NOTES
May, 1959
743
TABLE I to be -0.05, and this compares favorably with the RELATIVE A N D ABSOLUTE VISCOSITIESOF AQUEOUS SODIUM value of - 0.056 reported in this study. By virtue of its minimum solvation, the hydrated perchlorate PERCHLORATE SOLUTIONS AT 25' C, moles/l.
dno 1.0002 I . 0005 1,0009 1.0012 1,0021 1,0020 1.0038 1.0101 1,049 1.150
0.0008987 .004409 .01000
,01600 03600 ,06054 .loo0 .3048 1.0008 1.0975
poise
0,008905 .008907 .008911 .008914 .008922 ,008920
.008937 ,009003 ,009339 .01024
been analyzed using the Jones-Dole7 equation ?/VO =
1
+ A d / C + BC
*
(2)
ion is sufficiently small to loosen or disrupt locally the pseudo-tetrahedral structure of the water in the co-sphere about the ion and hence decrease the viscosity of the solvent about this ion. Such behavior is commonly observed only in water, and only for the few ionic species whose effective hydrated radii are minimal. Acknowledgment.-The assistance of Mr. H. D. Russell in measuring some of the densities of the sodium perchlorate solutions is acknowledged.
THE VIRIAL TREATMENT OF THE INTERhCTION O F GAS MOLECULES WITH SOLID SURFACES'
where r]/qo is the viscosity of the salt solution relaBY ROBERT S. HANSEN tive to that of the solvent, water, C is the molar Instilute for Alomic Research and Department of Chemistry, Tozaa State concentration, and A a.nd B are constants characC o l l e g e , Ames, Iowa teristic of the electrolyte. The A-coefficient repreReceiiwd Julu S I , 1068 sents the contribution from interionic electrostatic The virial treatment of gas-solid interactions forces and was first derived by Falkenhagen and Vernon.8 The B-coefficient appears to represent developed by Halsey and co-workers2-5 was exto interaction potentials based on the Lenthe contribution of the co-spheres of the i o i ~ s , ~ Jtended ~ although no satisfactory theoretical treatment has nardJones 6-12 potential for interniolecular atyet been given. This constant is a specific and traction, which is known to give a better representaapproximately additive property of the ions of a t>ionof gas second virial coefficients than does the strong electrolyte a t a given temperature.l' Re- rigid sphere model used by Halsey and co-work' this problem also has been arranging equation 2 and plotting (q/v0 - l)/dc e r ~ . ~ ,Independently, attacked by DeMarcus, Hopper and Allen8 and by us. the A-coefficient is the ordinate intercept, Freeman.9 This paper will therefore be limited to and the B-coefficient is given by the slope of the re- conclusions other than those reached by these sulting straight line. The experimental value for workers. the A-coefficient of G.8 X compares well with The second virial coefficient for gas-solid iiiterthe theoretical value of 6.4 X low3. The experi- action, including correction for quantum effect7** mental value for the B-coefficient is observed to be is given by +0.03 for sodium perchlorate at 25". Taking the value for the B-coefficient for the sodium ion to be +O.O86, 9,10 we calculate -0.056 as the magnitude of the B-coefficient for the perchlorate ion at this temperature. Recently, Gurney (ref. 9) has discussed the rela- where t(w) is the potential energy of a gas molecule tion between the viscosity B-coefficient for individ- at a distance n: from the solid, taken as a semi-infiual ions and the partial molar ionic entropy. This slab. The molecular 6-12 pot,entinl has been relation has been the basis for his selection of -5.5 taken in the form suggest,ed by Hirschfelder, Cure.u. as the absolute partial molar entropy of the hy- tiss and Bird' to fncilit,nte use of their tabular padrogen ion a t 250.12 In a separate study to be de- ramet,cr data scribed in another paper,13 it will be denionshrated P ( T ) = 4t" that a self-consistent set of radii for hydrated ions can correlate all the features of the viscosity Bcoefficient/ionic entropy relations. I n addition, if where p ( r ) is the interaction potential energy of two allowance is made for the configurational contribu(1) Work IVRS performed in the Ames Laboratory of the Atomic tions to the ionic entropy (e.g., rotational entropy), Energy Commission. ( 2 ) R'. A . Steelc and G . D. FIalsey, Jr., J . Chem. P h p . , 22, 979 the absolute partial molar ionic entropy can be (1954). shown to be a single linear function of the viscosity (3) W. A. Steele and G . D. Halsey, J r . , THISJOURNAL,59, 57 ionic B-coefficient. Using these relations, the B- (1955). ( 4 ) M . P. Freeman and G . D. Halsey, Jr., ibid., 69, 181 (1955). coefficient for the perchlorate ion has been estimated
z/c,
[(,)',
(7) G. Jones and M. Dole, J . A m . Chem. SOC.,6 1 , 2950 (1929). ( 8 ) H.Falkenhagen and E. L. Vernon, Physilc. Z . , 53, 140 (1932). (9) R. W. Gurney, "Ionic Process in Solution," McGraw-Hill Book Co., Inc., New York, N. Y.,1953, p. 160 ff. (10) M.Kaminsky, 2. Naturforsch., Ma, 424 (1957). (11) W. M. Cox and J. H. Wolfenden, Proc. Rov. SOC.( L o n d o n ) , A146, 475 (1934). (12) K. H. Laidler, Can. J . Chem., 54, 1107 (1956). (13) E.R. Nightingale, Jr., t o be published.
(97
(5) G. Constaharis and G. U. Halsey, Jr., J . Cliem. P h y s . , 2'7, 1433 (1957). ( 6 ) R. €1. Fowler and E. A. Guggenheim, "Statistical Thermodynatnics," T h e University Press, Cambridge, 19.19. (7) J. 0.Hirschfclder, C. F. Crtrtiss and R . B. Bird, "Molecular Theory of Gases and Liquids," John Wiley and Sons, Inc., New York, N . Y.. 1954. (8) W.C. DehIarcus, E. H. Hopper and A. M. Allen, A.E.C. Bulletin Ii1222 (1955). (9) B4. P. Freeman, THIs JOURNAL, 62, 723 (1958 I
NOTES
744
Vol. 63
& (DeMarcus, et al.)
=
e*
- (Freeman) = kT p
zo (DeMarcus, et al.) = SO(Freeman) =
-11 (DeMarcus, et al.)
=
0.3849 p
(4)
'lau
= BAS - (Freeman) = (;)'I G * (p)
A so A2
i
*
I 0 3 { L O G , ~ G-[O.I470p -0.3771)
5
1 2 (DeMarcus, et al.) = __ 15v'' R ( p )
't
J !
0
4 3
FREEMAN DEMARCUS ET AL
Asymptotic equations for G ( p ) and R ( p ) can be based on the fact that, if f(.x) has n strong maximum a t xo
lomv g(x) e v
(2)
I dx
-
g(zo) e w ~ x oI )
which lends to
90
1
*
-20 I
valid for large p. The series 43, and 4b are slowly convergent; the following empirical analytical forms were therefore developed for convenient representation
*
O
2
3
4
5 P
6
7
8
1 log G ( p ) = 0 . 1 6 7 1 ~- - log p 2
4
Fig. 1.
molecules (in this case, one in the gas, one a molecule of the solid). By integrating this expression over all molecules in the semi-infinite solid, we obtain, where p is the number of molecules per unit volume in the solid
log
G(p) =
G = 0.421(p
4-9.9
X
to perform the indicated integrations to obtain convergent power series expressions for the second virial coefficient. These are Lim P+O (N$
-V)
= uA IC(,)
- f R(p)\
(4)
where
with
M
= NOTYZ = molecular weight of the gas, UA = 108u = u in hgstrom units. Results of DeMarcus, et U Z . , ~ and Freeman9 are in a somewhat different form because different forms of the 3-9 potential eq. 3 were used; their parameters and quantities compare with those in the present work as
- 0.377 =k 0.002 for 8.2 < p < 17.4
- 2.029) + 3.94 X lO-'(p - 2.020)' l O - S ( p - 2.029)3 f 0.01, 2 < p < 6.5
(7b) (7~)
+ 0.1721 + 0'420 -f 0.001, 12 < p < 26 (Sa) pz f 0.001, log R(p) = 0.1294 + 0.1981p - 6.76 X log R ( p ) =
If this potential is substituted in eq. 1, it is possible
0.1470p
120 - 0.1756 + f 0.003 M for 16 < p < 30 (7a)
1 0.1671~ log p P
1< p
< 14
(8b)
The first three terms on the right in equations 7a and Sa derive in each case from the asymptotic expressions (6a) and (6b) ; the terms in l / p are empirical corrections. Equations 7% and Sa have been tested to the upper limits indicated, but may be expected to apply reasonably well above these limits. Figure 1 compares calculations of the present work, those of DeMarcus, et al., and those of Freeman in difference plots against eq. 7n, 7b, 7c, permitting accurate corrections to these empirical equations if desired. Equation 6b is particularly useful in the evaluation of parameters; many of the data of Hnlsey and eo-workers is in the range covered by this formula, so that the slope of the linear portion of a plot of log Lim ( N k T / P - V ) against 1/T diP+O
vided by 0.1470 gives the ratio of p to T , Le., To = (10/3)'/2 X 7re0pu3/k. Omission of the quantum correction (term in R ( p ) ) causes little error in the surface.area calculated by this trent,rnelit; the correctlon in Lim
(NlcT/P
-
P-+O
V ) amounts to 7% for the system He-
May, 1959
745
NOTES
carbon black2 a t 78°K. and 3% for the same system a t 194°K. The deviations due to quantum effects should, however, be observable. Data represented by equation 4, omitting the term in R ( p ) , in the range 8 < p < 17 will be almost equally well represented by the treatment of Steele and Halsey based on a rigid sphere - inverse r3 potential model, relevant parameters corresponding as follows: e*/k (Steele and Halsey) = 0.422T0; (ADo) (Steele and Halsey) = 1.605 ( A u ) . The range of agreement can be extended toward lower values of p by choosing different geometric volumes in the two treatments. Accurate high tempernture data will resolve between the two models since the model based on the 6-12 potential leads to zeros in Lim (NlcTIP - V ) a t p = 2.029 and at p = 0 P-0
with an intervening minimum a t p = 0.166 (G(0.166) = -0.546), while the model of Steele and Halsey leads to a single zero at p = 3.4 and a monotone decrease to -ADO at p = 0. Treatments of gas imperfections due to gas-solid interactions based on eq. 4 or on the model of Steele and Halsey determine the products A a or A D , and evaluation of the surface area A requires the independent evaluation of the parameters u or D. Previous workers2-5t9 have estimated these parameters from the I uf ~ ( 2 0 ) will depend importantly on A, pf, coy, and af,i . e . , on parameters of the film. The presence of such a film on carbon surfaces is to be strongly suspected from the work of Anderson and Emmett’o unless rigorous outgassing procedures have been followed. Acknowledgment.-I am indebted to Dr. Mark P. Freeman for calling my attention to the work of DeMarcus, Hopper and Allen, for furnishing me a copy of his own paper on tkis subject in advance of publicat8ion,and for stiniulating oral and written discuaion. (10) R. B. Anderson and P. €1. Emmett, THISJOURNAL,51, 1308
-
(1947).
A METHOD FOR THE CALCULATJON OF BOND MOMEKTS FROM ELECTRONEGATIVITY DATA’ BY RICARDO DE CARVALHO FERREIRA Escola Superior de QuimLca, University of ReciJe, Brast1 Received Nauember 10, 1968
The purpose of this note is to describe a method for the calculation of the polarity of simple bonds based on the principle of electronegativity equalization of Sanderson.2 We will show that this principle is completely general and may be applied for the determination of the charge distribution in . simple molecules. Sanderson’s principle may be derived as follows : let us suppose that the electronegativity of a certain atom A is greater than the electronegativity of another atom B, XA(O) > XB(O). If these atoms combine to form a molecule AB, then after the sharing of one electron-pair, A will acquire a partial negative charge and B will acquire a partial positive one. The electronegativity of A will thus decrease and that of B will increase until the charge distribution in the molecule AB is such that A-a and B+a have the same electronegativity. At this point it will be interesting to examine the meaning of the concept of electronegativity in the light of Sanderson’s principle. Electronegativity as defined by P a ~ l i n gM , ~~ l l i k e i iGordy5 ,~ and others,*,’ has the dimensions of energy. However, chemists have always used the concept of electronegativity as a characteristic atomic potential: the power of an atom in a molecule to attract electrons to i t ~ e l f . ~ Nom, this is precisely what is borne out by the principle of electronegativity ( 1 ) This work has been slipported by t h e Conselho Nacionel de Pesquisas, Rio de Janeiro. (2) R. T. Sanderson, Science, 114, 670 (1951). (3) L. Pading, J . A m . Chem. Soc., 54, 3570 (1932). (4) R. S. Mulliken, J . CAem. Phya., 2, 782 (1934). ( 5 ) W. Gordy, Phys. Rev., 69, GO4 (1946). (G) T.L. Cottrell and L. E. Sutton, Proc. Roy. Soc. (London), A207, 49 (1951). (7) H.0.Pritohard and H. A. Skinner, Chem. Reus., 55, 745 (1955). (8) L. Pauling, ”The Nature of the Chemical Bond,” Cornell University Press, Ithaca, N. Y.,1939, p. 58.
