Nov., 1957
APPROXIMATE SITEDISTRIBUTION FUNCTION FROM ADSORPTION DATA
Carbon Corporation fellowship for 1955-1956. Thanks also are due to the Council on Research
1551
of the Pennsylvania State University for a grant for the purchase of some equipment.
AN APPROXIMATE SITE DISTRIBUTION FUNCTION FROM ADSORPTION DATA; APPLICATION TO ADSORBED HELIUM1 BY WILLIAM A. STEELE
6
Contribulion No. 91 from the Cryogenic Laboratory of the College o Chemistry and Physics, The Pennsylvania State University, University Park, ennsylvania Received June 1 , 1967
A relatively simple method is developed for obtaining an approximate site distribution function for adsorption on a heterogeneous surface. This method requires that the adsorbate have negligible lateral interaction, and that the total energy of the film be known as a function of the pressure. A site distribution function is obtained which is a step function, with the number and spacing of the steps contingent upon how close a fit of model to experiment is desired. This method of analysis is used to determine distribution functions from helium adsorption data measured on several surfaces at temperatures of 10-20'K. The isotherms and isosteric heats for these systems are calculated from the model and compared with experiment. Some discussion of the results of this treatment of the data 18 given.
Introduction Only a few of the many studies which have been made of the heats of adsorption and adsorption isotherms of simple gases on solids have been discussed in terms of a quantitative model. This is due to the fact that most of the adsorbents discussed have shown heterogeneous surfaces, resulting in a complex situation in which a suitable model for adsorption must take into account the occurrence of various adsorption energies in an unknown distribution over the surface. Furthermore, if the system is one in which interactions between neighboring molecules in the adsorbate must be taken into account, the location of the various adsorption energies in the surface must be known as well as the number distribution of the energies. At the present time, it seems impossible to give a complete characterization of the distribution of these energies with respect to number and to location from a knowledge of the macroscopic thermodynamic properties of the adsorbate. If lateral interaction can be neglected, the problem is greatly simplified, since the properties of the adsorbate no longer depend upon the location of the adsorption energies. Previous attempts to relate monolayer adsorption isotherms for such a system to a site distribution function1-* have resulted in equations which are complex and difficult to solve even in the idealized cases in which the amount adsorbed is given by a simple function of the pressure. Since experimental results cannot, in general, be represented by simple analytic functions, these expressions are of little use in the treatment of real data. In this paper, a simple method is presented for obtaining an approximate site distribution function from the total energy of an adsorbed film which conforms to a model in which lateral interaction is neglected. The partial molal thermodynamic properties of such a film can then be calculated from (1) Thia research warn carried out under Contract DA-36-061ORD-509 of the Office of Ordnance Research, U. 8. Army. (2) 8. Roginsky, Doklady Acad. Nauk, S.S.S.R., 46, 61, 194 (1944). (3) R. Sipa, J . Chem. Phus., 16, 490 (1948); J. M. Honig, ibid., 28, 1557 (1955): 22, 851 (1954).
the model and compared with experiment. This method is used to analyze data for adsorbed helium on several surfaces a t temperatures of 10-20OK. The results of this analysis are presented, and some discussion of the method and its limitations also is given.
Theory In considering a film having no lateral interaction adsorbed on a heterogeneous surface, the problem can be greatly simplified if it is assumed that the surface is composed of a number of discrete sets of sites. The number of sites in the i-th set is given by Ni,the potential energy of interaction of an adsorbate molecule with a site in the set is given by ei, and the molecular partition function of an atom on a site in the set is given by j i . One can then treat the problem as one of adsorption on n independent, uniform portions of the surface and apply any one of a number of formalisms to calculate the thermodynamic properties of the atoms adsorbed on each portion, and from this, the total thermodynamic properties of the film. For example, the total amount of gas adsorbed, va, is obviously V,
Ni ei
= i
where ei, the coverage on the i-th set, is calculated from an appropriate isotherm equation (Langmuir, B.E.T., or other). By means of similar equations, all the thermodynamic properties of the film can be calculated. The distribution function of site energies will be a step function given by the Ni corresponding to the ei chosen to be characteristic of the surface. It is necessary to obtain some quantity from experiment which is a sensitive function of the Ni and ~ i and , then to develop a method which gives Ni which are a reasonable approximation to the actual distribution function. The total energy of the adsorbed layer, Ea, calculated as a function of the pressure of the film, is a suitable function for such an analysis, and is easily obtained from the experimental partial molal heats of desorption,
WILLIAMA. STEELE
1552
qst.4 (The sign of the isosteric heat qst is ordinarily taken to be positive, and this convention is followed in this paper. However, a positive heat implies that heat is taken up by the system and the process in which heat is absorbed is actually desorption. In agreement with this, qat is defined as Q g
- Ra.1
where Rgis the molal heat content of the gas, and the PV term of the adsorbed film is considered to be negligible. If E a is known a t various coverages, it can be shown that a set of linear equations is obtained. These equations can be solved to give Ni corresponding to an assumed set of ci. It is necessary to assume a specific model for adsorption in order to compute all the necessary parameters (in particular, the coverage 6i); for this reason, the detailed development of the theory is best discussed in terms of a specific example. Thus, the case in which the adsorbed phase is monomolecular will now be discussed (analogous equations for a multilayer film corresponding to the B.E.T. formalism are given in the appendix). The isotherm equation for adsorption on one set of sites is 1
Vol. 61
tional degrees of freedom. At sufFiciently high temperatures, j i becomes
where vix, viy and vi. are the frequencies of vibration on a site in the i-th set. Hill6 has suggested that the frequency of vibration be proportional to e'/*. On substitution, one obtains ai = AE'/sT-~/s
(9)
where A is an unknown constant. One can then set b In a J t ) ( l / T ) = - '/zT for use in equation 7. It is not possible to estimate the absolute value of A ; therefore, it is necessary to leave ai as an adjustable parameter to this extent. However, A contains no reference to surface properties; ie., if this model is applied to rea,l systems, A should be roughly the same for a given rare gas adsorbed on any surface, provided the temperature of measurement is such that equation 8 is valid. Ni 0i = va, equation 7 can now be writSince i
ten
In a system where the isosteric heats and coverages have been measured as a function of the pressure at a common temperature, one proceeds by choosing values for A and a number of ei, and calculating where j , ( T ) is the internal partition function of a the 0i for pressures corresponding to known values gas molecule. It would be desirable to be able to of qat dv,. The appropriate terms in equation calculate an a priori value of ai; however, since 10 are then calculated. Thus one obtains a set of there is no way to estimate j i , it is necessary to linear equations which can be solved for the Ni. leave ai as an adjustable parameter (possible limita- This procedure ensures a distribution function tions on the choice of ai are discussed below). exactly fits the plot of Ea versus p at a number The total energy of the film can be calculated which of points equal t o the number of sets of sites chosen. byS If a better fit is required, the equations can be solved using a larger number of terms, and a distri(5) bution function obtained which gives a fit to experiIf it is assumed that the gas phase is ideal, €& can ment a t a correspondingly larger number of points. The choice of ei is, of course, limited by the fact be written as that a bad choice will result in physically unrealistic values of the Ni ( i e . , negative numbers of sites for some sets). By combining equations 2 , 5 and 6, one obtains A distribution of sites made up of such a limited number of sets could not be expected to fit the experimental data a t all points; however, it is of interest to ascertain how well the various thermoThe last term on the right-hand side of equation 7 dynamic functions computed from the model agree is easily obtained from experimental data; if a with experiment. A comparison of the calculated numerical value can be obtained .for Ni 0i partial molal thermodynamic functions with exi periment will give a rigorous test. As is well (b In a& (l/T)), the two terms can be summed to known, adsorption isotherms are actually detergive a linear function of the Ni. The evaluation of the partial molal free energy Faof the of b In ai/b ( l / T ) is simply done if b In ( j g / $ ) / minations b ( l / T ) is known. The use of a rare gas as adsorb- adsorbed phase as a function of coverage. Thus, ate simplifies the estimation of the j ' s consider- it seems desirable to be able to reproduce the exably. I n this case, j,(T) = 1 , and j i ( T ) becomes perimental adsorption isotherms with the model. the partition function for the motion of an ad- In addition, the partial molal heat of desorption is sorbed atom relative to the surface. I n systems of interest. The partial molal energy of the film is where the adsorption is sitewise, it is reasonable to found by differentiating equation 5 assume that an adsorbed atom will have three vibra-
ei
=
1
+ ai exp(-ei/RT)/p
,
(3)
sovD
(4) (5)
T. L. Hill, J. Chem. Phys., 17, 520 (1949). T.L. Hill, kbid., 17, 762 (1949).
c
I I
~
I
APPROXIMATE SITEDISTRIBUTION FUNCTION FROM ADSORPTION DATA
Nov., 1957
700
k
i
i
6 00 qst
(cal./rnole).
500
400
0 ' s m
I
300
1553
0
\
Q\
200
100 IO
0
30
20 va
40 (CC.-STP/g.),
50
60
Fig. 1.-Isosteric heats of adsorption of helium in calories per mole plotted as a funct,ion of the coverage in cc. of gas a t standard conditions per gram of adsorbent: a, experimental heats on bare titanium dioxide (0.4 = 0)- a, experimental heats on Ti08 plus 19.7 cc. of argon per ram (0.4 = 0.36); 0, experimental heats on Ti02 plus 30.7 cc. of)argon per gram (0.4 = 0.60); 0 , experimental heats on T i 6 plus 53.6 cc. of argon per gram (eA = 1.05). The solid lines are calculated from eq. 14 using the parameters given in Table I. The calculated isosteric heats extend to a coverage of unity in each case; the arrows are located a t the helium monolayer capacity for each system considered.
=
bv.
dP
(12)
av,/ap
&/dp is calculated from equations 1 and 3. Since qat Rg- E,, one can calculate qst from the equation qst
a In ai
aiNiei2aiexp( - e i / R T )
R-
+
P'
N i m i exp( -€i/RT)
a(1/T)
(13)
i
The technique of analysis presented here gives a method for the determination of the monolayer volume Vm since it is clearly equal to the sum of the N ~ .It is interesting to note that this determinais independent of the isotherms; howtion of ever, as long as A is to be chosen arbitrarily, it would seem advisable to ensure that a given set of N~ will give isotherms and isosteric heats in agreement with experiment before the Ni can be considered acceptable. Application Recently, isosteric heats of adsorption were reported for On number Of surfaces which had been prepared by preadsorbed varying amounts
of argon on a sample of titanium dioxidea6 These systems showed a large variation in heterogeneity, as indicated by the measured heats. During the course of this work, a number of helium isotherms TVeredetermined at temperatures of 1O-2o0K* It has been shown'*8 that isotherms of helium measured under conditions similar to those used in this work are compatible with the Langmuir model for adsorption on a heterogeneous surface. Intuitively, one would expect that helium adsorption data in this temperature range would conform closely to this model. No appreciable multilayer formation would be expected (the critical temperature of liquid helium is 5.25'K.) and, furthermore, the hdiUm-helium interaction energy is quite small compared to the surface-helium interaction. Thus it would seem that these data would be susceptible to analysis by the method outlined above. The heats Of adsorption were measured On bare titanium d i ~ x i d e , and ~ . ~ on titanium dioxide plus 0.36) 0.60 and 1.05 layers of argona6 The points (6) W. A. Steele and J. G. Aston, J . Am. Chem. Soe., 79, 2393 (1957). (7) J.
M. Honig, J . Chem. Phye., 24,
log) 0. (9)
Phys.,
J.
R.
J. Tykodi and
510 (1956).
W. A. Steele, THIS JOURNAL, 69,
Aston. 8. V. R. Mastrangelo and R. J. Tykodi, J. Chsm.
as, 1633 (1965).
1554
WILLIAM A. STEELE
40
% (CC-STP/g
1.
30 eA-06 0 , T = 12 4OK.
20
Vol. 61
and minimum partial molar energies, and then taking as many intermediate values as seemed necessary to obtain a good fit. Equation 10 was then solved using this provisional selection of parameters. If the resulting N i were all positive numbers, the isotherms and isosteric heats were calculated. If these functions did not agree with experiment, A or the ~i were adjusted to improve the fit. As an illustration, a sample calculation for the system helium on Ti02 plus 1.05 layers of argon is given here: a graphical integration of the isosteric heats gave 1 / 2 v,RT qSt dv, = 1.140 cal. a t va = 10 cc. (the numerical values of all extensive quantities are for one gram of adsorbent), and 2.200 cal. at va = 20 cc. At 14.6'K., the isotherm pressure was 155 mm. for v, = 10 cc., and 1660 mm. (extrapolated) for va = 20 cc. If one chooses ~i = 225 cal./mole, e2 = 275 cal./mole and A = 1072, one can calculate the terms in equation 10
+
10
0 100
200
300
400
P
5 00
I m m Hgl
Fig. 2.-The amount of helium adsorbed on various surfaces in units of cc. of gas a t standard conditions per gram of adsorbent plotted as a function of the pressure. The temperature of measurement and the coverage of preadsorbed argon in the system are given adjacent to the appropriate curves. The solid curves were calculated from eqs. 1 and 8 using the parameters of Table I, and the experimental data determined for these systems are given by the points.
sr
1.140 = (0.285)(225)NI 2.200 = (0.810)(225)N1
+ (0.628)(275)N~ + (0.948)(275)Nz
where the Si are the first factors in the parentheses and the ei the second. Solving, NI = 5.60 X lom4 mole = 12.56 cc., and N z = 4.52 X mole = 10.13 cc. v, and qs. were then calculated for the values of the pressure given above, giving va = 9.9 in Fig. 1 show these data plotted as a function of cc. and 20.5 cc., and qat = 236 and 222 cal./mole, coverage. Curves were drawn through the ex- as compared to the experimental v, of 10 and 20 perimental points (not shown), and graphical inte- cc., and qat of 236 and 218 cal./mole. Only two values of E were used in the treatment grations of qst were carried out for several coverages in each case. The temperatures of measurement of the systems with 0.60 and 1.05 layers of argon, were 17.1'K. for the bare titanium dioxide and since these surfaces appeared to be nearly homoge12.5'K. for the other systems studied. Aston, neous. Four evenly spaced B were taken for the Mastrangelo and Tykodi also determined isotherms system with 0.36 layer of argon. Four values of E were also used in the analysis of the data for the for helium on the bare surface at 14.0 and 20.3'K. The heats and isotherms on the bare surface have bare surface, but the low energy members of this been discussed previously,8 but it seemed interest- set were spaced more closely than the high, since ing to reanalyze the data in the light of this theory, it was apparent from the isosteric heat curve that particularly since the assumptions concerning ji a relatively small portion of the surface was comare not the same in both cases. In addition to the posed of high energy sites. Table I shows the values of E used, and their acisotherms on the bare surface, the following isotherms were available for analysis: at tempera- companying N's. Also, the calculated monolayer tures of 12.0 and KOOK. on TiOz plus 0.36 layer of volumes are tabulated for each system, along with argon; a t temperatures of 12.5 and 15.7'K. on the final value of A used in each calculation. Ti02 plus 0.60 layer of argon; at temperatures of TABLE I 10.0, 12.5 and 14.6'K. on TiOz plus 1.05 layers of Ni vm si argon. (eo. (eo. (oal./ System A mole) STP/g.) STP/g.) In order to apply the theory of the preceding 813 225 28.72 63.54 section, it is necessary to have an accurate iso- Bare titanium dioxide (6A 0) 370 26.18 therm a t the temperature of measurement of qat; 530 7.83 however, the. only isotherms having suffcient ac705 0.81 curacy a t low coverages were those measured at Ti02 + 19.7 cc. STP/g. 912 220 11.10 38.00 the highest temperatures. The error introduced of Argon (BA = 0.36) 293 19.75 by using these isotherms to compute the pressure 366 6.45 dependence of qat is negligibly small, due to the 439 0.70 small temperature differences involved. Values for A and the ~i for these systems were TiOz + 30.7 cc. STP/g. 457 220 14.88 30.23 chosen as follows: a provisional value for a i was 285 15.35 Of Argon ( 6 ~ = 0.60) obtained by fitting the isotherms for the most uniTi02 + 53.6 cc. STP/g. 1072 225 12.56 22.69 form surfaces to the Langmuir theory. This pro275 10.13 of Argon (BA = 1.05) cedure gave a = 7 X lo6 a t T = 15'K., E = 210 cal./mole. A was then calculated and the ai obIn Fig. 1, the points are the measured isosteric tained from equation 9. ei were selected by taking heats. The curves were constructed from the values of E roughly equal to the observed maximum model by calculating the coverage and the isosteric
Nov. , 1957
APPROXIMATESITEDISTRIBUTION FUNCTION FROM ADSORPTION DATA
100
200
3 00
40 0
500
600
700
1555
800
P ( m m . Hg),
Fig. 3.-Helium isotherms are shown here which are similm to those presented in Fig. 2 except that the temperatures of measurement given adjacent to the curves in this figure are higher than those for the corresponding systems in Fig. 2.
heat for a given pressure using the parameters of Table I. The isosteric heats a t zero coverage and at a full layer were calculated by taking the zero and infinite pressure limits of equation 13. Figures 2 and 3 show the calculated isotherms and the experimental points for all the systems considered. It should be noted here that the B.E.T. nitrogen monolayer volume of the Ti02 used by Aston, Mastrangelo and Tykodi was about 10% larger than the v, for the sample used in this work, in spite of the fact that the two samples were from the same lot. This is not too surprising, since Mastrangelo, Tykodi and Astonlo have reported that the area of this powder decreases by 43% upon tighter packing. In order to keep their isotherms and heats on a scale consistent with the other data, the coverage scale has been reduced by 10% for their data on the bare surface. The nitrogen monolayer volume for this sample was 51.4 cc. STP/g. The coverages listed for preadsorbed argon were calculated using this v, value, and are thus only approximate.
Discussion The only significant discrepancies between the calculated thermodynamic properties and experiment are found a t high coverages, particularly in the more heterogeneous systems. This is not too surprising, since the basic assumptions of monolayer adsorption with negligible lateral interaction will not hold as well a t high as a t low coverages. In addition, the experimental data in this region are suspect owing to the large corrections for the gas in the dead space of the container. Little is known about the effect of a surface upon the equation of state of a gas in contact with it. Measure(IO) 8. V. R. Mastrangelo, R. J. Tykodi and J. G. Aston, J . Am. Chrm. Soc., 16, 5430 (1953).
ments made a t high temperatures have shown that there is an appreciable effect for high area powders,l1Il2but no method exists for dealing with this phenomenon in the temperature range of these experiments. The success of an analysis depends upon the number of sites chosen; for instance, the urn obtained for the bare surface system considered here varies as much as 20y0 depending upon the selection of the ei, whereas the urn’s for the other systems were much less dependent upon the choice of ei, presumably because the approximation to a continuous distribution was better in the latter cases. In the previous analysis of the data for the bare surface,8 an arbitrary site distribution was chosen, and a cruder assumption was used for the partition function of an adsorbed atom. The results obtained were qualitatively the same as those of this work, but the numerical results differed somewhat due to the approximations and assumptions mentioned above. For instance, the distribution function was assumed to be as follows: 65.201, sites with 180 cal./mole, 27.4% with 360 cal./mole, 7.2y0with 540 cal./mole, 0.2% with 720 cal./mole, and O . O l ~ o with 900 cal./mole. The resultant monolayer capacity was 78.1 cc. STP/g. (corrected to the coverage scale for the sample used in this work). The fact that two different distribution functions give an adequate fit to experiment is due to the approximate nature of the distribution. Indeed, the arbitrarily chosen function of Aston, Tykodi and Steeles seems intuitively more realistic in allowing the fraction of high energy sites to decrease rapidly with increasing energy. The num(11) W. A. Steele and G. D. Halsey, Jr., J . Chem. Phys., 22, 979 (1954). (12) M. P. Freeman and G. D. Halsey. Jr., THISJQWRNAL, 59, 181 (1956).
1556
WILLIAM A. STEELE
ber of sites of a given energy obtained in the calculation made in the present paper is quite dependent upon the choice of site energy and thus the site distribution function gives only a qualitative indication of the actual distribution. Of course, as the number of sets of sites used increases, the distribution function calculated by this method must approach the true distribution for the surface, if the experimental data are measured with an adsorbate which shows no lateral interaction. I n a previous publication,s the isosteric heats of Fig. 1 were discussed qualitatively, and it was concluded that the high energy sites could be ascribed to surface faults, cracks or similar topographical anomalies which allow an adsorbed atom to be in contact with the surface on two or more sides. This treatment of the data allows one to draw an additional conclusion: that the preadsorption of argon causes a large decrease in the monolayer capacity for helium adsorption. Clearly, the preadsorption of another gas would be expected to fill cracks and surface faults in the adsorbent, destroying heterogeneity, and also reducing surface area, since an argon atom adsorbed at any point in a narrow gap removes the site which it occupies from further adsorption. Indeed, the narrowest cracks which will admit helium atoms will be too small to admit argon atoms. Preadsorbed argon which has lodged in the mouth of such a crack will effectively prevent any helium from being adsorbed within. This would explain the result that 20 cc./g. of preadsorbed argon removes 25 cc./g. of helium adsorption sites (see Table I). It is to be noted that the monolayer capacity continues to decrease even after the topographical heterogeneity is removed. This effect can be ascribed to the filling of larger irregularities by adsorbed argon. A simple analogy is found in a sawtoothed surface in which teeth of the order of 50 A. high meet the basal plane a t 45' angles. Such a surface will present an area 1.4 times as large as a planar surface of the same dimensions but will appear as an essentially uniform surface to adsorbed helium. Any high energy sites occurring in smaller roughnesses in the surface will be filled in by relatively few argon atoms, and further argon adsorption will merely decrease the surface available for adsorption by filling in the gaps between the teeth. I n keeping with this, it has been shown1* that adsorption on graphitized carbon occurs in shallow depressions, although the surface of this material appears quite homogeneous with respect to heats of adsorption. Conclusions A reasonably simple method has been developed for correlating experimental data with a formalism in which lateral interaction can be neglected. This method is applied to the case of sitewise, monolayer adsorption (the appropriate equations for the B.E.T. formalism are given in the appendix). Drain and Morrison'4 have given an alternate method for obtaining a site distribution function (18) J. G . Aston and J. Greyson, TEISJOURNAL, 61, 613 (1957). (14) L. E. Drain and J. A. Morrison, Trona. Fomday Soc., 48, 216 (1952).
Vol. 61
from the isosteric heats. Their procedure consists in extrapolating the measured heats to O O K . At this temperature, a site distribution can becalculated directly from the slope of the curve of isosteric heat versus coverage. In practice, this procedure is also approximate, since a rigorous application requires a knowledge of the heat capacity of the film from OOK. to the temperature of measurement at all coverages. The technique of analysis presented in this paper is complementary to that of Drain and Morrison, since their method is most useful when heats have been measured a t relatively low temperatures where adsorption occurs on the highest energy sites available, whereas this method is particularly applicable to systems where heats have been measured a t temperatures such that adsorption is occurring on several kinds of sites simultaneously. The preadsorption of argon on high-area Ti02 has the effect of changing a highly irregular surface to one which is increasingly planar. Accompanying this effect is a loss of heterogeneity and a large decrease in surface area. No such decrease was noted by Singleton and Halsey16 who also determined adsorption isotherms on preadsorbed gases. However, the substrates used in their measurements were of relatively low specific surface area, and were also less heterogeneous than the Ti02 used in this work. It is clear that the rate of loss of surface area and the efficiency of removal of topographical heterogeneity will depend upon the nature of the adsorbent and also upon the choice of preadsorbed gas. Thus it appears that studies of adsorption systems containing various preadsorbed gases will give much valuable information concerning the degree and nature of the heterogeneity found in most high area solids.
Appendix The equations analogous to equations 3, 5 and 7 are given here for multilayer adsorption on a heterogeneous surface without lateral interaction (i.e., the B.E.T. model). This problem is greatly simplified if the thermodynamic functions of the adsorbed phase are considered relative to the liquid phase. If one proceeds by dividing the surface into i sets of sites, one can write that Bi, the coverage on the i-th set, is given by
I
4 I
u
where 6 = ji/& exp(ei - qi,)/RT, II: = p/po and po is the vapor pressure of the pure liquid a t the temperature of measurement. The total energy of the film is given by -E,, =
C e i w i [ei - Ra Inji/a(l/T)] i
+
@i'.**nNiAiq
i
(15)
where Efiq is the molal energy of the liquid, &'Ni is the amount adsorbed in the first layer on the i-th set of sites, and Bi2."n Ni is the amount adsorbed in the second and higher layers on the i-th set. Using the method of Dole,'Bit is easy to show that (16) J. € Singleton I. and G. D. Halsey, Jr., Tare JOURNAL, I S , 330 (1954). (16) M. Dole, J . C h m . Phys. 16, 25 (1948).
TEMPERATURE DEPENDENCE OF SORET COEFFICIENT OF AQUEOUS KC1
Nov., 1957
1557
It seems desirable to leave the value of $/jliq in the equation for ei as an adjustable parameter, analogous to the constant ai in the Langmuir forcixa e i 2 * * * n= = eix (17) malism. However, the choice of ji/jliq is restricted (1 - ~ ) ( 1 x Cix) by the generally accepted hypothesis that it Thus should be of the order of magnitude of one in most -Ea = ( 1 - 3) OiNi [ ~ i Rd In ji/a(l/T)] + cases. The valueof b lnji/a(l/T) can be estimated, i as in the Langmuir case, by assuming a likely configx eiNiBliq (18) i uration for the state of theadsorbed particle and takThe analogous equation for the energy of a B.E.T. ing the temperature derivative of the partition funcfilm on a uniform surface has been given by Hill." tion for an atom in such a state. Ea is obtained Remembering that Bi N; = ZJ,, equation 18 can from equation 2, and &'Eq N AHfi, Rg. Thus, the i parameters necessary to compute the terms on the be rearranged to give right-hand side of equation 19 can be obtained from Ea - X V S l i q + vaR 3 In j i = the experimental data for various values of 2, and sieiNi (19) 1-x W/T) i the Ni can be calculated by following the general procedure outlined previously. (17) T. L. Hill, J . Chem. Phys, 1'7, 772 (1949). Oil
=
cix
1 -x+cix
=
ei(l
- 3)
(16)
+
+
~
THE TEMPERATURE DEPENDENCE OF THE SORET COEFFICIENT OF AQUEOUS POTASSIUM CHLORIDE BY L. G. LONGSWORTH Contribution from the Laboratories of the Rockefeller Institute for Medical Research, New York, N. Y . Received June 3,19.57
Using a twin-channel cell filled with solution and solvent, respectively, through which a downward flow of heat is maintained, the resulting thermal diffusion of the solute has been determined with the aid of Rayleigh interferometry. Aqueous solutions of KC1 have been studied a t concentrations from 1 to 4 m over the temperature interval from 10 to 50" in 10" steps. The increase in the migration of this salt to the cold plate with rising temperature is quite marked.
Thermal diffusion in liquids is the systematic relative motion of the components of a solution that is coupled with the convection-free flow of heat. For example, if an aqueous solution of potassium chloride is brought into contact with a warm metal plate above and a cool one below the salt migrates to the cold plate until a balance is attained between the separating effect of thermal diffusion and the mixing effect of ordinary diffusion. The movement of salt may be described by the relation mass flow = -D'm(dT/dh)
- D(dm/dh)
(1)
in which D and D' are the ordinary and thermal diffusion coefficients, respectively, m the concentration in moles per 1000 grams of solvent, dT/dh the temperature gradient and dm/dh the concentration gradient resulting from thermal diffusion. The Soret coefficient u is defined as the fractional change in concentration for unit difference of temperature, (l/m)dm/dT, when the system is in a steady state. The mass flow is then zero and equation 1 becomes u =
(l/m)dm/dT = -D'/D
(2)
a relation that defines D'. Previous work's2 has suggested that the Soret coefficients of aqueous salt solutions increase in magnitude with rising temperature but no systematic study of the effect of both this variable and concentration has been reported. It is the purpose of this paper to describe a new Soret cell, in which the (1) C. C. Tanner, Trans. Faraday Soc., 28, 75 (1927). (2) K. F. Alexander, 2. phyrik. Chem., 208, 213 (1954).
concentration changes are observed in situ with the aid of Rayleigh interferometry, and to report measurements on aqueous KC1 at concentrations from 1 to 4 m in 10' intervals over the temperature range from 10 to 50'.
Experimental In the partially exploded view of the Soret cell shown in Fig. 1 the glass frame is clamped between silver plates provided with connecting annular channels through which water is circulated in maintaining these plates a t the desired temeratures. Not shown in Fig. 1 are the two lightly greased oroseal gaskets of 0.2 mm. thickness that are interposed between the glass frame and the silver plates or the two 1 mm. gaskets between the silver and Bakelite plates that complete the annular channels. The assembled cell is provided with a spirit level8 having a sensitivity of 5 minutes of arc and is dismantled for cleaning between experiments. A novel feature of the cell of Fig. 1is the central glass partition that permits both solution and solvent to be exposed to a given temperature gradient simultaneously. The crosssection of each of the two channels thus formed is 17.5 X 50 mm. The height of the channels in Fig. 1 is 10 mm. but a second frame of 15 mm. height is available and has been used to test the dependence of the relaxation time e on the square of this dimension. The concentration changes accompanying the heat flow have been followed with the aid of the same Rayleigh interference optical system as that used in this Laboratory for the measurement of isothermal diffusion.' As shown in Fi monochromatic light from the illuminated vertical slit made conver ent by the lens L and is split, after passing through the foret cell, into two beams by the vertical slits in the mask M that straddle the cell partition. The spherical lens 0 is focussed on the cell whereas the cylinder lens C,
%
k2
( 3 ) 9. Prager, J . Chem. Phys., 83, 1742 (1955). (4) L. G . Longsworth, J . A m . Chsm. Soc., 74, 4155 (1952).
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