<sup>M</sup>* - 377.1 and ( , ) *<sup>o</sup> Sm <sup>g</sup> AlF* = 3.840· *<sup>M</sup>* - 361.8; and **(b)** series of acylferrocenes in the condensed (F, *FF*, AF, and POF) and ideal gas states (F', FF', AF', and POF') where the straight lines correspond to equations: ( , ) *<sup>o</sup> S cond AcF <sup>m</sup>* = 1.392· *<sup>M</sup>* - 50.68 and ( , ) *<sup>o</sup> Sm <sup>g</sup> AcF* <sup>=</sup> 2.664· *M* - 144.6

of the entropy of vapor compression calculation over the liquid phase because there are no vapor pressure data over the crystals. The verification of reliability of the other ( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t values was analyzed on the basis of the additivity principle for extensive properties in homologous series of compounds.

As is seen from Table 9, the discrepancies between ( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t and ( ) *<sup>o</sup> Sm <sup>g</sup>* values of EF, *n-*BF, and FF are 5-10 times larger than the errors of these values. To find the reasons for these discrepancies, the linear correlations between the absolute entropies and molecular weights of the compounds were analyzed in homologous series of alkyl-, ( ) *<sup>o</sup> S AlF <sup>m</sup>* , and acyl-, ( ) *<sup>o</sup> S AcF <sup>m</sup>* , derivatives of ferrocene (Fig. 13).

The root-mean-square deviation [RMS] of the calculated ( , ) *<sup>o</sup> S cond AlF <sup>m</sup>* values from experimental ones was *s* = 29 JK- <sup>1</sup>mol-1. Besides, the entropy of ethylferrocene deviates by the value of 10.5 % from the linear dependence of the other homologues. These deviations are several times larger than the uncertainty of ( ) *<sup>o</sup> S cond <sup>m</sup>* values and can be explained by errors in calorimetric heat capacities, obtained in (Karyakin, et al., 2003). After excluding of this value, the coefficient *R2* of linear correlation increased from 0.9147 to 0.9992, and the *<sup>s</sup>* value decreased to 4 JK-1mol-1. Thus, the value ( ) *<sup>o</sup> S cond <sup>m</sup>* for EF is unreliable. The ( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t entropies for *n*-propylferrocene and *n*-butylferrocene deviate from the smoothed straight line (Fig. 13) by only (3.9 and 3.2) %, respectively, nevertheless the RMS deviation of calculated ( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t values from initial ones *s* = 29 JK-1mol-1 is fairly large. After replacing the absolute entropy of *n*-BF, obtained from experimental data, on the ( )( ) *<sup>o</sup> Sm <sup>g</sup> calc* value, calculated by the empirical method of group equations, the *s* deviation

Thermodynamics of the Phase Equilibriums of Some Organic Compounds 629

The method of extrapolation of the vapor pressure was tested with standard substances 1,1,1-trifluoro-2,2-dichloroethane and *n*-decane for which precision *pT* parameters and heat

Fig. 14. Deviation *sp p p* {( ) / } 100 *p recom extr recom* of extrapolated values *pextr* of the vapor pressure of *n-*decane from experimental recommended values *precom* as functions of temperature range of extrapolation *Textr* ,: *1* - calculation by a system of equations (23); *2*

Fig. 14 presents comparison of the extrapolation capabilities of equations (4) and the system of equations (23) by processing the precision *pT* data of *n*-decane in moderate range of pressure and the *Cp*,*m* differences in the neighbourhood of triple point *Ttp =* 243.56 K. The deviations of the extrapolated *p* values from the recommended experimental vapor pressures of *n*-decane (Boublik et al., 1984) and most reliable calculated values (Ruzicka & Majer, 1994) were evaluated using equation (4) (curves 2 and 3) and the system of equations (23) (curve *1).* Curves *2* and *3* were obtained for the complete and shortened temperature ranges of the *pT* data *T* = (80 and 50) K, respectively. In so doing, the deviation of the

dependence on the temperature range of extrapolation , *Textr* , from (50 to 120) K. The appropriate deviations for the system of equations (23) are much smaller and equal from (2

The results of approximation of *pT* data for l,l,l-trifluoro-2,2-dichloroethane (Weber, 1992) by individual equation (4) and by system of equations (23) are compared in Table 10 . The data were processed by the least squares method using the vapor pressure in the temperature range from 256 to 299 K and the heat capacities of the ideal gas (Frenkel et al.,

In so doing, the temperature range of extrapolation of vapor pressure from initial temperature 256 K to *Ttp* = 145.68 K equals *Textr* = 111 K. Tentative errors of approximation coefficients were evaluated only for revealing the change of extrapolation prediction in going from equation (4) to that of (23). As is seen from Table 10, errors of

1994) and liquid (Varushchenko et al., 2007) in the range from 150 to 240 K.

*p* , increase for equation (4) from (5 to 40) % in

and *3* - calculation by equation (4) using full and shortened ( *T* = 50 K) interval of

*pT* parameters

calculated *p* values from experimental ones,*s*

to 4) % for this range of extrapolation.

capacities of the ideal gas and liquid are available in a wide temperature range.

decreased to 6.2 JK-1mol-1 and appropriate deviations of ( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t values for *n-*PF and *n-*BF from the smoothed straight line did not exceed ~ 1 %.

Correlations for the absolute entropies of acylferrocenes obtained for the condensed and ideal gas states (Table 9) are shown in Fig. 13, b. The RMS deviation *s* = 6 JK-1mol-1 of the experimental ( , ) *<sup>o</sup> S cond AcF <sup>m</sup>* entropies from the smoothed straight line conforms to the errors of these values. Appropriate deviation for the ideal gas entropies ( ) *<sup>o</sup> Sm* e x p t A c F is substantially larger, *s* = 23.4 JK-1mol-1. Besides, the deviation of the entropy of formylferrocene (Table 9) from smoothed straight line, 27.5 JK-1mol-1, is ~10 times larger than the experimental error, which can be explained by errors in vapor pressure and the entropies of sublimation for FF. After replacing this ( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t value by the ( )( ) *<sup>o</sup> Sm <sup>g</sup> calc* entropy obtained by the method of statistical thermodynamics (Table 9), appropriate deviations of the ideal gas entropies from smoothed *AcFg* ),( *<sup>o</sup> Sm* line decreased to 1.4 % for the all compounds of the *Ac*F – series. This deviation is close to the measurement errors, which indicate to unreliability of the ))(( e x p t*<sup>g</sup> <sup>o</sup> Sm* value for FF.

A critical analysis of the absolute entropies of the alkyl and acyl derivatives of ferrocene shows that the ( )( ) *<sup>o</sup> Sm <sup>g</sup> calc* and ( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t values for *n*-propylferrocene, *iso*butylferrocene, acetylferrocene, propionylferrocene, and *iso*-butyrylferocene are reliable to within errors. The entropies of ethylferrocene in the condensed state and *n*-butylferrocene and formylferrocene in the ideal gas states are not reliable because of errors in EF heat capacity measurements and errors of the vapor pressures and enthalpy of vaporization and sublimation for *n-*BF and FF, respectively. On the basis of the critical analyses of the absolute entropies of the ferrocene derivatives, the recommended reliable ( )( ) *<sup>o</sup> Sm <sup>g</sup> recom* values have been presented in Table 9.

### **4.2 Extrapolation of vapor pressure to entire range of liquid phase**

A saturation vapor pressure of substances for entire liquid region was obtained by calculation methods on the basis of the experimental *pT* data of moderate "atmospheric range" from 2 to 101.6 kPa. Extending the *pT* parameters to the region of low pressures is performed by simultaneous processing the vapor pressure and differences between the heat capacities of the ideal gas and liquid. The vapor pressures above 100 kPa are calculated by the empirical bimodal equation obtained by processing the *pT* parameters and densities of liquids using the one-parameter corresponding states law in Filippov's version.

The extrapolation of vapor pressure to the low-temperature region down to the triple point is carried out by a system of two equations:

$$\begin{cases} \ln(p\,\!\!/\,} \, \text{ln}(\!\!\!/\, \text{ } \!\!\/ \, \text{ } \!\!\/) = A\!\!\!\!+B\!\!\/\, \text{ } \!\!\/\, \text{ } \!\!\/+\!\!\/\, \text{ } \!\!\/+\!\!\/\, \text{ } \,\!\/\, \text{ } \,\!\/\, \text{ } \,\!\/\, \text{ } \,\!\/\, \text{ } \,\!\/\, \text{ } \,\!\/\, \text{ } \,\!\/\, \text{ } \,\!\/\, \text{ } \,\!\/\, \text{ } \,\!\/\, \text{ } \,\!\/\, \text{ } \,\!\/\, \text{ } \,\!\/\, \text{ } \,\!\/$$

where <*p>* is the vapor pressure at the average temperature of experimental range, and *A', B',* C', and *D'* are coefficients of simultaneous processing of the *pT* data with *Cp*,*m* differences.

Mathematical processing by a system of equations (23) requires the use of heat capacities of the ideal gas and liquid in the neighbourhood of the triple point, where the linear temperature dependence of *Cp*,*m* is valid.

Correlations for the absolute entropies of acylferrocenes obtained for the condensed and

experimental ( , ) *<sup>o</sup> S cond AcF <sup>m</sup>* entropies from the smoothed straight line conforms to the errors of these values. Appropriate deviation for the ideal gas entropies ( ) *<sup>o</sup> Sm*

formylferrocene (Table 9) from smoothed straight line, 27.5 JK-1mol-1, is ~10 times larger than the experimental error, which can be explained by errors in vapor pressure

( )( ) *<sup>o</sup> Sm <sup>g</sup> calc* entropy obtained by the method of statistical thermodynamics (Table 9), appropriate deviations of the ideal gas entropies from smoothed *AcFg* ),( *<sup>o</sup> Sm* line decreased to 1.4 % for the all compounds of the *Ac*F – series. This deviation is close to the

A critical analysis of the absolute entropies of the alkyl and acyl derivatives of

butylferrocene, acetylferrocene, propionylferrocene, and *iso*-butyrylferocene are reliable to within errors. The entropies of ethylferrocene in the condensed state and *n*-butylferrocene and formylferrocene in the ideal gas states are not reliable because of errors in EF heat capacity measurements and errors of the vapor pressures and enthalpy of vaporization and sublimation for *n-*BF and FF, respectively. On the basis of the critical analyses of the absolute entropies of the ferrocene derivatives, the recommended reliable ( )( ) *<sup>o</sup> Sm <sup>g</sup> recom*

A saturation vapor pressure of substances for entire liquid region was obtained by calculation methods on the basis of the experimental *pT* data of moderate "atmospheric range" from 2 to 101.6 kPa. Extending the *pT* parameters to the region of low pressures is performed by simultaneous processing the vapor pressure and differences between the heat capacities of the ideal gas and liquid. The vapor pressures above 100 kPa are calculated by the empirical bimodal equation obtained by processing the *pT* parameters and densities of

The extrapolation of vapor pressure to the low-temperature region down to the triple point

where <*p>* is the vapor pressure at the average temperature of experimental range, and *A', B',* C', and *D'* are coefficients of simultaneous processing of the *pT* data with

Mathematical processing by a system of equations (23) requires the use of heat capacities of the ideal gas and liquid in the neighbourhood of the triple point, where the linear

e x p t

*s* = 23.4 JK-1mol-1. Besides, the deviation of the entropy of

e x p t

e x pt*<sup>g</sup> <sup>o</sup> Sm* value for FF.

e x p t

values for *n*-propylferrocene, *iso*-

values for *n-*PF and

= 6 JK-1mol-1 of the

e x p t A c Fis

value by the

(23)

decreased to 6.2 JK-1mol-1 and appropriate deviations of ( )( ) *<sup>o</sup> S g <sup>m</sup>*

ideal gas states (Table 9) are shown in Fig. 13, b. The RMS deviation *s*

and the entropies of sublimation for FF. After replacing this ( )( ) *<sup>o</sup> S g <sup>m</sup>*

measurement errors, which indicate to unreliability of the ))((

**4.2 Extrapolation of vapor pressure to entire range of liquid phase** 

liquids using the one-parameter corresponding states law in Filippov's version.

ferrocene shows that the ( )( ) *<sup>o</sup> Sm <sup>g</sup> calc* and ( )( ) *<sup>o</sup> S g <sup>m</sup>*

*n-*BF from the smoothed straight line did not exceed ~ 1 %.

values have been presented in Table 9.

is carried out by a system of two equations:

temperature dependence of *Cp*,*m* is valid.

*Cp*,*m* differences.

substantially larger,

The method of extrapolation of the vapor pressure was tested with standard substances 1,1,1-trifluoro-2,2-dichloroethane and *n*-decane for which precision *pT* parameters and heat capacities of the ideal gas and liquid are available in a wide temperature range.

Fig. 14. Deviation *sp p p* {( ) / } 100 *p recom extr recom* of extrapolated values *pextr* of the vapor pressure of *n-*decane from experimental recommended values *precom* as functions of temperature range of extrapolation *Textr* ,: *1* - calculation by a system of equations (23); *2* and *3* - calculation by equation (4) using full and shortened ( *T* = 50 K) interval of *pT* parameters

Fig. 14 presents comparison of the extrapolation capabilities of equations (4) and the system of equations (23) by processing the precision *pT* data of *n*-decane in moderate range of pressure and the *Cp*,*m* differences in the neighbourhood of triple point *Ttp =* 243.56 K. The deviations of the extrapolated *p* values from the recommended experimental vapor pressures of *n*-decane (Boublik et al., 1984) and most reliable calculated values (Ruzicka & Majer, 1994) were evaluated using equation (4) (curves 2 and 3) and the system of equations (23) (curve *1).* Curves *2* and *3* were obtained for the complete and shortened temperature ranges of the *pT* data *T* = (80 and 50) K, respectively. In so doing, the deviation of the calculated *p* values from experimental ones,*s p* , increase for equation (4) from (5 to 40) % in dependence on the temperature range of extrapolation , *Textr* , from (50 to 120) K. The appropriate deviations for the system of equations (23) are much smaller and equal from (2 to 4) % for this range of extrapolation.

The results of approximation of *pT* data for l,l,l-trifluoro-2,2-dichloroethane (Weber, 1992) by individual equation (4) and by system of equations (23) are compared in Table 10 . The data were processed by the least squares method using the vapor pressure in the temperature range from 256 to 299 K and the heat capacities of the ideal gas (Frenkel et al., 1994) and liquid (Varushchenko et al., 2007) in the range from 150 to 240 K.

In so doing, the temperature range of extrapolation of vapor pressure from initial temperature 256 K to *Ttp* = 145.68 K equals *Textr* = 111 K. Tentative errors of approximation coefficients were evaluated only for revealing the change of extrapolation prediction in going from equation (4) to that of (23). As is seen from Table 10, errors of

Thermodynamics of the Phase Equilibriums of Some Organic Compounds 631

The derivative *d pd T* ln( ) / (1 / ) was determined by differentiation of the equations ln( ) ( ) *p F T* for investigated substances. A random sampling of 14 compounds, having reliable data on the vapor pressure, density, and critical parameters was used to demonstrate (Varushchenko et. al., 1987) that 1) the employed algorithm is capable to

originally developed for compounds with the similarity criterion 1 *<sup>A</sup> <sup>F</sup>* 4 is suitable for a wider class of compounds, including polyatomic molecules with the criterion 0.5 *A*

A prediction of the thermophysical properties by the OLCS of Filippov (Filippov, 1988)

The saturation vapor pressure are extrapolated to the critical region using the empirical

employed for curvilinear extrapolation of saturated vapor pressure using only two pairs of the *pT* data. The formula was derived by the similarity conversion, namely using

substances. As a result, the family of curves reduces to a single curve which has segments of vapor pressure in the supercritical region. The pair of the *P T* parameters is of importance for remote point on the binodal curve, which enables one to calculate the vapor pressure of a substance in the region from the normal boiling temperature to the critical point. The error of the pressure calculation is (3 to 5) % depending on the temperature

Therefore, analysis of equations (4) and (23) demonstrated that equation (4) obtained by approximation of vapor pressure of the "atmospheric range" gives precision results when it used as an interpolation equation. It is further employed for extrapolation of the vapor pressure in the temperature range of 50 K with an error of (1 to 2) %. Extending the vapor pressure to the entire liquid phase from the critical to the triple points temperatures is performed by means of one-parameter law of corresponding states with errors from (3 to 5) % and, respectively, by simultaneous processing of the *pT* parameters and the differences between low-temperature heat capacities 0 0 () ( ) *C C g C liq pm pm pm* ,, , with uncertainties of 10 %, respectively. The data on extrapolation of the vapor pressure are suitable for many

The fundamental thermodynamic investigations of the phase equilibriums of many functional organic compounds were carried out by experimental and calculation methods in Luginin's Thermochemistry Laboratory of the Moscow State University. Modified setups have been created for precise determination of the saturation vapor pressures by

log( / ) lo *P p a T T T T b cT T* g( / ) [ / 1] [ ( / ], (25)

within ±1—2%; the appropriate errors of *P* , *c* are higher and amount to ±3—5%.

binodal equation with pseudocritical parameters *T T <sup>c</sup>* and *P P <sup>c</sup>* :

Two-parameter formula (25) weakly depends on similarity criterion *A*

 g( ) 

*F* and 2) the calculation algorithm

*<sup>F</sup> ;* therefore, it may be

*F* . The errors of the latter calculation are

which have the same slope for different

*F* 4.

produce adequate results in computing *T* , *c V* , *c* and *A*

requires following basic quantities: *T* , *c V* , *c* and *A*

where *a =* 3.9726, *b =* 0.3252, and *c =* 0.40529.

range of *pT* data extending and their reliabi lity.

superposition of curves log( ) / lo

technological calculations.

**5. Conclusion** 

coefficients of equations (23) are approximately ten times lower than those of equation (4), which explains increasing the extrapolation capabilities of the system (23). The resultant value of the vapor pressure at triple point *ptp* is about 25% lower than that of equation (4). Reducing the temperature range of initial *pT* data, *T* , from (25 to 50) K (accordingly, extending the range of extrapolation of vapor pressure to ~ 130 K) leads to the variation of *ptp* value by only 4-6%.


Table 10. The coefficients of equations (4) and (23) and saturated vapor pressure of 1,1,1 trifluoro-2,2-dichloroethane at the triple point, *Ttp* = 145.68±0.02 K (Weber, 1992)

Analysis of calculated data for *n*-decane and l,l,l-trifluoro-2,2-dichloroethane revealed that the use of the system of equations (23) for extrapolation of the vapor pressure in a wide temperature range *T extr* (120 to130) K enabled one to obtain *p*( ) *Ttp* values with errors of 10%.

Extending the saturation vapor pressure to the critical region and computing the critical parameters of the substances were performed by the corresponding states law of Filippov's version (Filippov, 1988). A large class of normal (unassociated) liquids obey the oneparameter law of corresponding states [OLCS] which manifests an existence of the oneparameter family of surfaces of liquids. For the saturation curve, the OLCS equation has the form (, ) 0 *A* or ( , ), *A* where *A* is the similarity criterion. The latter is assigned by ordinate with abscissa  *=* 0.625 on the reduced curve of saturated vapor pressure, *A* 100 *F* at 0.625.

Few literature data are available for critical parameters of organic compounds. The law of corresponding states in Filippov's version enables one both to obtain the required critical parameters using precise *pT* data and density of liquid and to calculate numerous thermophysical properties on the basis of known values of *T* , *c V* , *c* and *A F* . Generalized equations for the calculation of the critical parameters and of the thermophysical properties were obtained by empirical method using the array of experimental data on the appropriate properties for studied compounds (Filippov, 1988). The critical parameters were calculated by algorithms (Filippov, 1988) and (Varushchenko et. al., 1987), in which the following pairs of input data were used:

$$\begin{array}{ccccc} T\_{\textit{n},b.} & p = 101.325 & & & T\_{\textit{n},b.} & p = 101.325 \\ T\_{\textit{2}} & p\_{\textit{2}} & & & & & \\ & \hfil\textit{T}\_{\textit{3}} & \hfil\textit{P} & & & & \\ & \hfil\textit{T}\_{\textit{3}} & \hfil\textit{P} & & & & \\ & \hfil\textit{T}\_{\textit{3}} & \hfil\textit{P} & & & & \\ & & & & & & \\ & & & & & & \\ \end{array} \tag{24}$$

coefficients of equations (23) are approximately ten times lower than those of equation (4), which explains increasing the extrapolation capabilities of the system (23). The resultant value of the vapor pressure at triple point *ptp* is about 25% lower than that of equation (4). Reducing the temperature range of initial *pT* data, *T* , from (25 to 50) K (accordingly, extending the range of extrapolation of vapor pressure to ~ 130 K) leads to the

Equation *A A*/ (/ ) *B B* (/ ) *C C* <sup>3</sup> ( / ) 10 *D D ptp* ,Pa

Table 10. The coefficients of equations (4) and (23) and saturated vapor pressure of 1,1,1-

Analysis of calculated data for *n*-decane and l,l,l-trifluoro-2,2-dichloroethane revealed that the use of the system of equations (23) for extrapolation of the vapor pressure in a wide temperature range *T extr* (120 to130) K enabled one to obtain *p*( ) *Ttp* values with

Extending the saturation vapor pressure to the critical region and computing the critical parameters of the substances were performed by the corresponding states law of Filippov's version (Filippov, 1988). A large class of normal (unassociated) liquids obey the oneparameter law of corresponding states [OLCS] which manifests an existence of the one-

Few literature data are available for critical parameters of organic compounds. The law of corresponding states in Filippov's version enables one both to obtain the required critical

equations for the calculation of the critical parameters and of the thermophysical properties were obtained by empirical method using the array of experimental data on the appropriate properties for studied compounds (Filippov, 1988). The critical parameters were calculated by algorithms (Filippov, 1988) and (Varushchenko et. al., 1987), in which the

14.58016±1.97 (10%)

10.76986±0.093 (1%)

surfaces of liquids. For the saturation curve, the OLCS equation

( , ), *A* where *A* is the similarity criterion. The latter is

 *=* 0.625 on the reduced curve of saturated vapor

of liquid and to calculate numerous

*F* . Generalized

(24)

17.7902±3.56

9.0292±0.23

(20%) 0.155

(2.5%) 0.114

6061.251±273.0 (4%)

5677.915±29.87 (0.5%)

trifluoro-2,2-dichloroethane at the triple point, *Ttp* = 145.68±0.02 K (Weber, 1992)

thermophysical properties on the basis of known values of *T* , *c V* , *c* and *A*

variation of *ptp* value by only 4-6%.

(4) 98.82786±11.09

(23) 78.44263±0.496

errors of 10%.

parameter family of

assigned by ordinate

pressure, *A* 100

 (, ) 0 *A* or

has the form

(10%)

(0.6%)

 

*F* at

following pairs of input data were used:

0.625.

parameters using precise *pT* data and density

 

with abscissa

The derivative *d pd T* ln( ) / (1 / ) was determined by differentiation of the equations ln( ) ( ) *p F T* for investigated substances. A random sampling of 14 compounds, having reliable data on the vapor pressure, density, and critical parameters was used to demonstrate (Varushchenko et. al., 1987) that 1) the employed algorithm is capable to produce adequate results in computing *T* , *c V* , *c* and *A F* and 2) the calculation algorithm originally developed for compounds with the similarity criterion 1 *<sup>A</sup> <sup>F</sup>* 4 is suitable for a wider class of compounds, including polyatomic molecules with the criterion 0.5 *A F* 4. A prediction of the thermophysical properties by the OLCS of Filippov (Filippov, 1988) requires following basic quantities: *T* , *c V* , *c* and *A F* . The errors of the latter calculation are within ±1—2%; the appropriate errors of *P* , *c* are higher and amount to ±3—5%.

The saturation vapor pressure are extrapolated to the critical region using the empirical binodal equation with pseudocritical parameters *T T <sup>c</sup>* and *P P <sup>c</sup>* :

$$\log(P^\ast \;/\; p) = a \cdot \log(T^\ast \;/\; T) + \left[T^\ast \;/\; T - 1\right] \cdot \left[b + c(T^\ast \;/\; T)\right] \,, \tag{25}$$

where *a =* 3.9726, *b =* 0.3252, and *c =* 0.40529.

Two-parameter formula (25) weakly depends on similarity criterion *A <sup>F</sup> ;* therefore, it may be employed for curvilinear extrapolation of saturated vapor pressure using only two pairs of the *pT* data. The formula was derived by the similarity conversion, namely using superposition of curves log( ) / lo g( ) which have the same slope for different substances. As a result, the family of curves reduces to a single curve which has segments of vapor pressure in the supercritical region. The pair of the *P T* parameters is of importance for remote point on the binodal curve, which enables one to calculate the vapor pressure of a substance in the region from the normal boiling temperature to the critical point. The error of the pressure calculation is (3 to 5) % depending on the temperature range of *pT* data extending and their reliabi lity.

Therefore, analysis of equations (4) and (23) demonstrated that equation (4) obtained by approximation of vapor pressure of the "atmospheric range" gives precision results when it used as an interpolation equation. It is further employed for extrapolation of the vapor pressure in the temperature range of 50 K with an error of (1 to 2) %. Extending the vapor pressure to the entire liquid phase from the critical to the triple points temperatures is performed by means of one-parameter law of corresponding states with errors from (3 to 5) % and, respectively, by simultaneous processing of the *pT* parameters and the differences between low-temperature heat capacities 0 0 () ( ) *C C g C liq pm pm pm* ,, , with uncertainties of 10 %, respectively. The data on extrapolation of the vapor pressure are suitable for many technological calculations.

#### **5. Conclusion**

The fundamental thermodynamic investigations of the phase equilibriums of many functional organic compounds were carried out by experimental and calculation methods in Luginin's Thermochemistry Laboratory of the Moscow State University. Modified setups have been created for precise determination of the saturation vapor pressures by

Thermodynamics of the Phase Equilibriums of Some Organic Compounds 633

properties were performed by comparison of the absolute entropies, obtained by the third

The agreement between these quantities confirms their reliability, as well as all experimental

We are grateful to Professor O. Dorofeeva for providing Gaussian programs and assistance in quantum-chemical calculations of the ideal gas thermodynamic functions. Many thanks to Professor S. Verevkin for helping in determination of the vapor pressures of some derivatives of ferrocene and Dr. L. Pashchenko for taking part in determination of the vapor pressures of some freons. Special thanks are to Post-graduate student E. Tkachenko for

This work was financially supported by Russian Foundation for Basic Research under

 *<sup>T</sup>* ( )Tln-lnT <sup>21</sup> ln <sup>Δ</sup> 

2/ )]ln(ln <sup>2</sup> 2/ <sup>2</sup> [ *TTTTTT* <sup>3</sup>

Coefficients of equation (A1) are calculated by LSM using orthogonal functions (Kornilov &

<sup>0</sup>, <sup>1</sup>,... *<sup>m</sup>* they may be replaced by linear combination of the orthogonal functions

;... ... <sup>12102022</sup> ; <sup>01011</sup> ; <sup>00</sup>

 *<sup>k</sup> <sup>s</sup>* 

are orthogonal if the following condition is satisfied:

*TTT*

The vapor pressures in dependence on temperature are treated by a relationship:

, and calculated by statistical thermodynamics, ( ) *<sup>o</sup> S stat <sup>m</sup>* .

=0 at *sk* (A2)

 

*pq <sup>q</sup> pp qpq*

1 0

, (A3)

*<sup>i</sup>* sign was replaced by . If there are *m* functions

, are defined from equation (A3) and the orthogonal

(A1)

thermodynamic law, ( )( ) *<sup>o</sup> S g <sup>m</sup>*

**6. Acknowledgment** 

**7. Appendix** 

Vidavski, 1969). Two functions *<sup>k</sup>*

*p* ,...

condition as:

e x p t

and calculation data used in computing process.

providing illustrative materials of the chapter.

Grants No. 96-02-05445 and No. 05-02-17435.

**7.1 Mathematical processing of saturation vapor pressure** 

*<sup>T</sup> HpRT*

where )ln(/( <sup>1</sup> *<sup>T</sup> pRT <sup>T</sup>*

 and *s* 

According to Gauss, the

<sup>1</sup> , <sup>0</sup> using Shmidt method:

where coefficients of orthogonality, *pq*

*H* .

comparative ebulliometry, the enthalpies of vaporization by evaporation calorimetry, and the low temperature heat capacity and phase transitions by vacuum adiabatic calorimetry.

The saturation vapor pressures were determined in moderate range of pressure 2 (*p*/kPa) 101.6 with accuracy of the temperature 0.01 K, and pressure, 26 Pa, which correspond to the modern precision levels. The temperature dependences of the saturation vapor pressure, ln( ) ( ) *p F T* , and the enthalpies of vaporization *Hvap f* ( ) *T* were obtained by mathematical processing of the *pT* parameters by equation (1), derived on the basis of Clapeyron equation using LSM with orthogonal functions. The latter allow one to calculate the errors of *Hvap* values, which are urgent problem because the indirect method is the main source for determination of the enthalpies of vaporization. An agreement of the *Hvap* values obtained by direct (calorimetric) and indirect (calculation) methods proves their reliability.

The precise saturated vapor pressure data are extended to entire region of the liquids under study. Extrapolation of the *pT* parameters down to the triple point temperature are carried out by simultaneous processing the vapor pressures and low-temperature differences 0 0 () ( ) *C C g C liq pm pm pm* ,, , , which are the second derivatives of the vapor pressure upon the temperature. Extending the *pT* parameters to the critical region and calculation of the critical quantities are performed by Filippov's one-parameter law of the corresponding states. The latter enables us both to calculate the critical parameters on the basis of more readily available *pT* -data and density of liquids and to predict numerous thermo-physical properties of the equilibrium liquid – vapor by means of the known critical quantities*T* , *c V* , *c* and criterion of similarity *A F* .

The low temperature heat capacity in the temperature region (5 to 373) K, molecular motion in crystal and metastable phases, and solid state transitions and fusion were investigated by the adiabatic calorimetry. Uncertainties of the*Cp*,*m* measurements are on the average ~0.2 %

which correspond up-to-date precision level.

An accurate calorimetric study of the solid states of the functional organic compounds revealed different polymorphic modifications of the molecules, order – disorder transitions involving orientational and conformational disorder, glass-like transitions, and plastic crystalline phases with anisotropic and isotropic reorientations of the molecules. For interpretation of these transformations, the X-ray crystallography, infrared and Raman molecular spectroscopy were got involved in investigation.

The main thermodynamic functions in three aggregate states: the absolute entropy by the 3d law of thermodynamics, the changes of the enthalpy and free Gibbs energy are derived on the basis of the heat capacity and vapor pressure measurements. A critical analysis and verification of the reliability of obtained data are very significant parts of the thermodynamic investigation. With this in mind, the experimental thermodynamic functions are compared with calculated ones by additive principles and by statistical thermodynamics coupled with quantum mechanical (QM) calculation on the basis of DFT method. The QM calculation are performed on the level B3LYP/6-31G(d,p) by Gaussian 98 and 03 software packages.

A qualitative analysis of thermodynamic properties in dependence on some parameters responsible for intermolecular interactions and short range order of the liquid phase has been carried out for verification of the mutual consistency of the properties in homologous series and the series of the same type of compounds. A quantitative verification of the properties were performed by comparison of the absolute entropies, obtained by the third thermodynamic law, ( )( ) *<sup>o</sup> S g <sup>m</sup>* e x p t , and calculated by statistical thermodynamics, ( ) *<sup>o</sup> S stat <sup>m</sup>* . The agreement between these quantities confirms their reliability, as well as all experimental and calculation data used in computing process.
