Application of the Kirkwood Buff theory of solutions to acetonitrile + amide binary mixtures by using inversion procedure and regular solution theory

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1 J. Chem. Sci., Vol. 11, No. 3, May 009, pp Indian cademy of Sciences. pplication of the Kirkwood Buff theory of solutions to acetonitrile + amide binary mixtures by using inversion procedure and regular solution theory NIL KUMR NIN Department of Chemistry, Dyal Singh College (University of Delhi), New Delhi ak_nain@yahoo.co.in MS received 17 September 008; revised 3 March 009 bstract. he Kirkwood Buff (K B) integrals play an important role in characterizing the intermolecular interactions in liquid mixtures. hese are represented by the K B parameters, G, G BB, and G B, which reflect correlation between like like and like unlike species in the mixture. he K B integrals of binary mixtures of acetonitrile (CN) with formamide (F), N,N-dimethylformamide (DMF), N-methylacetamide (NM) and N,N-dimethylacetamide (DM) at K and at atmospheric pressure have been computed from the experimental data of ultrasonic speed and density. We have used the similar inverse procedure (as proposed by Ben Naim) to compute the K B Parameters of the mixtures, in which thermodynamic information on mixtures such as partial molar volumes, isothermal compressibility, and experimental data of partial vapour pressures are used. new route has been incorporated by using regular solution theory in the computation of excess free energy for obtaining the partial vapour pressures of binary liquid mixtures. he low values of excess entropy (S E 0) obtained for these mixtures indicate the applicability of regular solution theory to these mixtures. he results obtained regarding intermolecular interaction in all the four mixtures under study from this new procedure are in good agreement with those obtained from the trends exhibited by the excess functions of these mixtures. Keywords. Kirkwood Buff integrals; binary liquid mixtures; acetonitrile; amide; intermolecular interactions. 1. Introduction heoretical prediction of the physicochemical properties of multi-component liquid systems involves an interdisciplinary interest and proved to be an excellent qualitative and quantitative tool for elucidating the molecular structure and nature and extent of interactions in liquid systems. large amount of experimental data exists, in the literature, on the physicochemical, thermodynamic, transport, acoustic and spectroscopic properties of multi-component liquid systems But in contrast to this vast experimental work, there exist very few theoretical works on the elucidation of molecular structure and estimation of thermodynamic properties of liquid mixtures. Recently, researchers in this field have been focusing their interest more sharply on the molecular structure along with some representative macroscopic properties that serves to characterize it. 11,1 Kirkwood and Buff 13 theory of solutions relates the radial distribution functions of various molecular species in a mixture to the derivatives of the thermodynamic properties of the species. his is a general statistical mechanical theory of solutions which is applicable to all types of intermolecular interactions and is valid both classically and quantum mechanically. It is one of the most accepted theories of solutions that directly correlate the thermodynamic quantities with the solution structure without any assumptions. K B theory, being a powerful tool for liquid state theory, has not received much attention in thermodynamic literature, as it deserves. Only few examples of practical applications 14 8 to binary mixtures can be found in which this theory has been to be used only for aqueous-alcohol 14,18 binary mixtures and aqueous solutions containing electrolytes. 3,4 o the best of our knowledge, very few studies have been made in the literature on nonaqueous binary mixtures. 5 8 he main objective of the present study is to apply the K B theory to the systems of varied nature in order to extract new information on the interactions between the species existing in these systems on molecular level. 361

2 36 nil Kumar Nain In this paper, the K B theory has been extended to binary mixtures of acetonitrile (CN) with formamide (F), N,N-dimethylformamide (DMF), N- methylacetamide (NM) and N,N-dimethylacetamide (DM) at K and at atmospheric pressure. Some new routes for predicting various parameters/terms involved in K B theory have also been incorporated and effectively used. he experimental data required for this purpose are taken from our previous studies. 9,30. heory he solution theory proposed by Kirkwood and Buff 13 involves the theory of the grand ensembles on the one hand to relate compositional fluctuations to the derivatives of the chemical potentials of the components and on the other hand to relate them to the integrals of the radial distribution functions of the several type of molecular pairs present in the solution. he K B theory describes thermodynamic properties of a solution in the whole concentration range using the values of G αβ, expressed as K B integrals, given by 0 Gαβ = [ gαβ ( r) 1]4π r d r, (1) where g αβ (r) is the angle averaged pair correlation function and the integral extends over the entire range of intermolecular distances between the pair of molecules of species α and β. he interpretation of these parameters is best obtained by considering the product of number density, ρ* and G αβ, i.e., ρ α G αβ. or ρα[ gαβ( r) 1]4πr dr, where ρ α is the number density of the α species. 14 he quantity G αβ conveys information of the average affinity of the α molecules towards β molecules and vice-versa. In the present study we have incorporated a new method, in the inversion procedure proposed by Ben Naim, 14 for the computation of partial vapour pressures of the mixtures for which these vapour pressure data are not available. he composition dependence of G αβ values provides valuable insight into the molecular structure and nature of interactions in the multi-component liquid mixture..1 Inversion procedure to compute G αβ s he K B equations 13,14 for the binary mixture of species and B can be written as η= ρ + ρ + ρ ρ ( G + G G ), () B B BB B ξ = 1 + ρg + ρbgbb + ρρ B( GGBB GB ), (3) where η and ξ are constants, ρ and ρ B are the number densities of and B, respectively. he isothermal compressibility, k, the partial molar volumes, V m, and V m,b of the two components and derivatives of the chemical potentials are given by k = ξ kη, (4) / V m, = ρb GBB GB [1 + ( )] / η, (5) V m,b = + G GB [1 ρ ( )]/ η, (6) μ = ρ k/ ρ η V, (7) B μ = ρ k/ ρ η V, (8) BB B μb = μb = k/ η V, (9) where k is the Boltzmann constant, is the absolute temperature and µ αβ is given by µ α µ αβ =. (10) N β, p, Nα We have three thermodynamic relations between the number densities, chemical potentials and partial molar volumes of the two components of the mixture, given as ρμ + ρbμ B = 0, (11) ρμb + ρbμ BB = 0, (1) ρ ρ. (13) V m, + BV m,b = 1 herefore, we have three independent equations in the three independent unknown G αβ s. he derivatives of the chemical potentials have been obtained using the vapour pressure data, assuming that the vapour above the mixture at room temperature may be treated as an ideal gas. hus, for component, we can write µ = µ +kln P, (14) o

3 pplication of the Kirkwood Buff theory of solutions to acetonitrile + amide binary mixtures 363 where P is the partial pressure of component over the given mixture of and B. If x is the mole fraction of in the mixture, then we get the relation x p x ρ * = η. (15) he number density, ρ*, of the mixture is calculated from the partial molar volumes, V m, and V m,b of the components and B, respectively, in the mixtures. 1 ρ* = ρ + ρ B = ( xv m, + xbv m,b). (16) he values of V m, and V m,b in the mixtures are calculated from the experimental density data by using the procedure described elsewhere. 30 From (15) we can obtain η from the data on partial vapour pressures of either or B in the entire composition range. he partial vapour pressures are calculated from the activity coefficients, which are related to the excess Gibbs free energy of the mixture. he excess free energy of mixtures is given as E E E G = H S. (17) he excess enthalpies, H E and excess entropies, S E are calculated from the internal pressures, π i and free volumes, V f of the mixtures by using the modified relations proposed by Hildebrand et al. 31,3 E H = π V [ x π V + x π V ], (18) E i m i, m, B i,b m,b S = R[ x lnv + x lnv ln V ], (19) f, B f,b f,m where π i of the mixtures are calculated using the thermodynamic equation of state E P α p π i = P P, V = = (0) V k where α p is the isobaric expansivity of the mixture evaluated from temperature dependence of density data. For most of the liquids, the thermal pressure coefficient multiplied by absolute temperature, i.e. (α p /k ) is very high so that the external pressure P becomes negligible in comparison, 33,34 therefore it may be neglected in the (0) in the present calculations. he V f of the mixtures are calculated from the relation 31,3 V f R = ( P + π ), (1) i since P is very small as compared to π i, it has been neglected in the (1) in the present calculations. he systems are assumed to be regular mixtures, for which G E are given by G E = x x Nw () B, where w is a constant computed from (), which may depends on temperature but is independent of composition. he activity coefficients in a regular mixture are given by ln i (1 i) / γ = x wk. (3) he partial vapour pressures of the components in the mixtures are calculated using the activity coefficients. Using the mole fraction dependence of partial vapour pressure of either or B, η over the entire composition range can be easily obtained from (15). he values of ξ are evaluated from (4), in which the isothermal compressibility, κ, is calculated using the well-known thermodynamic relation k Vα = ks + C p, (4) where k s is isentropic compressibility and C p is the heat capacity. he C p values for the mixtures have been calculated by using the relation Cp = xcp, + xbcp,b. (5) Using the values of η, ξ, V m, and V m,b, the G B is calculated using the relation V V G m, m,b = ( ξ η B )/ η. (6) Once G B is obtained, (5) and (6) are used to calculate G and G BB. nother quantity proposed by Ben Naim, 14 Δ B, which is a measure of the degree of similarity between the two components of the mixture, has also been calculated by using the following equation Δ = G + G G. (7) B BB B he condition Δ B = 0 signifies symmetrical ideal solutions. he magnitude of Δ B can be used to indicate the extent of deviation from ideal behaviour. he above procedure has been used to calculate the K B parameters for CN + F, + DMF, + NM and + DM binary mixtures directly from ultrasonic speed and density data at K.

4 364 able 1. nil Kumar Nain Values of ρ, u, p, k s, k, α p, and C p for pure liquids at K used in the calculation of K B parameters. ρ u p k s k α p C p Liquid (kg m 3 ) (m s 1 ) (N m ) (10 10 m N 1 ) (10 10 m N 1 ) (10 3 K 1 ) (J mol 1 ) CN F DMF NM DM able. Values of ρ, u, ρ*, H E, S E, ln(p ), η, G B, G, G BB and Δ B for CN + amide mixtures at K. ρ u ρ* H E S E η G B G G BB Δ B x (kg m 3 ) (m s 1 ) (m 3 ) (J mol 3 ) (J mol 3 ) ln(p ) (m 3 ) (10 4 m 3 mol 1 ) CN + F CN + DMF CN + NM CN + DM

5 pplication of the Kirkwood Buff theory of solutions to acetonitrile + amide binary mixtures Results and discussion he values of K B parameters for the abovementioned binary systems have been calculated as a function of mole fraction, x of component (CN) at K. he values of various parameters of pure liquids used in the calculations are listed in table 1. he values of K B parameters, viz. G, G BB, G B and Δ B along with some other parameters for the binary systems investigated as function of mole fraction, x of CN for each system, are listed in table. he variations of G, G BB, G B and Δ B Figure 1. Plots of G, G BB, G B and Δ B against mole fraction, x of CN for CN + F mixtures at K. Figure. Plots of G, G BB, G B and Δ B against mole fraction, x of CN for CN + DMF mixtures at K.

6 366 nil Kumar Nain against mole fraction, x of CN for each mixture are shown graphically in figures 1 4. he low values excess entropy (S E 0) obtained for these systems (table ) clearly indicate that these mixtures can be considered as regular solutions. 31,3 Figure 1(a) shows the behaviour of CN-CN affinity over the entire composition range. able and figure 1(a) indicate that G values are negative and initially decreases with increase in mole fraction up to x 0 55 and then increases after this point and finally become positive, suggesting decreasing correlation between CN molecules and this correla- Figure 3. Plots of G, G BB, G B and Δ B against mole fraction, x of CN for CN + NM mixtures at K. Figure 4. Plots of G, G BB, G B and Δ B against mole fraction, x of CN for CN + DM mixtures at K.

7 pplication of the Kirkwood Buff theory of solutions to acetonitrile + amide binary mixtures 367 tion increases after x 0 55 may be due to strong dipole dipole interactions between CN molecules at higher CN concentration. Figure 1(b) indicates that G BB values are positive and initially decrease to negative values with increase in mole fraction till x 0 and then increases slightly after this point, suggesting decreasing correlation between F molecules on increasing the amount of added CN in the mixture, which leads to breaking of the associations present between F molecules. Figure 1(c) indicates that that G B values are negative and increases initially and exhibit a maximum at x 0 4 and then decrease with increase in mole fraction of CN, suggesting increasing correlation between CN and F molecules on increasing the amount of CN in the mixture till x 0 4, which then decreases with further increase in mole fraction of CN. his behaviour is in agreement with the conclusions drawn in our earlier study, 9,30 wherein V E m and Δk s vs x curves exhibit a minimum at x 0 4, indicating maximum interaction between CN and F molecules near this composition. lso, Δ B values (figure 1(d)) are positive and decrease initially to negative values exhibiting a minimum at x 0 4 and then again increase to positive values, indicating the dissimilarity between the unlike molecules in the mixtures and negative deviations from the ideal behaviour. 14 Figures 4 also depict similar trends in G, G BB, G B and Δ B values against mole fraction, x for CN + DMF/NM/DM mixtures as shown by CN + F mixtures. close perusal of table indicates that the magnitude of G B varies in the order: F > NM > DM > DMF, suggesting that the affinity or interactions between the CN and amide molecules follow the order: F > NM > DM > DMF, which is in agreement with the conclusions drawn regarding the intermolecular interactions from the trends exhibited by the excess functions of these mixtures. 9,30 he work on extending this procedure for calculation of K B integrals to a variety of other binary mixture containing combination of nonpolar nonpolar, polar nonpolar and polar polar liquids is in progress. We feel that in future this method may prove to be a valuable tool for the calculation of K B integrals for all mixtures, in general. cknowledgement he author is thankful to Dr I S Bakshi, Principal, Dyal Singh College (University of Delhi), New Delhi for providing facilities and encouragement. References 1. Iloukhani H, Zoorasna Z and Soleimani R 005 Phys. Chem. Liq George J and Sastry N V 004 Fluid Phase Equilib Giner B, rtigas H, Carrion, Lafuente C and Royo F M 003 J. Mol. Liq Nain K 006 J. Solution Chem Nain K 007 Phys. Chem. Liq Nain K 007 J. Chem. hermodyn Nain K 008 J. Mol. Liq Nain K 008 Fluid Phase Equilib li, Nain K, Chand D and hmad R 006 Bull. Chem. Soc. Jpn li and Nain K 00 Bull. Chem. Soc. Jpn Oswal S L, Maisuria M M and Gardas R L 003 J. Mol. Liq Gonzalez J, Riesco N, Mozo I, De La Fuente I G and Cobos J C 007 Ind. Eng. Chem. Res Kirkwood J G and Buff F P 1951 J. Chem. Phys Ben-Naim 1977 J. Chem. Phys Shuglin I L and Ruckenstein E 006 J. Phys. Chem. B Perera, Sokolic F, lmasy L and Koga Y 006 J. Chem. Phys Gonzalez J, Mozo I, Villa S, Riesco N, De La Fuente, I G and Cobos J C 006 J. Solution Chem Donkersloot M C 1979 J. Solution Chem Newman K E 1994 Chem. Soc. Rev Newman K E 1988 J. Chem. Soc., Faraday rans Vergara, Paduano L, Capuano F and Sartorio R 00 Phys. Chem. Chem. Phys Roux H and Desnoyers J E 1987 Proc. Indian cad. Sci. (Chem. Sci.) Banerjee D, Laha K, Chatterjee P and Bagchi S 1995 J. Solution Chem Covington K and Newman K E 1988 J. Chem. Soc., Faraday rans Pandey J D and Verma R 001 Chem. Phys Guha and Ghosh N K 006 Indian J. Chem Matteoli E 1997 J. Phys. Chem. B Marcus Y 006 J. Solution Chem Nain K 006 Bull. Chem. Soc. Jpn Nain K 006 J. Chem. hermodyn Hildebrand J H and Scott R L 196 Regular solutions (New Jersey: Prentice Hall) 3. Hildebrand J H 1947 J. Chem. Phys Dack M R J 1975 Chem. Soc. Rev Pal and Kumar H 004 Indian J. Chem. 43 8

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