J. Phys. Chem. 1982, 86, 1469-1472
TABLE V: The Variation of the Rate Constant at 70 "C in the AOT-Cyclohexane Reversed Micelles with
Fez+Concentration
rate constant, min-' concn, M
5.0 X
1.0 X
pH 1.0 8.0 X 1.5 X 10.'
pH 2.0 1.3 X 1.8 X 10.'
pH 3.0 2.4 X 2.9 X
preted in terms of an immobilization by increased hydrogen bonding of the micellar water. The increase of hydrogen bonding between water molecules in a thin layer neighboring to surfactants may form a field where easy transfer of electrons from Fe2+ to Fe3+ occurs by a tunnelling effect. The excess electrons will be consumed in aqueous solution to produce OH- in the presence of dissolved oxygen as H 2 0 + e + O2 OH-. Sunamoto et al.81~ gave evidence that the metal ions are made "naked" due to the strong hydration of surfactant molecules by determining solvochromism and other properties of the AOT-CC4 reversed micelle containing Cu2+solution. In our cases, a higher degree of hydration of Na+ will also be relevant to producing such "naked" iron ions. These versatile factors may be considered to cause the acceleration of the reaction. From the data in Table 11, we can calculate the activation energy of the overall oxidation reaction by the Arrhenius equation as 330 kJ mol-' for the AOT-cyclohexane reversed micelle with the Fez+ solution of the original pH 3.0. It is noted also that the activation energy decreases with the lowering of the original solution pH: 176 kJ mol-' at pH 2.0 and 120 kJ mol-l
-
(24) J. Sunamoto and T. Hamada, Bull. Chem. SOC.Jpn., 51, 3130 (1978).
1469
at pH 1.0. These somewhat high values of the activation energy and a dependency on pH reflect complicated processes during the changes in the reversed micelles including solid-phase formation.
Conclusion The apparent rate of oxygenation of ferrous ions by air in the water pools of the AOT reversed micelle and the potassium oleate reversed microemulsion was faster by approximately 102-103-fold of that in bulk solution. The appearance of a turbid state and precipitates after oxygenation of Fe2+ in the reversed micelle and reversed microemulsion, respectively, suggests an alkaline environment in the aqueous cores. The rise of pH and the resulting acceleration of oxidation reaction would be caused by the following factors: (1) the interaction between electrolyte ions (Fe2+,Na+, and K+) and anionic surfactant molecules provides a particular electrostatic field formed by the accumulated cations in the aqueous cores of reversed micelle and reversed microemulsion; (2) a dense layer of water molecules in the restricted water pools helps easier electron transfer from Fez+to Fe3+,and (3) the two factors mentioned above cause hydrolysis and tend to raise the pH of the water pools in both the micellar and microemulsion systems, leading consequently to the acceleration of oxidation of Fez+and its polynuclear complex formation. Acknowledgment. We wish to express our gratitude to Professor J. Sunamoto (Nagasaki University) and Professor A. Kitahara (Tokyo Science University) for their helpful suggestions during this work. The electron micrographs were kindly taken by Professors K. Terao (Figure 7) and K. Kobayashi (Figure 8) of this University, for which the authors' thanks are due.
Quantum Corrections to Second Viriai Coefficients for the Diatomic Lennard-Jones Potential S. H. Lingt and M. Rlgby" Chemistry Bpartment, Queen Elizabeth College, London W8 7AH, United Kingdom, and School of Chemical Engineering, Cornel1 Universlty, Ithaca, New York 14853 (Received: October 16, 1981)
Values are reported for the first-order translational and rotational corrections to the second virial coefficients of molecules interacting through the diatomic Lennard-Jonesintermolecular potential. The changing importance of the translational and rotational terms with variation in the elongation of the molecules is considered, and a corresponding states analysis of this behavior is presented. Typical magnitudes of the corrections for Nz and C02 are estimated.
Introduction In recent years the diatomic Lennard-Jones (DLJ) intermolecular potential function has been extensively used to model the behavior of simple linear molecules in the solid, liquid, and gaseous states.'" Although it is not to be expected that this intermolecular potential represents accurately the interactions of any real system it can nevertheless provide very useful information concerning the Department, Queen Elizabeth College, London. *Author to whom correspondence should be addressed at Queen Elizabeth College. t Chemistry
0022-3654/82/2086-1469$01.25/0
relationship between molecular shape and the properties of matter. Thus, for example, the crystal structures adopted at low temperatures by diatomic molecules have been correlated with their elongation by using this potential,' and the consequences of similar changes in shape (1) C. A. English and J. A. Venables, R o c . R. SOC.London, Ser. A , 340, 57 (1974).
(2) W B. Streett and D. J. Tildesley, R o c . R . SOC.London, Ser. A, 355, 239 (1977). (3) D. J. Evans, Mol. Phys., 34, 103 (1977). (4) J. R. Sweet and W. A. Steele, J . Chem. Phys., 47, 3029 (1967). (5) D. J. Evans, Mol. Phys., 33, 979 (1977).
0 1982 American Chemical Society
1470
on the transport properties of dilute gases have also been ~ t u d i e d .The ~ classical second virial coefficients for the DLJ potential have been calculated by several groups4+ and have been used to show the connection between molecular anisotropy and the extent of deviations from the simple principle of corresponding states.6 Studies of the second virial coefficients of spherical molecules have shown that results at very low temperatures are of particular value in discriminating between several proposed intermolecular potential function^.^ However, at such temperatures, particularly for fairly light molecules, the classical second virial coefficients are not sufficient and must be supplemented with the quantum correction^^*^ based on the Wigner-Kirkwood expansion of the configuration integral in powers of h2. These terms have been incorporated into calculations which have led to accurate intermolecular potentials for Ne’O and Ar.” As progress is gradually made toward the development of accurate potentials for diatomic and polyatomic molecules similar calculations are needed for diatomic and polyatomic molecules similar calculations are needed for these systems also. Although there have been some calculations of quantum corrections to the second virial coefficients of molecules which have permanent electrical m~ltipolesl~-’~ and applications have been made in particular to H2013-15 there seems to have been no general study of the effect of molecular shape on these corrections. (The calculations of Wang Chang16 were restricted to a particular model for H2,and the inclusion of shape effects in the work of Singh and Datta12was based on a simple model which does not adequately describe large deviations from spherical shape.) A survey of the relationships between molecular shape and the quantum corrections to the second virial coefficient is reported here for the DLJ potential and a wide range of molecular elongations. It is hoped that these results will prove useful in the search for accurate intermolecular potentials for diatomic or linear triatomic molecules by providing estimates of the likely magnitude of the quantum corrections for translational and rotational motion.
Calculations It was shown by Wang Chang16that for linear molecules the Wigner-Kirkwood expansion for the second virial coefficient may be represented as B(l7 = ~
Ling and Rigby
The Journal of Physical Chemistry, Vol. 86, No. 8, 1982
\ - ,
BcI
+ -Bir h2 m + h2 I
+
(
:)’B’ir
+
(
where Ba represents the classical second virial coefficient, and the subscripts tr and rot identify corrections resulting from quantum effects in translational and rotational motion for molecules of mass m and moment of inertia 1. Pompe and Spurling15have reported also the occurrence of a further second-order term, B’\r,rot,dependdnt on (6)G. C. Maitland, M. Rigby, E. B. Smith, and W. A. Wakeham, “Intermolecular Forces”, Clarendon Press, Oxford, 1981, Chapter 3. (7) M. Klein and H. J. M. Hanley, Trans. Faraday SOC.,64, 2927 (1968). (8) E.A. Mason and T. H. Spurling, ‘The Virial Equation of State”, Pergamon Press, Oxford, 1969. (9) J. 0. Hirachfelder, C. F. Curtias, and R. B. Bud, “Molecular Theory of Gases and Liquids”, Wiley, New York, 1964. (10) J. A. Barker, Chem. Phys. Lett., 14, 242 (1972). (11) J. A. Barker in “Rare Gas Solids”, M. L. Klein and J. A. Venables, Ed., Academic, London, 1976, Chapter 4. (12) Y. Singh and K. K. Datta, J . Chem. Phys., 53,1184(1970). (13) M.McCarty and S. V. K. Babu, J.Phys. Chem., 74,1113 (1970). (14) T. B. MacRury and W. A. Steele, J. Chem. Phys., 61,3352 (1974). (15) A. Pompe and T. H. Spurling, A u t . J. Chem., 26, 855 (1973). (16) C. S. Wang Chang, see ref 9, pp 434-7.
h4/mI. For molecules other than the very lightest, such as H2, or those with very small moments of inertia, such as H20, the expansion is rapidly convergent, and the second-order terms may be neglected. The expressions for the quantum corrections in terms of the intermolecular potential were worked out by Wang Chang. For intermolecular potentials characterized by a length parameter, u,and an energy parameter, E , it is convenient to represent the expansion to first order in reduced units:
B*(T*) = B(kr/e)/bo= B*cl + A*2BMtr(T*)+ 6’*rotB‘rot(T*) (1) where bo = 2rNu3/3, = h2/a2me, = kO,,/e, and Orot = h2/8r21kis the characteristic rotational temperature of the substance. Then B*cl =
B‘rot
-s
J[exp(-u*/T*)
-
l]r*2 dr* d 0
=
-&(E)+ A(
+ sin2 e,
dr* d 0
where u* = u / t , r* = r / a , and, for linear molecules with end for end symmetry
... dQ =
In these expressions the intermolecular energy, u, is a function of the separation of the molecules, r , and of the relative orientation defined by 01, e,, d2 - r$l. The DLJ model consists of two interaction sites on each molecule, separated by a distance R*a. The energy, u , is the sum of four site-site interactions, each of which has the Lennard-Jones 12-6 form. Hence we may write u*.. lJ =
2= ; 5 €
[ (t>” ( 31
a=l@=l
-
where CY and /3 represent the sites on molecules i and j , respectively. When R* = 0 this model reduces to the normal atomic LJ form, but for R* > 0 the energy becomes a function of relative orientation in a manner which seems to reflect reasonably the interactions of linear molecules. The anisotropy of the potential increases as R* is raised to its maximum plausible value of 1, and for this model the reduced virial coefficients B*cl, B Mtr, BMrot, are functions both of the reduced temperature, T* (= k T / t ) ,and of R*. Several different numerical procedures have in the past been used in calculating the second virial coefficients and quantum corrections for nonspherical systems.12-14 Expansions of the Buckingham-Pople type have been employed and also methods based on spherical harmonic expansions. For moderate anisotropy of the potential these methods have the advantage of economy, but when the anisotropy is large the relatively slow convergence of the expansions can be troublesome. As it was our intention to study the whole range of R* from 0 to 1 it was found most convenient to perform the four-dimensionalintegrals numerically, using a nonproduct algorithm.17J8 The in-
Quantum Corrections to Second Virial Coefficlents
The Journal of Physical Chemistry, Vol. 86, No. 8, 1982 1471
TABLE I: Second Virial Coefficients and Quantum Corrections for the DLJ Potential T*
B*,.l
R*=O B'*e
0.30 -27.88 0.40 -13.80 0.60 -6.20 0.80 -3.734 1.00 -2.538 1.20 -1.836 1.40 -1.376 1.60 -1.052 1.80 -0.812 2.00 -0.628 3.00 -0.115
R* = 0.3 B*cl
24.96 -17.20 6.99 -9.34 1.673 -4.399 0.729 -2.613 0.416 -1.701 0.274 -1.151 0.197 -0.784 0.151 -0.523 0.121 -0.328 0.100 -0.178 0.051 +0.241
R* = 0.6
B'*t, B'*,ot 12.25 4.10 1.185 0.574 0.350 0.242 0.181 0.142 0.116 0.098 0.052
B*cl
12.20 -11.18 3.88 -6.31 1.056 -2.869 0.492 -1.520 0.292 -0.806 0.198 -0.386 0.145 -0.073 0.113 t0.138 0.091 0.295 0.076 0.416 0.039 0.752
tegrations were performed by using several sets of sample densities and the final values are believed to have typical uncertainties of around 0.02 at the lowest temperatures and 0.001 or less at high temperatures. The classical virial coefficients calculated by using this program agreed within the quoted error with the results of Sweet and Steele4 and Evans6 and with the values obtained earlier by a product Gaussian integration scheme.lg After suitable modification the program was used to calculate the quantum corrections for the case of the Stockmayer potential. Comparisons with the results of McCarty and Babul3 showed good agreement, when allowance was made for the different reduction procedure used by these workers.
Results and Discussion A summary of the results obtained for the classical second virial coefficient and for the two first-order quantum corrections is given in Table I. More complete data are available from the authors. The temperature range covered was chosen to exceed the experimentally accessible lower l i i i t and to extend to temperatures sufficiently high that the quantum corrections were negligible. Results for the atomic LJ limit, R* = 0, are included to facilitate comparisons. (There are several errors in the expansion coefficients and calculated quantum corrections in Table 1-E of ref 9.) It is seen that as might be expected the size of the rotational correction increases with increasing anisotropy of the potential for fixed reduced temperature, P. The translational correction, on the other hand, decreases as R* is increased at constant P. Such a comparison is a little misleading, however, since the actual corrections for a given substance are obtained by multiplying the values which in Table I by characteristic factors, A*2 and depend both on the physical characteristics of the molecule under investigation, such as its mass or moment of inertia, and also on potential parameters, e and u. The values of these parameters will depend on the value of R* which is used to describe the molecule, and hence the reduced temperature, T*, appropriate to the experimental temperature will also vary as R* is changed. These complications may be dealt with by the use of an alternative procedure for studying the effects of changes in molecular shape. This uses the principle of corresponding states, in conjunction with characteristic parameters derived from the Boyle temperature. Such an approach has been used to good effect in studying the effects of changes in the intermolecular potential on the virial (17) C. H. J. Johnson and T. H. Spurling, A u t . J . Chem., 24,1567 (1971). (18) A. H. Stroud, 'Approximate Calculation of Multiple Integrals", Prentice Hall,Englewood Cliffs, NJ, 1971. (19) Reference 6, Appendix 8.
B'*b
B'*rot
6.54 2.57 0.887 0.474 0.307 0.222 0.171 0.137 0.114 0.097 0.054
38.4 14.58 4.83 2.514 1.602 1.140 0.869 0.695 0.574 0.486 0.267
B*cl
R* = 0.8 B"*b B'*,t
R * = 1.0 B'*t, B'*,t
B*cl
-9.30 4.99 58.6 -8.20 -5.24 2.11 24.2 -4.55 -2.224 0.789 8.74 -1.755 -1.007 0.439 4.76 -0.606 -0.355 0.292 3.12 t0.013 t0.046 0.214 2.264 0.395 0.318 0.167 1.753 0.654 0.512 0.136 1.418 0.838 0.656 0.114 1.184 0.974 0.767 0.098 1.012 1.079 1.070 0.056 0.572 1.036
4.10 83.0 1.84 36.4 0.727 14.0 0.416 7.89 0.281 5.28 0.208 3.89 0.164 3.05 0.134 2.48 0.113 2.09 0.097 1.793 0.056 1.030
TABLE 11: Reduced Boyle Parameters for the DLJ Potential
R*
TB*
0 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1 .o
3.418 3.207 2.761 2.318 1.956 1.677 1.463 1.173 0.995
VU * 0.811 0.852 0.956 1.098 1.262 1.439 1.625 2.008 2.389
coefficients and transport properties of spherical molecules: and is readily extended to the present problem. The Boyle temperature, TB, is the temperature at which the second virial coefficient becomes zero, and the characteristic Boyle volume, VB,may be defined V B= TB(dB/ dT)T,. Experimental values of these quantities may be derived, at least for simple molecules, from tabulated data for second virial coefficients and, if available, Joule Thomson coefficients. For a model intermolecular potential we may define a reduced Boyle temperature and volume TB* = kTB/c VB* = VB/bo and these may be derived from tables of second virial coefficients and their temperature derivatives. Values for the D U potential for R* = 0 to R* = 1are given in Table 11. At the relatively high Boyle temperature the quantum corrections for all but the lightest gases are very small and the values of TB* and VB* are based on the classical virial coefficients. We may define a reduced second virial coefficient, B+, for a corresponding state comparison, as B / VB,and represent it as a function of a reduced temperature, Tc,equal to TITB. The expansion, eq 1, may be rewritten in such a way as to separate properties which depend solely on the physical characteristics of the molecule and those which depend on the intermolecular potential
B + = - = - B* = B+cl + A+2B+tr+ O+rotB+,ot VB vB* where B+~J= B*a/ VB*,B',, = B " r t , T ~ * / V ~ *B+rot 1/3, BNrOtT~*/VB* and
Changes in B+cl, Bftr,and B+,otat a given reduced temperature, Tc,will now reflect the variation in the form of the intermolecular potential in a way which is unbiased by a selection procedure for potential parameters c and u. The variation of the classical second virial coefficient with temperature is shown for several values of R* in
1472
The Journal of Physical Chemistry, Vol. 86, No. 8, 1982
Ling and Rigby
T/TB
.3
4
Flgure 1. Corresponding state comparison of classical second virial coefficients for the W potential: (-) R = 0; (---) R = 0.3; R' = 0.6; (' * ) R' = 0.8. (..a)
+
Figure 1. It is seen that as the elongation increases B+ becomes more negative at low reduced temperatures. At temperatures above 0.5TB the differences become very small. This behavior is consistent with that generally expected on the basis of the extended principle of corresponding states, as formulated by Pitzer.20 In Figure 2 we show the reduced quantum corrections, defined according to eq 2 and calculated by using the results in Tables I and II. Two pointa emerge directly from this graph. For both the translational and rotational contributions an increase in the value of R* gives an increase in the quantum correction at a given reduced temperature. This is in contrast to the apparent trend of the translational term shown in Table I. Secondly, the effects of changes in R* on the translational correction are much less than those on the rotational term. (For R* = 0, the latter is of course zero.) The changes in the rotational correction may be quite large, as for example when going from R* = 0.3 to R* = 0.6 at Tc = 0.3, where an increase by a factor of 7 is observed. In order to assess the absolute magnitude of the quantum corrections it is necessary to consider specific molecules, since the corrections depend on molecular parameters. We have chosen to consider N2 and C02 as typical examples. For Nz a D U potential with R* around 0.3 has been commonly used,2l and for C02, despite its triatomic structure, the D U potential with R* around 0.8 has been employed.22 For N2 reliable second virial coefficients extend down to 75 K,23a reduced temperature, Tc,of 0.23. At this temperature, for R* = 0.3, the quantum corrections for translation and rotation are 3.7 and 1.5 cm3 mol-', respectively, which is to be compared with an experimental (20)K.S. Pitzer, J. Am. Chem. Soe., 77,3427,3433 (1955). (21)P. S. Y. Cheung and J. G. Powles, Mol. Phys., 30, 921 (1975). (22)M. Suzuki and 0. Schnepp, J . Chem. Phys., 66, 5349 (1971). (23)J. H.Dymond and E. B. Smith, "The Virial Coefficients of Pure Gases and Mixtures", Clarendon Press, Oxford, 1980.
10560t
1-
I .6 4 .2
.4
T/ TI3 Figure 2. Translational and rotationel quantum corrections for the W potential. Symbols are the same as in Figure 1.
virial coefficient of -275 cm3 mol-'. The quantum correction thus makes a small but not negligible contribution to the total. For C02 the lowest temperature at which reliable second virial coefficients are available is 220 K,23 giving P 0.31. A t this temperature, and with R* = 0.8, the combined quantum corrections amount to about 2 cm3 mol-', less than 1% of the experimental value. This small correction results in part from the low rotational temperature (large moment of inertia) of C 0 2 , and also from the relatively high value of the lowest experimental reduced temperature. These estimates based on the D U potential suggest that in calculations aimed at defining an accurate intermolecular potential for N2 the effects of quantum corrections on the second virial coefficient should be taken into account, while for C 0 2 they may probably be neglected without introducing substantial errors.
-
Acknowledgment. M.R. thanks K. E. Gubbins and W. B. Streett for their hospitality during a visit to Cornell University, where part of this work was carried out.
