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A new theoretical approach for estimating excess internal pressure
Journal of Molecular Liquids 124 (2006) 121 – 123 www.elsevier.com/locate/molliq
A new theoretical approach for estimating excess internal pressure Ranjan Dey *, A.K. Singh, J.D. Pandey Department of Chemistry, University of Allahabad, Allahabad, 211002, India Received 13 April 2005; accepted 5 September 2005
Abstract Over the years, internal pressure has proven to be an important tool for the study of intermolecular interactions in binary and multicomponent liquid mixtures. Hence prediction of this property and its corresponding excess parameters assumes paramount significance in the light of intermolecular interactions. In this regard, the assumption of mole fraction additivity of internal pressure for the ideal mixture is somewhat erroneous. In this context, a corrected and modified expression for a recently proposed relation has been presented and a comparative study carried out thereof. For this, three binary mixtures of dimethylsulphoxide(DMSO) + 1-alkanol(1-butanol, 1-hexanol and 1-octanol) at 303.15 K have been put to test. D 2005 Elsevier B.V. All rights reserved. Keywords: Internal pressure; Excess; Intermolecular interactions
Assuming the thermodynamic equation of state,
1. Introduction The study of internal pressure of binary liquid mixtures has been carried out by several workers [1– 3]. It has been extensively used as a tool to study the intermolecular interactions, internal structure, clustering phenomenon, ordered structure, etc. [4– 6]. Correlations with the liquid state model in case of binary mixtures have been also made [7 – 9], and this investigation has been further extended to multicomponent mixtures in recent past [10]. 2. Formula derivation Internal pressure of the mixture may be defined in terms of the influence of the volume on the internal energy of the liquid. Thermodynamically, the isothermal internal energy and molar volume coefficients of real fluids measure the internal pressure( P int) as Pint ¼ ðBU =BV ÞT
ð1Þ
where U is the internal energy, V the volume and T the temperature. * Corresponding author. E-mail address: [email protected] (R. Dey). 0167-7322/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2005.09.005
ðBU =BV ÞT ¼ T ð BP=BT ÞV P
ð2Þ
where P is the pressure. Employing the expression of thermal expansion coefficient (a) and isothermal compressibility(K T), internal pressure can be expressed as [11] aT Pint ¼ ðBU =BV ÞT ¼ P ð3Þ jT where all the symbols have their usual meaning. The excess internal pressure, P intE of liquid mixture is given by X E id Pint ¼ Pint Pint ¼ Pint Pint i Ixi ð4Þ Where x is mole fraction and i, id denote the ith component and the ideal mixture, respectively. The quantity calculated by Eq. (4) is sometimes called the deviation of internal pressure [12]. Very recently, Marczak [13] proposed an idea for computing excess internal pressure. In his approach, he computed ideal internal pressure by making use of mole fraction additivity in terms of thermal expansion coefficient (a) and isothermal compressibility (j T). This is evidently erroneous since both thermal expansion coefficient and isothermal compressibility
R. Dey et al. / Journal of Molecular Liquids 124 (2006) 121 – 123
are very well known volume fraction additive quantities [14]. Since such a wrongful assumption results in major computational error, the aim of the present investigation is an attempt to rectify the error and obtain a corrected and modified expression for the computation of internal pressure of a thermodynamically ideal mixture. This ideal value has then been employed for computation of excess internal pressure. For the calculation of P id int , various authors have assumed [6,15 – 17] that X id ¼ Pint i Ixi ð5Þ Pint which is erroneous and id a T id Pint ¼ P jid T
id P int
can be given as
jid T ¼
X
200.00 0.00 -200.00
0.00 0.19 0.37 0.53 0.67 0.80 0.93 1.00
-400.00 -600.00
x1
-800.00
eqn-4
ð6Þ
600.00
eqn-8
DMSO+1-hexanol
400.00 200.00
/ i ai /i jT i
0.00 -200.00
0.00 0.19 0.36 0.52 0.67 0.80 0.92 1.00
-400.00 -600.00
x1
-800.00
xi V i /i ¼ X xi Vi
eqn-4
ð8Þ
c 600.00
eqn-8
DMSO+1-octanol
400.00 200.00 PintE(atm)
where / i is the volume fraction. Now from Eq. (6) we have X /i ai T id Pint ¼ X P ð7Þ ui jT i using Eqs. (3) and (7) in Eq. (4) we have X ! /i ai T aT E Pint ¼ P X P jT /i jT i X ! / a T i aT i E Pint ¼ X jT /i jTi
DMSO+1-butanol
400.00
PintE(atm)
X
600.00
b
where aid ¼
a
PintE(atm)
122
0.00 -200.00
0.00 0.19 0.36 0.52 0.66 0.80 0.92 1.00
-400.00 -600.00 -800.00
x1
ð9Þ
where thermal expansion coefficient and isothermal compressibility have been computed through the recently developed relation [18 – 20]. 3. Results and discussion Excess internal pressure of three binary systems as dimethylsulphoxide (DMSO) + 1-butanol, DMSO + 1-hexanol and DMSO + 1-octanol have been calculated using Eqs. (4) and (9) at 303 K. Necessary data needed for computation have been taken from the literature [21]. The excess internal pressure obtained via the two approaches have been plotted graphically against the mole fraction and the graphical representations have been shown in Fig. 1(a – c). In all the three systems under investigation we find that the plots show contrasting trends. While the excess internal pressure values obtained from the previous method (Eq. (4)) show negative values over the entire mole fraction range, those given by the proposed method show positive values over the entire concentration range. This trend is seen in all the systems under investigation.
eqn-4
eqn-8
Fig. 1. Excess internal pressure of DMSO + 1-butanol (a), DMSO + 1-hexanol (b) and DMSO + 1-octanol (c) at 303.15 K vs mole fraction.
In all the three systems under investigation, the excess internal pressure values obtained via the corrected method are found to be positive. It is a well known fact that while negative values of excess functions indicate strong interactions and are observed when intermolecular complexes are formed, positive values are indicative of weak interactions. Previously reported values [21] of variation of excess velocity, u E, excess specific acoustic impedance, Z E and excess adiabatic compressibility, b SE, point out towards weak interactions between the component molecules and this interaction shows a decrease with increase in the carbon chain length from 1-butanol to 1-octanol. These literature values further strengthen the validity of the modified and the corrected expression for evaluation of excess internal pressure as compared to the one proposed earlier [13]. Further, it also proves that the excess internal pressure computed with the mole fraction additivity (Eq. (4)) is definitely erroneous, since the negative values of internal pressure obtained thereof indicate presence of strong interac-
R. Dey et al. / Journal of Molecular Liquids 124 (2006) 121 – 123
tions which are in complete contradiction to literature values [21]. The proposed relation receives further credence from the fact that positive values of excess properties can be interpreted as being due to dispersion forces. Experimental findings and literature survey reveals that structure breaking effect and weak interactions between unlike molecules predominate in all the DMSO + 1-alkanol (1-butanol, 1-hexanol and 1-octanol) systems due to rupture of hydrogen bonded chains of the alkanols and loosening of dipolar interactions and decrease in strength of interaction between unlike molecules. The observed excess free energy of activation of viscous flow, G E, and excess viscosity values, g E values (reported earlier) both suggest that dispersion forces are operative between unlike molecules for all the binary systems under investigation. Thus we conclude that the proposed relation in the present investigation definitely proves its suitability and credibility for computation of excess internal pressure over the previously developed methods. References [1] M.R.J. Dack, Aust. J. Chem. 281 (1975) 643. [2] J.D. Pandey, R.L. Mishra, Acustica 33 (1981) 200.
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
123
J.D. Pandey, J. Chem. Soc., Faraday Trans. II 76 (1980) 1215. J.D. Pandey, B.R. Chaturvedi, N. Pant, Chem. Scr. 18 (1981) 224. J.D. Pandey, U. Gupta, Z. Physik. Chemie. (Leipzig) 268 (1987) 477. J.D. Pandey, R. Dey, Acoust. Lett. 24 (2000) 105. S.V. Subrahmanyam, V. Hyder Khan, K. Malakondaiah, Acustica 45 (1980) 196. J.D. Pandey, N. Dubey, D.K. Dwivedi, R. Dey, Phys. Chem. Liq. 39 (2001) 781. Amalendu Pal, Harsh Kumar, Phys. Chem. Liq. 41 (2003) 415. J.D. Pandey, A.K. Shukla, Neelima Tripathi, G.P. Dubey, Pramana J. Phys. 40 (1993) 81. J.A.R. Renuncio, G.J.F. Breedveld, J.M. Prausnitz, J. Phys. Chem. 81 (1977) 324. T.M. Aminabhavi, V.B. Patil, K. Banerjee, R.H. Balundgi, Bull. Chem. Soc. Jpn. 72 (1999) 1187. W. Marczak, Phys. Chem. Chem. Phys. 4 (2002) 1889. M.H. Blandemer, M.J. Davis, G. Douheret, J.C.R. Reis, Chem. Soc. Rev. 30 (2001) 6. V. Rajendran, Ind. J. Pure Appl. Phys. 31 (1993) 812. L. Acevedo, G.C. Pedrosa, M. Katz, J. Solution Chem. 19 (1990) 911. I.M. Aralaguppi, T.M. Aminabhavi, R.H. Balundgi, S.S. Joshi, J. Phys. Chem. 95 (1991) 5299. J.D. Pandey, R. Dey, M. Upadhyaya, Acoust. Lett. 21 (6) (1997) 120. J.D. Pandey, R. Verma, Chem. Phys. 270 (3) (2001) 429. J.D. Pandey, R. Dey, R. Verma, Phys. Chem. Liq. 14 (2) (2003) 145. A. Ali, K. Tiwari, A.K. Nain, V. Chakravortty, Phys. Chem. Liq. 38 (2000) 459.
A new theoretical approach for estimating excess internal pressure Ranjan Dey *, A.K. Singh, J.D. Pandey Department of Chemistry, University of Allahabad, Allahabad, 211002, India Received 13 April 2005; accepted 5 September 2005
Abstract Over the years, internal pressure has proven to be an important tool for the study of intermolecular interactions in binary and multicomponent liquid mixtures. Hence prediction of this property and its corresponding excess parameters assumes paramount significance in the light of intermolecular interactions. In this regard, the assumption of mole fraction additivity of internal pressure for the ideal mixture is somewhat erroneous. In this context, a corrected and modified expression for a recently proposed relation has been presented and a comparative study carried out thereof. For this, three binary mixtures of dimethylsulphoxide(DMSO) + 1-alkanol(1-butanol, 1-hexanol and 1-octanol) at 303.15 K have been put to test. D 2005 Elsevier B.V. All rights reserved. Keywords: Internal pressure; Excess; Intermolecular interactions
Assuming the thermodynamic equation of state,
1. Introduction The study of internal pressure of binary liquid mixtures has been carried out by several workers [1– 3]. It has been extensively used as a tool to study the intermolecular interactions, internal structure, clustering phenomenon, ordered structure, etc. [4– 6]. Correlations with the liquid state model in case of binary mixtures have been also made [7 – 9], and this investigation has been further extended to multicomponent mixtures in recent past [10]. 2. Formula derivation Internal pressure of the mixture may be defined in terms of the influence of the volume on the internal energy of the liquid. Thermodynamically, the isothermal internal energy and molar volume coefficients of real fluids measure the internal pressure( P int) as Pint ¼ ðBU =BV ÞT
ð1Þ
where U is the internal energy, V the volume and T the temperature. * Corresponding author. E-mail address: [email protected] (R. Dey). 0167-7322/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2005.09.005
ðBU =BV ÞT ¼ T ð BP=BT ÞV P
ð2Þ
where P is the pressure. Employing the expression of thermal expansion coefficient (a) and isothermal compressibility(K T), internal pressure can be expressed as [11] aT Pint ¼ ðBU =BV ÞT ¼ P ð3Þ jT where all the symbols have their usual meaning. The excess internal pressure, P intE of liquid mixture is given by X E id Pint ¼ Pint Pint ¼ Pint Pint i Ixi ð4Þ Where x is mole fraction and i, id denote the ith component and the ideal mixture, respectively. The quantity calculated by Eq. (4) is sometimes called the deviation of internal pressure [12]. Very recently, Marczak [13] proposed an idea for computing excess internal pressure. In his approach, he computed ideal internal pressure by making use of mole fraction additivity in terms of thermal expansion coefficient (a) and isothermal compressibility (j T). This is evidently erroneous since both thermal expansion coefficient and isothermal compressibility
R. Dey et al. / Journal of Molecular Liquids 124 (2006) 121 – 123
are very well known volume fraction additive quantities [14]. Since such a wrongful assumption results in major computational error, the aim of the present investigation is an attempt to rectify the error and obtain a corrected and modified expression for the computation of internal pressure of a thermodynamically ideal mixture. This ideal value has then been employed for computation of excess internal pressure. For the calculation of P id int , various authors have assumed [6,15 – 17] that X id ¼ Pint i Ixi ð5Þ Pint which is erroneous and id a T id Pint ¼ P jid T
id P int
can be given as
jid T ¼
X
200.00 0.00 -200.00
0.00 0.19 0.37 0.53 0.67 0.80 0.93 1.00
-400.00 -600.00
x1
-800.00
eqn-4
ð6Þ
600.00
eqn-8
DMSO+1-hexanol
400.00 200.00
/ i ai /i jT i
0.00 -200.00
0.00 0.19 0.36 0.52 0.67 0.80 0.92 1.00
-400.00 -600.00
x1
-800.00
xi V i /i ¼ X xi Vi
eqn-4
ð8Þ
c 600.00
eqn-8
DMSO+1-octanol
400.00 200.00 PintE(atm)
where / i is the volume fraction. Now from Eq. (6) we have X /i ai T id Pint ¼ X P ð7Þ ui jT i using Eqs. (3) and (7) in Eq. (4) we have X ! /i ai T aT E Pint ¼ P X P jT /i jT i X ! / a T i aT i E Pint ¼ X jT /i jTi
DMSO+1-butanol
400.00
PintE(atm)
X
600.00
b
where aid ¼
a
PintE(atm)
122
0.00 -200.00
0.00 0.19 0.36 0.52 0.66 0.80 0.92 1.00
-400.00 -600.00 -800.00
x1
ð9Þ
where thermal expansion coefficient and isothermal compressibility have been computed through the recently developed relation [18 – 20]. 3. Results and discussion Excess internal pressure of three binary systems as dimethylsulphoxide (DMSO) + 1-butanol, DMSO + 1-hexanol and DMSO + 1-octanol have been calculated using Eqs. (4) and (9) at 303 K. Necessary data needed for computation have been taken from the literature [21]. The excess internal pressure obtained via the two approaches have been plotted graphically against the mole fraction and the graphical representations have been shown in Fig. 1(a – c). In all the three systems under investigation we find that the plots show contrasting trends. While the excess internal pressure values obtained from the previous method (Eq. (4)) show negative values over the entire mole fraction range, those given by the proposed method show positive values over the entire concentration range. This trend is seen in all the systems under investigation.
eqn-4
eqn-8
Fig. 1. Excess internal pressure of DMSO + 1-butanol (a), DMSO + 1-hexanol (b) and DMSO + 1-octanol (c) at 303.15 K vs mole fraction.
In all the three systems under investigation, the excess internal pressure values obtained via the corrected method are found to be positive. It is a well known fact that while negative values of excess functions indicate strong interactions and are observed when intermolecular complexes are formed, positive values are indicative of weak interactions. Previously reported values [21] of variation of excess velocity, u E, excess specific acoustic impedance, Z E and excess adiabatic compressibility, b SE, point out towards weak interactions between the component molecules and this interaction shows a decrease with increase in the carbon chain length from 1-butanol to 1-octanol. These literature values further strengthen the validity of the modified and the corrected expression for evaluation of excess internal pressure as compared to the one proposed earlier [13]. Further, it also proves that the excess internal pressure computed with the mole fraction additivity (Eq. (4)) is definitely erroneous, since the negative values of internal pressure obtained thereof indicate presence of strong interac-
R. Dey et al. / Journal of Molecular Liquids 124 (2006) 121 – 123
tions which are in complete contradiction to literature values [21]. The proposed relation receives further credence from the fact that positive values of excess properties can be interpreted as being due to dispersion forces. Experimental findings and literature survey reveals that structure breaking effect and weak interactions between unlike molecules predominate in all the DMSO + 1-alkanol (1-butanol, 1-hexanol and 1-octanol) systems due to rupture of hydrogen bonded chains of the alkanols and loosening of dipolar interactions and decrease in strength of interaction between unlike molecules. The observed excess free energy of activation of viscous flow, G E, and excess viscosity values, g E values (reported earlier) both suggest that dispersion forces are operative between unlike molecules for all the binary systems under investigation. Thus we conclude that the proposed relation in the present investigation definitely proves its suitability and credibility for computation of excess internal pressure over the previously developed methods. References [1] M.R.J. Dack, Aust. J. Chem. 281 (1975) 643. [2] J.D. Pandey, R.L. Mishra, Acustica 33 (1981) 200.
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
123
J.D. Pandey, J. Chem. Soc., Faraday Trans. II 76 (1980) 1215. J.D. Pandey, B.R. Chaturvedi, N. Pant, Chem. Scr. 18 (1981) 224. J.D. Pandey, U. Gupta, Z. Physik. Chemie. (Leipzig) 268 (1987) 477. J.D. Pandey, R. Dey, Acoust. Lett. 24 (2000) 105. S.V. Subrahmanyam, V. Hyder Khan, K. Malakondaiah, Acustica 45 (1980) 196. J.D. Pandey, N. Dubey, D.K. Dwivedi, R. Dey, Phys. Chem. Liq. 39 (2001) 781. Amalendu Pal, Harsh Kumar, Phys. Chem. Liq. 41 (2003) 415. J.D. Pandey, A.K. Shukla, Neelima Tripathi, G.P. Dubey, Pramana J. Phys. 40 (1993) 81. J.A.R. Renuncio, G.J.F. Breedveld, J.M. Prausnitz, J. Phys. Chem. 81 (1977) 324. T.M. Aminabhavi, V.B. Patil, K. Banerjee, R.H. Balundgi, Bull. Chem. Soc. Jpn. 72 (1999) 1187. W. Marczak, Phys. Chem. Chem. Phys. 4 (2002) 1889. M.H. Blandemer, M.J. Davis, G. Douheret, J.C.R. Reis, Chem. Soc. Rev. 30 (2001) 6. V. Rajendran, Ind. J. Pure Appl. Phys. 31 (1993) 812. L. Acevedo, G.C. Pedrosa, M. Katz, J. Solution Chem. 19 (1990) 911. I.M. Aralaguppi, T.M. Aminabhavi, R.H. Balundgi, S.S. Joshi, J. Phys. Chem. 95 (1991) 5299. J.D. Pandey, R. Dey, M. Upadhyaya, Acoust. Lett. 21 (6) (1997) 120. J.D. Pandey, R. Verma, Chem. Phys. 270 (3) (2001) 429. J.D. Pandey, R. Dey, R. Verma, Phys. Chem. Liq. 14 (2) (2003) 145. A. Ali, K. Tiwari, A.K. Nain, V. Chakravortty, Phys. Chem. Liq. 38 (2000) 459.