NOTES
2494
adsorbed on the metal surface. The first step in the polymer formation must be polymerization of the gas already adsorbed on the metal surface, subsequent layers being formed by adsorption of further gas prior to polymerization. Experimental measurements with a gas pressure of 5 X torr and a beam current of 5 pa. indicated that at all writing times down to 1 X see. the polymer film thickness decreased continuously with decrease in writing speed. This showed that a t least more than one layer of polymer must have been deposited, since a deviation from the smooth dependence of signal height on writing time would be anticipated when less than a monolayer was present on the surface. Also, the dependence of film thickness, a t a constant writing speed, on both pressure and current density was consistent with the kinetics previously deduced for formation of polymer on p ~ l y m e r . ~ Both these facts indicate that the po!yrr,er film formed at miting times down to 1 x sec. was composed of more than one layer. It appears therefore that under the conditions of the experiment, 1 X sec. must represent an upper limit of the time required for complete polymerization of the gas already adsorbed on the metal surface. Consider nom what may be inferred from this conamp. is equivaclusion. A beam current of 5 x lent to 3 X 1013electrons striking the surface per second. Assuming R cross section area of 5 X cmS2for butadiene, there would be about molecules adsorbed in a close packed monolayer with an area equivalent to that of the electron beam (about 5 X cm.2). During the interwl of time necessary to form the polymer layer, 3 x 10” electrons would have collided with the surface. It seems, therefore, that one electron is sufficient to cause polymerization of about 30 adsorbed molecules. This compares with approximately 100 electrons/molecule required for polymer formation on a polymer surface, with a similar current and electron energy of about 0.3 kv. Preliminary experiments indicated very little or no dependence of the rate of polymer formation on electron energya3 The very large difference in the number of electrons required to form a single layer of polymer in the two cases must therefore be related to the difference in degree of bonding of the adsorbed gas to the underlying material and the consequence of this on the electron distribution in the adsorbed molecuiz. This is not unusual, since in a number of caaes it has heen shown that the absorption spectra of molecules are appreciably altered when they are chemisorbed on surfaces. It seems, therefore, that the monomer molecules adsorbed on a metal surface are in an energy state from which they can very easily enter into a chain reaction and form polymer molecules containing initially up to 30 monomer units. This initial layer must then form the base from which the polymer film qrows by the mechanism already described. SCALING AND THE VIRIAL THEOREM BY G. HUNTER, D. G. RUSH,AND H. 0. PRITCHARD Chemistry Department, Universitg of Manchester, Manchester 13, England Receised M a y 3, 196.9
I n some recent LCAO calculations on Hzf, de Carlo and Griffingl found that the virial theorem (2T
+
Vol. 67
t 2.06
2.04
s
i 2.02 ,g 6
2.00
1.98
1.96
I
I 1.0
I 1.2
I
I
I
1.4
1.6
2.
Fig. 1.-Variation
of Rminwith 2 for Pia+.
V = 0) was best obeyed for their 3-term wave function a t R = 2 a.u. when the scale parameter 2 was 1.2, whereas a much higher Z of 1.45 gave the best energy (see Tables V and IX of ref. 1). Dr. Griffiiig asked us if we could investigate this apparent anomaly further, and in so doing, we have found that the situation is somewhat complicated by the fact that, for severely truncated basis sets, the calculated equilibrium bond length is quite sensitive to the scale parameter. There is no anomaly, however, because neither ( E T ) nor (E T R dE/dR) ever become zero when Z is around 1.2. Pritchard and Sumner2 investigated the energy of Hz+ a t R = 2 a.u. for nine values of the scale parameter 2. Since all the integrals involved are actually functions of RZ, it is a simple matter to reinterpret these so that we can choose a fixed value of 2 and calculate the energy for nine different values of R. The minimum in the potential curve for each Z was then located by interpolation or extrapolation of the differences between our calculated energies and the exact energy function derived from the results of Bates, Ledsham, and S t e ~ a r t . The ~ values of R m i n for 2, 3, 4, and 10 basis functions are shown in Fig. 1; for 10 functions with Z = 1.415, Rmin = 1.997 a.u., and coincides with the value given by the exact solution. Figure 2 shows the variation of the virial function (E T ) with scale parameter for internuclear separations R = Rmin. In the range accessible to us, these
+ +
+
+
(1) V. de Carlo and V. Gliffing, J . Phys. Chcm., 66, 845 (1962), and personal communication. (2) H. 0. Pritchard and F. H. Sumner, abzd., 67, 641 (1961). (3) D. R. Bates, K. Ledsham, and A. L. Stewart, Phil. Trans Roy. Xoc. London, A146,215 (1953).
Nov., 1063
23%
KOTES
2.0
'Y
1' = 2
.Y
c c,c,(2 .+ 2&i)-+(2 +
x
'1 RIODII'IEI) IIEIjYE TIIEORY OF SALTING OF XONELECTROLYTES I N ELECTROLYTE
0
SOLUTIOKS
j d
131- 31. G~rozr,Y. MARCUS, ASD -If. Sm,orr
c n '
+
Radiochemislry Depnrlmcnt. Soreq Research Eslablishmenl, Israel Atomic Bneryy Commission, Rehovolh, Israel
2 0 0,
2tj,)--l'*
1 = 1 j=l
1.0
Recezked d l a y 17, 1963
- 1.0
Consider a system of three components: water, an electrolyte, and a nonelectrolyte. The Debye theory gives the activity coefficient of the noiielectrolyte as' -2.0
I
I
1
1.0
1.2
I
I
1.4
1.6
z.
+
Fig. 2.-Vari:~tion of the virinl function ( E 7') with Z for I2 = ft,,,,, solid lines; variation of the vinal function ( E T R dE/dR) with Z for R = 2 ax. and N = 3, dotted line.
+ +
curves are of a cubic type, and the points at which they cross the axis correspond to extremes in the energy function. For a n expansion length of N = 2, there are three extremes; a maximum at about 2 = 1.25 (cf. Table I of ref. 2 ) and two minima a t 2 = 1.65 and 0.97. For other expansion lengths, there is only one real root in (13 T ) , correspoiidiilg to minima in the energy a t Z = 1.59, 1.55, and 1.413 for N = 3, 4, and 10, respectively. Since the three basis functions used by de Carlo and Grifhig' are equivalent to the first three members of the basis set used by Pritchard and Sumner,2 the two sets of results should agree. Table IX of ref. 1 shows the agreemcnt between the energies for X = 3, but our extrapolated best scale parameter of Z = 1.39 does not agree with their value of 1.45; however, our higher value is confirmed by some unpublished calculations by S o ~ e r s . On ~ the other hand, me do coiifirni a very small numerical magnitude for ( E 5") near Z = 1.2 ( c j . Fig. 2 ) , but (E T) never becomes zero in this region, which is the necessary condition for an energy minin~urn.~Maxima and minima in ( E 3") only correspond to inflections in the energy-scale paramrter curve. Our results for N = 10 show that tho best energy obtainable is 0.602607 a.u. a t R = 1.907 a.u. Thus, it is only when me have a reasonable expansion length that both the energy and Rminconverge sunultaneously to the correct valurs. The relevant formulas for the potential and kinetic enrrgies, which were not given in ref. 2, are, in a.u.
+
+
+
+
(4) 0.Sovers, personal communication. (5) P.-0. Ldwdin, J . Mol. Spectry., 8,46 (1959).
where NoisAvogadro's number, C* are theconcentrations of the respective ions of the electrolyte in moles per liter, and Jkare functions described below. We shall c o n h e ourselves to the case C+ = C- = C2: extension to other cases, or to mixtures of electrolytes (with ZJC instead of the binary sum), is obvious. The functions J may be written approximately as where R is a characteristic length discussed below, z is the charge of the ions in terms of the electronic charge e, and b* are the radii of the ions. Debye has shown that R4 may be calculated from
.E4= Noe2[~3(bD/bni)., vl(dD/bnd., 1/8aRTv1D2 (3) where R is the gas constant, T the absolute temperature, the molal volume per molecule i (v, = Vi/N,,),D the dielectric constant of the solution, and ni the number of It is the difference in the brackets moleculesi per which determines whether there will be salting in or salting out. We shall depart here from Debye's treatmeiit,l by consideriiig in a different manner the dependriiw of the dielectric constant on the composition. .it moderate concentration of electrolyte it is usually found that2 G',
D
= Di
-
- 83C3
(4) where D1 is the dielectric constant of pure water, a2 is the molar dielectric decrement of the electrolyte, and as that of the nonelectrolyte. Considering the derivative (bD/dnl),, in eq. 3, it may be written as (bD/bC2). (dC2/bnl), where the first factor, if C3 is sufficiently small, may be equated with -& from eq. 4. The second factor is
(bC?/dnl)
=
82C2
-1000
v1/(VzQ
+ hzV1)
( I ) 1'. I k b j e , Z. p l r l p k . Chem., 1S0, 56 (1027). (2) P. Debye and J. McAulay. Phveik. Z.,26. 22 (1925).
(5)
