Langmuir 1990,6, 1729-1733
1729
The Case for Langmuir’s Case VI Douglas H. Everett Department of Physical Chemistry, University of Bristol, Bristol BS8 1 TS, U.K. Received February 15, 1990. In Final Form: May 15, 1990
Attention is drawn to the equation (case VI) derived by Langmuir in 1918 for multilayer adsorption, which has been largely neglected. When derived by a statistical mechanical route as an alternative to Langmuir’s kinetic derivation,the physical significance of the parameters is clarified. With the assumption that all adsorbed layers after the first have similar properties, but differ from the bulk liquid, the resulting equation, in the form of a modified BET equation, extends the applicability of the BET equation to high relative pressures. This has been tested by using recent precise measurements on a range of adsorption systems, and illustrative examples are presented. The superiority of the case VI equation over the conventionalBET equation is clearly demonstrated. The soundness of the theoretical basis of this equation requires tabulated data over a range of temperature, but no published data are available.
Introduction Langmuir’s 1918 paper’ is remembered mainly for his case I and the resulting “Langmuir isotherm”. Only rarely does one find reference to the other cases he considered. In particular, case VI dealing with multilayer adsorption has been almost completely negle~ted.~B In this, Langmuir derived by a kinetic argument a general equation for multilayer adsorption in which the properties of each successive adsorbed layer were supposed to vary with distance from t h e ~ u r f a c e .By ~ assuming t h a t the properties (in particular the lifetime of an adsorbed molecule) are the same for layers other than the first, as shown by Jones? Langmuir’s case VI equation takes on the same form as the BET equation and becomes identical with it if the molecules in the second and subsequent layers have the same lifetimes as those a t the surface of the bulk liquid. de Boer@presented a kinetic argument essentially the same as that of Langmuir. In effect, in the equations of Langmuir and de Boer the quantity po, the vapor pressure of the liquid, in the BET equation is replaced by a parameter, called in the present paper p * , which need not be equal to PO. Anderson7 adopted a similar approach by assuming that molecules in the second and higher layers had an enthalpy of adsorption numerically less than that of liquefaction; his equations include a pgrameter k = p o / p*. Brunauer et a1.8 obtained the same equation, but by using somewhat different arguments. The underlying assumptions of Langmuir’s case VI are revealed more clearly in the statistical mechanical treatments of Hill? Dole,loand Guggenheim: which again leave open the question whether the properties of molecules in the multilayers are the same as those in bulk liquid. Cassie’sll statistical mechanical derivation introduces the BET assumption a t the outset. What all these “modified BET equations” imply is that (1) Langmuir, I. J. A m . Chem. SOC.1918,40, 1361. ( 2 ) Jones, D. C. J. Chem. SOC.1951,126.
( 3 ) Guggenheim, E. A. Applications of Statistical Mechanics; Oxford University Press: London, 1965; Chapter 11. (4) Compare: Baly, E. C. C. Proc. R. SOC.(London) 1937, A160, 465. (5) Brunauer, S.; Emmett, P. H.; Teller, E. J. Am. Chem. SOC.1938, 60, 309. (6) de Boer, J.
H.The Dynamical Character of Adsorption; Oxford University Press: London, 1953; p 61. (7) Anderson, R. B. J. Am. Chem. SOC.1946,68, 686. (8)Brunauer, S.; Skalny, J.; Bcdor, E. E. J. Colloid Interface Sci. 1969, 30, 546. (9) Hill, T. L. Statistical Mechanics; McCraw-Hill: New York, 1956; App. 5. (10) Dole, M. J. Chem. Phys. 1948, 16, 25. (11) Cassie, A. B. D. Trans. Faraday SOC.1945, 42,450. 0743-7463/90/2406-1729$02.50/0
t o the BET parameters V m and C a third adjustable parameter p* must be added. That this parameter might be used to extend the range of the BET equation (usually limited to relative pressures p / p o in the range 0.05-0.35 or less) was first considered by Anderson: who showed that in a number of instances an appropriate choice of k enabled adsorption isotherms of various gases on several substrates to be fitted up to relative pressures of 0.8. The surface areas derived in this way were between 6% and 19% higher than those obtained by using the conventional analysis. Anderson’s work was largely ignored since it seemed to be based on the unrealistic assumption that the energy of adsorption in the higher layers is numerically less than that of liquefaction. In fact, Anderson commented that entropy effects might be important, though he did not pursue this point. A comparison of the Anderson equation with other BET variants was made by Carman and Raal.12 de Boer et al.13 also showed that the range of validity of the BET equation can be extended to p / p o = 0.75 if p * / p o is taken as about 1.3. They comment, however, that “this result has no significance” since they supposed it to depend on Anderson’s unrealistic assumption. Similarly, Brunauer et aL8 found that for nitrogen adsorption on silica, titania, manganese dioxide, and zirconium silicate the range of the equation was greatly extended if their “modified BET equation” was used with values of l / k between 1.19 and 1.26; the corresponding values of u m were some 5 % higher than the conventional values. They then made the cryptic remark “Although the new 0, values are believed to be correct, the surface areas calculated from the old u, values remain correct.” This they rationalized by saying “the BET areas .., have received so many independent confirmations that there is no reason to doubt their correctness”; hence, the two methods of analysis can be brought together by associating a molecular area for nitrogen of 15.4 A2 with the modified equation. The contention that conventional BET areas, despite their undoubted usefulness in the characterization of solid surfaces, have an absolute significanceis open to argument. The exact values derived from experimental data often depend on the range of p / p O used in the analysis. Moreover, it is rare for laboratories carrying out comparative tests on carefully sampled batches of the same material, analyzed over the same pressure range, to agree
I
(12) Carman, P. C.; Raal, F. A. Trans. Faraday SOC.1953,49, 1465.
(13)de Boer, J. H.; Lippens, B. C.; Linsen, B. C.; Broekhoff, J. C. P.; van den Heuvel, A.; Osinga, Th. J. J. Colloid Interface Sci. 1966,21,405.
@21990 American Chemical Society
1730 Langmuir, Vol. 6, No. 12, 1990
Everett
to within f5%.14 Nor, despite the assertion of Brunauer et al., are there many examples of independent checks on the absolute values of BET surface areas. Gregg and Sing15state that a, calculated from the BET equation with a,(Nz) = 0.162 nm2agrees to within &20%,and sometimes better, with geometrical or other independent estimates of a,. Brunauer et al. conclude that “It would make no sense to attempt to evaluate u m ...from a modified BET equation, and then use 15.4 Az for the area of a nitrogen molecule. The result would be the same as using the BET-Um with the old area.” The fact remains that the Langmuir case VI equation (or the modified BET equation) fits the experimental data over a far wider range of p / p o ,makes use of many more experimental points, and does not raise in the same acute form the question of the limits between which the BET equation can be applied. Nicolaon and Teichner16 also employed Brunauer’s modified equation to analyze their data for the adsorption of argon on nonporous solids for which k = 0.71 ( p * / p o =.1.41), and a good fit was obtained between p / p o = 0.08 and 0.80. However, they accepted Brunauer’s contention that the monolayer capacity should be calculated from the conventional BET equation. GuggenheimH besides giving a statistical mechanical derivation of the Langmuir case VI equation, examined a few sets of experimental data to illustrate its application and again found values of p * / p o in the range 1.2-1.25. In the present paper, we apply this equation to the adsorption of nitrogen and argon by several nonporous solids a t 77 and 85 K. The shortcomings of the BET equation are brought out clearly. We confirm the general validity of case VI for oxides up to p / p o about 0.7. For carbons, the limit is between 0.3 and 0.4, which is, however, far higher than that at which the BET equation breaks down: 0.14-0.25. Some preliminary comments are made on the theoretical basis of case VI, and it is compared with the FrenkelHalsey-Hill (FHH) equation1618for multilayer adsorption.
Use of the Langmuir VI Equation Guggenheim’s equations for case VI may be written in terms of the amount adsorbed na, expressed as mol (g of solid adsorbent)-’: nla = na(l - p / p * )
(1)
where nla is the amount present in the first layer and n, is the monolayer capacity. The parameter C is defined as
c = 91/q*
(3)
while
P*/PO= q0/q* (4) Here 41, q*, and qo are the molecular partition functions for, respectively, molecules in the first layer, in succes-
s.
(14) Everett, D. H.; Parfitt, G. D.; Sing, K. W.; Wilson, R. J. Appl. Chem. Biotech. 1874,24,199 (numerical data are given in National Physical Laboratory, I.M.S. Internal Note 66, Project 2, 1973). (15) Gregg, S.J.; Sing, K. S. W .Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: London, 1982. (16) Nicolaon, G.A.; Teichner, S. J. J. Colloid Interface Sci. 1972,38, 172. (17) Frenkel, J. Kinetic Theory of Liquids; Oxford University Press: London, 1946. (18) Halsey, G. J. Chem. Phys. 1948, 16,931.
sive layers, and in the bulk liquid. According to eq 2, p/nla when plotted against p should be linear with a slope (S) and intercept (0 given by S = (C - l)/Cnm; Z = p*/Cnm
(5)
c = (S/Z)p*+ 1
(6)
so that
n, = (C - l)/(CS) (7) It is useful when making the analysis to include values of nla since, according to these equations, nlatends to n, as p goes to p*. The best value of p* is that which satisfies simultaneously the linearity of a plot of eq 2 over as wide a range of p / p oas possible and at the same time provides values of nla which approach n, at high values of p , thus providing a check on the self-consistency of the analysis. In those cases in the literature where data are available only in terms of p / p o , then eq 1 becomes
while both sides of eq 2 are divided by po. Then @/po)/nla is plotted against p / p o . The slope is again given by eq 5 while the intercept is
I’ = (p*/po)/Cnm
(9)
c = (S/Zyp*/pO)+ 1
(10)
Then
and as before n, is given by eq 7. As will be seen later, deviations from these equations occur rather sharply at the limit of their applicability at high relative pressures and indicate a change in adsorption mechanism, often to capillary condensation. In most cases, deviations, presumably caused by heterogeneity of the substrate, also occur at relative pressures below about 0.1. Application to Nonporous Solids Ander~on,~ Carman and Raal,12Brunauer et al.,s Nicolaon and Teichner,l6 and Guggenheim3present examples in which the Langmuir case VI equations give linear graphs when plotted in terms of eq 2. While illustrating the applicability of the equation, it is not clear whether the values of p * / p o (or k) were optimized to obtain the best fit; nor, except for Guggenheim’s work, was the behavior of nla investigated. We have examined several sets of data in detail, including the “standard isotherms” for Nz and Ar on nonporous silica (Fransil) given by Gregg and Sing,1s-20*z1 Nicolaon and Teichner’s results for Ar on nonporous silica (Aerosil), the experimental t-curve of Broekhoff and Linsen,22 the measurements of Drain and Morrisonz3on NZand Ar adsorbed on rutile, and a selection from the isotherms for Nz on silica and graphitized carbon obtained in the collaborative study reported by Everett et al.14 Recent very precise data for the adsorption of NZon un(19) Hill, T. L. J. Chem. Phys. 1949, 17, 590, 668 Adu. Catal. 1952, 4, 236. (20) Bhambhani, M. R.; Cutting, P. A,; Sing, K. S. W.; Turk, D. H. J. Collord Interface Sci. 1972, 38, 109. (21) Payne, D. A.; Sing, K. S. W.; Turk, D. H. J.Colloid Interface Sci. 1973, 43, 287.
(22) Broekhoff, J. C. P.; Linsen, B. G. In Physical and Chemical Aspects of Adsorbents and Catalysts; Linsen, B. G., Ed.; Academic Pres: London, New York, 1970; Chapter 1, p 33. (23) Drain, L. E.; Morrison, J. A. Trans. Faraday Soc. 1952,48,&10; 1953, 49, 654.
The Case for Langmuir's Case VI
V I ,
W.5 1
)
1
8
1
1
1
Y l l l l 0 l. 5 l , l l
Figure 1. Comparison of the BET and case VI equations:lower
part, BET and case VI eq 2 curves; upper part, adsorption in the monolayer (np)according to the BET and case VI equations (solid points, BET; open points, case VI). (a) N2/nonporous silica: standard isotherm,15same as that for Fransil;20u is in pmol m-2. (b) Ar nonporous silica: standard isotherm,15same as that for Fransi(21 (Y = u(p)/u(p/pO= 0.4). (c)Nz/TK800, laboratory B;14 u is in cm3 g-l. (d) de Boer t-curve;22t is in A. graphitized carbon24and of neopentane on silica and carbonz5 are also examined. Representative graphs, comparing the conventional BET equation with case VI, are shown in Figures 1 and 2. In each case, the lower part of the figure shows graphs of (p/po)/nla,while in the upper section rile is plotted as a function of p / p o . The shortcomings of the B E T equation are brought out most strikingly in the behavior of nla which falls off rapidly beyond the range of applicability of the equation. The case VI curves for nonporous oxides remain linear, and nla becomes asymptotic to n, up to relative pressures of a t least 0.7. For carbons, the limit is about 0.4. Beyond the limit of applicability of the case VI equation, the calculated rile values rise sharply, and the total adsorption na calculated by using the best-fit parameters shows a positive deviation from the observed values. T h e best-fit parameters obtained in this work are summarized in Table I. The precision with which p*/pO can be determined depends very much on the quality and range of the experimental data. Most of the values in the table are considered to be reliable to f0.01 and in some cases to f0.005. The standard deviations of the surface areas from the BET equation are about f0.5-1$: , while because of the extended range of p / p o and the much larger (24) Fernandez-Colinas, J.; Denoyel, R.; Grillet, Y .; Rouquerol, F.; Rouquerol, J . Langmuir 1989,5, 1204. (25) Carrott, P. J. M.; Roberts, R. A.; Sing, K. S. W. Langmuir 1988, 4. 740.
Langmuir, Vol. 6, No. 12, 1990 1731
Figure 2. Comparison of the BET and case VI equations,as in Figure 1. (a) Nz/Vulcan G, laboratory H;14u is in cm3gl. (b) Nz/ungraphitized Vulcan;24 na is in rmol g-l. (c) Neopentane/ as in Figure silica;25as in Figure lb. (d) Neopentane/carb~n;~~ Ib.
number of experimental points used in the analysis the standard deviations using the case VI equations are in most cases h0.39;. The close agreement between the experimental results and calculated curves is illustrated in the case of N2 on nonporous silica in Figure 3. In the range of p / p o from 0.1 to 0.75, the root mean square deviation between these quantities is 0.07 pmol m-2, whereas the tabulated data have been smoothed and rounded to the nearest 0.10 pmol m-2. The agreement is thus virtually perfect. Similar results are obtained with most of the other systems examined. Figure 3 also brings out clearly the influence of condensation on the isotherm a t p / p o > 0.75 and of heterogeneity of the surface at p / p o < 0.10. Hsu et a1.26 have shown that, using the modified equation to define adsorption on a uniform surface, the deviations a t low relative pressures shown in Figure 3 can be described quantitatively by ascribing to the surface a MaxwellBoltzmann-type distribution of adsorption energies. For the adsorption of Nz and Ar on oxide surfaces, p * / po lies in the range 1.30-1.64 and leads to surface areas between 8:;) and 20% higher than the conventional BET values. As illustrated by the results of the comparative studies for Nz/TK800 silica, the analysis is very sensitive to small differences between the sets of experimental data. If as seems probable these deviations from case VI result from the onset of interparticle condensation, the variability of p*/pO may reflect small changes in the packing of (26) Hsu, C. C.; Rudzineki, W.; Wojciechowski, B. W. J. Chem. SOC., Faraday Trans. 1976, 72, 453.
1732 Langmuir, Vol. 6, No. 12, 1990
Euerett
Table I. Comparison of Parameters and Range of Applicability of Case VI a n d B E T Equations case VI
type oxides
system
c
P*/PO
N2/Si02 (Fransil)
62.2
1.564
a,
nm"
pmol
mz g-l 1.17
m-2
11.97 am
Ar/SiO2 (Fransil)
N2/TK800 means
1.642
27.2
1.408 1.420
44.0 43.4
64.2 1.360 1.445 93.4 1.300 85.4 1.368 81.0 f0.07 f15.1
-
0.839
41.44
f0.89 am
neopentane/SiOn Nz/Rutile Ar/Rutile
1.32 1.49 1.37
10.2 0.841 39.1 25.77e 24.5 23.6ge
de Boer t-curvec de Boer t-curved
1.37 1.316
49.6 4.25 53.0 4.11
Nz/Sterling FT(G) Nz/Vulcan G Nz/Vulcan untreated
0.70 0.79
501
1.10
104
range
C
0.1-0.75
102.6 30.0
0.08-0.80 0.08-0.80
49
184.8 177.3 179.3 180.47 f3.5
0.1-0.80 0.1-0.75 0.06-0.75
91 101 98 96.7 f5.1
112.2e -
0.15-0.70 0.01-0.7 0.15-0.75
6
-
0.08-0.7 0.08-0.7
410
range
a, ratiob
refs
1.00
0.1-0.18
1.17
15, p 92; 20
-
0.1-0.26
1.13
15, p 99; 21
0.05-0.35
1.11
16
167.9 162.8 166.0 165.6 f2.6
0.04-0.20 0.04-0.20 0.03-0.20
1.10 1.09 1.08 1.09 fO.01
94.5' -
0.15-0.30 0.1-0.29 0.08-0.3
1.17
1.19 1.20
25 23 23
-
0.08-0.26 0.08-0.33
1.16 1.16
22, p 33 13
12.06 74.2
0.05-0.14 0.05-0.17
0.85 0.95
14, lab H 14, lab H
82.0
0.05-0.25
1.06
24
-
0.05-0.15
1.02
25
mg g-1
37.05
cm3 g-1 38.5 37.0 38.1 38.0 f0.6 am
0.719 21.69 16.6ge
14, lab H 14, lab B 14, lab G
tm
2.364 16.18
10.30 70.44
0.05-0.31 0.05-0.36
19.74
86.6
0.05->0.4
100
g-'
am
1.32
0.741
82.5 3.67 3.54 cm3 g-1 2.77 323 17.04 347
cm3
neopentane/ carbon
10.26
a, m2 g-1
am
tm
carbons
rima pmol m-2
0.1-0.80
mg g-' 41.1
41.34 cm3 g-1 42.43 40.71 41.18
BET
-
0.712
18.83 am
0.005-0.7
130
0.699
an, is given variously as indicated in units of pmol (where the surface area has been estimated by the conventional BET equation), cm3 g-l, CY,,, = y / [ u ( p / p o= 0.4)], or t the thickness of the adsorbed layer (A). Ratio: case VI/BET. Calculated by the authors. Calculated in this work. e Based on sample weight of 39 g. -2
0
P 0.5
/I" I
Figure 3. (a) Comparison of the experimental and calculated adsorption isotherms for adsorption of NPon nonporous silica: points and full line, experimental; dotted line, calculated from eq 2 with p*/pO = 1.564, C = 62.2,n, = 11.97 pmol m-2. (b) Deviation between the experimental and calculated values. The dotted lines indicate the precision (h0.05pmol m-2) with which the experimental data are tabulated.
experimental samples or they may be caused by systematic errors in the measurements themselves. More extensive data would be needed to decide whether the values of p * / po are equal for N2 and Ar. The results for carbons present interesting features. It
is well known that for very homogeneous graphitic surfaces stepwise isotherms, indicating a layer by layer mechanism, are found. This cannot be represented by the BET or any other theory in which higher layers begin to form before the first layer is complete. Sterling FT(G) represents the most homogeneous graphitized carbon power available: its nitrogen isotherm exhibits some evidence of incipient stepwise behavior, and the BET graph shows slight curvature throughout the usual BET range. The BET area obtained by Lab H using the curve between p / p o = 0.05 and 0.17 is 12.06 f 0.04 m2 g-l. The case VI equation with p * / p o = 0.70 extends the linear range to 0.31 and leads to a lower surface area of 10.30 m2 g-l, a 15% reduction compared with the BET value. Vulcan G is also believed to be largely homogeneous, though less so than Sterling FT(G), and with p * / p o = 0.79 the linear curve between 0.05 and 0.36 yields a surface area only 5 % lower than that from the BET equation. Untreated Vulcan, which is nongraphitic and thought to be heterogeneous, requires a value of p * / p oof 1.10 and yields a surface area now 6% higher than the BET value. These observations lead to speculation as to the reasons for the difference of behavior between carbons and oxide surfaces. Particularly noteworthy is the very wide range of p/pO over which the system neopentane/carbon follows the case VI equation and the fact that p * / p ois the same for the adsorption of neopentane on both silica and carbon. Other Multilayer Equations In recent years, there has been increasing emphasis on the use of the Frenkel-Halsey-Hill (FHH) e q u a t i ~ n ' ~ - ' ~ for multilayer adsorption: In (PIPo) = - A ( ~ Z " ) - ~ (11) where the parameter B is related to the rate of fall-off of
Langmuir, Vol. 6, No. 12, 1990 1733
The Case for Langmuir's Case VI the adsorption potential with distance from the surface and A to the difference between the constants of the van der Waals potential for interaction of a molecule in the liquid film with the solid substrate and that between two molecules in the liquid. For van der Waals forces, B is expected to have a value of 3, although in practice values close to 2.7 are frequently observed. According to Carrott, McLeod, and Sing27 the FHH equation when applied to N2 adsorption by nonporous oxides represents the data well in the relative pressure range 0.3-0.4 up to 0.7-0.92. It is therefore of interest to establish the relationship between case VI and FHH, both of which appear to fit experimental data in the multilayer region. To do this, we have calculated "synthetic" FHH isotherms for a series of values of B and analyzed these isotherms in terms of case VI. With an appropriate choice of p * / p o ,good linear graphs were obtained in the relative pressure range 0.010.75. There is a smooth relation between B and the bestfit values of p*/pO as shown in Figure 4. The close similarity in form between the FHH equation and case VI over much of the relative pressure range seems to be fortuitous since there is no obvious algebraic link between them nor between the models on which they are based. An alternative modification of the BET equation limits the number of layers which can form on the surface.28By varying this number, a series of isotherms are obtained which are superficially similar to those corresponding to different values of p * / p o . However, since this limitation is supposed to be imposed by the pore structure of the adsorbent, this model is inappropriate in the case of nonporous adsorbents.
Discussion The above analysis establishes the case VI equation, at least as an empirical multilayer adsorption isotherm, as greatly superior to the conventional BET equation. Todecide whether it is has more than empirical value, it is necessary to establish whether the parameters have a physical significance. To do this, one needs accurate adsorption measurements over the full range of relative pressures and a wide range of temperature so that enthalpic and entropic contributions can be assessed. Unfortunately, there do not seem to be any suitable numerical published data for nonporous solids which both extend into the multilayer region and cover a sufficient range of temperature. The theoretical basis of the case VI equation has been criticized in that p * / p o > 1.0 is said to imply that the energy of adsorption in the higher layers is smaller than (27) Carrott, P.J. M.; McLeod, A. I.; Sing, K. S.W. In Adsorption at the Cas-Solid and Liquid-Solid Interfaces; Rouquerol, J.; Sing, K. S. W., Eds.; Elsevier: Amsterdam, Oxford, New York, 1982; p 403. (28) Bruneuer, S.;Deming, L. S.; Deming, W. E.;Teller, E. J . Am. Chem. SOC.1940,62, 1723.
,6" -
2t
P"
.
Figure 4. Relationship between p*/pO from case VI and B of the FHH equation.
the energy of condensation of the liquid and that this is physically implausible. However, as Anderson pointed out,' this argument ignors the influence of entropy factors, which can only be calculated if one knows the temperature dependence of p * / p o . If the entropy of molecules adsorbed in multilayers is less than that of molecules in bulk liquid, then this will militate against multilayer formation. The fact that p * / p o > 1.0 means that the formation of an infinitely thick multilayer (i.e., capillary condensation to the liquid) can occur only at p p* > po. However, case VI refers to adsorption on a plane surface, and a change of mechanism to capillary condensation caused by curvature of the surface of the adsorbed layer usually intervenes, so that the case VI model breaks down well below p*. According to the present analysis, this occurs for many systems in the range of relative pressures between 0.7 and 0.8. It is interesting to note that this corresponds very roughly to the point at which the total adsorption is about twice the monolayer value. As will be shown in the following paper,29for porous solids the breakdown of the case VI equation occurs near the relative pressure a t which the hysteresis loop closes, and this again suggests that capillary condensation is the reason for its failure. The status of the FHH equation must also be examined. Although it is asymptotically correct for adsorption on a plane surface close topo, its applicability over a wider range of relative pressures is probably fortuitous. Thus one to two molecular layers are hardly likely to be modeled as a slab of liquid. Moreover, for the systems considered here the index B varies from system to system, which would imply a different form of the adsorption potential as a function of distance from the surface. As will be shown in the following paper, B for adsorption on porous solids is also temperature dependent, which, if B has the significance attributed to it in the theoretical derivation, would mean that the exponent of the van der Waals potential changes with temperature; this seems physically unlikely.
-
(29) Burgess, C.C. V.; Everett, D. H.; Nuttall, S. Langmuir, following paper in this issue.
