NOTES
3072
I n the pressure-resistance relation, hysteresis effects larger than would be tolerable in a sensitive pressure transducer were observed. After a pressure cycle of 0-12,000 kg./cni.2 one sample showed a 10% reduction in thickness.
the distinction between the two usual Gibbs dividing surfaces, the surface of tension (SOT) and the T = O ” surface (17.~’~ I n particular, the Frenkel potential @ for the twophase system liquid ( 2 ) and vapor (1)is given by
Discussion
(1) where N , p , P, T , u, and A have their usual significance. It will be shown that a specific choice of dividing surface is required to make @ a thermodynamic potential consistent with the equilibrium condition
From our data (Fig. l), it can be seen that in these samples, the log of the resistivity is not a strict linear function of p”‘. If only that part of the curve between p”’ = 50 and p l / ’ = 100 is considered, one might argue whether the points form a straight line. However, if a larger range of pressure and the changes in the sample dimensions are considered, it is clear that the resistivity-pressure relation is not that required by the proposed theory3 and is leveling off with increasing pressure. It is now accepted5 that results obtained from measurements on compacted powders do not represent the bulk properties of the material under investigation. The “sample-electrode” effect, discussed in this note, clearly demonstrates that other factors in addition to the bulk resistivity contribute to the observed resistance. The results obtained in these experiments indicate again6 that electrical ch,aracteristics observed on compacted powders are a complex function of the bulk properties, electrode contacts, and intergrain effects, and simple interpretations based only on the intrinsic properties of materials can be misleading. Acknowledgment. We wish to express our thanks to J. R. Ladd for the preparative work and to F. P. Bundy for help with the high pressure work. (5) D. D. Eley and J. D. Parfitt, T r a n s . Faraday Sac., 51, 1629 (1955) (6) C. M. Huggins and A. H. Sharbaugh, J . Chem. P h y s . , 38, 393 (1963).
@
+
= A~lPl(TP1) N2P2(PlrT)
(b@)T.P1,Ntot,l
=
+ uA
0
(2)
and that this choice is not consistent with commonly employed experimental techniques for the determination of U. From the Gibbs theory, in the form prior to a choice of dividing surface, for a spherical interface5
+ pl(Pi,T)dNl + + udA + Cdo (3)
dl/Ttotal= TdStotai - PidVi - PdVz p:’(PZ,T)dN:,I- pdN8
Integrating in the usual fashion3 a t constant curvature
Ut
=
+
TSt - PlV1 - PZVZ Nlpl
+ NW + + uA p5N6
(4)
For an incompressible droplet pz(P2,T)
=
P2(Pl,T)
+
%@2
- P1)
(5)
whence @ may be written @ = l7.i
- TSt
+ PlVl + P2Vz -
p5Ns
(6)
Combining (6) with dli, from (3) and noting the equilibrium condition Pl(P1,T)
=
P2(PZrT) =
P5
= P
one obtains
+ (Pi - Pz)dV2 + VdP1 N,dp5 + p(dN1 + dNz) + udA + Cdc
d@ = -StdT The Frenkel Approximation in the Thermodynamic Theory of Surfaces
by H. Saltsburg’
The conditions d T that
=
0, dP1 = 0, and dNt
(PI -- Pz)dV, - d(Nsp,)
=
(7)
0 require
+ udA + Cdc = 0
(8)
Department of Chemistry, Boston V n i z e r s i t y , Boston, 114assachusetts (Receited February 3, 1.9641
and thus (8) is a t variance with the result of the exact
The thermodynamic equilibriuin between a droplet and its vapor is of fundamental importance in theories of vapor nucleation.2 It is the purpose of this note to demonstrate that the thermodynamic potential employed by Freiike12b to examine this equilibrium is an approximation3 and, as commonly employed, ignores
(1) General Dynamics/General Atomic, San Diego, Calif. ( 2 ) (a) >I, Volmer, “Kinetik der Phasenbildung.” Theodor Steinkopff, E d . , Dresden and Leiden, 1934: (b) J. Frenkel, “Kinetic Theory of Liquids,” Clarendon Press, Oxford, 1946, p. 368. (3) (a) F. P. Buff, J . Chern. P h y s . , 19, 1591 (1961); (bj “Handbuch der Physik,“ Vol. X , Springer-Verlag, Berlin, 1960, p . 281. (4) J. W , Gibbs, “Collected Works,” 1’01. I, Longmurls Green and Co., S e w York, S . Y., 1931, p. 219. (5) J . W. Gibbs, ibid., p . 225.
T h e Journal of Physical Chemistry
NOTES
3073
Gibbs theory unless d(p,N,) = 0. This is equivalent to the choice of the r’ surface. If however one makes this choice of dividing surface then the equilibrium condition is correctly given by
(Pi- Pz)dVZ + UdA
+ Cdc = 0
(9)
and if the theoretically or experimentally determined values of u are calculated from such an equation, n o modification of the thermodynamic theory is required, curvature dependence having already been included. If the defining equation for u is the more commonlly employed forin (Pi - P2)dVZ
+ UdA
XN =
0
(10)
Buff and J. Kirkwood, J . Chem. Phys., 18, 991 (1950).
The Electronegativity of Perhalo Groups
10.71
+ 14.446~
where 6 is the partial charge on the atom (6 = 0 corresponds to the singly occupied orbital of Jaff6, and the value of x when 6 = 0 may be considered the ‘‘inherent electronegativity” of the atom). The electronegativity of fluorine is XF =
=
which refers to the SiOT then one cannot simply include curvature corrections to u in the Frenkel theory. To do so is to neglect the distinction between the SOT and the r surface. Consequently, the explicit inclusion of curvature terms in the formal thermodynamic framework must be done to make @ EL potential but the value of u and C must be appropriate to the I’ surface, a consideration usually overlooked. This difficulty in the usual Becker-Doring type nucleation theory was first pointed out by Buff and I G r k w ~ o d . ~The present analysis demonstrates explicitly that this difficulty is inherent in the Frenkel approach without considering the logical difficulty of choosing a dividing surface unless one can clearly have particles in either one phase or the other, a difficulty in the operational definition of droplet radius in nucleation theories. (6) F. P.
electronegativity to be estimated. This simplifies the problem of calculation of group electronegativities. a The orbital electronegativity of nitrogen containing 80% p-character4is (Mulliken’s scale)
12.18
+ 17.366~
Combining3 t’hese equations gives the group electronegativity XNF, =
11.63
+ 5.426
or in Pauling units XNF,
=
3.70
+ 1.826
Since the inherent electronegativity (3.70) is somewhat higher than that proposed by Ettinger, other perhalo groups were checked. Trichboromethyl Group. From the infrared absorption of the phosphoryl group in diethyl trichloromethylphosphonate (7.83 p) ,5 the electronegativity of this group may be estimated by the method of Bell and coworkerse6 This niethod yields an estimate of 2.68. From the orbital electronegativity method2 one obtains an estimate of 2.84. TriJEuoromethyl Group. From the phosphoryl group infrared absorption on tris(trifluoroniethy1)phosphine oxide (7.53 M ) , ~ the group electronegativity can b e calculated6 to be 3.29. The orbital electronegativity method gives a value of 3.45. Fluoroxy Group. Although the calculations for this group are far more uncertain than for the previous
by James E. Huheey Division of Chemistry, Worcester Polytechnic Institute, Worcester, Massachusetts (Receiued February 3, 1964)
Recently, Ettingerl has discussed the electronegativity of the difluorarnino group andt arrived at a value of 3.25-3.30 (Pauliing’s scale) by three independent methods. He also pointed out thai if partial charge is neglected, this it3 the orbital electronegativity of nitrogen hybridized to give 84% p-character as calculated by Jaff6.2 However, neglect of ionic character in fluorine compounds is often a serious oversimplification. Fortunately, the orbital electronegativity method of Jaff6 allows the effect of partial charge on
(1) R. Ettinger, J . P h y s . Chem., 67, 1558 (1963). (2) (a) J. Hinee and H . H . Jaff6, J . Am. Chem. Soc., 84, 540 (1962); (b) J. Hinse, M. A. Whitehead, and H. H. JaffB, ibid.,85, 148 (1963); (e) J. Hinze and H. H. Jaff6, J . P h y s . Chem., 67, 1501 (1963). (3) Jaff6 and co-workers2b described a method for the calculation of group ilectronegativities from orbital electronegativities. I n the present work, a modification of JaffB’s method was used utilizing two premises: (1) the orbital electronegativities of all of the atoms in the group become (and remain) equal; (2) the sum of partial charges residing in the orbitals must equal zero for a neutral group, plus one for a monovalent cation, etc. The complete method and results for several groups will be published in the near future. (4) Estimated from bond angle in NF3. Small changes in p-character do not change the results appreciably. (5) C. E. Griffin, Chem. I n d . (London), 1058 (1960). (6) J. V. Bell, J. Heisler, H. Tannenbaum, and J. Goldenson, J . Am. Chem. Soc., 76, 5185 (1954). (7) R. C. Paul, J . Chem. Soc., 574 (1955).
V o l u m e 68, Number 1 0 October; 1964
