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0. LEVY
Limiting Behavior of Alkylammonium Salts in Benzene by 0. Levy Department of Chemistry, University of the Negev, Beer-Sheva, Israel
(Received June 21, 1071)
Publication costs assisted by the Department of Chemistry, University o j the iVegev
The behavior of several alkylammonium salts ((CI~H&NHX, where X = C1, clod, FeCl4, and FeBr,) a t infinite dilution in benzene was studied by a molecular solution theory. Second virial coefficients were evaluated using a model which assumes two main contributions to the average potential between the particles : (i) repulsion due to their own molar volume and (ii) attraction resulting from dipole-dipole interactions. Good agreement was found between the calculated activity coefficient values and those obtained from vaporpressure-lowering measurements. Several molecular distance parameters were evaluated and compared with those obtained from dielectric constant and conductivity data on the same systems.
I. Introduction Previous dielectric1 and osmometric2z3 investigations on binary systems of long-chain alkylammonium salts in nonpolar organic solvents were interpreted in terms of deviation of the systems from ideal behavior due to the aggregation of the solute. It has been found that the extent (number) and degree (size) of the molecular associates depend considerably on the anionic part of the alkylammonium ion pair. Additionally, the osmometric data were also treated in terms of nonspecific nonideality expressed through the activity coefficients of the solutes via the Gibbs-Duhem relat i ~ n s h i p . ~ The - ~ curves representing the activity coefficient dependence on solute concentration indicateas do the aggregation constants-a pronounced departure from ideal behavior, and again the extent of nonideality depends on the anion. The deviation from ideal behavior increases in the order5 C1- < Br- < NO-3 < Clod- 2 FeC14- < FeBr4- in agreement with the increase of the anionic radii and other related properties.6 Various oligomerization models were tested for these systems. The choice of a certain model was dictated by the mathematical computation method, chemical intuition, and comparison with other literature data. I n the case of trilaurylammonium salts with both simple and complex anions, the basic aggregation unit is the dimer. An angular shape can be inferred from dielectric constant measurements. An attempt is made here to treat such binary systems where the interactions are essentially physical in nature by a statistical mechanical approach in order to predict thermodynamic data useful in interpretation of systems having one polar component. Virial coefficient calculations were made by Kozak, et ~ 1 . ~for 7 a large number of organic compounds. The effects of soIute size were anaIyzed by using the Iattice theory and the llcMillan-i\/Iayer theory for sohtions. The intermolecular forces considered depend on the type of the system studied-self-association and The Journal o j Physical Chemistry, Vol. 76, N o . 12, 1072
structural changes for alcohol-water mixtureq8 van der Waals and hydrogen-bond interactions for other mixturesg including those with hydrophobic bonding.10 In alkylammonium salt-organic solvent systems a dipole interaction was considered as the main factor governing the self-aggregation in addition t o a possible solute-solvent interaction. 11,12 We choose now to treat the limiting behavior at infinite dilution of these solutions using a model based on the McMillan-Nlayer theory13 and to compare the predicted activity coefficients of the solutes with those determined experimentally.
11. Theory For an imperfect gas the equation of state can be expanded to
p/kT
=
P
+ &(T)p2 + B3(T)pa+ . . .
(1)
where p and T are the pressure and temperature of the (1) 0 . Levy, G. Markovits, and A. S. Kertes, J . Phys. Chem., 75, 542 (1971) (2) A. S. Kertes and G. Markovits, ibid., 72, 4202 (1968). (3) 0. Levy, G. Markovits, and A . S. Kertes, ibid., 74, 3568 (1970). (4) A . S. Kertes and G. Markovits in “Thermodynamics of Nuclear Materials,” I. A. E. Agency, Vienna, 1968, p 227. ( 5 ) G. Markovits, Ph.D. Thesis, Jerusalem, 1969; 0. Levy, Ph.D. Thesis, Jerusalem, 1970. (6) A. S. Kertes, H. Gutmann, 0. Levy, and G. .Markovits, Israel J . Chem., 6 , 643 (1968). I
(7) J. J. Kozak, W. S. Knight, and W. Kauzmann, J . Chem. Phys., 48, 675 (1968).
(8) F. Franks and D . J. G. Ives, Quart. Rev., Chem. Soc., 20, 1 (1966). (9) D. Stigter, J . P h p . Chem., 64, 118 (1960). (10) G. Nemethy and H. A. Scheraga, J . Chem. Phys., 33, 3302, 3401 (1962). (11) Y . Marcus and A. S. Kertes, “Ion Exchange and Solvent Extraction of Metal Complexes,” Wiley-Interscience, London, 1969, Chapter 10. (12) Yu. G. Frolov, A. V. Ochkin, and V, V. Siergievsky, Atom. Energ. Rev., 7, 71 (1969). (13) T . Hill, “Introduction to Statistical Thermodynamics,” Addison-Wesley, London, 1962.
LIMITING BEITAVIOR OF ALKYLAIIMONIUM SALTS IN BENZENE gas, p is the number density of the particles, and k is the Boltzmann constant. The virial coefficients B,(T) express the departure of the system from ideality. They are temperature dependent and often related to the various molecular interactions. By defining the activity coefficient y of the gas and using the virial expansion one obtains13 In Y = --CPR(T)Pt k>l
+
+ In p + In y
being the chemical potential in the standard state chosen. Therefore, using the virial expansion, one obtains (4)
In a very dilute solution, at a first approximation, all the terms in eq 4 can be neglected but the first with the second virial coefficient; then eq 4 becomes (5)
+2B*zp
Generally, the second virial coefficient is expressed in terms of W(T, Q ) , the reversible work necessary to bring the molecules together from r = 03 to r in the solvent of the given proper tie^'^^'^ B*z(T)
>
B* =
(-1)4nr2dr -
l / 2 r
0
(exp[-w(r, Q)/kT] - l)4nr2dr (8)
l/zJd*Jc
The repulsive term (integrand 1) can be equated with half the volume of the particle. In order to evaluate the attractive term (integrand 2)) we have to look for an adequate expression for W ( T , n). According to Keesom,14 for rigid spheres containing point dipoles W(T, 0) =
PZ -[ 2 COS 61 COS 62 -
er
(3)
p*O
In y
(for r < a ) and (ii) an attractive energy component which (neglecting the Lennard-Jones potential) includes the dipole-dipole interaction (for r a ) . Equation 6 then becomes
(2)
where Pi, is defined as Pr = - [ ( k l)/k]Pk+l. According the the McMillan-Mayer theory for solutions, an analogy can be made between a dilute solution (solute in solvent) and “gas in vacuo,” and consequently the virial expansion can be used for any suitable property of the solution. The chemical potential ( p ) will be expressed by p*/kT = p*O/kT
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sin 61 sin 62 cos(+z -
-1/21mJ
(9)
where p is the dipole moment of the solute, 61 and e2 are the angles between the direction of dipoles relative to their connecting line, 41 and cpZ are the relative rotational angles, and e is the dielectric constant of the solvent-the latter being introduced to account for the effect of the medium’2s14 Now the expression for Bz*can be written explicitly as B* =
l/zl 4.rrrZdr -
‘ / 1 6 T 1
4 n r z d r 1sin 6 l d 6 1 l sin 62d6212k
d ( h - +d(expI--(r,
Q)lkTI - 1) (10)
where w(r, Q ) is expressed by eq (9) and normalization factor.
=
+I)]
‘/16T
is a
(exp[-w(r, O)/kT] - 1)4nrzdrd0 (6)
111. Results and Discussion The above integrands were calculated with a CDC
where the second intergrand expresses the angledependent contributions due to the interactions of dipoles. For the hard-sphere hypothetical model
6400 computer using the trapezoidal rule and the experimental data obtained previously. Dipole moments were calculated from dielectric constant measurements,’ and for the values of a (the closest approach between two molecules) we have some clues from dielectric, conductivity,6,16and density data. 1,6 B2* was calculated by minimizing its value wit,h respect to a. Parameter a was varied in steps of 0.1 A till the minimum was reached, and the obtained values represent, the “reversible work necessary to bring two solute molecules together from r = 03 to r = a in the given so1vent”13(Table I).
w(r, Q )
