STATISTICAL ASPECTS OF CHEMISORPTION BY A. R. MILLER Received 7th March, I950 The basis of the statistical mechanical account of adsorbed monolayers is discussed, and the way in which the properties of the monolayer can be de- duced is indicated. The physical distinction between immobile and mobile monolayers of particles adsorbed on localized or fixed sites is emphasized and important differences in the properties of these two types of films are examined. Tbe way in which the experimental results for adsorption on to clean sur- faces can be interpreted in the light of these ideas is described. For oxygen films adsorbed on tungsten it is shown that these ideas lead to a consistent picture of the way in which successive layeIs of the film can be built up. The application to hydrogen films adsorbed on tungsten is also examined.More recent measurements of the variation of the heat of adsorption of these films suggest that the earlier interpretation of them was oversimplified, although the basic ideas are vindicated. It now appears that hydrogen films adsorbed on tungsten pass through an intermediate region when the surface is about three-quarters covered, and their behaviour changes from that of an immobile film to that of a mobile film adsorbed on localized sites. 1. Introduction.-The classical experiments of Langmuir showed that when a clean surface of a solid or a liquid is exposed to a gas or a vapour, a monomolecular layer of the gas is often formed on the surface. An adsorbed monolayer of this sort is of considerable importance in the study of surface propertics and surface reactions, as well as in the study of the interchange of energy between a solid surface and a gas.The purpose of this paper is to give a review of some statistical aspects of chemisorbed monolayers. Many of these topics are dealt with in my monograph to which the reader is referred for the mathematical details of the argument. Here, I shall restrict myself to considering the basis of the statistical mechanical theory of adsorbed monolayers, and the inter- pretation of the available experimental data in the light of the theoretical deductions. In particular, I want to examine some of the ideas of the statistical theory in the light of experimental results which have become available since my book was written. 2.Physical Model.-Both experimental and theoretical studies have shown that a solid surface provides a periodic Fotential field for an atom or molecule incident on the surface. The potential field near a crystal surface was determined in a number of cases by Lennard-Jones and his collaborator^.^ These calculations provide a potential map of the space as experienced by a gas molecule approaching the surface. The potential field consists of a periodic array of positions of minimum potential energy. The structure of this array is determined by the Langmuir, J . A m e r . Chem. SOC., 1912, 34, 1310; ibid., 1915, 37, 417; Miller, T h e Adsorption of Gases on Solids (Cambridge University Press, Lennard-Jones and Dent, Trans. Faraday SOC., 1928, 24, 92 ; Gen. Elect. Rev., 1926, 29, 153.1949). Lennard Jones, ibid., 1932, 28, 333. 5758 STATISTICAL ASPECTS O F CHEMISORPTION crystal structure of the surface. This work provides a firm basis for the idea of localized sites for adsorption which was advanced by Langmuir. More recently, it has also received direct experimental confirmation. The experiments of Crawford and T~mpkins,~ for instance, on the ad- sorption of a number of gases on barium fluoride crystals have shown that the amount of gas adsorbed in a complete monolayer is independent of the particular gas. Even though ammonia and nitrous oxide mole- cules differ in cross-section by about 50 %, the same amount of each was adsorbed in a complete monolayer. The number of molecules in a mono- layer depends not on their size, for they do not form a close-packed structure on the surface, but on the localized sites for adsorption defined by the potential field provided by the solid surface.This idea of the adsorption of molecules on localized sites is the basis of the examination of adsorbed monolayers using the methods of statistical mechanics. This was initiated by Fowler 6 who derived the adsorption iso- therm statistically on the assumptions that (i) there is a definite number per unit area of localized sites €or (ii) one gas molecule or atom is adsorbed on each site, (iii) the vibrational states of any adsorbed particle are independent (iv) thcre is no interaction between the adsorbed Farticles. adsorption, of the occupation of neighbouring sites, and I t was clearly necessary to extend the theory by taking account of the interactions between the adsorbed particles.6 In this case, assumption (iv) is discarded, but then the regular arrangement of the sites becomes an essential feature of the argument.3. Statistical Theory.-In using the methods of statistical mech- anics to study adsorbed monolayers, the equilibrium between the gas phase and the adsorbed phase is considered. We first consider the case covered by assumptions (i), (ii), (iii) and (iv) of 0 2. If there are N , particles adsorbed on the surface and each is in its lowest state, the energy of the surface layer is - Nux, where x is the difference in energy between the ground state of a particle in the adsorbed phase and the lowest state in the gas phase. If a particle adsorbed on a particular site can occupy a set of states of energy E~ and weight wr, its vibrational partition function is v,(T) = 1 w,.exp (- r,/kT). In the absence of interactions between the adsorbed particles, the energy of the adsorbed phase depends only on the number of adsorbed particles and not on their arrangement on the surface. The partition function of the monolayer can then be written r Let there be N , sites for adsorption on the suriace. where g ( N , , Na) is the number of ways in which N , adsorbed particles can occupy the N, sites available for adsorption. That is g(N,, N J is the weight of the adsorbed state of the whole assembly of energy - hT,x ; for it, we have N , ! g(N3' = N,! ( N , - Na) 1' The contribution of the adsorbed layer to the Helmholtz function of the assembly is Fa = - kT log S.Crawford and Tompkins, Trans. Faraday SOC., 1948, 44, 698. Fowler, Proc. Camb. Phil. SOC., 1935, 31, 260. 6 (a) Fowler, ibid., 1936, 32, 144 ; (b) Peierls, ibid., 1936, 32,471 ; (c) Roberts, Proc. Roy. SOC. A , 1937, 161, 141.A. R. MILLER 59 Consider the gas phase containing N , particles in a volume V in equilibrium with the adsorbed monolayer. The Helmholtz function of the gas phase is where v,(T) is the partition function for the internal degrees of freedom of a particle in the gas phase and the other symbols have their usual meanings. The condition for equilibrium between the gas phase and the mono- layer is This leads immediately to the adsorption isotherm, which can be written in the form 0 h3 exp cxlkl’) u,(T) I - e - ( 2 .r r m ) 3 i z ( ~ ~ ) 5 / 2 * urn * PJ * (4) -- * where 6 = NJN,, has been introduced €or the fraction of the surface covered. So far we have neglected the interactions between the adsorbed mole- cules. Suppose that the interaction between two particles adsorbed on a pair of closest neighbour sites has a fixed value, and neglect the inter- actions between more distant adsorbed particles. Theories of this sort will be referred to as fixed interaction theories. The energy of adsorption of a layer of particles is then where V is the interaction energy between two particles adsorbed on a pair of closest neighbour sites and X is the number of such pairs. The interaction energy now depends on the arrangement of the particles on the adsorbing surface.In other words, the weight of the state having a particular energy depends on the arrangement of the particles on the surface ; and the partition function is - N,x -j- X V , . * ( 5 ) 8 = z g ( N , , No, X ) exp {(N,x - XV)/kT)(v,(T)}Na. * (6) 5 In this expression, g(N,, N,,, X ) has been written for the number of arrangements of No particles on the N, available sites for adsorption so that there are X closest neighbour pairs. An approximate value can be found for S by assuming either (i) that the interacticn energy has its average value@,, for the number of adsorbed particles, or (ii) by using the method of local configurations 6 * ~ 0 in order-disorder theory. Fowler using the forrner approximation, and Peierls using the latter, showed that critical adsorption could occur only when there were strong attractive forces between the particles in the monolayer.I f we use the former approximation, the adsorption energy becomes This expression has not been evaluated exactly. - Na(X + 4ev). The only term in 8 which involves X is then g(N,, N,, X ) and E can be evaluated since The equilibrium states are again determined by eqn. ( 3 ) , and the ad- sorption isotherm is60 STATISTICAL ASPECTS OF CHEMISORPTION Fowler showed that his results were in qualitative agreement with the critical phenomena observed by Cockroft ' in studying the deposition of metallic vapours on to insulating surfaces. However, in systems in which chemisorption occurs, the forces between the adsorbed particles are generally repulsive. Once the equilibrium conditions have been determined in this way, other quantities of interest can be calculated.The heat of adsorption and the way in which it varies as the fraction of the surface covered changes, has been studied widely. The heat of adsorption per molecule is defined as the fall in the energy of the system when one molecule is adsorbed. It is given, as a function of the fraction of the surface covered, by the relation In this case, critical conditions do not arise. in which zt is the energy of a molecule in the gas phase and U(0) is the total energy of the adsorbed film. If 2 is the average number of closest neighbour pairs for a given value of 0 then up) = - N,eX + x(e)v. . - (10) Consideration of the experimental results in relation to simple theor- etical considerations led to an important distinction between the different types of monolayer that can be adsorbed on an array of localized sites.It is necessary to examine these types of adsorbed film, before calculating the variation of the heat of adsorption. 4. Types of Adsorbed Mono1ayers.-The idea of the distinction between immobile and mobile films adsorbed on fixed or localized sites was introduced by Roberts8 In his experiments on the adsorption of hydrogen and oxygen on tungsten wires, Roberts found heats of adsorption of the order of 2 x 105 joule/mole. Taking the vibrational frequency for the adsorbed particles to be of the order of 1olS sec.-l, the frequency of evaporation at room temperature is of the order of I O - ~ ~ sec.-l. These films must therefore be very stable at room temperature.On the other hand, for heats of adsorption of the order of 104 joule/mole. the rate of evaporation would be of the order of 10l0 sec.-l, and quite different behaviour would be expected. In this case, the evaporation of the film, and consequent re-condensation, is so rapid that the particles ad- sorbed on the surface could be expected always to assume an equilibrium (Boltzmann) distribution. For an intermediate value of the heat of adsorption, say, 8 x 104 joule/mole, the rate of evaporation is 0-1 sec.-l. When the heat of adsorption of a system is in this neighbourhood, it is in a transition state between the two extreme cases represented by the other values which have been considered. These orders of magnitude illustrate the two kinds of film adsorbed on localized sites.They can be given formal definitions 8 in the following way. By a mobile film is meant one in which the energy of activation necessary to enable an adsorbed particle to move from a given site to a. neighbouring vacant site is much less than kT, so that the particles move freely from one site to another. This free movement from one localized site to another ensures that the film takes up equilibrium configurations during the occurrence of any process. By an immobile film is meant one in which the energy of activation necessary to enable a particle to move from one site to another is so much greater than kT that, for the times which are concerned in any experimental procedure, the particles may be treated as remaining on the sites on which they are first adsorbed.The particles in such a film will not assume an equilibrium distribution ; Cockroft, Proc. Boy. Soc. A , 1928, 119, 293. Roberts, Proc. Camb. Phil. Soc., 1938, 34, 399 ; see also Miller, ibid., 1947, 43, 232.A. R. MILLER 61 the occupation of sites by the adsorbed particles will be random. It should be noted that not only free mobility on the surface, but also evapor- ation and re-condensation will also set up an equilibrium distribution. The conditions under which evaporation can occur are roughly the same as those under which the adsorbed particles are freely mobile over the surface from one localized site to another. As has already been pointed out there will be intermediate cases. I n these, the particles can be looked upon as being able to move from site to site on the surface but so slowly that there is an appreciable time lag in establishing equilibrium again if it is disturbed in any way. 5.Statistical Theory of the Heat of Adsorption.-The two ex- treme cases which have been specified in $ 4 can be dealt with theor- etically. The methods of statistical mechanics can be used to examine the equilibrium distribution in the mobile film and to examine some properties of the immobile film. Here, we shall determine the equilibrium conditions and show how the variation of the heat of adsorption with the fraction of the surface covered can be determined. For an immobile film in which the particles from the gas phase are adsorbed on single sites, the heat of adsorption varies directly as the fraction of the surface covered.This follows immediately from the fact that in such a film there is a random occupation of single sites. In the adsorption of hydrogen cn tungsten, it is probable that each molecule occupies two sites, one for each atom in the molecule. An immobile film of this sort consists of a random distribution of pairs of sites on the surface. There is not an exactly linear relation between the heat of adsorption and the fraction of the surface covered in this case, but the departure from linearity is not very great. The heat of adsorption of a mobile film is quite different from this. The exact shape of the heat curve depends on the interaction between the adsorbed particles but in all cases it shows a rapid fall in the vicinity of 0 = 0.5.We now show how these results can be obtained by the use of the grand partition function. The grand partition function of the adsorbed layer can be written in the form where Aa is the absolute activity of the particles in the adsorbed phase. It is related to the partial or chemical potential pa by the relation The other symbols in eqn. (11) have the meanings which have been ascribed to them in earlier sections of this paper. Without serious error, as far as finding average values is concerned, the sum in eqn. (11) can be replaced by its maximum term. Denote the values of the parameters in the maximum term by asterisks. Then the grand partition function can be replaced by pa = kT loge Am 3* = g(N,, N,*, X*) exp (- X* V/kT)( A,v,(T) exp (x/kT))Nu* (12) = g(N,, N,*, X " ) exp (- X* V fkT)&va*, where The essence of the method of local configurations is to deal with a small group of sites in detail and to represent the effects of particles adsorbed on sites outside this group by average values.For definiteness, we shall consider a square array of sites, as is provided by the (100) plane of a tungsten crystal, and examine the occupation of a particular site (the central site) and its z closest neighbours (the first shell sites). We introduce average quantities in the following way. Let fav be the geometrical mean contribution per site of the array to the factor has been written for h,v,(T) exp (x/kT).62 STATISTICAL ASPECTS OF CHEMISORPTION let I; be the geometrical mean contribution per pair of occupied sites to the term involving the interaction energy ; and let y be the geometrical mean contribution per site of the array to the combinatory factor.Thus, we define * (13) tF; = IXaVa(T) exp ( x / k ~ ) > ~ a * , y" = g(Nm N,*, X * ) . I;*zNa = qX* = exp (- X*V/kT), Consider a particle adsorbed on a particular first shell site. If the central site is occupied, its interaction with that particle is allowed for by a factor T]. It occupies a site which is a closest neighbour also to z - I other sites ; its possible interactions with particles which may be adsorbed on these sites is represented by a factor P-l, obtained by taking the aver- age value defined in the second of eqn. (13). Each arrangement of par- ticles adsorbed on the group of z + I sites can be considered in detail in this way.Examining the detailed occupation of this group of sites, and using the average values defined in eqn. (13) with regard to the oc- cupation of the N , - z - I sites outside the selected group, the partition function is represented by "(1 + .lgI;"-l)" + (I + 5~Z-l)"]YNIY-Z-1 W - z Z 5 ' (14) where 5 has been written for Aav,(T) exp (x/kT). In considering different modes of occupation of the array, we are concerned only with the ratios of terms in eqn. (14). The last three factors in the right-hand member of this equation can therefore be omitted, without affecting the results which are obtained. Thus, the expression ((1 + s.1)" + (1 + El)", . (15) in which l has been written for @-l, is used for the partition function. This methods is equivalent to that used by Roberts.6c The argument given here is more formal than the physical considerations on which Roberts based his deduction of the essential results.Their equivalence becomes clear on reflecting that successive terms in the right-hand mem- bers of eqn. (II), (14) and (15) give the relative probabilities of different modes of occupation of the sites on the surface. Having constructed an expression for the partition function, the deriva- tion of any quantities of interest can be proceeded with. The equilibrium relations for particles on the surface are obtained by using the fact that each site must be an average site. If a fraction 8 of the sites on the whole array is occupied by adsorbed particles, then the average occupation of both the central and the first shell sites must also be 8.This leads to relations (17) the detailed derivation of which can be found in 0 2.3 of m y monograph. When equilibrium is established with a gas phase in contact with the surface, the absolute activity of an adsorbed particle must be equal to that of a particle in the gas phase. The latter is given by h3 " = (2 mz)% ( kT)610ua (T)' while the former is given by 'a = I exp (- x/kT)/v.(T). 9 Guggenheim, Proc. Roy. Soc. A , 1938, 169, 134.A. R. MILLER 63 When equilibrium between the adscrbed and gas phases is reached, A,, is therefore proportional to the pressure in the gas phase. Eqn. (16) and (17) then give the adsorption isotherm. The value of x is determined by the average occupation of the first shell sites when the central site is occupied.This can be found from the first term of relation (15) to be ~ E J ( I + TCJ. If we use eqn. (9) and (10) with this result, we obtain finally for the heat of adsorption If the adsorbed particles are diatomic molecules which dissociate on adsorption so that a mobile film of adsorbed atoms is formed, the only difference in the expression for the heat of adsorption is that the factor 4 must be omitted from the right-hand member cf eqn. (18). The way in which the heat of adsorption varies with the fraction of the surface covered, according to eqn. (IS), is shown by curve (a) of Fig. I. - R -4.0 FIG. I.-Variation of the heat of adsorption with the fraction of the surface covered, calculated from the statistical theory. Curve ( a ) is for a mobile film on localized sites, and curve ( b ) is for an immobile film.This has been calculated for V = 2-3 x 10-l~ erg per pair of particles and 300OK. This interaction energy corresponds to the difference be- tween the heat of adsorption on a bare surface and on an almost filled surface for hydrogen adsorbed on tungsten. Now let us examine the case in which each particle from the gas phase occupies a pair of closest neighbour sites when it is adsorbed. The grand partition function is again replaced by its maximum term, and we consider the same group of z + I sites as in the other case. Each adsorbed particle can occupy either the central site and one first shell site, or a first shell site and an outer site, or a pair of outer sites. Geometrical mean contributions are introduced by equations which are formally.the same as eqn. (13), but now N: refers to the number of pairs of closest neighbour sites that are occupied by adsorbed particles, and X* refers to the number of closest neighbour interactions between difleerent adsorbed particles. Again, in considering the ratios of terms in the partition function, the last three factors can be omitted without affecting the results which are obtained. Thus, the expression The maximum term is then represented by [((2z-1(1 + q&2i-3)2-1 + (I + 5522-3)z]rNs-Z-lSN~-~-l5)~Ns-22~. AV .%(I + ?)%)*-l 3- (1 + El)., * (19) 10Wang, Proc. Roy. Soc. A , ., 1937, 161, 127. 11 Roberts and Miller, Proc. Camb. Phil. Soc., 1939, 35, 293.64 STATISTICAL ASPECTS OF CHEMISORPTION in which co has been written for t<22-1 and c1 has been written for 552z-3, is used for the partition function.To determine the heat of adsorption it is necessary to determine the probability that if the central site is occupied, a given first shell site is occupied by some other molecule. This is z - I 7)€1 -- I + 7)El' and the expression obtained l2 finally for the heat of adsorption is z - I 7 ) q 4 - 40 -=--- zv z I + 7)EI This formula applies where there is adsorption as molecules to form a mobile film and each molecule occupies a pair of closest neighbour sites on the surface. If an immobile film is formed by the dissociation of molecules, the sites are occupied in pairs, and formula (20) can be applied to this case. An immobile film of this sort corresponds to a random selection of pairs of closest neighbour sites ; a random distribution corresponds to the case 7) = I.The heat of adsorption of an immobile film is therefore given by This is shown by curve (b) of Fig. I, and it is seen that it departs only slightly from a linear relation and is far different from the heat curve for a mobile film. The heat of adsorption of an immobile film in which each particle from the gas phase occupies a pair of closest neighbour sites on the surface was first obtained by a numerical method using a model of the surface l3 and gave a curve in good agreement with eqn. (21). 6 . Structural Discontinuities in the Final Immobile Film.-There is another feature of immobile adsorbed films of the kind which we have been considering which must be examined before we consider the inter- pretation of the experimental results.This is the existence of structural discontinuities or gaps l4 in the immobile film when adsorpticn into the first layer is completed. We have seen that when each particle from the gas phase occupies a pair of closest neighbour sites on the surface to form an immobile film, there is a random distribution of fiairs of sites. As the surface becomes covered, there will be some sites on it which remain bare while all of their closest neighbour sites are occupied. In an immobile film of this sort, such unoccupied sites can never become available for adsorption into the first adsorbed layer, and remain bare when adsorption into the first layer is complete. A surface film of this sort therefore necessarily has structural discontinuities or gaps in it.The extent of these gaps can be determined either numerically using a model of the surface l6 or statistically.'l In the statistical determin- ation it is necessary to determine the probability that the central site is unoccupied while each of its first shell of neighbours is occupied. It is found that the number of single unoccupied sites completely surrounded by occupied sites, in the final film is about 8 yo of the total number of sites. This means that about gz yo of the surface can become covered in the first surface film. l2 Miller, ibid., 1947, 43, 232. l3 Roberts, ibid., 1935, 34, 399. 3 4. J4 Roberts, Nature, 1935, 135, I037 ; P~oc. ROY. SOC. A , 1935, 152, 445 ; ibid., 1937, 161, 141 ; Proc.Camb. Phil. SOC., 1938. 34, 399. Roberts, Proc. Ray. SOC. A , 1935, 152, 464, 5.A. R. MILLER 65 Structural discontinuities of a similar kind occur in any adsorbed film in which each adsorbed particle occupies more than one site on the surface. Another possible case is that in which the occupation of a site precludes the occupation of its shell of closest neighbours by any other particle. In this case,la the coverage in the final film is reduced by about 20 yo. The properties of adsorbed films which have been obtained in this and the preceding section have prcved particularly useful in the inter- pretation of the experimental results. We now consider the data on hydrogen and oxygen films formed on clean tungsten surfaces. 7 . Oxygen Films Adsorbed on Tungsten.-An oxygen film adsorbed on tungsten has a very high heat of adsorption.That on a bare surface is about 500,000 joule/mole of oxygen; and it would be expected from the considerations advanced in 0 2, that here we have to deal with an immobile film. We have seen that the final film will then have gaps in it. These might be expected to provide preferential sites for adsorption into a second film. There would be no such preferential sites if the film adsorbed immediately on the metal surface were a mobile film adsorbed on localized sites. The detailed structure of the adsorbate built up of successive layers can then provide information about the nature of the film. The structure of the oxygen film adsorbed on tungsten was deter- mined by the measurements of Morrison and R0berts.l' The basis of their method is the sensitiveness of the accommodation coefficients of neon and helium against tungsten to traces of adsorbable impurity.Changes in the accommodation coefficient thus provide a sensitive test of the presence of adsorbed films on the surface and also of any changes in them. The results which were obtained for oxygen adsorbed on tungsten can be indicated briefly. The accommodation coefficient of neon for a bare tungsten surface is 0.057. When oxygen is admitted, it rises to a final steady value which depends on the partial pressure of oxygen in contact with the surface. This can be plotted as a function of pressure and extra- polated to low pressures of oxygen ; this gives a value 0.226, which apparently corresponds to a layer of oxygen adsorbed on the tungsten.The higher steady values of the accommodation coefficient, which depend on the partial pressure of oxygen, are apparently due to the adsorption of a second layer of oxygen. The rise in the final value of the accommoda- tion coefficient with the partial pressure of oxygen, indicates that when adsorption into the second layer is completed, the accommodation co- efficient has reached a value 0.36. It is also found that the accommodation coefficient varies with the pressure in much the same way as the fraction of the surface covered does ; it can therefore be used as a measure of the fraction of the surface covered in the second adsorbed layer. The adsorption isotherm deter- mined from the experimental results in this way agrees well with that calculated statistically.16 This agreement justifies the determination of the coverage in the second layer from the accommodation coefficient by the formula which Morrison and Roberts used.There is one other point to be noted. When the wire carrying the adsorbed layer corresponding to an accommodation coefficient of 0.226 is heated to IIOO' K, the accommodation coefficient falls to 0.177, but no further evaporation occurs until it is heated to 1700OK. This in- dicates that the first layer is itself composite and consists of two distinct films. Furthermore, comparison of the changes in the accommodation coefficient suggests that the fall in it from 0.226 to 0-177 corresponds to the evaporation of much less than a complete monolayer. This would not be possible if we were dealing with mobile monolayers on an array Roberts, Proc.Camb. Phil. Soc., 1938, 34, 577, 5 7 ; see also Miller, ref. 2, eqn. (6.4). Roberts, ibid., 1930, 129, 146 ; 1932, 135, 192 ; 1933, 142, 518. C l7 Morrison and Roberts, Proc. Roy. Soc. A , 1939, 173, I.66 STATISTICAL ASPECTS OF CHEMISORPTION of localized sites; but it is possible if the more stable film in the first layer is immobile. Its intense stability to heating up to 1700~ K lends weighty support to this view, as also do the very high heats of adsorption which are observed. The first adsorbed layer of oxygen on tungsten consists of a very stable immobile film. This film is probably atomic and formed by the dissociation of an oxygen molecule and the adsorption of its two atoms on a pair of closest neighbour sites.It therefore has 8 yo vacant sites ; and these provide preferential sites for further adsorption. The first layer is completed by the ad- sorption of molecules in these gaps; corresponding to this, the ac- commodation coefficient rises from 0.177 to 0.226. On top of this composite layer a second layer is then adsorbed. It probably consists of molecular oxygen adsorbed on the array of localized sites determined by the tungsten substrate, and its coverage depends on the partial pressure of oxygen. These experiments about oxygen films on tungsten provide conclusive evidence for the existence of mobile and immobile films adsorbed on localized sites. The differences in the behaviour of these two kinds of film and of their structure indicate the importance of the distinction.The application of these ideas in the present instance, as it was developed largely by the late Dr. J. K. Roberts, indicates also their power and utility in the interpretation of otherwise complex data. 8. Hydrogen Films Adsorbed on Tungsten.-The heat of adsorp- tion of hydrogen on tungsten wires which were cleaned by flashing at about zoooo C before each experiment, has been measured by Roberts.140 The heat of adsorption varied from about 180,000 joule/mole of hydrogen for a bare surface to 80,000 joule/mole of hydrogen for a filled surface. The relative values of the heat of adsorption for successive admissions of hydrogen to the surface could be determined much more accurately than the absolute values which would be needed to compare the results obtained in different experiments.This uncertainty can, however, be overcome by plotting relative values of the heat of adsorption Q/Qo. where Q o is the heat of adsorption for a bare surface. Accordingly, I have re-plotted Roberts' results for the adsorption of hydrogen on tungsten and they are shown by the points in Fig. 2. The values ob- tained in each separate run are shown by a different symbol. The full line shown in Fig. 2 is the curve obtained theoretically for the variation of the heat of adsorption of the immobile film which is obtained when a diatomic molecule dissociates on adsorption, and its two atoms occupy a pair of closest neighbour sites on the surface, given by eqn. (21). The agreement between the experimental points and the theoretical curve is good.The important point is that these experimental results could not be fitted to a curve such as (u) of Fig. I, which is the other possible theoretical curve. These experiments appear to establish that the hydrogen film adsorbed on tungsten is an immobile film on localized sites, as we have defined it in 0 4. Similar results have been obtained by Beeck and Wheeler lS who have examined the adsorption of various gases on evaporated metal films at liquid air and room temperatures. Their results for hydrogen adsorbed on nickel show a variation of the heat of adsorption from 130,000 joule/mole of hydrogen for a bare surface to 60,000 joule/mole of hydrogen for the filled surface. The experimental points (Fig.3) show an almost linear variation over this range, and this suggests that the hydrogen film adsorbed on nickel is also an immobile film, formed by the adsorption of the two atoms of a hydrogen molecule on a pair of closest neighbour sites. We are led to the following consistent picture. 19 Beeck and Wheeler, J . Chem. Physics, 1939, 7, 631 ; Beeck, Rev. Mod. Physics, 1945, 17, 61.A. R. MILLER 67 The question is not, however, quite so clear cut as these results suggested at first. So far, we have considered what I have called a fixed interaction theory. However, the interactions between the adsorbed I 200,OOL la.4 m FIG. 2.-Experimental values of the relative heats of adsorption of hydrogen on tungsten (Roberts) as a function of the fraction of the surface covered compared with the theoretical curve for an immobile film.particles ensure that for a surface which is only partly covered, the ad- sorbed particles will be dimlaced from the Dotential minima Drovided by the surface. When thiLfactor between two particles with their distance apart are taken into ac- count, it can be expected that the heat curves will bz changed. These effects have been con- sidered only for a linear chain.20 The calculations are tedious and there is no immediate extension to the two-dimensional case. As far as they go, they indicate that there is a less marked difference between the heat curves for mobile and immobile films than is ob- tained with the fixed interaction theory. However, the heat curve for the immobile film is still much more nearly linear than that for a mobile film.The theoretical curve for the immobile film represents the experimental results for hy- drogen films adsorbed on tungsten and nickel better than does the curve for a mobile film. More recently, Trapnell *1 has obtained experimental results for the adsorption of hydrogen on tungsten which differ from those obtained bv Roberts. As he is and the ;ariation of the interaction I %I Fruchn of Surfice Covered B lorno ,0*2 10.4 ,O-6 ,0081 FIG. 3.-Experimental values of the heat of adsorption of hydrogen on nickel (Beeck and Wheeler) as a function of the fraction of the surface covered, compared with the theoretical curve for an immobile film. to Dresent his results later in this Discussion, I shall say merely that A they suggest that in Roberts' Miller and Roberts, PYOC.Camb. Phil. SOC., 1941, 37, 82. 21 Trapnell, this Discussion.68 STATISTICAL ASPECTS OF CHEMISORPTION experiments the surface of the wire was never more than about seven- tenths covered. If this be so, it would require a reconsideration of the data. This is reinforced by the fact that at his highest coverages, Trapnell obtains a flattening of the heat curve, which is more typical of a mobile film adsorbed on localized sites than it is of an immobile film. It should also be noted that at these highest values of 8, the heat of adsorption has fallen to a value at which one would expect appreciable evaporation and re-condensation in the time required to make a measure- ment. It may be that in this region, the behaviour of the film is changing from that typical of an immobile film to that typical of a mobile film ; this is, in fact, the intermediate region spoken of in 0 4.The point is that the two types ~f adsorbed film which we have described are charac- teiized by heats of adsorption which differ by a factor of four or five. But the change in the heat of adsorption of hydrogen on a tungsten sur- face as it becomes fully covered is sufficiently great, and it is in such a range of values, that it overlaps regions typical of each kind of film. The nature of the hydrogen films on tungsten must at present be re- garded as obscure, and the original view that it forms an immobile film needs some qualification. While it has not been refuted, it is perhaps an over-simplification. FIG.4.-Drop in the heat of adsorption in the neighbourhood of a region in which a film adsorbed on localized sites changes from immobile to mobile in character. The full line represents the likely shape of the heat curve if this change takes place at about 8 = 0.75. It is possible that the new results which appear to cast some doubt on the original interpretation of Roberts’ measurements, strongly vindic- ate the essential ideas-which have been developed in the statistical theory given in earlier sections-on which it was based. Consider what happens if both the factcrs which have just been mentioned are operative. That is, we assume that the lowest heats of adsorption of hydrogen on tungsten measured by Roberts referred, for some reason not at present known, to a film covering seven-tenths of the surface.At this stage, the heat of adsorption is about 80,000 joule/mole, and evaporation and re-condensation are approaching the rate at which a mobile film would be set up. If it be assumed that there is a sudden change at this point from an immobile to a mobile film, there would be a sudden drop in the heat of adsorption as is shown by the dotted vertical line in Fig. 4. Allowing for the fact that the change in behaviour could not be ab- solutely sudden, and that the film would pass through a region of inter- mediate behaviour, the complete heat curve would show a drop of this sort but rounded off a bit. A curve such as that shown by the full line in Fig. 4 would then result. This is very like the results obtained by Trapnell.A. R. MILLER 69 This is why these latter results which at first sight appear to cast doubt on the detailed interpretation of Roberts’ results for hydrogen films on tungsten, also appear to vindicate the basic ideas on which it was based. For only a view of adsorption phenomena which contains the possibility of two kinds of behaviour would appear to have much hope of explaining these results. On this view, even the later measure- ments provide support for the basic ideas which have been developed in the statistical theory. 9. Conclusion.-The conclusion to be drawn from all these experi- mental results is, I think, as follows. For adsorption on an array of localized sites, the distinction between mobile and immobile films is important. These two kinds of film show markedly different behaviour as far as the variation of the heat of adsorption is concerned, and the statistical theory leads to ideas about the structure of immobile films which can play a major role in the interpretation of the experimental data about oxygen films adsorbed on tungsten, and which make it possible to give a consistent picture of the way in which successive layers are built up on the surface. While the original interpretation of the be- haviour of hydrogen films adsorbed on tungsten appears now to be only partly correct, the essential validity of the ideas on which it was based appears to be vindicated. Cawendisk Laboratory, Cam bridge.