indicates the energetic effects in solutions of simple molecules can be described by the partition function with the aid of one mixture parameter, the value of which is consistent with that for the pure component interaction (Winnick, 1972). The description of polar-nonpolar liquid mixture enthalpy is not quite so successful as that for polar-polar mixtures. However, for those systems with n-eakly repulsive unlike interaction energies (negative m12’,Table I), the excess Gibbs energy prediction agrees with experiment. These mixtures are much less ordered than those with strong mutual attraction. Thus, the semi-random theory used here accurately represented the excess entropy. The failure of both the polar-polar and 1)olar-nonpolar treatments for the chloroform-alcohol systems is due in part to the induction effect of’ each species on the other. T o include the 1-2 and the 2-1 induction effects as well as the permanent dipole interaction would require two additional terms, similar to 20, in eq 16. It is felt the extra empiricism involved is not justified a t this time. Acknowledgment
Joseph Kong helped with the calculations. literature Cited
Adcock, D. A,, McGlashan, 11.L., Proc. Roy. SOC.,Ser. A 226, 266 (1954). Bae, J. H., Reed, T. RI., 111, IND.ENG.CIXM.,FUSDIM.6, 67 (1967).
Blanks, R. E., Prausnitz, J. M., IND. ENG.CHEM.,FUNDAM. 3, 1 11964). Brown, I., Fock, W., Smith, F., Aust. J . Chem. 17, 1106 (1964). de Leew, H. L., 2. Phys. Chem. 77, 284 (1911). 11, 38 Duncan, A. G., Him, 31. J., IKD.EKG.CHEM.,FUNDAM. i1972). Franks,’F., Ives, D. J. G., Quart. Reu. Chem. SOC.20, 1 (1966). Hirobe, H., J . Fac. Sci. Tokzio Cniv. 1, 155 (1926). Jones, H. C., Bingham, E. C., Amer. Chem. J . 34, 481 (1906). Kappallo, W., Schafer, K., Z . Elektrochem. 66, .ill (1962). Kister, A. T., Waldnian, I). C., J . Phys. Chern. 62, 245 (1958). Kurtz, S. S., et al., J . Chem. Eng. Data 10, 330 (1965). Lama, 13. F., Lu, B. C., J . Chem. Eng. Data 10, 216 (1965). I‘Iaffiolo, G., Vidal, J., Iienon, H., IND.EKG.CHEM.,FUKDAM. 11, 100 (1972). RlcGlashan, 11.L., Prue, J. E., Sainsbury, I. E. J., Trans. Farad a y SOC.50, 1284 (1954). Xicholson, D. E., J . Chem. Eng. Data 5, 309 11960). Paraskevopoulos, Q. C., llissen, R. W., Trans. Faraday SOC.58, 869 (1962). Prigogine, I., “Molecular Theory of Solutions,” Chapter 14, North-Holland, Amsterdam, (1957). Rowlinson, J. S., Mol. Phys. 1, 414 (1935). Scatchard, G., ef al., J . Amer. Chem. SOC.74, 3721 (1952). Scatchard, G., Satkiewicx, G., J . Amer. Chem. SOC.86, 130 (19644). Scott, It. L., J . Chem. Phys. 25, 193 (1956). Stavely, L. A. K., Tupman, W. I., Hart, K. R., Trans. Faraday SOC.51, 323 (19%). Weast, It. C., “Handbook of Chemistry and Physics,” 50th ed: p D189, Chemical Rubber Co., Cleveland, 1969. Winnick, J., IND.ENG.CHEM.,FUKDLM. 11, 239 (1972). Winnick, J., Prausnitz, J. M., Chem. Eng. J . 2, 233 (1971a). Winnick, J., Prausnitz, J. AI., Chem. Eng. J . 2, 239 (1971b). RECEIVED for review May 17, 1972 ACCEPTED February 2, 1973 This work was carried out under the partial sponsorship of the Department of Defense (Project Themis).
Enthalpies of Mixing of Halogenated Methanes and Their Interpretation. 1. Dihalogenated Methanes Vittorio Ragaini, Cesare Giannini Zstitufo di Chimica Fisica, C-niversita di X i l a n o , Milan, Italy
Sergio Carra Zstituto d i Chiwiica E’isica e Spettroscopia, Cniversita di Bologna, Bologna, Italy
Enthalpies of mixing for binary mixtures of three dihalogenated methanes (CH2C12, CH2Br2, and CHAd at 2 5 ° C have been measured. The three systems show an endothermic behavior. The results of the excess enthalpies have been theoretically interpreted through the evaluation of the potential energy of the mixture following the procedure proposed by Hildebrand and Wood and employing the radial distribution function of hard spheres and a perturbing potential.
K n o w i e d g e of enthalpies of mixing of two or more liquids is important for many eiigineering calculations. Actually such data, for polar and noiipolar mixtures a t different cornpositions and temperatures, are not always easily available and therefore many attempts are in progress for the correlation or prediction of such thermodyiiamic data on the basis of theoretical (Rowlinson, 1969) or semiempirical models (Ahselineau and Reiion, 1970; Papadopoulos and Derr, 1959; Redlich, etal., 1959). In order to verify the therniodyiiamic behavior of mixtures of some halogenated hydrocarbons, a set of six biliary mix-
tures of halogenated methalies has been studied. I n this paper the experimental heats of mixiiig a t 25OC of dihalogenated methanes arid their interpretabion on the basis of intermolecular forces are reported. I n a subsequent paper (Ragaini and Carrh, 1973) the results for binary mixtures among CI12C12, CHC13, aiid CCl, \vi11 be given. The considered systems are: system 1, CH2C12-CH213r2; system 2, CH2C12-CHnIy; and system 3, CH21h-2-CH212. I11 the explored molar fraction field (0.1 5 z 5 0.9) the three systems show a n endothermic behavior, with maxima in H E which decrease in the folloiving order: system 2, about Ind. Eng. Chem. Fundam., Vol. 12, No. 2, 1973
209
400
t
Table II. Values of the Coefficients for Eq 1 Determined by the Method of least Squares System
Cl
c2
ca
1 2 3
619,64 3965.3 1384.0
- 128.98
- 342.68
1683.6 308.67
6,9699 -440.33
Table 111. Experimental Results for the Densities C * (g/cm3) of Mixtures of Dihalogenated Methanes as a Function of Mole Fraction x l r at 25°C CHzCI2(1)-CHzBr2(2)
x1
0
2
4
1.
0.437 0.497 0.746
P*
1.9969 1.9256 1.6320
CHzCIz(1 )-CHh(2)
P*
Xl
0.244 0.508 0.753
2.8898 2.3982 1.8903
CHzBrA 1 )-CHzh(Z)
x1
P*
0.253 0 511 0.752
3.1050 2.9048 2.6993
Xl
Figure 1. Experimental enthalpy of mixing as a function of mole fraction of component 1 at 25°C. Solid lines from eq 1: A, system 1, CH2C12 (l)-CHzBrz(2); 0, system 2, CH2CI2(1 )-CH212(2);0,system 3, CH2Br2(1)-CH2I2(2)
Table I. Experimental Results for the Excess Enthalpies of Dihalogenated Methanes, HE in J mole-’, as a Function of Mole Fraction, x l r at 2 5 ° C CHzClz(1 I-CHzBrz(2) x1 HE
0.100 0.222 0.279 0.435 0.542 0.725 0,900
51.38 99.42 125.2 144.5 152.8 110.9 36.45
CHzCIz(1 )-CHz12(2) XI HE
0.150 0.271 0.336 0.390 0.469 0.535 0.625 0.691 0.851 0.890
377.4 560.4 779.2 875.8 984.0 1057.0 990.2 934.0 690.1 526.0
CHzBrz(1)-CHzlz(2) XI
0.132 0,280 0.395 0.479 0.574 0.704 0,823 0.886
H”
121.0 219.8 302.8 352.9 347.1 308.7 190.6 140.4
1000 J mole-’; system 3, about 350 J mole-’; and system 1, about 150 J mole-’. The obtained results have been theoretically interpreted through the evaluation of the potential energy of the fluid mixtures by means of the procedure originally proposed b y Hildebrarid and Wood (1933). Such a procedure has been applied by employing a perturbation technique in which the molecular distribution in the fluid has been simulated by means of the rigid spheres approximation. The employed method allows the estimation of the different contributions of the intermolecular forces, Le., nonpolar aiid polar, to the excess enthalpies. Experimental Section
A Tian-Calvet microcalorimeter with two elements (Calvet and Prat, 1956), was used with a Sefran recording galvanometer. Among the different methods by which it is possible to calibrate a calorimeter we have chosen a traditional one based on the comparison with the heat evolved or absorbed by well-studied systems, following the recommendations of McGlashan (1962). The calibrating systems are : benzenecyclohexane and benzene-carbon tetrachloride, and the data are the ones recently reported b y Murakami and Benson (1969). 210 Ind. Eng. Chem. Fundam., Vol. 12, No. 2, 1 9 7 3
The cell in which the two liquids were mixed is described by Ragaini aiid Cavenaghi (1969). It allows the mixing of liquids, reducing to the utmost the dead space above the mixture. It was verified by means of blank runs that the heat evolved in the rupture of the glass bulbs 1% hich contained one of the two liquids and in the stirring procedure contributed less than ly0 to the total heat involved. Materials. Cyclohexane and benzene were commercial products (Carlo Erba) with purity (M t %) >99 and >99.5%, respectively. Carbon tetrachloride was a 1Ierck product “for spectroscopy.” The dihalogenated methanes were purchased from Fluka; they have the following purity (wt 70) : CH2C12,99%; CH2Bi-2, 99.5%; CH212,>98%. The densities of the dihalogenated rnethanes a t 25°C from our determinations are: CH2C12, 1.3158; CH213r2,2.4803; CH&, 3.2985 g/cm3. Experimental Results
I n Table I the experimental values of the excess enthalpies HE/J mole-’, are reported (+0.2%) ; these values are illustrated in Figure l where the curves are obtained by interpolations of the experimental points by means of the equation n
HE = zi(l - xi)
C
c,(l - 221)’-’
(1)
p= 1
whose parameters are reported in Table 11. The densities of the three systems a t 25OC for some mole fractions have also been measured (+0.01%). These results are summarized in Table 111. From the data of this table we can calculate for instance an excess volume of mixing VE for 2 1 N 0.5 for the three systems as follows: system 1, z1 = 0.497, VE = 0.011 cm3/mole; system 2, xi = 0.508, VE = -0.181 cm3/mole; system 3, z1 = 0.511, VE = -0.142 cm3/mole. Theoretical Analysis
The description of the thermodynamic behavior of a liquid mixture can be performed by means of interpretive approaches in which a n approximate structure of the system is employed (Mansoori and Canfield, 1970). These approaches are also called lattice theories since the structure of the mixture is customarily assumed to resemble the lattice of crystalline solids. Actually, liquids “offer a serious challenge to the viability of interpretative approaches because their
structure is continually changing” (Mansoori and Canfield, 1970). I n the second approach emphasis is given to the process by which intermolecular forces determine the molecular distribution in the liquid phase. This theory involves the employment of the distribution functions which specify the probability of finding sets of molecules in particular configurations. I n a mixture such an approach can be developed by employing a set of radial distribution functions, g i j , corresponding to each pair of molecules i and j. The potential configuration energy, Em, for the mixture can be expressed as follows (Hildebrand and Wood, 1933)
Table IV. Critical Data and Dipolar Moments Tc, “K
CHzClz CH2Br2 (33212
Ew
=
En, - (E1
+ E?)
(3)
E l and E? being the potential configuration energy of the single components 1 and 2, respectively. By combining eq 2 and
193.0 235.0 285.0
p, D
1,57 1.43 1.14
The last term a t the right-hand side of eq 7 represents the mean value of the dipolar interaction (Rowlinson, 1969, p 274). The intermolecular potential parameters have been evaluated by means of the following mixing rules
CiJ
the intermolecular potential of the pair i-j molecules at distance r13.For a binary liquid mixture the energy change b y mixing, E M ,is given by pi3 is
Vc, cmS mole-‘
506.39 61.39 744.09
= 6i5
A
€
j
The former rule is consistent with the fact that in our model for r 5 u the molecules behave as rigid spheres. According to London’s theory the parameter 6 c j depends on the values of the ionization potentials, I , and collision diameters, U , as follows.
3 and neglecting the excess volume it derives
(4) being 1, =
J0
(5)
and Q i = z i V i / c i z i V are i the volume frac,tioris (for the other symbols see the Somenclature section). The evaluation of the integrals which appear in the previous equations can be performed by means of the perturbation met’hod (Zwaiisig, 1954). Such a method allows the description of a real fluid by assuming that it’sequilibrium configuration is the one correspoiiding to a system of hard spheres with‘ radial distribution function go(r), h’ow a small perturbing potential is applied to each pair of molecules; i t is assumed that the packing of the molecules, essentially due to hardsphere repulsion, should not be affected significantly. The following intermolecular potential, up, has been assumed for r i j < u i j (rigid spheres) p ( r i j ) for r i j > u i j m
Up(rij)
Each integral present’in eq 4 can be evaluated as
The previous equation has been applied to the system CH2C12-CH2Br2 for which the necessary data for ionization potentials are available (Venedeyev, et d.,1966) (ICHGlS = 11.4 eV; I c H ~ = B ~10.8 ~ eV). Otherwise the values of ui and U , are related to the critical volumes of the species under consideration (see Table IV). It follows that uij = 0.9964; therefore, taking also into account that the critical volume of CH212is close to the values of the other two compounds, i t seemed to be justified to attribute to ut, a unitary value. Throop and Bearman (1965) have given a n extensive tabulation of the radial distribution function g o ( [ , pu3) for pure liquids as a function of the intermolecular reduced distance and density number, eu3. According to the one-fluid model the radial distribution function for the mixture has been evaluated applying the quation
where gijO is still obtained by the tabulation given by Throop and Bearman (1965). It is advisable to split the energy change by mixing (mixing energy) E M ,into two terms, one due to the Lennard-Jones contribution, E L J ,and another to the dipolar contribution, ED,as follows.
E M = ELJ (6) Since the systems taken into consideration are made of polar molecules the intermolecular potential must be written as a sum of two terms
+ ED
(11)
where
ELJ = A ( z b i j J ~ ~ , ij
ED =
-
A(2CijJ~,ij
- bjjJLJ,jj)
biiJLJ,ii ciiJD,ij
-
CjjJD,jj)
(12) (13)
The symbols are defined by A
=
being
27rNZ(xiVi
b i.j. = cij
and JLJ,ij
=
4
=
+
~jVj)$i$j
(14)
eijcij3/ViVj
(15)
xijgij3/ViVj
(16)
la
([-’*
-
[-‘)gij”[’d[
Ind. Eng. Chem. Fundam., Vol. 12, No. 2, 1973
(17) 211
Y
Table V. Parameters for the Density (g/cm3) from Eq 19 and Molar Volumes (cm3 mole-') at 25OC
CHzClz CHzBrz CHzL
0
b X 10-8
3.988 3.275 2.348
-1.887 -2.739 -5.152
X 10-6
Molar vol
-1.33 2.89 5.67
64.567 69.981 80.977
c
.
.w
s
Y
-
c
;
1000
Table VI. Comparison between Some Values of E / & ( O K ) and u (A)Estimated b y Different Methods for Some Polar Molecules From viscosity (Reid and Sherwood, 1966)
s/k U
e/k U
t/k U
t/k U
340.2 355 5.31 5.389 356.3 483 4.52 4.898 560.2 4.600 362.6 4.530
From virial (Blanks and Prausnitz, 1962)
1060
and 21
600
Substance 200
I
4.878
Chloroform
454.2 4.536
} 463'1 ) 4.323 456'7 4.673
Methylene chloride Acetone
Figure 2. Comparison between experimental enthalpies of mixing (symbols as in Figure 1 ) and calculated energies of mixing. Solid lines from eq 1 1
Table VII. Comparison between the Parameters E& ( O K ) and uti (A)Calculated and Optimized
Ethanol
i
(1/3)
1-
E-'silOEZdt
(18)
The parameters necessary for the application of eq 11-18 have been evaluated as follows. Molar Volumes (Vi, Vj). Timmermans' d a t a (1965) for densities p* have been interpolated by means of the equation p* = a
160
From
2.98
JD,U =
0
eq 2 0
481.3
479 3.67 620 2.45
,
+ bt + ct2(g/cm3)
(19)
( t , "C), whose parameters are reported in Table V with the calculated molar volumes. The densities calculated by means of eq 19 are in good agreement with our determinations for the dihalogenated methanes a t 25OC. Critical Data (Tc, VC). T h e critical temperature Tc and t h e critical volumes V C have been evaluated b y means of additive rules (Lydersen, 1955) (see also Reid and Sherwood, 1966). This procedure has been adopted since such d a t a are not available for all the three components and therefore i t seemed preferable t o choose the same method for their evaluation. T h e critical temperatures have been obtained by means of t h e method of Gulberg-Lydersen (Lydersen, 1955) employing the following normal boiling point, Tb, for the three substances: CHzClz, 339.95'K; CHZBrz; 370.05OK; CH212,453.15'K. The critical volumes instead were calculated by means of the method proposed by Schuster (1926) (see also Reid and Sheraood, 1966). The results of these calculations are collected in Table I V in which also the values of the dipole moments, b, of the molecules are given. A comparison between those data and those reported from Landolt-Bornstein (1960) reveals a more accurate estimation of critical temperatures. From the critical volumes and temperatures the values of the intermolecular parameters can be evaluated. Particularly for dipolar forces the employment of the following relationship has been suggested
B / K = 0.897Tc, OK 212 Ind. Eng. Chem. Fundam., Vol. 12, No. 2, 1 9 7 3
(20)
CHzClz CHzBrz CHJz
Eq 2 0
454.2 548.3 667.5
eiilk Optimized
369.2 527.4 926.0
'Jii
Eq 21
4.536 4.844 5.166
b
c = 0.785V~'C'/~,
Optimized
4.657 4.811 4.946
(21)
which give the values listed in Table VI. Actually a comparison of the values of the intermolecular parameters obtained from critical data with the ones obtained from physical properties of fluids (second virial coefficient, viscosity) reveals some significant discrepancies. For instance, for chloroform, methylene chloride, acetone, and ethanol the values summarized in Table VI come out. For this reason the intermolecular parameters derived from eq 20 and 21 have been adjusted by means of a nonlinear regression analysis applied to the total absolute relative error between experimental and calculated values of enthalpy of mixing. This comparison is possible since, owing to the small value of VE(as can be derived from Table 111)) it is justifiable to put EM= (HE)calc,jwhere (HE)calcd represents the calculated excess of mixing enthalpy. The results are reported in Table VII. The calculated values of E M ,E L J ,and ED as a function of the mole fraction are summarized in Table VIII, where the experimental results (HE) are also reported. I n Figure 2 a graphical comparison between HE and E M for the three systems is given. On the whole the agreement between experimental and calculated data is satisfactory, with a mean relative error of 10% on each determination while with the parameters calculated from eq 20 and 21 (see Table VII) the mean relative error is 81.5%. The analysis of Table VI11 allows the comparison of the two contributions ELJ and ED to the total energy of mixing. For the three systems the ratio E L J I E Dis in the range from 6 to 8. The data given in Table VI1 show that the collision diameters resulting from the optimization are very close to the
table VIII. Calculated Values at 25°C of EM, E M , and €D (J mole-’) as a Function of Mole Fractions, and Experimental Results, HE (J mole-’) CH2C12(1 I-CHzBrz(2) x1
EM
0,100
55,OO 0,222 103.4 0,279 119.2 0,435 142.3 0.542 141.4 0.725 110.4 0,900 48.49
ELJ
48.74 91.00 104.5 123.5 121.8 93.81 40.63
CHZBrz( 1 )-CH21~(2)
CHzCIz(1)-CHzIz(2)
E?
6.259 12.43 14.70 18.81 19.59 16.56 7.845
HE
XI
Esi
ELJ
ED
51.38 0.150 427.7 374.0 53.68 99.42 0.271 687.5 602.5 84.98 125.2 0.336 787.3 690.8 96.48 144.5 0,390 857.0 752.7 104.3 152.8 0.469 919.5 808.8 110.7 110.9 0.535 937.3 825.5 111.8 36.45 0.625 9p9.2 802.1 107.1 0.691 846.8 748.1 98.74 0.851 530.3 469.9 60.37 0.890 415.2 368.2 46.99
ones obtained from eq 21. Some differences appear between the values of optimized parameters elk and the ones calculated by means of eq 20,but in any case such differences are of the order of the approximation limits of eq 20 also taking into account the errors involved in the additive calculations of critical data. It is significant that the values of t’he intermolecular parameters obtainid in the preseiit paper are consistent with the ones derived from viscosity data. I n fact, i t is well known that t’he collision diameters obtained from Tirial coefficients for polar gases are unrealistically small (CarrB aiid Zanderighi, 1967). For what coiiceriis the application of the model described in the preseiit paper for a predictioii of the eiithalpies of mixing of polar and noiipolar systems it seems that such a possibility is essentially coniiected with the availability of an adequate set of iiitermolecular parameters. Such a possibility requires an aiialysis of more experimental data, and for this reason the method has been applied also with satisfactory results to the set of binary mixtures among CH2C12,CHC13, and CC14 (Ragaini and Carrh, 1973).
= p1 = pl* = Tc = VC = i , ~=
fill
HE
XI
Ebi
ELJ
ED
HE
377.4 0.132 122.3 104.9 17.36 121.0 560.4 0.280 234.6 204.1 30.50 219.8 77g.2 0.395 295.9 259.9 35.96 302.8 875.8 0.479 327.7 290.0 37.69 352.9 984.0 0.574 341.0 304.1 36.89 347.1 1057.0 0.704 315.7 284.3 31.38 308.7 990.2 0.823 238.6 216.7 21.91 190.6 934.0 0.886 172.6 157.4 15.20 140.4 690.1 526.0
dipolar moment molar density density critical temperature critical volume (subscript) mean molecule or substance i , ~
literature Cited
Asselineau, L., Renon, H., Chem. Eng. Sci. 25, 1211 (1970). Blanks, 11. F., Prausnitz, J. M., A.Z.Ch.E. J. 8 , 86 (1962). Calvet, E.,Prat, H., “Microcalorinietrie,” Masson, Paris, 1956. Carrh, S., Zanderighi, L., A7uovo Cimento, Ser. X 49, 133 (1967). Hildebrand, J. H., Wood, S. E., J. Chem. Phys. I , 817 (1933). Landolt-Bornstein, “Zahlenwerte und Funktionen,” SpringerVerlag, Berlin, 1960. Lydersen, A. L., “Estimation of Critical Properties of Organic Compounds,” College of Engineering, University of Wisconsin, Engineering Experimental Station Report 3, Madison, Wis.,
1955.
Mansoori, G. A., Canfield, F. B., Ind. Eng. Chem. 62 (8), 12
(1970).
McGlashan, M. L., “Experimental Thermochemistry,” Vol. 2, H. A. Skinner, Ed., Interscience, New York, N. Y., 1962. Murakami, S.,Benson, 0. C., J . Chem. Z’hermodyn. 1,559(1969). Papadopoulos, M. N., Derr, E. L., J . Amer. Chem. SOC.81, 2285
(195Q).
Nomenclature
escess enthalpy (enthalpy of mixing) excess volume (volume of mixing) potential energy of a misture molar volume intermolecular potential radial distribution function radial distribution function for hard spheres distance bet*een the centers of two molecules mole fraction ’ Aivogadro’snumber 13oltzmaiiii’s number energy of mixing potential energy of pure substance i volume fraction perturbing intermolecular potential potential parameter potential parameter reduced distance for intermolecular potential ’
Ragaini, V., Carrh, S., unpublished data, 1973. Ragaini, V., Cavenaghi, C., Chim. Ind. 51,370 (1969). Redlich, O.,Derr, E. L., Pierotti, G. J., J. Amer. Chem. SOC.81,
2283 (1959).
Reid, R. C., Sherwood, T. K., “The Properties of Gases and Liquids,’’ 2nd ed, pp 72,632,McGraw-Hill, New York, N. Y.,
1966.
Rowlinson, J. S., “Liquids and Liquid Mixtures,” 2nd ed, Butterworths, London, 1969. Schuster, F., 2. Elektrochem. 32,191(1926). Tbroop, G. J., Bearman, R., J. Chem. Phys. 42, 2408 (1965). Timmermans, J., “Physico-Chemical Constants of Pure Organic Compounds,” Elievier, Amsterdam, 1965. Venedeyev, V. I., Gurvich, L. V., Koubrat’yer, U. N., Medveded, V. A., Franckvich, Ye. L., “Bond Energies, Ionization Potentials and Electron Affinities, Edward Arnold, London, 1966. Zwanzig, R. W., J. Chem. Phys. 22, 1420 (1954). RECEIVED for review May 26,1972 ACCEPTEU November 13, 1972 The authors are indebted to the Italian Consiglio Nazionale delle Ricerche for financial aid.
Ind. Eng. Chem. Fundam., Vol. 12, No. 2, 1973
213
