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Journal of Molecular Liqui

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

1. Introduction

The study of internal pressure of binary liquid mixtures has

been carried out by several workers [1–3]. It has been exten-

sively 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(Pint) as

Pint ¼ BU=BVð ÞT ð1Þ

where U is the internal energy, V the volume and T the

temperature.

0167-7322/$ - see front matter D 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.molliq.2005.09.005

* Corresponding author.

E-mail address: drranjan@hotmail.com (R. Dey).

Assuming the thermodynamic equation of state,

BU=BVð ÞT ¼ T BP=BTð ÞV � P ð2Þ

where P is the pressure.

Employing the expression of thermal expansion coefficient

(a) and isothermal compressibility(KT), internal pressure can

be expressed as [11]

Pint ¼ BU=BVð ÞT ¼ aTjT

� P

��ð3Þ

where all the symbols have their usual meaning.

The excess internal pressure, PintE of liquid mixture is given

by

PEint ¼ Pint � Pid

int ¼ Pint �X

Pint iIxi 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 (jT). This is evidently erroneous since both

thermal expansion coefficient and isothermal compressibility

ds 124 (2006) 121 – 123

www.elsev

DMSO+1-butanol

-800.00

-600.00

-400.00

-200.00

0.00

200.00

400.00

600.00

-800.00

-600.00

-400.00

-200.00

0.00

200.00

400.00

600.00

-800.00

-600.00

-400.00

-200.00

0.00

200.00

400.00

600.00

0.00 0.19 0.37 0.53 0.67 0.80 0.93 1.00

Pin

tE(a

tm)

Pin

tE(a

tm)

Pin

tE(a

tm)

eqn-4 eqn-8

DMSO+1-hexanol

0.00 0.19 0.36 0.52 0.67 0.80 0.92 1.00

eqn-4 eqn-8

DMSO+1-octanol

0.00 0.19 0.36 0.52 0.66 0.80 0.92 1.00

x1

x1

x1

eqn-4 eqn-8

a

b

c

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.

R. Dey et al. / Journal of Molecular Liquids 124 (2006) 121–123122

are very well known volume fraction additive quantities [14].

Since such a wrongful assumption results in major computa-

tional 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 thermodynami-

cally ideal mixture. This ideal value has then been employed

for computation of excess internal pressure.

For the calculation of Pintid , various authors have assumed

[6,15–17] that

Pidint ¼

XPint iIxi ð5Þ

which is erroneous and Pintid can be given as

Pidint ¼

aidT

jidT

� P

��ð6Þ

where

aid ¼X

/iai

jidT ¼

X/ijTi

/i ¼xiViXxiVi

where /i is the volume fraction. Now from Eq. (6) we have

Pidint ¼

X/iaiTXuijTi

� P ð7Þ

using Eqs. (3) and (7) in Eq. (4) we have

PEint ¼

aTjT

� P

���

X/iaiTX/ijTi

� P

! ð8Þ

PEint ¼

aTjT

�X

/iaiTX/ijTi

! ð9Þ

where thermal expansion coefficient and isothermal compress-

ibility 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.

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, uE, excess specific

acoustic impedance, ZE and excess adiabatic compressibility,

bSE, point out towards weak interactions between the compo-

nent 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) 121–123 123

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) sys-

tems 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, GE, and excess

viscosity values, gE 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.

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