Indian Journal of Chemistry Vol. 45A, March 2006, pp. 593-598

Partial molar quantities and thermodynamic interaction parameters for some binary mixtures: CH2Cb + CS2, CH30H + CCI4, CH30H + C6H6

and CH30H + CHCh systems A Guha* & N K Ghosh

Department of Chemistry. University of Kalyani , Kalyani. Nadia 741 235 , West Bengal. India

Email : profasit_guha @yahoo.co. i n

Received 21 luly 2005; accepted 1< lalll/arv 2006

The molecular structures and the solute-solvent il1leraction play vital role in finding the correlation between the like and unlike molecules in a mixture. The systems CH "Ci2 + CS2• CH,OH + CC I4• CH30H + C6H6 and CH, OH + CHCI, have been investigated over the whole composition range. Stati stical thermodynamics shows a way to calculate the partial molar qual1lities, isothermal compressibilities and interaction parameters through a few paramelers like density. hard sphere diameter. temperature. The interacti on is represented by Kirkwood-Buff parameters Cu, Cjj and C jj • representing the like- like and like-unlike molecular il1leractions. The partial molar volumes. partial vapour pressures and partial isothermal compressibilities have also been also calculated from statistical thermodynamics. The interaction parameters at differenl mole fraction s are discussed and compared with other experimental results of partial vapour pressures.

Kirkwood-Buff solution theoryl has widely been used as an accepted theory of solutions. This is because it is a statistical mechanical theory, which directly correlates the thermodynamic (macro) quantities with the molecular structure (micro) without any assumption. The general features of this theory are: (a) it is valid for any kind of particles not necessarily spherical, (b) no assumption of additivity of potential introduced, and (c) special pair correlation function appears in the relations. In particular, this theory is preferred in the treatment of complex fluids such as water and organic liquids.

The Kirkwood-Buff (K-B ) parameters are defined by for any pair of species of molecules i and j as :

Gij (r) = f[g ji (r) - I] 4n/dr ... ( I )

where gij (r) is the angle averaged radial distribution function of the j-th species around the i-th species and is dependent only on the scalar distance . r' between the pair of type ij. The physical significance of Gij

may be understood in such a fashion that Pj Gij (Pj is the number uensity of the j-th species) represents the average excess number of molecules j in the whole space around a molecule i with respect to the bulk average, i.e. , Gij determines the average affinity of the i-th species for the j-th speci es.

In case of a binary mixture, the parameters G ii, Gli and Gij appear in this theory . As gil r) is an observable quantity by X-ray or the neutron diffraction method . Gij is also an observable quantity by thi s method in principle. But, practically each value of g;;( r) cannot be determined separately . Hence. it is of much more importance to determine G ii, Gjj and Gij vaiues from microscopic quantities such as partial molar volume. partial pressure, isothermal compressibility. etc.

Ben Naim2 presented a procedure to calculate the K-B parameters in an inverse way as gij determined by X-ray and neutron diffraction technique are severely criticised' . The method was the use of some thermodynamic quantities to calculate the p,u·ameters. Literature survey shows nn direct method to evaluate K-B parameters. Pandey and VermaJ calculated the K-B parameters fo r some binary mixtures using the compressibility and partial molar volume. But th at method is applied for regular mixtures only.

Statistical mechanics enables us to calculate the partial molar quantities and the compressibility using the hard sphere diameters and densities. We have used this method to calculate the partial pressures and compressibi lity. So, the method is a direct one and the calculation of the activity coefficients is not necessary . The K-B parameters calculated from the partial molar volume. isothermal compressibility and partial pressure are interpreted according to the principles of solution chemistry.

594 INDIAN J CHEM, SEC A, MARCH 2006

Theoretical According to K-B theory for a binary mixture with

two components i andj, the partial molar volumes (Vj

and Vj ) and isothermal compressibility {Jr can be expressed as,

Vj = [1 + pj (Gjj - Gij}/rJ

V,Y j = [~- rJ Gij]/rl

... (2)

. .. (3)

.. . (4)

T he two thermodynamic quantItIes, rJ and ~ that appear in the K -B theory can be defined by Eqs (5) a d (6), respecti vely.

.. . (5)

and

f3r = ljrKTrJ) ... (6)

where Pj is the partial pressure of the j-th species

whose mole fraction within the mixture is Xj' P is the total number density , f3T is the isothermal compressibility of the mixture, K is Boltzmann constant and T is the absolute temperature.

j ow, rom K-8 integrals, one can calculate the 4 ' quantity .J:

. . . (7)

where NN is the number density of the mixture and gij(r) 's the radial distribution function of the i-j pair.

!..Nj i ~ (\ measure of the difference between the distribution of j-molecules surrounding an i-molecule and the distribution of j-molecules surrounding a j -molecule. Now, t1Nj can be related to Gij. Gjj and G jj, from Eqs (I) and (7) as:

Mlj = Xj NN f [{g ij (r) - I} - {gjj (r) - I}] 4n-ldr = Xj NN f [g ij (r) - 1] 4n/dr - xj NN f [gjj(r) - 1]

4n/dr = Xj NN Gij -Xj NN Gjj

= Xj P r G ij - Gj)

Similarly,

t1Nj = Xj P rGu - Gjd

.. . (8)

.. . (9)

where t1N; is a measure of the difference between the distribution of i-molecules surrounding a j-molecule and the distribution of i-molecules surrounding an i­molecule.

Assuming each solute and solvent molecule to be a hard sphere of diameter ,cr, the isothermal

compressibility (f3rl can be expressed as,

f3 = (1-y)6

. T pKT(l + y + /)(1 + 2y )2 .. . (10)

where y = 7T/6 p(x; a; + Xj a/ and it is called packing fraction of the mixture.

The partial molar volumes of a particular species in a binary mixture can be c .lkulated from the followin g

. 7 expressIOns .

Vffj = C (X3 + 3ax2 + 3a{3X + af3y)

V.;vo; = C (X3 + 3ax2 + 3af3X + af3y)

... (lla)

.. . (11 b)

1- },2 1 y 0: = --, 13 = ---'------,--

1+2y 1+2y - i

1- y y=-,

y

C = (1 +3a+3af3+af3yr i

[k := i, j].

Here, V; and Vj are the partial molar volumes, V;I)

and Vij are the molar volumes8 of i-th and j-th components respectively in a binary mixture considering i-th component as a solute and j -th component as a solvent and vice-versa, ao and a s are solute and solvent diameters assuming them as hard spheres, M; and Mj are molecular weigh ts of the i-th and j-th species, respectively and y is packing fraction of the mixture given in Eg . (10).

Partial vapour pressure of the individual component can be calculated from the equations9

:

.. . (1 2)

. . . (13)

GUHA & GHOSH: THERMODYNAMIC PARAMETERS FOR SOME BINARY MIXTURES 595

where Pi. Pj are the partial vapour pressures of the i-th and j-th components and pF. pf are that for pure

components respectively. vF V/. l/>;. l/>j' 8; and ~ are the molar volumes, volume fractions and sol ubility parameters of the i-th and j-th components, respectively . R is universal gas constant and T is absolute temperature.

The above equations can be applied for nearly ideal mixtures only . But, the following equations l0

-12 can be

used for the calculation of partial vapour pressures, P; and Pj, for ideal mixtures as well as when strong deviations from Raoult ' s law is observed,

. . . (l4a)

and

_ { {l ( I - .I)I' JP o Pj - Xj e j ... (l4b)

Here, f3 = 0 for purely ideal mixtures. But, for non­ideal mixtures it has been found that:

f3= Y2 (K + KI)

K ' = 8/~ , where 8; and 8j are the solubility parameters for i-th and j-th components, respectively and are

given by ll :

and

where, l1H" and V are the heat of vaporization and molar volumes of the components, respectively . For endothermic mi xing f3 should be positive and for

exothermic mixing f3 should be negative. The heat of mixing of all the four mixtures has positive values. Therefore, f3 should be positive for these systems. It has been found that our calculated total vapour pressure is very close to the experimentally 10

observed vapour pressures for CH2Ch (j) + CS2 (i) mixture (Fig. J) . Using same procedure we have calculated partial vapour pressures for other three

mixtures and calculated dlnpj/dxj at different Xj. The calculated total vapour pressures are al so shown in Fig. 2 as a function of mole fraction of the second component (Xj) for other three systems.

Mixtures for study Fisher and Lago l 3 first applied the perturbation

theory to calculate some molecular properties of mixtures CS2 + CCl4 . But, the results are poor. Araci l et al. 10 measured the bulk properties of CS2 + CCI4

syste.m and found that the excess propel1ies are independent of temperature and pressure in contrast to the results obtained by Cal ado et al l 4

. So, we have chosen this mixture to calculate the interaction parameters.

Similarly , the CH2CI 2+CS2 system was studied experimentally by Aracil et al. lo This shows 110n­ideality for the system. The other two components studied were cyclohexane-methanol and carbon tetrachloride-methanol system. Both of those component mixtures show unusual behaviour.

Results and Discussions The theory and method of calculation are presented

earlier. For this purpose, the required parameters for pure compounds are given in Table 1. The partial molar volumes are calculated according to Lee 's7 method and tabulated in Table 2. The partial pressures have been calculated according to Eq. (14) and plotted as shown in Figs I and 2. With these data G;;. Gu• Gjj,

l1N; and l1Nj have been calculated with the help of Eqs (2)-(9) for the entire range of composition.

Dichloromethane + carbon disulphide system Figure 3 shows the values of Gu• G;; and Gjj for the

CH2Ch+CS2 system throughout the range of composItIon. The sudden decreased correlation between dichloro-methane and carbon di sulphide molecules are observed in the Gu values at Xj = 0.6.

Table I - Parameters for pure components at 25°

Components p (J pO 8 (g, cm·3) (A) (KP,) (M P,~)

Carbont etrachloride 1.58452' 5.18c 15.18' 17.6Q8

Benzene 0.87368" 5.02c 12.80' 18.7Q8

Methanol 0.78654" 3.59c 16.92' 30.5Q8

Chloroform 1.4770b 2.98c 25 .90f 18.9Q8

Dichloromethane 1.3255b 4.759d 58.08d 19.8Q8

Carbon disulphide 1.26 lOb 4.438d 48. 12d 20.40g

a: Ref. 15; b: Ref. 16; c: Ref. 17; d: Ref. 10; f: Ref. 18; g: Ref. 9

596 INDIAN J CHEM, SEC A, MARCH 2006

Table 2 - Calculated partial molar volumes of different systems for different mole fractions

Vi V Vi V Systems Xj (cm~morl) 1 J I

(cnl' mor ) Xj (cm3mor l) 3 .I

(cm' mor l

CH"CI2(j) 0.1 55.98 + 0.2 57 .64

0.3 59.25 CS~ (i ) 0.4 nO.90

0.5 62.53

CH, OH(j) 0.1 89.87 + 0.2 89.50 CCI4(i) 0.3 87.62

0.4 84.25 0.5 79.37

CH3OH(j) 0.1 83.88 + 0.2 82.60 C6Hr,( i) 0.3 81.00

0.4 79.00 0.5 73 .25

CH, OH(j) 0.1 53.00 + 0.2 61.45 CHCI,(i) 0.3 65.65

0.4 66.83 0.5 67.04

rig. l - - Composition dependence o f total vapour pressures for C H1CI:: (j ) + CS1 (i) mixture. The curves 1 and 2 indicate the calculated and experimental result s respectively .

Similarly, the abrupt increase in Gjj values at Xj ""

0.5 suggests an increased correlation between dichloromethane molecules at Xj "" 0.5. ,1Nj and ftNi 'alues are shown in Fig. 4. The correlation between like molecules is maximum at equimolar composition. ~ t also suggests that like molecules are prefelTed than unlike molecules as ,1Ni and ,1Nj values arc negati ve.

Applying Clapeyron equation that applied to azeotrope can be written as 10

68.95 0.6 64.15 60.30 67.36 0.7 65.77 59.25 65 .77 0.8 67.36 57.64 64. 15 0.9 68.95 55.98 62.53

53. 12 0.6 73.75 84.25 60.62 0.7 67.37 87.62 67.37 0.8 60.62 89.50 73.75 0.9 53.12 89.87 79.37

50.50 0.6 68.00 79.25 57.00 0.7 63.25 81.00 63.25 0.8 57.00 82.60 68.00 0.9 50.50 83.88 73.25

58. 12 0.6 68.12 66.83 63.31 0.7 66.09 65.65 66.09 0.8 63.31 61.45 68.12 0.9 58.12 53.00 67.04

Mole fraclion (x,»)

Fig. 2 - Composition dependence of total vapour pressure for methanol containing binary mixtures. The curves I, 2 and 3 are indicated for CH30H (j) + CHCI3(i), CH,OH(j) + CCI4(i) and CH30H(j) + C6Hr,(i) mixtures respectively .

dPa. t..H --' =

elT Tt1V

where Paz is the pressure at the azeotrope. The graph of Gij in our calculations aiso shown the azeotrope at xj=0.6 at 298.15 K. The calculations can be extended ro other temperatures also.

GUHA & GHOSH: THERMODTNAMIC PARAMETERS FOR SOME BINARY MIXTURES 597

Methanol containing binary mixtures ·The mixtures chosen are: (i) methanol + carbon

tetrachloride, (ii) Methanol + benzene, and (iii) methanol + chloroform systems. The values of G jj, Gij and Gjj are shown in Table 3. In all cases, it is found that Gij has a minima indicating the decreased correlation between the CH30H and CCl4 or C6H6 or

Fig. 3 - Plot of K-B parameters vs. mole fraction of CH2CI2 U) + CS2 (i) mixture . The curves 1,2 and 3 are indicated for Gjj• G;; and G;l respectively.

CHCh. The minima occurred for the compositions Xj

::::: 0.5. Xj = 0.6 and Xj = 0.7 for the three systems mentioned above. Gjj also shows similar nature at Xj = 0.5, 0.6 and 0.7 which confi rms the decrease in correlation between the CH30H molecules. Thi s again is supported by the observation of vapour pressures which shows maxima at the same compositions. However, the G;; values show no ablUpt change

-44-------------~r-------------_r-----0.0 0.4 0.8

Mole fraction (Xi)

Fig. 4 - Plot of neighbouring excess number of molecules (MV) versus mole fraction of CH2C1 2 U) + CS2 (i) mixture. The curves I and 2 are indicated for MV; and MVj respectively.

Table 3 - KB parameters and .1N values for different systems

System

Methanol U) + Carbontetrachloride (i)

Methanol U) + Benzene (i)

Methanol U) + Chloroform (i)

Xj

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-Gij (cm3mor l)

52.30 44.80 62.85 108.40 127.70 343.25 230.50 218.75 181.45 141.40 42.45 58.45 96.70 137 .60 171.85 181.80 173.45 153.40 120.20 -82.45 -30.25 33.50 90.50 133.75 143.80 178.75 128.50 81.90

-G 1 11

(cm' mor l)

67.50 50.75 60.95 8315 97.70 208.15 159.80 157.75 139.50 121.90 48.55 57.05 74.90 97.55 116.90 128.95 126.85 118.30 102.30 -66.70 -26.35 13.95 49.70 73.61 83.35 99.45 77.70 49.40

-G;; -.1Nj x ]0.23 . .11\'; x J 0.13

(cnr' morl)

56.00 -0.150 -0.330 92.80 -0.320 -2.884 9 1.03 0.265 -1.572 84.55 0.583 1.290 77.03 1.047 2.653 20.55 6.417 15.328 23.75 4.433 8.642 0.140 4.867 7.476 - 12.75 4.111 4.758 -20.75 2.369 2.190 85.90 -0.0438 -2.805 83.90 0.0211 -1.537 72.85 0.530 1.012 72.15 1.393 3.416 51.30 2.624 5.756 32.00 3.171 5.992 17.55 3.539 5.075 1.85 4.510 3.667

-3.80 2.110 1.624 -55 .75 -0.1285 -1.960 -24.70 -0.069 -0.395 13.95 0.565 1.319 6.05 1.693 5.257 13.85 3.341 6.660 27.05 4.298 5.534 22.20 6.988 5.912 32.50 5.416 2.559 29.40 4.1 I 0.731

598 INDlAN J CHEM. SEC A, MARCH 2006

except for CH30H + CHCI3 system which shows a

weak minima at Xj "" 0.3. This is due to decreased cOlTelation between chloroform molecules.

Excess number OJ' molecules !1N; and !1Nj (Table 3) show a sudden decrease at Xj = 0.5, 0.6 and 0.7 for these system. !1N shows negative values which indicates the preference of like molecules over the un like molecules.

The presented method shows calculation of the interaction parameters Gij and the vapour pressure of the binary mixture without depending on the measurement of vapour pressure at different mole fraction and non-ideality can be interpreted' s. The method reported by Pandey and Verma) assumed the regular solution which cannot be applied as non­ideality is prevailing in these mixtures.

Kirkwood-Buff parameter which is an convoluation integral, can be calculated from radial distribution function gij(r ). The radial distribution function can be obtained from X-ray scattering data ''), but the experilllentalllleasurement of the distribution function is very difficult and there is a controversy of this determination at low angle20

. As Gij depends on the radial distribution function gij. it is to be seen whether the radial distribution function can be back calculated to find gij without USll1g perturbation technique.

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