Feb., 19.55
IKTERACTION OF PAIRS OF GASATOMSWITH SURFACES
tween these two terminal values of F ( s ) may be used as a measure for the energetic assymmetry in the binary system. This measure is considerably more sensitive than, for example, the deviation of the maximum of the AHM curve from the equiatomic composition. Two particularly simple cases occur when we have for all concentrations or
dF/dx = 0
(a)
dF/dx = constant
(b)
The former case corresponds to a completely symmetrical heat of mixing curve (“regular” solution of components of similar volumes) , while the second case corresponds to the case discussed very recently by Hardy18under the name “sub-regular” solution. (18) H. K. Hardy, Acta Met., 1, 202 (1953).
181
We note that the lead-tin system does not completely fulfill any of these very simple criteria, although either of them would represent a suitable rough approximation. Acknowledgments.-The author is indebted to Professors J. W. Stout and N. H. Nachtrieb for reading the major part of the present manuscript and particularly for suggesting valuable improvements in the part dealing with evaluation of experimental data. He also wishes to acknowledge that the forged section of 2s-Aluminum which was used for making the thermostat jacket was a gift from the Aluminum Company of America through Dr. W. L. Fink. This work was supported in part by the Office of Naval Research through Contract No. N-6ori-02004 with the University of Chicago.
THE INTERACTION OF PAIRS OF GAS ATOMS WITH SURFACES1s2 BY MARKP. FREEMAN’
AND
G. D. HALSEY, JR.
Department of Chemistry, Universily of Washington, Seattle, Washington Received August IC,le64
The interaction of rare gases with high surface area powders has been measured iri the temperature range where the important contributions come from isolated atoms and isolated pairs of atoms only. The pair term governs the variation of the apparent volume of the system with pressure. The experimental values of dV/dP are compared with a crude model based on hard sphere repulsion between the surface and the atoms, and a cube law attraction between each atom and the surface. There are no adjustable parameters. The agreement between calculated and observed slopes is adequate. The systems studied are argon and neon on saran charcoal, argon on carbon black, and krypton on alumina.
Introduction The problem of adsorption a t an interface can be treated, in principle, from the same standpoint as the theory of a bulk liquid. That is, relationships can be written down that would ultimately yield a distribution function for the adsorbed molecules. Wheeler4 has pointed out how such a calculation can be made and Ono6 has presented some theoretical solutions. However, as Wheeler has shown, the application of this theory to the usual adsorption isotherm, even with certain approximations, involves very lengthy numerical computations. Steele and Halsey6have analyzed the interactions of gas atoms with surfaces in the temperature region above the critical point of the adsorbate where the only important interactions are between isolated gas atoms and the surface. In the present paper the experiments and the analysis are extended to the next higher order interaction, that between two gas atoms and the surface. Wheeler has pointed out that this case can be written down explicitly.
The gas itself in the absence of a surface will be assumed to remain effectively ideal. Development of the Mixed Third Virial Coefficient.-We shall follow in outline the simple treatment of Mayer and Mayer.? The total potential energy, U , due to interactions can be written as the sum of interaction energies, u i , between each molecule and the surface and the interaction energies, ujk, between two molecules in the gas
(1) This research was partially supported by Contract AF19(604)247 with the Air Force Cambridge Research Center. (2) Presented a t the lZGth National Meeting of the American Chemical Society, New York City, N. Y., September 12-17, 1954. (3) Presented in partial fulfillment of the requirements for the Ph.D. degree. (4) A. Wheeler, 120th National Meeting, American Chemical Society, New York City, N. y., September 3-7, 1951. ( 5 ) 8. Ono, Memoirs of the Faculty of Engineering. Kyushu Vniv., 10, 195 (1947); J . Chem. P h y s . , 18, 397 (1950); Mem. Fac. Eng., Kyushu Univ., 1 2 , 9 (1950); J . P h y s . SOC.Japan, 6 , 10 (1951). ( 6 ) W. A. Steele and G . D. Halsey, Jr., J. C h s m . P h y s . , 22, 979 (1954).
The second product can be expanded to give
N
u‘=
c u i i=
1
+
N N-1 j>
k k=
Ujk
(1)
1
(2)
-1
(3)
1
We now define two functions
fi
= exp {-ui/kTI
-
and fjk
= esp I-ujklkT]
Then
-l
This expression is then integrated over the coordi(7) J. E. Mayer and M. G. Mayer, “Statistical Mechanics,” John Wiley and Sons, Inc., New York, N. Y., 1940,p. 203,
MARKP. FREEMAN AND G. D. HALSEY, ,JR.
182
nates, T ~ of , all the particles throughout the available volume, Vge,, to yield the configuration integral
VOl. 59
as defined by the ideal gas law, rather than the number of molecules N V
=
N(kT/P)
(14)
If we make this substitution in (13), we find At zero pressure V becomes its limit, Vo, and we find VO
We now define two integrals
dV/dP
Notice that if the proper irreducible integral (ref. 7, p. 287)
GAS =
- Vgeo
= BAS
(16)
the expression used by Steele and Halsey.6 In the experiments reported here, we have measured the slope dV/dP, which by differentiation of (15) is
svge0
Jfifzfi2d7d7~
=
+
l/(dP/dV) = V3CA~s/(2kT(Vgeo BAS)^ kTV(Vge0 BAS)'] (17)
+
Near zero pressure where V approaches Vgeo BAS,(17) becomes dV/dPlp=o
=
CAAS/~T
+
(18)
(9)
This is the result we require. Crude Model for CAAS.-we shall extend the crude model of Steele and Halsey6 in as simple a CAAS = C'AAS f (2BAS vgeo)p (10) way as possible in order to compute the integral where CAAS. We shall retain the assumption that a t the hard sphere distance DGSthe solid-gas interaction B = fidn (108) is E* and that this energy then falls off with the power of the distance. We shall assume in However, if the gas is ideal, p is negligible and it is third addition that the gas atoms have a hard sphere diimmaterial whether CAASor C'AAS is evaluated ameter for self repulsion DGGand that other than since they are equal. We shall therefore not break this they do not interact. Thusfiz (see eq. 3) is - 1 down the contribution in CAAS. Note that it is when the spheres overlap and zero elsewhere. We analogous to a gaseous mixed third virial coefficient shall therefore perform our integration between the between two atoms of a species A and one of B. limits of contact of the sphere with the value -1 I n terms of these defined integrals, the integra- for flz. DGG and 6, the ratio between DGSand DGG, tion leads to are the only new parameters we introduce. &7 = ( V g e o BABY The coordinate system we use is as follows (Fig. 1): z1 and z2 are the perpendicular distances of the atoms from the surface; L is the distance between these and the angular orientation UrselP has shown that for large N this type of ex- of gas perpendiculars; atom 2 with respect to gas atom 1 we denote pansion is correctly approximated by the form by e, a cyclic coordinate. If for drl, we use dA dxl and for dr2 we use L dL de dxz, then the integral N2 CAAS log Q7 = N log ( V g e o BAS) -2- (vgeo+ (12) (see eq. 2,8) becomes (rememberingflz = - 1) from which is defined, then
+
svge0
+
+
+
-
L dL de dzt dA dxl (19) (13)
The area is held constant during this differentiation. This corresponds t o holding the quantity of the second component (the solid surface) constant. Also, the integral CAAShas been treated as if it were independent of volume, whereas eq. 10 indicates that only the proper irreducible integral C'AABis truly independent. Thus, a term proportional t o 0 has been neglected. This neglect is possible because the relatively low pressure and small volume compared to the large surface area used in our experiments make contributions from bulk gas imperfections totally negligible. It is convenient to express the experimental results in terms of the apparent volume of the system (8) See R. H. Fowler, "Statistical Mechanics," Cambridge y n i yereity Press, Cambridge, 1936,p. 241.
where the limits of x1 are from DGSto 'Vge,/A m, z2 goes from DGSto z1 DGGwhen DGS> (xl DGG)and otherwise the lower limit is the latter expression, L goes from o to v'D~GG - (21 - z z ) ~ , and 0 from 0 to 2n. If the integral is rewritten in terms of reduced lengths where DGGis 1 (see Fig. 1) and the ratio DGS/DGG is 6, then the integral becomes
+
=
- D4GG $*
JUI
(l/u13
Ju*JUJ
+ l/d)\
exp
UL
duL dB dA dm
dm (20)
where E * / Z ~ ~= - E ~ / D ~ G and S the limits are nom: u1 goes from 6 to V g e o I A D G G ; q goes from 6 when 6 is greater than u1 - 1 and otherwise from C T ~ 1 to g1 1; 8's limits are as before; and UL goes
+
Feb., 1955
INTERACTION O F PAIRS O F
Fig. 1.-Schematic representation of the coordinate system.
from 0 to 2/1 - (u1 - a2)2. After the analytic integrations are performed, me have
+
exp { 6 3 ~ * / k T ( l / u 1 3
which becomes CAAS= --aD4cc
su2
l/u23)}
GASATOMSWITH SURFACES
183
pressure is about 200 mm., the apparent volume was shown to be effectively independent of pressure. In the present work the temperature range has been extended down to an e*/kT value of about eight. Over this range the apparent volume becomes a linear function of pressure. Therefore V was determined as a function of P, and the results fitted to a straight line by the method of least squares. The slope of this line, dV/dP, was then compared with the calculated values of CAMby means of eq. 18. I n Fig. 3 the experimental data for krypton on alumina are shown; it is clear from these data, which are typical, that the degree of experimental accuracy is not high. The hysteresis shown on reversing the order of taking the points is within the experimental error, however, which confirms the applicability of an equilibrium theory to these measurements, The crudeness of the theory, the uncertainty in the assignment of a value to DGG, and the absence of any adjustable parameter combine to make the degree of accuracy adequate.
dui duz (21)
su,
2.92
exp { 6 3 e * / k T u ~ 3 ]
exp {6%*/kTuza)l/l
- (SI - d2dm du.2
2.91
(22)
This integral was evaluated graphically. The dimensionless quantity log 1CAM/ AD4cc) is plotted as a function of ~*/lcTfor various values of the parameter 6 in Fig. 2. This reduced form of representation will be employed below to compare the experimental data with the results of the crude theory.
2.90
2 2.89
v
bb -?
2 2.88 2.87
12.0 6
rU
2 10.0 3 d1
8.0 2.23
F
50
4
6.0
150 200 250 300 Pressure, mm. Fig. 3.-Interaction of krypton with alumina at 0' (top) and 30.2': 0 , increasing pressure; 0 , decreasing pressure; , theoretical slope based on viscosity DGG; ------, theoretical slope based on critical volume DGG.
-
4.0
4.0
100
5.0
7.0 8.0 9.0 e*/KT. values of log (CAAS/AD&)for various 6.0
Fig. 2.-Calculated values of the parameter 6.
Experimental Results.-The apparatus and general procedure have been described in a previous paper.6 The rare gases were obtained in sealed bulbs from the Air Reduction Company. I n the earlier paper the data consisted of isosteres, performed over a range of temperatures such that the quantity e*/iiT varied from unity to a maximum of about six. Over this range, when the total
The solids have been investigated previously; the carbon black and alumina are described,6 and the high surface area saran charcoal is de~cribed.~ ~ given The parameters used for calculating C A Aare in Table I ; all values are from the previous work except DGG. For this distance the tabulated values'o of hard sphere diameters estimated from viscosity data have been used. I n the case of krypton (9) W. A. Steele and G. D. Halsey, Jr., THIS JOURNAL, 59, 57 (1954).
(10) Joseph 0. Hirsohfelder, R. Byron Bird and Ellen L. Spotz, Chem. Revs., 44, 205 (1949).
R . 0. MACLAREN AR'D N. W. GREGORY
184 TABLEI Adsorbent
Carbon black Alumina
Gas
koal.
Argon Krypton
4.34 3.46
262 141
DGG.
A.
3.42 3.61 (3.06)" Saran S-85 Argon 3.66 1030 3.42 Neon 1.28 1135 2.80 a Calculated from the critical volume.
'
M
c
I
DQE,
A.
2.75 1.9!1 2.90 2.55
we have also made calculations using the rather widely differing value based on critical volume to illustrate the effect of uncertainties in DGG. I n Fig. 3 the calculated slopes are drawn in with the intercept defined by the least-square line. The agreement is surprisingly good. In Fig. 4 the data for argon on carbon black are compared with the 12
I
I
A, In.*/g.
e*,
VOl. 50
6.0 6.0 Fig, 5.-Argon
TARLEI1 Adsorbent
I
Carbon black
9
8 7 6.0
7.0 8.0 e*/KT. Fig. 4.-Reduced plot of data for argon on carbon black compared with theoretical line.
theoretical line in reduced form. It is clear that the temperature dependence of CAASis correctly accounted for by the theory. The results for neon and argon on saran charcoal are similarly presented in Fig. 5. The numerical comparison of the leastsquares slopes and the theory is given in Table 11. Conclusions.-If one considers the crudeness of the model, the close agreement between theory and experiment found here is surprising. A similar (and
7.0 8.0 e*/KT. (left) and neon (right) on Saran charcoal. Temp., Slope (oc./g.)mm.IIg X I O 4 OC. Calcd. Obsd.
Gas
Argon
0.00 27.71 57.85 84.3 0.00
62.1 G0.2 13.1 16.4 5.29 6.82 2.25 2.70 Alumina Krypton 1.51 2.17 (1.84)a 30.2 0.528 0.628 (0.635)" Saran S-85 Argon 0.00 34.0 32.1 5.09 2G.4 24.2 31.0 10.1 13.6 Neon -195.7 806.0 538.0 -194.7 679.0 645.0 -182.1 87.3 83.4 a Based on Daa calculated from the critical volume.
apparently fortuitous) agreement has been observed with the crude theory for B A S . $ The introduction of a Lennard-Jones potential between the gas atom and the surface led to less reasonable values for the area than did the crude theory. It is possible that refinements in the model for CAAS will cause similar difficulties. It is clear, however, that the theory of imperfect gases is suitable for the interpretation of the interaction of gases with solids.
THE VAPOR PRESSURE OF IRON(I1) BROMIDE BY R. 0. MACLAREN AND N. W. GREGORY Contribution from the Department of Chemistry, University of Washington,Seattle, Washingten Received Aueust 10,1964
The vapor pressure of FeBrg has been measured using the effusion, transpiration and diaphragm methods, covering the temperature interval 350 to 900'. The accommodation coefficient is found to be unity.
I n connection with equilibrium studies on the iron-bromine system, it became necessary to know the vapor pressure of FeBr2. We have made measurements between 400 and 640' using the transpiration method with argon as the carrier gas. The quantity of FeBrz transported was determined by a colorimetric technique which enabled convenient determination of vapor pressures as low as 10-3 mm. At the lower temperatures have been compared wlth data obtained by the effusionmethod
(350445'). Pressures were also measured between 600 and 900" in a quartz diaphragm gage. An independent determination of the triple point was made by differential thermocouple cooling curve analysis.
Experimental FeBrt was prepared by reaction of anhydrous bromine with iron powder (Eimer and Amend, electrolytic). The product was sublimed as FenBrs in a bromine atmosphere a t 450'; the condensate, a mixture of iron(I1) and iron(II1) bromides, was then heated a t 120' in vacuo to decompose
. I
P
