THEHILLEQWATION FOR ADSORPTION ON UNIFORM SURFACES ethanol, and 2-propanol,21 respectively, at 77°K. It can therefore be seen that not even in the case of 2-propanol can abstraction by thermal H atoms be of significance in the radiolysis mechanism. The formation of glycol from radiolysis at 77”K, at least in the case of ethanol, has been shown to proceed exclusively ‘Ombination Of during warmup;’8 the present study indicates that these radicals must orig-
1955
inate either by hot radical reactions or via an ionic mechanism. It furthermore seems likely that formation of aldehydes in the above radiolysis proceed, at least in part, by disproportionation reactions of “freely diffusing” H atoms with “trapped” alcohol radicals. (21) R. H. Johnsen and D. H. Becker, J . Phys. Chem., 67, 831 (1963).
The Hill Equation for Adsorption on Uniform Surfaces by Conway Pierce Department of Chemistry, University of California, Riverside, California 92602 (Received January 29, 1968)
Isotherms are reported for carbon tetrachloride adsorption by a uniform-surface graphite at temperatures from 0 to -41.5’. Below the freezing point, second-layer adsorption is inhibited by decrease in temperature, as previously observed for benzene. Isotherms for four temperatures are used to test the Hill equation for mobile adsorption with lateral interactions. At coverages below 0.5, the equation provides the best description of the isotherm yet proposed, but it is not applicable when 0 > 0.5. It is shown that the equation is not sufficiently sensitive to the value of V , to be used for estimating this quantity.
I n 1946, Hill’ developed an isotherm equation for mobile adsorption, based on the use of a two-dimensional van der Waals equation of state for the adsorbed vapor. This equation may be written in the form
where p is the equilibrium pressure, 8 = V/V,, KL,is a constant related to the heat of adsorption, and K1 is defined in terms of the two-dimensional van der Waals a2 and b2 by
Later, de Boer2 made an extensive study of this equation and showed how relative effects of surface forces and lateral interactions affect the shape of the isotherm. Neither Hill nor de Boer had available a t the time good data for first-layer adsorption on a uniform surface to use for testing the equation, but they were able to predict that the isotherm for a uniform surface would be initially convex to the p axis and would rise steeply with increasing coverage, a shape quite different from that of the concave BET-type isotherms obtained from all of the adsorbents in use a t the time. Later, when uniform surfaces became available for study, it was found that the characteristic isotherm did have the convex shape predicted,
particularly when measured at sufficiently low temperatures that effects of thermal agitation do not mask those due to lateral interactions. Ross and associatesa were the first to make quantitative tests of the Hill-de Boer equation by fitting it to experimental data. This was done by rearranging to a linear form
e
e
KIB - In Kb = -+lni-e i - e
-
In P = W P ,e> (3) so that a plot of W os. 6 is a straight line whose slope is K1 and whose intercept a t zero coverage is -In k b . In tests with various isotherms on graphite surfaces, they found that a W-8 plot is indeed linear up to a coverage of about 0.5. Avgul and associates4 have also shown that their graphite isotherms for CC14 and C(CH3)4give a W-0 plot that is linear. (1) T.L. Hill, J . Chem. Phys., 14,441 (1946). (2) 3. H. de Boer, “The Dynamical Character of Adsorption,” Clarendon Press, Oxford, 1953. (3) W.D.Machin and S. Ross, Proc. Roy. SOC.,A265, 455 (1962); 8. Ross and W. W. Pultz, J . Colloid Sci., 13, 397 (1958);S.Ross and W. Winkler, ibid., 10, 319,330 (1955); S. Ross and J. P. Oliver, “On Physical Adsorption,” Interscience Publishers, Inc., New York, N. Y., 1964. (4) N. N. Avgul, A. V. Kiselev, I. A. Lygina, and E. A. Mikhailova, Bull. Acad. Sci. USSR, Div. Chem. Sci., 717 (1962); Trans. Faraday SOC.,59,2113 (1963).
Volume 7.8, Number 6 June 1968
CONWAY PIERCE
1956 At first sight these results appear to provide confirmation for the validity of the equation, but on closer examination some questionable points are raised. (1) The values of K I reported by Rossafor various temperatures show far greater variation than is expected if the relation between K 1 and T is that of eq 2. (2) Both Avgul and Machin used what appear to be unreasonably high values for Vm in computing 8. Avgul finds the best linear plot for W when V , for CCL is taken as 6.1 pmol/g, whereas his BET value is only 4.5 pmol/g. Machin's isotherms for carbon tetrachloride were not carried to completion of the first layer, but comparison of his computed V , values with our isotherms shows that they are much larger than our point B values. (3) Machin's lower temperature isotherms (which are the most important ones for testing theory, since they are least affected by thermal agitation) appear to be somewhat in error. When isoteric heats are computed for various coverages, no consistent values are obtained and some are obviously quite unrealistic. It is probable that the errors are due to use of RIcLeod gauge pressure measurements, an error that was not recognized a t the time the work was done. As shown more recently,6 the pumping effect of streaming mercury vapor can cause rather large errors under some conditions. I n view of these uncertainties in previous tests of the applicability of the Hill equation to experimental data, it seems desirable to make further tests, using a diaphragm gauge6 that appears to give more reliable pressure measurements than RIcLeod gauges.' Isotherms of carbon tetrachloride on graphitized carbon black of highly uniform surface are used for this study. This adsorbate was selected for two reasons: (1) its large spherelike molecules should yield isotherms free of orientation effects, and (2) the previous data of Machin and Ross3 and of Avgul and associates4 are available for comparison with the present results. Experimental Section The adsorption system, shown schematically in Figure 1, is the same as previously used.* It consists of two separate regions, the manifold, M, between valves 1 and 6 and the sample-tube region, S, to the right of valve 6. A 30-torr gauge, G2,is used for M ; a 3-torr one, GI, for S. The reference sides of both gauges are kept a t a high vacuum by continuous pumping. Valves 1-6, which come into contact with adsorbate vapor, are a brass-bellows# type with Viton O-ring seals. Conventional glass stopcocks are used in those parts of the system where there is little or no contact with organic vapors. Bulb V has a standard volume of 114.9 ml to use in calibration if needed. Normally, however, the manifold is calibrated in terms of milligrams of vapor per torr by adsorbing measured doses in a charcoal-filled bulb attached a t valve 2and weighing, The Journal of Physical Chemiatry
DlWERERTlAL VAPOR PRESSURE THERMOMETER
8lMPLt
Figure 1. Schematic diagram for adsorption systems. The insert, lower left, shows the design of a differential vapor pressure thermometer for temperature control.
a procedure which eliminates the need for gas-law corrections. The sample tube and a po bulb filled with pure adsorbate are immersed in a constant-temperature bath of petroleum ether (bp 30-75") cooled by aspirating liquid nitrogen through a copper coil, as described by GrahamlO and Wood." The differential vaporpressure thermometer which activates a Thermocap relay is shown at the lower left of Figure 1. An essential feature is a crossover between the two arms of the manometer, which is left open until the bath is a t the proper temperature, then is closed. When NH3 or SOZ is used as the vapor, the cross connection may be a greased stopcock, but for organic vapors a mercury seal is preferred, even though adjustment of this is somewhat more dificult than adjustment of a greased stopcock. The bath temperature is monitored by frequent po readings on Gz. Regulation to keep p o constant within 0.01 - 0.03 torr is readily achieved. In operation each dose of vapor is measured in M then distilled into S by surrounding T2 with liquid nitrogen and opening valve 6. If there is a measurable pressure at this stage (due to leakage of air), the system is pumped. Valve 6 is then closed and the nitrogen bath removed from Tz. During adsorption the pressure changes are followed by readings of GI. At pressures above 0.1 torr or so, a steady-state pressure is reached in 15-60 min. In case of doubt that the steadystate represents true equilibrium, a small amount of (5) C. Meinke and G. Reich, Vacuum, 13, 579 (1963); P. H.Carr, ibid., 14, 37 (1964). (6) MKS Instruments, Inc., Burlington, Mass. (7) N. G. Utterback and T. Gritlith, Jr., Rev. Sci. Instrum., 37, 866 (1966). (8) C. Pierce and B. Ewing, J . Phys. Chem., 71, 3408 (1967). (9) Granville-Phillips Co.,Boulder, Colo. (10) D. Graham, J . Phys. Chem., 66, 1815 (1962). (11) T.A. Wood, J . Chem. Educ., 44,423 (1967)
THEHILLEQUATION FOR ADSORPTION ON UNIFORM SURFACES vapor is removed by desorption to M, and the equilibrium pressure is then approached from the opposite direction. The amount of vapor desorbed to M is, of course, subtracted from the total of the doses admitted to s. At very low pressures, attainment of equilibrium may require a long time, since the mass transfer of vapor a t these pressures is a slow process. Even standing overnight may fail to reach the true equilibrium pressure, although a steady state appears to be reached. Consequently, the final readings a t such pressures are always obtained after desorbing some vapor from S, but even so there is often an uncertainty of 1-3 1.1 in the measurements. The right-hand portion of Figure 1shows the arrangements used for introduction of helium (for calibration) and other tank gases. The tube dipping into a mercury reservoir is a safety device serving also for rough pressure measurements. The adsorbent for the present study is the same sample of graphitized carbon black MTg used in previous work.* Provision of a wide-bore sample tube eliminates need of a correction for thermal transpiration. The carbon tetrachloride used as adsorbate was purified in a preparative vpc column12 and shows no side peaks when tested in an analytical column. Isotherm temperatures and the observed PO values are given in Table I. The absolute values of both temperatures and pressures may be slightly incorrect but neither appears to be seriously in error, since when they are used to compute heats of vaporization for CCh we obtain self-consistent values for the various temperatures and good agreement with heats computed from published vapor pressures. Table I Temp, "C
PO, torr
0
-19.7
32.7 10.00
-25.2 -41.5
6.75 1.90
eI
*
1957
0
C
i%/B
0.2
o -41.5'
-
p(forr) 0.4
0.6
0.8
1.0
Oi2
PIP0 Oi4
25.8'
Oi6
pared with calorimetric values given by AvguL4 Plots of our values computed from the 0 and -19.7' isotherms (open circles) and his values (solid line) are given in Figure 2E. The agreement shows that our data are quite accurate. Similar agreement is found for the two lower temperature isotherms, but the spread of points is somewhat greater due to the uncertainty in pressure measurements for the -41.5' isotherm.
Results and Discussion The temperatures used were chosen so as to give two isotherms above the melting point of pure adsorbate and two below. This was done in order to ascertain whether or not there is an effect similar to that previously observed for benzene* in multilayer adsorption a t subfreezing temperatures. Plots of the experimental data are given in Figures 2A-D, as actual pressures for the first-layer region and reduced (relative) pressures for the total isotherms. Both sets of isotherms were tested for accuracy by using the pressure ratios a t two temperatures to compute isosteric heat vs. coverage curves,18 which were com-
Since isotherms of the shape of those obtained for CC14 do not give reliable BET plots, it is necessary to estimate V , by visual inspection. Our estimated values are 39 pmol/g for the 0 and - 19.7' isotherms and (12) Provided by courtesy of Professor Corwin Hansch, Chemistry Department, Pomona College, Claremont, Calif. (13) This is a very sensitive test which we use routinely for both our own data and those of others when isotherms are available a t two temperatures. The general shape of the heat-coverage curve for uniform surfaces is well known and consequently errors due to incorrect isotherms are readily detected. Such errors have been noted in many published isotherms that otherwise appear to have the proper shape. The discovery of an anomalous peak in our own heat curve for cyclohexane first made us aware of the serious error that may occur in McLeod gauge measurements.
Volume 7'8, Number 6 June 1968
CONWAY PIERCE
1958 40 pmol/g for the two at lower temperaiures. The latter corresponds to a cross section of 38 A2/molecule if we use an area of 9.1 m2/g for MTg, as previously discussed.6 Using the conventional nitro en area of 7.65 m2/g leads to a cross section of 31.5 f 2 . We feel that the higher figure is the more reliable. If so, it indicates a sufficiently loose packing at completion of the first layer to permit free rotation, as discussed by Avgul. The isotherms of Figure 2D for adsorption below the melting point show an unexpected effect of temperature on second-layer formation. I n view of the spherelike shape of the ccl4 molecule, it was anticipated that in the multilayer region the isotherms would behave like those of Prenzlow and Halsey14 for argon adsorption at subfreezing temperatures. Their isotherms for various temperatures cross at each tread and each step, with the relative pressure at the midpoint of the second-layer step remaining the same for all temperatures. The cCI4 isotherms are more like those we found for benzene,s with a retardation of second-layer adsorption as the temperature is lowered. The midpoint of the CCh second-layer step shifts from a relative pressure of 0.75 at -25.2' to about 0.825 at -41.5'. Apparently there is the same kind of imperfect-crystal effect in formation of a solid second layer as there is for benzene. Using the procedure of Rossa and A ~ g u la, test ~ of the applicability of the Hill equation was made by using the data to construct a W-8 plot, according to eq 3. Results for the four isotherms of Figure 2 and for a 20" isotherm, not shown, are given as open circles in Figure 3A. We find that (1) the data for all temperatures fit straight lines up to B 0.5 but not at higher coverage and (2) except for the somewhat questionable one for the isotherm at -41.5", the W plots at all temperatures are, within experimental error, parallel to one another. Since K1 is measured by the slope of the W plot, this means that K1 does not change with temperature, as indicated in eq 2. The values found for the constants Kb and K1, which were obtained by using the slopes and intercepts of the W plots are given in Table 11.
-
Table I1 Temp. O C
Kb
20 0 -19.7 -25.2 -41.5
7.4 2.39 0.67 0.41 0.13
K:
5.80 6.20 6.30 6.20 6.60(?)
Another test, suggested by Machin and Ross, is the variation in constant Kb with temperature. They showed that if eq 1 is valid, the limiting isosteric heat The Journal of Physical Chemistra,
Figure 3. (A) W-e plots by eq 3: open circles, from isotherm data; open triangles, by fictitious V , for 0 and -19.7' isotherms: crosses, from the fictitious isotherm of Figure 2C. (B) Variation in Kb with T: closed circles, present values; open circles, from Machin and Ross.
of adsorption at zero coverage (where there are no lateral interaction contributions to the heat) must be given by
(4) (the minus sign is inadvertently omitted in their publication). From this it follows that a plot of In Kb (or log Kb) os. 1/T must be a straight line, since pst should be essentially independent of temperature. Using our values for Kb, shown as solid circles in Figure 3B, we find that the plot is indeed linear, with all points falling on or very near to the line. Moreover, Machin's values, shown as open circles, fall near the same line. The slope of the plot gives a limiting heat of 9.0 kcal/mol, a figure in good agreement with the value of 8.5 obtained by extrapolating the curve of Figure 2E to zero coverage. As a final test, theoretical isotherms for the various temperatures were calculated by eq 1, using the values of K1 and Kb computed from the W plots. The calculated points for the 0 and - 19.7"isotherms are shown as solid triangles in Figure 2A (results for the isotherms at other temperatures are similar). I n every case the calculated points are in quite good agreement with experimental up to about 8 = 0.5, but, as expected from the behavior of the W plots, the agreement does not hold at higher coverage. At all temperatures the calculated isotherm starts to level out below the experimental curve, and by the time 8 reaches 0.8 or so, the calculated pressure for a given coverage is several times the observed. Actually, eq 1 cannot be expected to apply at very high coverage, since it requires that p approach infinity as 8 approaches unity. It must, therefore, cease (14) C. F. Prenalow and G. D. Halsey, Jr., J . Phya. C h m . , 61, 1168 (1967).
THEHILL EQUATION FOR ADSORPTION ON UNIFORM SURFACES to be exact at a coverage somewhat less than 0 = 1. Our results for CC14 and the former ones of Ross and Winkler3 suggest that the practical limit of applicability may be near one-half coverage (at a point about midway of the first-layer step) but no definite value can be selected at this time until data are obtained for other adsorbates. It would appear, however, that application of the Hill equation at coverages high enough to reach the sharp bend near completion of the first layer, as has been done by others, is probably incorrect. On the basis of our results and the others cited it appears that the Hill equation provides the best d e scription of the isotherm for adsorption on a uniform surface of any yet proposed. It correctly describes the initial convex shape of the isotherm and the steep rise due to lateral interactions. When used in the linear form, it provides a simple means for evaluating the two constants by plots from experimental data. Theoretical isotherms computed by the equation fit experimental data quite closely. Most convincing of all, the equation correctly predicts how the constant KI, (and the isotherm) changes with temperature. There are, however, certain limitations that should be kept in mind in using this equation. (1) Despite the fact that the equation is based on a two-dimensional van der Waals equation of state for the adsorbed vapor, the excellent agreement between theoretical and experimental isotherms does not necessarily prove that the van der Waals equation is the correct one to describe the adsorbed phase. Our basis for this statement is the fact that KI is not temperature related as pre-
1959
dicted by eq 2, which is based on the van der Waals equation. (2) The equation fits experimental isotherms only at coverages below approximately onehalf, and attempts to apply at higher values may lead to false conclusions. (3) The equation is not sensitive to small changes in either the value of 0 or the equilibrium pressures of the isotherm at intermediate coverage, owing to the fact that in the linear form the function W is the sum of three separate terms. To investigate variations in e, we computed fictitious W values from the 0 and -19.7" isotherms by assuming a V , value of 50 pmol/g, in lieu of the observed 39 pmol/g. This change had practically no effect on either the slope or the intercept of the plot. We find that the fictitious points, represented by open triangles in the plots of Figure 3A, all fall on the same straight lines as the correct ones. It is obvious, therefore, that the equation cannot be used to evaluate V,, since quite large variations in this quantity have so little effect on the W plots. The effect of variations in pressure measurements was similarly investigated by using the fictitious -19.7" isotherm, shown as a dotted line in Figure 2C, to construct a W plot. These computed points, indicated by the symbol X in Figure 3A, lie just above the real plot, so as to give a line of essentially the same slope and a slightly changed intercept.
Acknowledgment. This work was supported in part by a grant from the Petroleum Research Fund administered by the American Chemical Society. Grateful acknowledgment is hereby made to the donors of this fund.
Volume 72,Number 6 June 1888
