A NEW INTERPRETATION OF THE ADSORPTION ISOTHERM G. E. CUNNINGHAM Department of Chemistry, Clarlcson College of Technology, Potsdam, New York Received June 14, 1934
To date, the most satisfactory theory which has been advanced to account for the relation of gas adsorption to pressure is that of Langmuir (1). This theory assumes that the time rate of condensation is proportional to the pressure and to the amount of surface remaining bare, and that the time rate of evaporation is proportional to the amount of gas adsorbed, which is in turn proportional to the amount of surface covered (assuming a monomolecular film). A mathematical equation is derived which is fairly simple for a surface made up of only one kind of elementary spaces, but becomes quite complex with an increase in the number of kinds of elementary spaces; so that, even for a surface comprising only two kinds of spaces, the evaluation of the constants in the mathematical equation becomes, in the words of Langmuir, “a somewhat laborious method of trial.” As Langmuir has pointed out, the above theory is not to be condemned simply because the equations are complicated. However, it is the purpose of this paper to propose a theory which is susceptible to a less complicated mathematical analysis and which, instead of requiring an intimate knowledge of the nature of the adsorbing surface in order to establish the mathematical function, will, if the theory be valid, give valuable information concerning the nature of the surface. This theory requires a slightly different picture of the kinetics of the adsorption process from that assumed by Langmuir. Let us assume, as does the Langmuir theory in its simplest form, that the adsorbing surface may be divided into elementary spaces, each capable of holding one gas molecule. That is, an elementary space may be defined as a space whose attractive, or adsorbing, force is entirely satisfied by the adsorption of one gas molecule. Contrary to the Langmuir theory, however, instead of assuming that an actual colEision between a gas molecule and such a space must take place entirely through the inherent kinetic energy of the molecule, let it be assumed that it is only necessary for the gas molecule, through its molecular motion, to come within a certain radius of attraction of one of the spaces. This seems to be a reasonable assumption, for, if a space on the surface has enough attractive force to hold a molecule after a collision, it surely can attract the molecule through a small 69
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distance. As soon as a gas molecule comes within this radius of attraction, it will be adsorbed and the attractive force will be temporarily satisfied. However, that will not prevent a second adsorbable gas molecule from coming within the required distance of the elementary space while the first is still held. As soon as the first molecule is released, the second will be adsorbed. At a given pressure, a certain fraction of the elementary spaces having a given adsorbing force will always have a t least one gas molecule within their radii of attraction, which, on the basis of this theory, is tantamount to saying that a t a given pressure a certain fraction of these spaces always have gas molecules adsorbed upon them. R o m this point of view, the kinetic treatment proposed by Langmuir does not hold for the reason that the time rate of desorption, that is, the number of spaces actually becoming bare per unit time, is not independent, of the pressure. The chance for a given elementary space to be covered depends upon the chance of its having an adsorbable gas molecule within its radius of attraction. This is proportional to the pressure. The theory may be developed mathematically in the following manner: Let the different kinds of elementary spaces on the adsorbing surface be classified merely on a basis of the pressures a t which they become saturated, the saturation pressure of a given kind being defined as the lowest pressure a t which there is always an adsorbable gas molecule within the range of attraction of every elementary space comprising that kind. Let it be assumed that the amount of gas adsorbed upon each kind of spaces is proportional to the pressure and to the total number of spaces comprising that kind. Let X = total amount of gas adsorbed a t pressure P, z1 = amount adsorbed on first kind of elementary spaces, q = amount adsorbed on second kind of elementary spaces, and zn = amount adsorbed on nth kind of elementary spaces. Then, up to P,, the pressure a t which the nth kind of space becomes saturated, xn = C,A,P where C, is a constant for the given kind of space, and A , is the total number of spaces comprising the kind, and is constant for the given quantity of adsorbent. That is, C A is a parameter constant for a given kind and different for different kinds. It will be referred to as the coefficient of adsorption for the given kind of spaces. dx Then, for the given kind of space, -2 = constant, and dP
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including all kinds of spaces not saturated a t pressure P. However,
e
# constant, but diminishes as the respective kinds of spaces become dP saturated. It should be theoretically possible to express the total number of elementary spaces belonging to each kind as a function of the saturation pressure. However, that would require, in order to establish the mathematical equation, a knowledge of the value of the constant, C , for each kind. The values of C are different for different kin& of spaces, because the attractive forces are different; if the attractive forces were not different, all spaces would become saturated a t the same pressure. In order to obviate the above difficulty, it is convenient to express CA, rather than A , as a function of the saturation pressure, that is, to let
CA = f ( P )
(3)
We then have
and dX = J p 8 f ( P ) d P where P, is the pressure at which the entire surface becomes saturated. At any given pressure P, X , the total adsorption, is the sum of the amounts adsorbed on all kinds of spaces up to pressure P. Therefore
X
=~
p
~
f(P)dPdP p 8
As a matter of convenience in demonstrating the application of the theory, let it be hypothetically assumed that the distribution of the adsorption coefficients wit'h regard to pressure follows the probability law (2). That is, let
where the pressure is measured positively and negatively from the value corresponding to the maximum value of C A , or
where the axes have been shifted to avoid negative values of P and P,
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G . E. CUNNINGHAM
is the new coordinate of the pressure corresponding to the maximum value of C A , Then
Since the probability equation may be integrated only by expanding into an infinite series and integrating each term separately, it is more convenient to integrate mechanically. Figure 1, curve 1, shows the distribution of Cd with regard to saturation pressure when K in equation 9 is arbitrarily assigned a value of 0.5. Theoretically, of course, P, = m .
R-G’€SSUX?€ /Pe/u//be)
FIG.1. HYPOTHETICAL CURVE^ Curve 1, distribution of adsorption coefficients with respect to saturation presdX sures. Curve 2, - versus P. Curve 3, adsorption isotherm, X versus P. dP
However, in the figure, the origin has been shifted t o that pressure a t which CA = 0 within the limitations of the scale. Figure 1, curve 2, is obtained by plotting against pressure, P, the area under curve 1 from P to P,, the pressure a t which the entire surface bedX comes saturated. It represents the relation of - to P. Figure 1, curve dP 3, is obtained by plotting the area under curve 2 from 0 to P. It represents the relation of X to P or, in other words, is the hypothetical adsorption isotherm. Only the central part of this curve resembles an actual isotherm curve, which means, of course, that the hypothetical assumption is not the correct one. By a method of trial, curve 1 doubtless could be altered in such a way as to make curve 3 more nearly resemble an isotherm
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73
curve. This is unnecessary, however, for it is more convenient to attack the problem in the following manner: Since curve 2 is obtained by integrating curve 1 and curve 3 by integrating curve 2, in any actual case curve 2 may be obtained from curve 3 and curve 1 from curve 2 by the reverse Drocess of differentiating. I n other dX words, in figure 1 curve 3 represents X versus P, curve 2 represents dP d2X versus P, and curve 1 represents - - versus P, the minus sign being dP2 dX introduced on account of the fact that - decreases as P increases. dP
PPESSUE€, Bum
FIG.2. ADSORPTION OF METHANEON GLASS (LARGMUIR) dX d2X Curve 1, adsorption isotherm. Curve 2, - versus P. Curve 3, - -versus P, dP d P2 Le., relation of adsorption coefficients to saturation pressures.
Figure 2, curve 1,represents the isotherm for the adsorption of methane on glass, taken from the data of Langmuir (1). Figure 2, curve 2, repredX versus P, the sents - versus P and figure 2, curve 3, represents dP dP2 differentiation being done meehanically in each case. This shows a maximum value of the adsorption coefficient (negative of second derivative) a t a saturationpressure of approximately 1.7 bars, which means that there is a maximum amount of adsorption on spaces becoming saturated a t that pressure. Since the curve is so steep on either side of that pressure, the glass surface may be regarded as composed of elementary spaces which are nearly all of the same kind. Figure 3 represents a similar treatment of the data of Langmuir for the
-
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G. E. CUNNINGHAM
adsorption of methane on mica. There are two maxima in the
- d2X dP2
versus P curve, indicating two principal kinds of elementary spaces on the mica surface. The same conclusion was reached by Langmuir by his method of treatment.
dX versus P. Curve 3, dP Le., relation of adsorption coefficients t o saturation pressures. Curve 1, adsorption isotherm. Curve 2,
-E xversus P, dP2
Figure 4 represents a similar treatment of the data of Langmuir for the adsorption of nitrogen on mica. Curve 3 again shows two principal kinds of elementary spaces on the mica surface, whereas the Langmuir equation is satisfied by assuming only one kind. Since the mica specimens for the two experiments were prepared in exactly the same way, it is not unreasonable to expect similar adsorption phenomena for two inert gases. ,
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If the theory proposed in this paper be correct, the CA-P distribution should be represented by the negative of the second derivative of any equation, theoretical or empirical, which absolutely satisfies the adsorption isotherm. The Langmuir equation applied to this case is of the form
X = - abP
1+aP
N/TPOGEN Off M/CA
;:+--PO
PPECj’SfJPE, 50ri
FIG. 4. ADSORPTION OF NITROGENON MICA (LANGMUIR) Curve 1, adsorption isotherm:
0,experimental
+,
values; Langmuir’s calculated d2X values. Curve 2, &?versus P. Curve 3, - - versus P, i.e., relation of adsorption dP d P2 coefficients t o saturation pressures. Curve 4, negative second derivative of Langmuir equation versus P. Curve 5, negative second derivative of Freundlich equation versus P.
where a and b are constants for the given case, and the second derivative of this equation simplifies to
dzX
- 2a2b
I
dP2
(1
+ aP)3
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G . E. CUNNINGHAM
whence d2X --
2a2b
dP2 - (1
+U P ) ~
The 0’s in figure 4, curve 1, represent experimental values, whereas the +’s represent the values calculated from equation 10 by letting a = 0.156 and b = 38.9, the values determined by Langmuir. Figure 4, curve 4, is obtained by plotting equation 12. By smoothing out curve 1 in accordance with equation 10, the inflection in curve 2 and the second maximum in the CA curve are lost entirely, and the first maximum in the CA curve is reached a t P = 0. It is thus seen that in the case under discussion the equation used by Langmuir is reasonably applicable to the present theory only at pressures greater than approximately 10 bars. (Curve 4 practically coincides with curve 3 a t pressures greater than 15 bars.) Moreover, this discussion brings to light the fact that the Langmuir equation fails also to agree with the Langmuir theory in an important respect, One of the principal claims for the Langmuir equation is that it explains the fact that the adsorption is practically a linear function of pressure at low pressures. Without reference to the present theory of adsorption, it is a mathematical fact that the second derivative of a rectilinear equation is zero. The second derivative of the Langmuir equation not only is not approaching zero at low values of P, but its negative actually has its maximum value of 2a2ba t zero pressure. It therefore seems, when the equation is analyzed in this way, that it does not justify this claim made for it, which, however, probably has been amplified in text books, more than the originator intended. On the other hand, the theory proposed in this paper is flexible in that respect, since the isotherm begins to bend downward a t the pressure a t which an appreciable number of elementary spaces become saturated, whether that pressure be zero or not. The accuracy of the interpretation in this and all other pressure ranges depends solely upon the accuracy of the experiments. The limited applicability of the empirical Freundlich equation,
X
=
KP”
(13)
may be represented as follows: From equation 13, d2X
Kn(1 - n) pa-n)
Letting K = 8.4 and n = 0.417, the values assigned by Langmuir, and plotting equation 14 gives curve 5, figure 4. From this equation, the negative second derivative is infinitely large a t zero pressure. A study of this curve indicates that by the proper evaluation of the constants the
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Freundlich equation may be made to conform approximately to the facts only a t pressures greater than that corresponding to the maximum value of the adsorption coefficient. This is a lower pressure in some cases than others, and therefore the empirical equation will apply over a wider pressure range in some cases than others. It will be most applicable in those cases in which a large number of elementary spaces become saturated a t low pressures. Fnrther use of the data of Langmuir has not been made for the reason that in the other cases cited by him the number of experimental values given in the tables is not considered sufficient to warrant an assumption as to the exact shape of the isotherm curve. The data chosen were believed to be the most accurate available, and it is believed that the present theory explains them better than does the Langmuir theory. The opportunity has not been available for a search for suitable data in other sources and the author therefore invites other workers dealing with adsorption isotherms to plot the negative second derivatives of those curves and criticise the theory accordingly. The treatment given in this paper is not regarded by the author as a finished product, but is offered in the hope that it may prove to be the nucleus of extended work in the field. SUMMARY
A new theory of the kinetics of gas adsorption has been proposed. It is assumed that, in order to be adsorbed, a gas molecule must only come within a certain range of attraction of the adsorbing surface, rather than actually collide with the surface through its molecular motion. The amount of gas adsorbed upon a given kind of elementary surface space is proportional to the pressure. Different kinds of elementary spaces become saturated at different pressures. A mathematical treatment has been derived which is in better agreement with published data than is the Langmuir equation. The applicability of the Langmuir equation to the present theory has been discussed and a discrepancy between the Langmuir equation and the Langmuir theory has been pointed out. The applicability of the empirical Freundlich equation to the present theory has been briefly discussed. REFERENCES (1) LANGMUIR, IRVING: J. Am. Chem. SOC.40,1361 (1918). (2) WILSON,E. B.: Advanced Calculus, p. 388. The Macmillan Company, New York (1912).
