3400
J. Phys. Chem. 1980, 84, 3480-3483
Measured Soret Coefficients for Simple Liquified Gas Mixtures at Low Temperatures D. Longree, J. C. Legros,' and 0. Thomaes Service de Chimie Physique E.P., facult6 des Sclences Appliq&es, Universit6 Libre de Bruxelles, B- 1050 Bruxelles, Belglum (Received: July 15, 1980)
The Soret coefficients for six systems, Ar-Kr, Ar-CH,, Kr-CH,, CF4-Ar, CF4-Kr, and CF4-CHI, have been measured between 82 and 105 K. A relation is found between the Soret coefficient and the molecular diameter difference. The results are compared with the Enskog theory for dense hard-sphere fluids and also with a Rice-Allnatt calculation using a constant radial distribution function or a distribution function calculated by Lebowitz in solving the Perkus-Yevick equation.
Introduction Many experiments have been done to determine accurately the Soret coefficients of various systems.l These measurements are very delicate because even slow movements of convection can destroy the small concentration gradient induced by the temperature gradient. This convection is set up if the temperature field is not rigorously vertical in the whole apparatus. Indeed, the Navier-Stokes equation for a nonisothermal layer of fluid admits a solution with a vanishing velocity only if the temperature gradient has no horizontal components. But such a condition is impossible to realize because the lateral boundaries never have the same thermal conductivity as the solution; for the latter, it is temperature and concentration dependent. Two things can be done: to reduce the convection velocity in such a manner that it no longer perturbs the concentration gradient (e.g., the slit cell of Korsching2) or to confine the convection in a small part of the cell in order to neglect it (e.g., the flowing cell with very large widthldepth used in our laboratory3 or Tyrrell's device4). On the other hand, very elaborate theories have been developed to calculate the Soret coefficients, but the agreement between calculated and measured values has never been satisfactory, even if particular theories are in agreement with one or a few measured values. This is due to the lack of a general theory of the liquid state, for solutions of complex organic molecules as well as for solutions of electrolytes. It seems to be necessary to study systems which correspond as closely as possible to the initial assumptions of the theories. For this reason, we start with measurements of the Soret coefficients of liquified mixtures of simple gases. As far as we know these are the first Soret coefficient measurements of such simple systems. The presented Kr-Ar mixture rigorously obeys the central forces assumption of the statistical theories. The other five systems, Ar-CH4, Kr-CH,, CF4-Ar, Kr-CF,, and CF4-CH4,provide good approximations to this symmetry condition. In the third part of this paper, we compare our measured values with the thermal diffusion ratio as calculated by means of Thorne's extension of the Enskog theory to dense fluidsS6 This comparison is also made with the RiceAllnatt theory, extended to mixtures by Wei and Davis! The agreement with the thermodynamic theory of Prigogine et al.' has also been tested. Measurements The Cell. The cell schematically drawn on Figure 1 has been designed to work at various constant temperatures, by means of a classical liquid-nitrogen cryostat. The temperature difference between the top and the bottom 0022-3654/80/2084-3480$01 .OO/O
of the cell is imposed by controlled electrical heating. The inside is filled with 125 vertical thin-walled stainless steel tubes, in order to protect the system against convection. The induced concentration difference is measured by the variation of the capacity, with the liquid solution as dielectric medium, at the top, between plates 6 and 12, and at the bottom, between plates 5 and 13. This is measured by a Wayne-Kerr capacitance bridge of high precision (B-331-MKI1, accuracy: sensitivity, reproducibility: lo4). The temperatures are read by means of platinum resistor thermometers (100 Q at 273 K) and a Mueller bridge from Leeds and Northrup. A calibration of the capacitors with respect to temperature and concentration allows us to follow the molar fraction gradient as a function of time, until the steady state is attained.1° Near equilibrium, the following linear relation is valid:
where k?/T = thermal diffusion ratio = ( D T / D ) x l x zwith DT/D being the Soret coefficient. Supposing the gradient constant permits us to write eq 1 as kT*
AX1 = --AT
T Equation 2 is used to calculate kT*/T, knowing AT and AX, with AX, = the molar fraction difference of component 1 between top and bottom. The Results. Table I summarizes the obtained values of the Soret coefficient as a function of the molar fraction for the six studied systems. From these values, it is possible to find a relation between kT*/T and the difference of molecular diameters. The thermal diffusion factor kT*/T includes as a factor the product XlX2 (see eq 1). This dependence on the composition is eliminated by multiplying by (XIXz)-l. Furthermore, the remaining expression (kT*/r) [ l/(xlxd] has K-l dimension; to reduce it to a dimensionless quantity, a factor e12/k is introduced, and, in Figure 2, the mean is plotted as a function value ( (kT*/T)(€12/k)[l/(XlXz)]) of (on - ~ ~ ~ ) 2 / ( + a , azz) , = Aa/a12 for the six studied mixtures. ( ) means average on different initial concentration mixtures. a and e are paramaters in intermolecular potential functions and c12 = (e11e22)1/2; u12 = (all + az2)/2. This function can be represented by eq 3. A plot of
(2%) ( ): = 3.727 exp 5.4094-
-1
(3)
(kT*/T)(e12/k) at the same initial concentration for the 0 1980 Amerlcan Chemical Society
The Journal of Physical Chemistry, Vol. 84, No. 25, 1980 3401
Soret Coefficients for Liquified Gas Mixtures
l---fkia
e Figure 1. The cell: (1) lateral boundary, cylinder in stainless steel; (2) filling by 125 thin-walled stainless steel tubes; (3) lower plate: (4) methane thermometer; (5-12) supports of the capacitors; (0-13) plates of the capacitors; (7) quartz spacer: 4 0.05 mm: (8) stainless steel wire 4 0.1 mm acting on spring 10; (9) insulator: (10) stainless steel plates, 0.08 mm thick, used as spring; (11) heating wire.
Figure 8. kTa/ Tvs. X, for the Kr-Ar system: 0experimental value: (-) Enskog theory: (---) Rice-Allnatt theory with ‘g(al)= Og(a2) = 1: (- -) Rice-Allnatt theory, Lebowltz radial correlation functlon.
-
I‘““”
Figure 4. k T * lT vs. Xk for the Ar-CH, system.
Figure 2. The reduced mean value ((kT*/T) [ E ~ ~ / ( ~ X , Xas ~ )a] ) function of the molecular diameter difference for the six studied systems.
different mixtures yields qualitatively to the same variation. Comparison with Theoretical Predictions The comparison has been done with two statistical theories: (i) with the extention by Thorne to dense fluids of the theory of Enskog6 and (ii) with the Rice-Allnatt theory extended to mixtures by Wei and Davis! The Enskog Theory. In the book of Chapman and Cowling,6the thermal diffusion ratio used in the Enskog theory is defined as (4)
with dlzEbeing the diffusive force. The quantity kTE has not the same physical meaning as the thermal diffusion ratio kT* of relation I,because the vector dlZEis still a function of the l;emper,aturegradient. Thus it is necessary, as done by Wei and Davis, to derive dliE, independent of the thermal gradient
so
with kTE = thermal diffusion ratio as expressed in eq 4 and QE = QE(T,ui,ni,pi,mi,Mi) with T the temperature, q the radius, ni = pi/rni the density number, pi the density, mi the mass, Mi = mi/rnowith mo = Cimi of the particles i, and with n = Eini as the density number of the whole
Figure 5. kTg/Tvs..XK, for the Kr-CH, system.
system. QE is a complicated and long expression which can be found in ref 5. This yields kT* k p B (7) T TQn
- - -- +
The Rice-AllnaCt Theory. Here too, we have to define dnRAby
ax1 ar
d12 = -nQM
-
d12’E
kTRA kT* = Q n’ kTM has a definition similar to eq 6 and now QM = QRA(T,bi,ni,pi,lTli~i,~,Og(Ui)) with two new parameters: ti the frictional coefficient and Og(q) the radial distribution function. If it is easy to identify f to BkT/DI2,’the calculation of O.g(ui)is rather more difficult. We first chose to keep it constant and equal to unity; secondly we used a relation calculated by Lebowitzs in resolving the Perkus-Yevick equation. The comparison of our experimental values of the thermal diffusion ratio, defined in eq 1with the values calculated with the Enskog theory and with the Wei-Davis equation, are reported in Figures 3-8. In order to study the relative influence of each of the parameters (q,mi,pi,Ji?g(ui)) on the thermal diffusion ratio, we have
3482
The Journal of Physical Chemistry, Vol. 84, No. 25, 1980
Longree et ai.
TABLE I: Experimental Values of the Soret Coefficient for Different Molar Fractions of the Six Studied Systems Ar-CH,,
Kr-CH,,
Kr-CF,, ( T )= 108 K, A T = 2.42 K A T = 4.86K A T = 2.39 K A T = 2.31 K A T = 4.14 K XAr 1 O 3 k ~ * / T x ~ r 1 O 3 k ~ * / T X K ~ 1 O 3 k ~ * / T XCF, 1 O 3 k ~ * / T X K ~ 1 0 3 k ~ * / T
(TI = 88 K,
0.1248 0.2025 0.2765 0.3771 0.4923
Kr-Ar,
(T)= 107 K,
1.39 0.89 1.13 3.33 2.13
0.1590 0.2244 0.3052 0.4460 0.4989 0.5464
(2’) = 95 K,
1.10 1.88 2.08 1.46 2.21 2.31
0.2158 0.3211
3.58 6.09
1
Cf4.Ar
I
t
CF,-Ar,
CF,-CH,,
(2’) = 82 K,
(T)= 98 K, A T = 3.29 K
0.2438 0.4077
26.1 33.4
0.4660
-0.141
Enskog
,
system.
Figure 7. kT*/ T vs. X,, for the Kr-CF, system. 1(;12.x10-2
2 . 1 10.2
,I
CF,.CN,(
1
\ I
I
A
Figure 8. k,*/Tvs. XW, for the CH4-CF4system: the calculated lnflnite separations encountered for two particular values of the initial concentration should be related to limit of vaildlty of the Enskog theory.’
computed numerical values of kT* J T for numerous different sets of these parameters; they show that it is always the component with the larger molecular mass which migrates toward the cold region; it is also worthy to note that the thermal diffusion ratio is quasi-independent of the frictional coefficient. These last results are reported elsewhere? Thermodynamical Theory of Prigogine et aL7 For molecules of rather equal dimensions, this theory based on an Eyring model yields for the Soret coefficient the following expression:
11.0
38.8
TABLE 11: Comparison of the Agreement of the Calculated Values by Enskog Theory or by Rice-Allnatt Theory with the Experimental Valuesa
Ar-CH, Kr-CH, Kr- Ar Kr-CF, CF, Ar CF, - CH, Figure 0. k T * /T vs. XcF, for the CF,-Ar
XCF, 103k,*/T
0.1600 0.2096
-
-++ + --
RiceRiceAllnatt Allnatt gQ= 1 gap Lebowitz
+ ++ ++ -
-
-_ --
++
-
--
a + + means agreement better than 30%; + means agreement between 30 and 65%;- means agreement between 65 and 100%;- - means bad agreement, more than 100% discrepancy.
with E: = the latent heat of vaporization of i and x = a numerical factor. When the sizes of the molecules are different, the direction of the concentration gradient can be predicted by the cohesion density (heat of vaporization per unit volume). It is the component with the largest cohesion density which migrates toward the cold plate. This criterion agrees with the measured values except for the Ar-CH4 mixture, for which this theory predicts that Ar migrates toward the hot plate in contradiction to the experiments. What is puzzling is on one hand that the thermal diffusion ratio is quasi-independent of the fractional coefficient? which contains the whole energetic contribution in the Wei-Davis theory, and on the other hand that thermodynamic theories, like Prigogine’s, are built on energetic arguments only.
Conclusions Our measurements permit a comparison with theoretically predicted values of the thermal diffusion ratio, for simple systems, corresponding rather well with the assumptions made in the statistical theories. It appears that the statistical theories always predict the good direction of separation and that it is the Rice-Allnatt theory with Og(a,) = 1which gives the numerical values with the best order of magnitude, as can be seen from Table 11. Acknowledgment. We thank Professors G. Nicolis and J. K. Platten for their stimulating discussions during the course of this work and Professor H. J. V. Tyrrell for helpful comments. This research was partially supported by the Fonds National de la Recherche Scientifique, Belgique. One of us (D. LongrBe) is indebted to I.R.S.I.A. (Institut pour l’Encouragement de la Recherche Scientifique dam l’hdustrie et I’Agricdture,Belgique) for a grant.
References and Notes (1) H. J. V. Tyrrell, “Dlffuslon and Heat Flow In Llqulds”, Butterworth, London, 1981. (2) H. Korschlng, Z . Naturforsch. A , 10, 242 (1955). (8) G. Thomaes, J. Chem. Phys., 58, 407 (1958); J. C. Legros, D. Rasse, and 0. Thomaes, Physlca, 57, 585 (1972); P. Pow, J. C. Legros, and Q. Thomaes, Z. Naturforsch. A , 29, 1915 (1974).
3483
J. Phys. Chem. 1980, 84, 3483-3485
(4) L. Guczi and H. J. V. Tyrrell, J. Chem. Soc., 6576 (1965). (5) S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases", Cambridge University Press, New York, 1960. (6) C. C. Wei and H. T. Davis, J. Chem. Phys., 45, 2533 (1966). (7) I. Prigogine, L. De Brouckere, and I?. Amand, Physica, 16, 577 (1950).
(8) J. L. Lebowitz, Phys. Rev. A , 133, 895 (1964). (9) D. Longree, Ph.D. Thesis, University of Brussels (U.L.B.), Brussels, Belglum, 1979; D. Longree, J. C. Legros, and G. Thomaes, Bull. Acad. R. Be&., in press. (10) D. Longfee, J. C. Legros, and G. Thomaes, Z.Naturforsch. A, 32, 1061 (1977).
Reanalysis of the ESR Spectrum of the Triethylarsine Dimer Radical Cation (Et3As'AsEt3+) Reggle L. Hudson* Department oi Chemistry, Eckerd College, St. Petersburg, Florkfa 33733
and Ffrancon Wllllams" Department of Chemistty, Universky of Tennessee, Knoxvlle, Tennessee 379 16 (Received: August 1, 1980)
The dimer radical cation As2Et6+has been generated by y irradiation of triethylamine, and its ESR spectrum interpreted as the full 16-line 75Aspattern expected when higher-order effects are important. The ESR parameters (All = 546 G, A, = 413 G, gl = 2.004, g, = 2.027) result in spin densities which establish that the unpaired electron is largely localized Letween the two arsenic atoms in As2Etg+.
Introduction Some time ago, the anisotropic ESR spectra of the dimer radical cations P2(OMe1)6+and A S ~ ( E ~(Me ) ~ += methyl and Et = ethyl) were presented and analyzed by first-order ESR theory to give the hyperfine parameters of their 31P and 75Asnuclei, respe~ctively.'-~However, a later study4 demonstrated the necessity and benefits of carrying out a higher-order analysis of the P2(OM&+ spectrum and suggested that a similar analysis for the &,(Et),+ spectrum was needed. It was noted in particular that large second-order ~plittings,~ ithe largest on the order of 200 G, should give rise to marked deviations from a first-order seven-line hyperfine splitting pattern for A S ~ ( E ~ )L9' ~ince +. arsenic dimer radical cations remain virtually unknown, we report here a higher-order analysis of the ESR spectrum of A S ~ ( E ~along ) ~ + with new experimental results. This information gives new rnagnetic resonance parameters for this radical and shows that the previous ESR parameters for A S ~ ( E ~are ) ~in + considerable error. Experimental Section Triethylarsine was purchased from Strem Chemicals Inc. (Newburyport, MA) and used without purification. Samples were prepared in Suprasil ESR tubes according to standard high-vacuum techniques. Samples which were cooled quickly in liquid nitrogen gave glmses, while careful warming of such glasses gave polycrystalline solids which were subsequently cooled to 77 K. The equipment used for y irradiation, photobleaching, and ESR measurements has been described previouslya6 Results Figure 1 shows the ESR spectrum of glassy AsEt3 at 85 K after y irradiation of the sample at 77 IC. While for the most part this spectrum strongly resembles that shown in it also Figure 3 of a previous study by Lyons and Sy~nons,~ reveals the presence of two additional wing features which are now identified as the outermost lineo of the parallel components for A S ~ ( E ~ )Stick ~ + . diagrams in our Figure 0022-36541aoi2oa4-34a3~01.OOIO
TABLE I : ESR Spectroscopic Transitions for As,(Et)l
-
1 line
(3, 3) (3, 2) (2, 2) (3,1)
(1,1) (3,O) (290) (1,O)
(090)
(39-1) (%--I)
(1,-1)
(3, - 2) (2, - 2) (3, -3)
exptl position: G 1876 2100 23 20 2403 2629 (-2756C d d d d 3443 3588 3743 3965 4353
calcd position,b G
difference, G
1876 2100 2319 2386 261 5 2764 2746 2984 31 36 3210 3193 3434 3585 3738 3963 4353
0 0 1 17 14
9 3 5 2 0
a Taken from the ESR spectrum of a polycrystalline sample (Figure 2). The uncertainty in the line position is t 4 G. Calculated line position using A 1 1 = 559 G, .A1 = 409 G, gii = 1.994, and g1= 2.029. The microwave frequency was 9116.5 MHz. Overlap too great t o resolve both lines. d Lines masked by other signals in the center of the spectrurn.
-
M show how the spectirum may be analyzed in terms of two overlapping 16-line patterns, one for parallel and the other for perpendicular hyperfine features. The positions of the lines in the stick diagrams were given by a matrix diagonalization calculations using All = 546 G, A, = 413 G, gil = 2.004, and g, = 2.027. While 7-line patterns would be expected for As2(Et)B+from first-order ESR theory, the 16-line patterns of Figure 1 are exactly what is predicted for hyperfine interaction involving two 75As( I = 3/2) nuclei with higher-order effects being important. These patterns are thus assigned to h2(Et)6+. Our analysis is corroborated by the ESR spectrum of the dimer radical cation in a sample of y-irradiated poly@ 1980 American Chemical Society
