Excess thermodynamic parameters for binary and ternary mixtures of {1-butanol (1) + cyclohexylamine (2) + n-heptane (3)} at different temperatures: A theoretical study

1 Introduction

This paper is a continuation of our earlier work related to the study of thermodynamic properties of binary and ternary mixtures of different groups of organic compounds (Rezaei-Sameti et al., 2009; Rezaei-Sameti and Iloukhani, 2009; Iloukhani et al., 2005, 2006; Iloukhani and Rezaei-Sameti, 2005a,b). Excess thermodynamic properties of mixtures are useful in the study of molecular interactions and arrangements. In the recent years, measurements of thermodynamic and transport properties have been adequately employed in understanding the nature of molecular systems and physicochemical behavior in liquid mixtures. The non-rectilinear behavior of the above mentioned properties of liquid mixtures with changing mole fractions is attributed to the difference in size of the molecules and strength of interactions (Henni et al., 2005; Domiınguez et al., 2000; Lj Kijevcanin et al., 2009).

In this work we used the experimental data of Kijevcanin et al. for binary and ternary mixtures formed by 1-butanol + cyclohexylamine + n-heptane at (288.15–323.15) K for calculating thermal expansion coefficients, α, excess thermal expansion coefficients, α^E, and isothermal coefficient of pressure excess molar enthalpy, ${(\partial H_{\text{m}}^{\text{E}}/\partial P)}_{\text{T}}$ . The values of α^E and ${(\partial H_{\text{m}}^{\text{E}}/\partial P)}_{\text{T}}$ for binary mixtures are calculated by using the Flory theory. All excess parameters for binary mixtures have been fitted by Redlich–Kister (Redlich and Kister, 1948).

and the ternary mixture has been fitted by the Cibulka (Domiınguez et al., 2000), Jasinski and Malanowski (Domiınguez et al., 2000), Singe et al. (Domiınguez et al., 2000), Pintos et al. (Domiınguez et al., 2000), Calvo et al. (Domiınguez et al., 2000), Kohler (Domiınguez et al., 2000), and Jacob–Fitzner (Domiınguez et al., 2000), equations to determine the deviation between experimental and theoretical values.

2 Results and discussion

The experimental data are given at various temperatures (288.15–323.15 K) and the results are fitted by the Redlish–Kister (Redlich and Kister, 1948) equation of temperature-dependence:

These parameters were obtained by the unweighted least-squares method. The parameters A_pq for all the binary mixtures are listed in Table 2.

In the ternary mixture the excess molar volumes, V^E excess thermal expansion coefficients, α^E, and isothermal coefficient of pressure excess molar enthalpy, ${(\partial H_{\text{m}}^{\text{E}}/\partial P)}_{\text{T,}}$ were fitted with the following temperature-dependant equations:

1. Cibulka equation:

   (3)
   $$Y_{\text{ijk}}^{\text{E}}=Y_{\text{m,bin}}^{\text{E}}+x_1{x}_2{x}_3(M_0+M_1{x}_1+M_2{x}_2)$$

2. Jasinski and Malonowski equation:

   (4)
   $$Y_{\text{ijk}}^{\text{E}}=Y_{\text{m,bin}}^{\text{E}}+x_1{x}_2{x}_3(M_0+M_1(2x_1-1)+M_2{(2x_1-1)}^2)$$

3. Singe et al. equation:

   (5)
   $$Y_{\text{ijk}}^{\text{E}}=Y_{\text{m,bin}}^{\text{E}}+x_1{x}_2{x}_3(M_0+M_1{x}_1(x_2-x_3)+M_2{x}_1^2{(x_2-x_3)}^2)$$

4. Pintos et al. equation:

   (6)
   $$Y_{\text{ijk}}^{\text{E}}=Y_{\text{m,bin}}^{\text{E}}+x_1{x}_2{x}_3(M_0+M_1(x_1^2+2x_1{x}_2-1)+M_2{(x_1^2+2x_1{x}_2-1)}^2)$$

5. Calvo et al. equation:

   (7)
   $$Y_{\text{ijk}}^{\text{E}}=Y_{\text{m,bin}}^{\text{E}}+x_1{x}_2{x}_3(M_0+M_1(2x_1+2x_2-1)+M_2{(2x_1+2x_2-1)}^2)$$ where $Y_{\text{m,bin}}^{\text{E}}$ are the contributions of binary mixture (i, j)
   (8)
   $$Y_{\text{m,bin}}^{\text{E}}=Y_{\text{m,}12}^{\text{E}}+Y_{\text{m,}13\text{,}}^{\text{E}}+Y_{\text{m,}23}^{\text{E}}$$ Every M_i ternary parameter is a function of temperature as expressed in Eqs. (6)–(8):
   (9)
   $$M_i={\displaystyle \sum _{q=0}^2}{B}_{\mathit{iq}}{T}^q$$ The parameters B_iq for the ternary mixture are listed in Tables 3–5 along with the standard deviation σ. Among the semi-empirical equations used to predict, V^E, α^E and ${(\partial H_{\text{m}}^{\text{E}}/\partial P)}_{\text{T,}}$ for the studied system, the best for prediction were those of the Calvo equation, and other semi-empirical equations erred significantly. Thus, it is up to the design engineer to consider, the advantages of relying on these relations for the description of these properties as continuous models for the whole range of ternary compositions.

2.1

2.1 Thermal expansion coefficient and their excess values

The temperature dependence of density of the pure components was fitted to the equation:

(10)

$$\rho (T)/g{\mathit{cm}}^{-3}={\displaystyle \sum _{i=0}^4}{a}_i{T}^i$$

The parameters a_i for the pure components are listed in Table 6. The thermal expansion coefficient, α, as in the case of pure components was obtained by analytical differentiation of the density fitting equation:

(11)

$$\alpha =-{\rho }^{-1}{(\partial \rho /\partial T)}_P$$

Table 6 Coefficients a_i of Eq. (10) of the pure components densities in the temperature range (288.15–323.15 K).

Component	a0	a1	a2	a3	a4
1-Butanol	−3.9 × 10−1	1.8 × 10−2	−9.4 × 10−5	2.1 × 10−7	−1.7 × 10−7
Cyclohexylamine	1.6	−7.0 × 10−3	3.2 × 10−5	−7.1 × 10−8	5.7 × 10−11
n-Heptane	1.7	−1.1 × 10−2	5.0 × 10−5	−1.1 × 10−7	8.6 × 10−11

The thermal expansion coefficients of pure components at different temperatures are presented in Table 1. The basic expression relating the molar volume of a mixture and its excess molar volume is, in accordance with the Eq. (1):

(12)

$$V={\displaystyle \sum _{i=1}^N}{x}_i{V}_i+V_{\text{m}}^{\text{E}}$$ where V_i and x_i correspond to the molar volume and to the mole fraction concentration of component i. By differentiating Eq. (12) and dividing by V and reversing the order of terms:

(13)

$$\alpha =1/V\left[{\left(\partial V_{\text{m}}^{\text{E}}/\partial T\right)}_{P\text{,}{x}_i}+\partial {\displaystyle \sum _{i=1}^N}(x_i{V}_i)/\partial T\right]$$

Table 1 Flory and related parameters of pure components at (288.15–323.15) K.

	T/K	104 α/K−1	ρ/g cm−3	V∗/(cm3 mol−1)	T∗/K	$\tilde{v}$	$\tilde{T}$
1-Butanol
	288.15	9.298	0.813373	74.296	5423.666	1.226	0.053
	298.15	9.403	0.809573	74.313	5427.915	1.232	0.054
	298.15	9.504	0.805762	74.336	5434.213	1.237	0.054
	303.15	9.609	0.801923	74.360	5440.291	1.243	0.055
	308.15	9.724	0.798053	74.376	5444.098	1.248	0.056
	313.15	9.857	0.794147	74.375	5443.756	1.254	0.057
	318.15	10.013	0.790231	74.348	5437.737	1.261	0.058
	323.15	10.201	0.786238	74.290	5424.401	1.269	0.059
Cyclohexylamine
	288.15	10.439	0.871290	91.099	5080.063	1.249	0.056
	298.15	10.478	0.866747	91.259	5109.430	1.253	0.057
	298.15	10.524	0.862207	91.414	5137.278	1.258	0.058
	303.15	10.574	0.857671	91.565	5164.096	1.262	0.058
	308.15	10.627	0.853138	91.716	5190.462	1.267	0.059
	313.15	10.681	0.848607	91.870	5216.903	1.272	0.060
	318.15	10.733	0.844073	92.031	5243.871	1.276	0.060
	323.15	10.782	0.839547	92.201	5271.949	1.281	0.061
n-Heptane
	288.15	12.249	0.687954	113.426	4670.356	1.284	0.061
	298.15	12.372	0.683733	113.574	4687.364	1.290	0.063
	298.15	12.512	0.679494	113.703	4701.826	1.297	0.063
	303.15	12.666	0.675229	113.820	4714.529	1.303	0.064
	308.15	12.830	0.670941	113.931	4726.326	1.310	0.065
	313.15	13.001	0.666621	114.043	4737.890	1.318	0.066
	318.15	13.175	0.662273	114.163	4749.926	1.325	0.067
	323.15	13.349	0.657896	114.297	4763.044	1.332	0.068

From this expression, the excess thermal expansion coefficient can be rewritten as:

(14)

$${\alpha }^E=\alpha -{\displaystyle \sum _{i=1}^n}{\varphi }_i{\alpha }_i$$ where

(15)

$${\phi }_i=x_i{V}_i/{\displaystyle \sum _{i=1}^2}{x}_i{V}_i$$ and ϕ_i is the volumetric fraction of the components of the mixture. This property can be described as the variation of density with temperature due to the no ideality of the mixture. The thermal expansion coefficient α, and excess thermal expansion coefficient α^E are for all binary mixtures recorded in Supplementary Tables and graphically represented in Fig. 1. The excess thermal expansions coefficient α^E, for {1-butanol  + cyclohexylamine} at temperatures 288.15–323.15 K are S-shaped and it is positive in the 1-butanol rich-region and increase with increasing temperatures from (288.15 to 323.15) K. The excess thermal expansion coefficients, α^E, are positive over the mole fraction range for the binary mixture of {1-butanol + n-heptane} and increase with increasing temperatures from (288.15 to 323.15) K and negative over the mole fraction range for a binary mixture of {cyclohexylamine + n-heptane} and decrease with increasing temperatures from (288.15 to 323.15) K. Positive values of α^E due to self-association of components in the mixtures, and the components of mixture are strongly associated in the pure state and in mixtures. Negative values of α^E reflected the association of components in the mixtures and are strongly associated in the mixtures. Thermal expansions’ coefficient is related to the fluctuation of cross term of enthalpy interaction and interaction volume of liquids and mixtures. The excess amount may reflect the molecular orientation and packing of mixtures. Molecular orientation effects are ascribed to the shapes of molecules for nonpolar solvents and dipole moments, higher multipole moments, and special interactions such as hydrogen bonds for polar solvents.

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Figure 1

Plot of excess thermal expansion coefficients α^E against mole fraction for {(a₀₁); 1-butanol + cyclohexylamine, (a₀₂); 1-butanol + n-heptane, (a₀₃); cyclohexylamine + n-heptane} at 288.15 K (■), 293.15 K (□), 298.15 K (●), 303.15 K (○), 308.15 K (▴), 313.15 K (△), 318.15 K (♢), 323.15 K (♦).

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2.2

2.2 Isothermal coefficient of pressure excess molar enthalpy

The isothermal coefficient of pressure excess molar enthalpy can be derived accurately from volumetric measurements by application of the following expression:

(16)

$${(\partial H_{\text{m}}^{\text{E}}/\partial P)}_{T\text{,}x}=V_{\text{m}}^{\text{E}}-T{(\partial V_{\text{m}}^{\text{E}}/\partial T)}_{P\text{,}x}$$

This quantity represents the dependence of the excess molar enthalpy of mixing with pressure at fixed composition and temperature. The isothermal coefficient of pressure excess molar enthalpy, ${(\partial H_{\text{m}}^{\text{E}}/\partial P)}_{\text{T}}$ , for all binary mixtures is recorded in Supplementary Tables and graphically represented in Fig. 2. The isothermal coefficient of pressure excess molar enthalpy, ${(\partial H_{\text{m}}^{\text{E}}/\partial P)}_{\text{T}}$ , for binary mixture {1-butanol + cyclohexylamine} is S-shaped and dependence on composition with negative values in the 1-butanol rich-region decrease with increasing temperatures from (288.15 to 323.15) K. The values of, ${(\partial H_{\text{m}}^{\text{E}}/\partial P)}_{\text{T}}\text{,}$ are negative over the mole fraction range for binary mixture of {1-butanol + n-heptane} and decrease with increasing temperatures from (288.15 to 323.15) K. The ${(\partial H_{\text{m}}^{\text{E}}/\partial P)}_{\text{T}}$ values are positive over the mole fraction range for a binary mixture of {cyclohexylamine + n-heptane} and increase with increasing temperatures from 288.15 to 323.15 K. The values of ${(\partial H_{\text{m}}^{\text{E}}/\partial P)}_{\text{T}}$ for all binary mixtures are due to the presence of significant interactions between solvents in these mixtures. With increasing temperature the interactions of molecular mixtures increased and so the ${(\partial H_{\text{m}}^{\text{E}}/\partial P)}_{\text{T,}x}$ values are more negative.

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Figure 2

Plot of isothermal coefficient of pressure excess molar enthalpy ${(\partial H_{\text{m}}^{\text{E}}/\partial P)}_{\text{T}}$ , against mole fraction for {(a₁₁);1-butanol + cyclohexylamine, (a₁₂); 1-butanol + n-heptane, (a₁₃); cyclohexylamine + n-heptane} at 288.15 K (♦), 293.15 K (■), 298.15 K (▴), 303.15 K (×), 308.15 K (△), 313.15 K (●), 318.15 K (○), 323.15 K (□).

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2.3

2.3 Excess partial molar volume

The excess partial molar volumes, ${\overline{V}}_{\text{m,i}}^{\text{E}}\text{,}$ of a component in a binary mixture can be determined from excess molar volume data using,

(17)

$${\overline{V}}_{\text{m,i}}^{\text{E}}=V_{\text{m}}^{\text{E}}+(1-x){\left[\frac{\partial V_{\text{m}}^{\text{E}}}{\partial x_i}\right]}_{P\text{,}T}$$ The excess partial molar volumes of ${\overline{V}}_{\text{m,1}}^{\text{E}}$ and ${\overline{V}}_{\text{m,2}}^{\text{E}}$ for all compositions can be calculated by using the Redlich–Kister coefficients, A_pq (Table 3) in Eq. (1). Results of excess partial molar volumes for all binary mixtures are also listed in Supplementary Tables and graphically represented in Fig. 3. The excess partial molar volumes ${\bar{V}}_i^E$ for all binary mixtures in this study show that with increasing mole fraction of one component, the ${\bar{V}}_i^E$ values are increased. In the above binary mixtures, expansions in volume are due to a decrease in the molecular order in the lattice. With increasing of the mole fraction of one component in the mixtures the possibility of electron donor–acceptor type (or charge-transfer) interactions between the mixture’s components is decreased, so the ${\bar{V}}_i^E$ values are increased.

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Figure 3

Plot of excess partial molar volume ${\overline{V}}_{\text{m,i}}^{\text{E}}$ , against mole fraction for {(a₂₁);1-butanol + cyclohexylamine, (a₂₂); 1-butanol + n-heptane, (a₂₃); cyclohexylamine + n-heptane } at 288.15 K (■), 293.15 K (□), 298.15 K (●), 303.15 K (○), 308.15 K (▴), 313.15 K (△), 318.15 K (♢), 323.15 K (♦).

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3 The Flory theory

The Flory (Flory, 1965, 1970; Flory et al., 1964a,b; Prigogine, 1957; Van and Patterson, 1982; Abe and Flory, 1965) model has been commonly employed to analyze the molar volume of the mixture and the excess molar volume parting from the equation of the state in a function of the reduced variables:

Theoretical values of $\left[\frac{\partial V_{\text{m}}^{\text{E}}}{\partial T}\right]$ were calculated from the Flory theory using:

In the present study the value of the reduced volume for the liquids and their mixtures $\stackrel{⌢}{v}$ was determined for α value of the mixtures using the Eq. (19). Table 1 shows the data used in the application of the Flory model. The excess function ${[\frac{\partial V_{\text{m}}^{\text{E}}}{\partial T}]}_P$ was computed by analytical differentiation of the Eq. (1) at (288.15–323.15 K) and $V_m^E$ of this work. we have also obtained isothermal coefficient of pressure excess molar enthalpy, ${(\partial H_{\text{m}}^{\text{E}}/\partial P)}_{\text{T}}$ , and excess thermal expansions coefficient α^E, according to the Eqs. ()()()()()(18)–(22). Values of the isothermal coefficient of pressure excess molar enthalpy, ${(\partial H_{\text{m}}^{\text{E}}/\partial P)}_{\text{T}}$ ,and excess thermal expansions coefficient α^E, for binary mixtures are reported in the supplementary data and are represented in Figs. 4 and 5. The results show a good agreement between experimental and theoretical values.

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Figure 4

Plot of excess thermal expansion coefficients α^E, against mole fraction for {(a₃₁);1-butanol + cyclohexylamine, (a₃₂); 1-butanol + n-heptane, (a₃₃); cyclohexylamine + n-heptane } 323.15 K. (o) Experimental; (……) calculated by using Flory theory.

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Figure 5

Plot of isothermal coefficient of pressure excess molar enthalpy ${(\partial H_{\text{m}}^{\text{E}}/\partial P)}_{\text{T}}\text{,}$ against mole fraction for {(a₄₁);1-butanol + cyclohexylamine, (a₄₂); 1-butanol + n-heptane, (a₄₃); cyclohexylamine + n-heptane} 323.15 K. (o) Experimental; (……) calculated by using the Flory theory.

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4 Conclusion

In this work thermal expansion coefficients α, their excess values α^E, isothermal coefficient of pressure excess molar enthalpy ${(\partial H_{\text{m}}^{\text{E}}/\partial P)}_{\text{T}}$ , partial molar volumes ${\bar{V}}_i$ and excess partial molar volumes ${\bar{V}}_i^E$ , were calculated from experimental densities. The results shown that: The diagrams α^E and ${(\partial H_{\text{m}}^{\text{E}}/\partial P)}_{\text{T}}$ for binary mixture {1-butanol + cyclohexylamine} are S-shaped and those values for other binary mixtures are positive or negative over the whole mole fraction due to the presence of significant interactions between solvents in these mixtures. This study shows that, the ${\bar{V}}_i^E$ values increase with increasing mole fraction of one component. In the above binary mixtures, expansions in volume and changes in other calculated parameters are due to a decrease in the molecular order in the lattice. Among the semi-empirical equations used to predict, V^E, α^E and ${(\partial H_{\text{m}}^{\text{E}}/\partial P)}_{\text{T,}}$ for the studied system, the best for prediction were those of the Calvo equation, and other semi-empirical equations erred significantly. The comparison between the experimental and Flory theories shows a good agreement.