J. Phys. Chem. 1984, 88, 3359-3363 TABLE VII:
s, and c,at 25 OC solute
s,/J K-'mol
argon
-25
krypton methane
-8.4 -2 1 -13 +34
ethane Cr(acac),
c,/JK-'mol 63
50 130
92 -110
of the solute from an ideal gas to solid. Calculated solubility is about three times larger than the observed one at 0 O C and about one fifth at 70 O C . But differences in the free energy corresponding to the differences in the observed and calculated values are only about 10 and 15% for 0 and 70 OC, respectively. Therefore, it can be said that the major part of the aqueous solubility is esimated by eq 8 for this temperature range. In the present calculation, al of water is assumed to be constant throughout the temperature range. If one changes al from 2.77 to 2.74 A, the solubility calculated at 70 O C agrees with the observed value, and to 2.785 8,the calculated one fits to observed one at 0 "C. The values of al may become smaller with increasing though the change is not significant in the present temperature range.13-38 Therefore, it is not appropriate to adjust the temperature dependence of the solubility only by changing a,. AB2(solid) from eq 12, given in Table 11, changes from endothermic to exothermic at 55 "C, which corresponds to a solubility minimum at 55 OC, while the solubility calculated by eq 8 lacks a minimum, which may be due to entropic factors ignored in the interaction terms. From Figure 2, it can be seen that Gdisp increases solubility linearly with decreasing temperature, while AGSubincreases it with increasing temperature. But the contribution of G,, exhibiting a small concave upward tendency, is not sensitive to the temperature in aqueous solution. This is different from the trend observed in organic solvents, where G, increases the solubilities monotonically with temperature and the-effects of Gdisp are overcompensated by the superposition of G, and Gsub. Then, the solubilities in organic solvents increase monotonically with temperature as expected from regular solution theory. These considerations on the temperature dependences of the solubility confirm that the specificity of aqueous solubility of the nonpolar (35) (36) (37) (38)
Ben-Naim, A.; Friedman, H. L. J . Phys. Chem. 1967, 71, 448. Mayer, S. W. J. Chem. Phys. 1963, 38, 1803. Tiepel, E. W.; Guggins, K. E. J . Phys. Chem. 1972, 76, 3044. Pierotti, R. A. J . Phys. Chem. 1967, 71, 2366.
3359
solutes depends almost entirely on the uniqueness of G, of water compared to G,of organic solvents. Moreover, it can be seen that the very low solubility of the nonpolar solute in water is due to the large positive value of G, in water compared to G, in CC14. In spite of the simple interaction model, the present calculation is rather successful in this temperature range for estimating thermodynamic parameters of aqueous solubility by using fixed parameters for a', a2, and t z / k . Hydrophobicity. Entropies of solution and heat capacities of solution (At? ) have been estimated, neglecting interaction terms and byq7 hs2 = - R In R T / v l a,RT (18)
(as2)
(si ci)
s, + si
ACp = C, + t?i
-R
+
+ 2c@T + RT2(dap/dT)p
(19)
When the same calculations are done for C r ( a ~ a c ) there ~ , are differences between observed and calculated values. The differences may be assumed to correspond to and Ci and are given for C r ( a ~ a c is ) ~positive and almost constant in Table VII. from 5 to 65 OC, while those for gas solutes are small and negative. The positive values for C r ( a ~ a c )mean ~ that, by introducing Cr(acac), in the cavity, it loosens the water structure around the cavity, though the degree of structure loosening seems to be constant for the temperature range. A similar statement can be Therefore if cavity formation terms made in consideration of could be estimated exactly, the parameters or may be useful as a measure of hydrophobicity by choosing hard-sphere solutes of the same sizes as reference solutes. A dipolar interaction of the water molecule with a central metal ion of a chelate complex was proposed for the explanation of variation of distribution coefficients of tris(acety1acetonato)s on solvent e x t r a ~ t i o n . ~A~ hydrogen-bonding interaction is also known with protic solvents.4s43 Such an interaction causes the positive values of by slightly modifying the solvation cage of the hard-sphere solute, which results in weakening the hydrophobicity of a polyfunctional solute of Cr(acac),.
si
si
ci.
si ci
si
Acknowledgment. The author is grateful to Prof. Y .Yamamoto for his suggestions and valuable discussions throughout the work. H e also expresses thanks to Mr. Tamai for generous assistance with the experiments. (39) Hopkins, P. D.; Douglas, B. E. Inorg. Chem. 1964, 3, 357. (40) Davis, T. S.; Fackler, Jr., J. P. Inorg. Chem. 1966, 5, 242. (41) Frankel, L. S.; Langford, C. H.; Stengle, T. R. J . Phys. Chem. 1976, 74, 1376. (42) Chan, S. 0.; Eaton, D. R. Can. J . Chem. 1976, 54, 1332. (43) Vigee, G. S.; Watkins, C. L. Inorg. Chem. 1977, 16, 709.
Evaluation of Velocity Correlation Coefficients from Experimental Transport Data in Electrolytic Systems Hansjurgen Schonert Institut fur Physikalische Chemie, RWTH Aachen, 51 Aachen, West Germany (Received: October 26, 1983;
In Final Form: January 31, 1984) For isothermal-isobaric binary electrolyte solutions and salt melts with two and three ionic constituents the relationships between the velocity correlation coefficients of the linear response theory and the experimental transport coefficients (electrical conductivity, transference numbers, inter- and intradiffusion coefficients) are derived. Introduction The theory of thermodynamics of irreversible processes provides for a rigorous framework to describe the transport processes in multicomponent systems on a phenomenological basis.' This (1) R. Haase, "Thermodynamics of Irreversible Wesley, London, 1969.
Processes", Addison-
0022-3654/84/2088-3359$01.50/0
theory has been successfully applied to the study of transport phenomena in electrolyte solutions' and in salt melts.24 However, (2) R. W. Laity, J . Chem. Phys., 30, 682 (1959). (3) H. Schonert and C. Sinistri, Ber. Bunsenges. Phys. Chem., 66, 413 ( 1962). (4) J. Richter, Z . Nuturfsch. A , 25, 373 (1970).
0 1984 American Chemical Society
3360 The Journal of Physical Chemistry, Vol. 88, No. 15, 1984 the interpretations of the transport coefficients of this theory in terms of a microscopic model are thus far limited to a few cases: for example, several a ~ t h o r s have ~ - ~ established the relationships between the Debye-Huckel-Onsager theory and the phenomenological equations of irreversible thermodynamics. Douglasdo3'' and H e r d 2 have pioneered in showing a new way of connecting the two types of descriptions: they, and other authors, have established the relationships between the velocity correlation functions of the linear response theory13 and the transport coefficients of the irreversible thermodynamics for binary solution~.'~,"'~ But there are some conflicting results in these papers: the diffusion coefficient for an electrolyte component in a binary electrolyte solution in terms of velocity correlation coefficients is different in the derivations of Her@ and Miller.lg Also, the question of how to incorporate the intra- (tracer) diffusion of the solvent into the solvent-fixed phenomenological transport equations seems not to be satisfactorily s01ved.I~ Therefore, it seems necessary to reconsider these relationships. To this end, a recently generalized derivation will be used.20 Also, the increasing number of data for salt melts and the growth of computer simulation experiments for these systems make it desirable to apply these considerations to these systems.
General Outline Before discussing in detail binary electrolyte solutions and salt melts with two and three ionic constituents, we want to give a description of the procedure to be followed and some of the reasoning behind it, because these are scattered in different papers.20-22 In this connection, two points need to be discussed: firstly, the reference frames for the fluxes in the linear response theory and in the measurement of transport coefficients are different; secondly, the concept of intra- (tracer) diffusion has to be incorporated into the irreversible thermodynamics in such a way that it conforms to thermodynamic principles, to the experimental observations, and to the ideas which lead to the Kubo relation in the linear response theory. Only if both of these points are carefully observed can one arrive at self-consistent results for the evaluation of velocity correlation coefficients from experimental transport data. Regarding the first point, the formulation of the phenomenological transport equations in irreversible thermodynamics is based on the terms in the entropy production, Tu = xoJ,Xl,where Ji are the fluxes and X, (i = 0, 1,2, ...,N ) are the generalized forces. In the derivation of this function for a continuous system one uses, beside others, the balance equations for energy and momentum. Because these are formulated in the barycentric reference frame, the fluxes are referred to this frame. Yet, in experiments the boundaries of the system studied exchange momentum with the solution and this depends on the design of the experimental setup. Therefore, the barycentric reference frame has to be replaced by another reference frame which takes account of the experimental boundary conditions and, by the same token, eliminates one re-
Schonert dundant flux and force. This is most conveniently done by choosing one constituent, the solvent, as reference substance, if one studies the transference numbers and conductivities of ionic constituents. In the measurement of interdiffusion the usual setup leads to the volume velocity as reference velocity. However, it is easy to transform the measured diffusion coefficient to the same solvent-fixed reference frame, so that a single common reference frame is used. In this way, in a system with N 1 constituents, one has N independent fluxes and forces and a N X N symmetric matrix of phenomenological coefficients Lik. On the other hand, in the linear response theory the system studied is in equilibrium: the momentum exchange with the boundaries is zero and hence the barycentric reference frame is the natural choice. Therefore, it is convenient to use a dependent set of N + 1 fluxes and forces with a ( N + 1) X ( N + 1) symmetric matrix i?,k for the linear flux-force relations. We now consider the transformations between these reference frame^.^^,^^ The fluxes ( J J 0of the solute constituents i = 1, 2, ...,N are referred to the velocity of the solvent i = 0. In electrolyte solutions the constituent i = 0 is the solvent proper, whereas in salt melts the solvent may be any one of the ionic constituent^.^,^ The phenomenological transport equations of irreversible thermodynamics are
+
N
(Ji)o=
k-1
Li&k
i = 1, 2,
..., N
or in matrix notation Jo = LX
(3)
where Jo = ((JI)o,(J~)o,...,(JN)o), x = ~ ~ J Z V - . Aand N ]L, = (Lik). In these notations, & and zk are the chemical potential and the signed charge number of constituents k, F is the Faraday constant, and 9 is the electrical potential. The flow is linear along the space coordinate x . The generalized conductivity coefficients Lik obey the Onsager reciprocity relations. These coefficients can be related to the conventionallyused coefficients: the specific electrical conductivity K , the solvent-fixed transference numbers ti, and the volume-fixed interdiffusion coefficients Dik.24-27 The equations are transformed into the barycentric reference frame. The result is19*20,23
nil = Qli =
N
N
i, 1 = 1, 2,
&&/mLk,,, N
Qoi
=
Qio
..., N
(4)
k-1 m = l
-E MiQil/M0 I= 1
=
N QOO
=
i = 1, 2,
...,N
(5)
N
1=1 k = l
(6)
MkMls21k/MOM0
Here, Mi is the molar mass of constituent i and (5) R. Haase, Z . Naturforsch. A , 29, 534 (1974). (6) J. Richter and U. Priiser, Ber. Bunsenges.Phys. Chem., 81,508 (1977). (7) J. Rastas, Acta Polytech. Scand., Chem. Incl. Metall. Ser., 50 (1966). ( 8 ) M. J. Pikal, J . Phys. Chem., 75, 3124 (1971). (9) R. Paterson, Faraday Discuss. Chem. Soc., 64, 304 (1978). (10) D. W. McCall and D. C. Douglass, J . Phys. Chem., 71,987 (1967). (1 1) D. C. Douglass and H. L. Frisch, J . Phys. Chem., 73, 3039 (1969). (12) H. G. Hertz, Ber. Bunsenges. Phys. Chem., 81, 656 (1977). (13) W. A. Steele in "Transport Phenomena in Fluids", H. J. M. Hanley, Ed., Marcel Dekker, New York, 1969, p 209. (14) R. Mills and H. G. Hertz, J . Phys. Chem., 84, 220 (1980). (15) H. L. Friedman and R. Mills, J. Solution Chem., 10, 395 (1981). (16) H. G. Hertz, K. R. Harris, R. Mills, and L. A. Woolf, Ber. Bumenges. Phys. Chem., 81, 664 (1977). (17) L. A. Woolf, J . Phys. Chem., 82, 959 (1978). (18) L. A. Woolf and K. R. Harris, J . Chem. Soc., Faraday Trans. 1,74, 933 (1978). (19) D. G. Miller, J . Phys. Chem., 85, 1137 (1981). (20) H. Schonert, Ber. Bunsenges. Phys. Chem., 87, 23 (1983). (21) H. Schonert, Ber. Bunsenges. Phys. Chem., 82, 726 (1978). (22) H. Schonert, Z . Phys. Chem. (Frankfurt am Main), 119,165 (1978).
(1)
Pik
N
Plk
=
6ik
- ciMk/(c
CjMj)
(7)
j=o
with ci the molarity of constituent i and 6jk the Kronecker symbol. This shows that the barycentric-fixed coefficients o i k can be evaluated if the set of conventional transport coefficients K , ti,,and Dik is known. Turning to the second point, we notice that in the linear response theory one discusses the so-called velocity correlation coefficient between different particles a and P of constituents i and k (23) J. G. Kirkwood, R. L. Baldwin, P. J. Dunlop, L. J. Gosting, and G. Kegeles, J. Chem. Phys., 33, 1505 (1960). (24) D. G. Miller, J . Phys. Chem., 64, 1598 (1960). (25) D. G. Miller, J. Phys. Chem., 70, 2639 (1966); 71, 616 (1967). (26) R. Haase and J. Richter, Z . Naturforsch. A , 22, 1761 (1967). (27) H. Schonert, Z . Phys. Chem. (Frankfurt am Main), 54,245 (1967). ( 2 8 ) H. Schonert, Electrochim. Acta, 27, 1043 (1982).
The Journal of Physical Chemistry, Vol. 88, No. 15, 1984 3361
Evaluation of Velocity Correlation Coefficients
L m ( u l a ( o )U k p ( f ) ) dt
(Ji90+ (J!)o Ji* = (J!)o - C!(J~)O/(CY+)!c (Jib =
i, k = 0, 1 , ..., N
and the so-called autocorrelation coefficient of particle a
(14)
Likewise, there is L m ( u l a ( 0u,,(t)) ) dt
i = 0, 1 ,
..., N
xi = (C,"Xi" + C,19X!,/(Ci" + c)!
For the evaluation of this integral, we should be able to follow the particle CY in space and time. This can only be done if it is labeled. Therefore, if we want to correlate this integral with a corresponding coefficient (Le., intradiffusion coefficient Dl*) in the transport equations of irreversible thermodynamics, we have to introduce the concept of labeled fluxes J," and JF into the irreversible thermodynamics, together with the concept of forces X l a and X!
a
Xlm = --(wlm ax
+ z,Fp)
m : a , ,8
of labeled constituents. If we restrict our considerations to systems in which each of the constituents i = 0, 1 , ...,N is composed of just two isotopes, the vector Jd of the fluxes is nowZ9
Jo' = i(Jo) o", (Jo)o';(
Ji )o",(Ji
)op; (JN)o", ( J N ) ~ ' ) 9s.;
(8)
and correspondingly for the generalized forces
X' = (xo"&)';x,"J~';.. .;x,a#Yh.Pj
(9)
Here, the reference frame, denoted by the subscript 0, is the total solvent, i.e., the sum of both isotopes LY and ,8. The phenomenological equations are hence J,' = L'X
+
-
3 = (Jo;Jo*,J]*,...,JN*)
x' 2 = (x$O*,xl*,...,xN*)
(11)
and lead to the result (in matrix form)
J =
A
These equations directly yield the following: in a conventional transport experiment the isotope ratio (cY/c!) remains constant in space and time. Hence, the tracer force Xi* equals zero and so eq 12 reduces to eq 3. On the other hand, in a tracer experiment the system is homogeneous throughout with respect to the constituents: thus (d/dx)(c: + )c! = 0. Therefore, the constituent force Xi vanishes, and by means of eq 12 the flux ( J J 0is zero too. Hence, we get from eq 12-15 i = 0, 1 , ..., N
J! = -Di*(a@/ax)
a1
(16)
which is Fick's law for intradiffusion in a homogeneous system. In all other experimental situations, for example intradiffusion in a nonhomogeneous solution, we have a superposition according to the foregoing relations. We have clarified the two points: the proper introduction of the transport of labeled constituents into the phenomenological equations of irreversible thermodynamics and the transformation of the matrix L from the experimental reference frame to the barycentric frame. There remain only a few further stepsZoto derive the desired results
+
are20-22,28 -+
(15)
(10)
where L' is a ( 2 N 1 ) X ( 2 N 1 ) matrix. However, the coefficients of this matrix, which obey the Onsager reciprocity relations, are interrelated by the fact that the isotopes a and ,8 of constituent i show "equal behavior", if one neglects the effects due to the small mass difference between them. This can be used to transform eq 10 into a form which shows convincingly the relationship with the conventional transport equations 1 , 3 and with Fick's law for intradiffusion. These transformations J,'
a
Xi* = XY - X! = -RT- In (cY/c!) ax
.,
O ) ?
(12)
ON
Thus, for constituents with only two isotopes the conductivity matrix decomposes into the matrix L , known from eq 3 and determinable from conventional transport measurements, and into N + 1 diagonal elements a, ( i = 0, 1 , ...,N), which are given by the intradiffusion coefficients D,*:
ui = D,*clac,~/[RT(cIa+ c?)]
(13)
In these equations, the solvent-fixed flux ( J J 0 ,which is the same as in eq 1 , and the newly introduced tracer flux J,*, which is independent of the solvent-fixed and volume-fixed reference frames, are given by (29) In the linear response theory we need to differentiate only between two particles CY and 0. Therefore, here two isotopes are sufficient. Systems, in which some constituents are composed of more than two isotopes can be dealt with in a slightly more complex notation.1°
0, 1 , ..., N (17)
and to regain the Kubo relation y 3 L m ( u i a ( 0 ui,(t)) ) dt = Di*
i = 0, 1 , ..., N
(18)
In these equations, No is the Avogadro number, Vis the volume of the ensemble, and ( ) denotes the ensemble average over the constituents i and k . Note that the Kubo relations for the constituents i = 0, 1 , ..., N are a product of our general procedure; 2o they need not be assumed to be valid a priori. Equations 17, 18 together with eq 4-7 and the appropriate equations for relating Ljk to conventional quantities below (eq 19 for electrolyte solutions and eq 28 and 36 for molten salts) now give the relationships between the conventional transport coefficients K , ti, Dik and Di* and the velocity correlation coefficients. This straightforward procedure, which has been shown to be valid for any liquid system13sZ0and which is free of ambiguities regarding the reference system, will be applied in the following sections. Binary Electrolyte Solution The constituents solvent, cations, and anions are denoted by the subscripts 0, 1, and 2. The relations between the phenomenological coefficients Lik and the conventional coefficients are given by19
Ljk = LViVk
+ tjtkK/(zjzkp)
i, k = 1 , 2
(19)
with the abbreviation L = c D V / { R T v [ l+ m(a In r / d m ) ] )= cZ)/RT
(20)
The stoichiometric numbers are denoted by vi (with v = v I + v2) and the signed charge numbers by zi for i = 1 , 2. The molarity and the molality of the electrolyte are c and m; y is the molality-based activity coefficient of the electrolyte. If we introduce the partial density by Pi
and the density by
=
CiMi
(21)
3362 The Journal of Physical Chemistry, Vol. 88, No. 15, 1984
Schonert
+ vl(1)’l
(A) s ~o(0)’O
The phenomenological equations (1) reduce to
( J d o = L I J l = -LllzlF(dcp/dx) (26) The phenomenological coefficient L , can be expressed in terms of the specific electrical conductivity K , which is defined by Ohm’s law Z = zlF(Jl)o = -~(dcp/dx) (27) for the electrical current density I . Hence =
K
LllZlZlp
(28)
Applying the transformation relations 4-7, we find =
Qll
=
001
K ( ~ o ~ o ) 2 / ( ~ l ~ M > 2
= -KV@JOMOMl/(Z1Fkf)2
= K(voM1)2/(z1FM)2
Qoo
where
+
M = vOMO
is the molar mass of the salt. Therefore, we have from eq 17
Therefore, it is possible to calculate all velocity correlation coefficients in this system, if the specific conductivity and the tracer diffusion coefficients are known. Of course, this applies also in the reverse direction: if we have from computer simulation experiments the velocity correlation coefficients, we can get the conductivity as well as the values of the tracer diffusion coefficients. Salt Melts with Three Ionic Constituents Two salts A and B dissociate into three ionic constituents 0, 1, and 2, where 0 is the common ion:
(A)
\
F!
V o ~ ( 0 ) ‘+ ~ Vi~(1)’~
(B) @
+ vzB(2)z2
VOB(O)‘O
The molarities obey the relations Ci
Cz
the equivalent conductivity A (eq 25). This procedure is the same A=
K/(VIZiC)
Salt Melts with Two Ionic Constituents The salt A dissociates into vo ions (0) and v l ions (1) according to the reaction
=
VZBCB
(33) co = VOACA + VOBCB Likewise, for the molar masses, the chemical potentials pi,and the charge numbers zi, there are vOAMO
+ VIAMl
MB =
VOBMO
+ vZBMZ
=
vOAPO
+
MA
(25)
as Mi1ler’s,lg except that the solvent isotope properties are now correctly incorporated. These relations, together with the Kubo equations (18), constitute a complete set to evaluate all velocity correlation coefficients from conventional transport coefficients. They agree with the results of Millerlg (except for typographical errors in his eq A l , A2, and A6). They are different from those of Hertz,IZ who uses another concept of the velocity of a salt molecule.
V~ACA
MA
=
PB = vOBPO
VIAhl
+ v2BP2
0=
VOAZO
+ VlAZl
0=
VOBZO
+ v2BZZ
(34)
If we choose the velocity of the common ion as reference velocity, we get from eq 1 three phenomenological coefficients, which are connected with the conventional transport coefficients as follows: 2-4
K/F
E
ZlZlL11
+ 2 Z i Z 2 L i z + ZZz2L22
The Journal of Physical Chemistry, Vol. 88, No. 15, 1984 3363
Evaluation of Velocity Correlation Coefficients
(35)
Although the specific electrical conductivity is independent of the reference frame, the ti are the transference numbers measured with respect to the movement of the common ion. D, is the interdiffusion coefficient which describes the diffusion of the two electrically neutral components A and B in the volume-fixed reference frame. The partial molar volume of component A is VA. These equations can be solved for the Lik: Lll
= .72ZZDO + t l t l K / ( z l z l ~ ! )
L I Z= -Z,Z2Do
+ fltZK/(Z1ZZp)
L22 = ZlZlDO
+ t2tZK/(ZzZ2m
(36)
These equations are quite similar to those for the binary electrolyte solutions; see eq 19. Next, we use the relations 4-7 for the transformation of the Lik into the barycentric Qik. After some lengthy calculations we find
all = z2*z2*Do+ t l * t l * K / ( z l z I p ) Ql2
= -Zl*Z2*Do
tl*tz*K/(ZlZ2P)
+
= zl*zl*Do t 2 * t 2 * ~ / ( z 2 z 2 P )
(37)
where we have introduced the following reduced variables in order to point out the analogy between the last two sets of equations: 21
*
= MAcOzO/ [VlA(CAMA+ cBMB)l
z2* = MBcOzO/ [V2B(cAMA+ CBMB)]
Furthermore, there are
This then gives a direct route for the calculation of the Qik from measured transport data, and hence by eq 17 the possibility of evaluating all velocity correlation coefficients. The inversion of these relations, Le., the calculation of the transport coefficients from velocity correlation coefficients, is most easily done with transformation equations given elsewhereSz0 Discussion Difficulties in some earlier papers in establishing the relationships between conventional transport coefficients and the velocity correlation coefficients stem from several sources. As noted earlier, the reference frames for the fluxes are different in the linear response theory and in the phenomenological equations of irreversible thermodynamics. In the linear response theory, the system studied is in equilibrium, and hence the barycentric reference frame is the natural choice. In irreversible thermodynamics, the solvent-fixed reference frame is natural and has the necessary and sufficient number of experimental coefficients. However, experiments may use other convenient reference frames. Therefore, it is essential to properly transform all quantities to the appropriate reference frames. Secondly, we are faced with the problem of how to introduce the concept of intra- (tracer) diffusion self-consistently into the two theories. It has been shown that this can be done unequivocally and in agreement with the experimental Fick's law for tracer diffusion by introducing the concept of fluxes and forces of labeled constituents into the irreversible thermodynamics.z0 Moreover, it can be shown that the Kubo relation is a consequence of a proper introduction of tracer fluxes into irreversible thermodynamics, not an a priori assumption. This then leads to the straightforward procedure outlined in the second section and the applications here represented. Finally, it is easy to show that the expression for the velocity correlation coefficients of a binary electrolyte solution is in agreement with the result of Miller and W o ~ l rather f ~ ~ than that of Hertz.lz The deviation is due to a different concept of the velocity of a salt molecule. For nonelectrolytes, we have previously verified that the result for the velocity correlation coefficients of Mills and Hertz14 is correct. Acknowledgment. The work was supported by the Deutsche Forschungsgemeinschaft via Sonderforschungsbereich 160. Thank Dr. D. G. Miller for helpful discussions and suggestions.
