 In this lecture, we will learn about surface adsorption and isotherms. As we all know that surface adsorption is one of the very critical phenomena in any particular processing or any particular catalysis, catalysis activities. So, in this particular place, we will define adsorption. Adsorption is defined as the accumulation of ions or molecules on a surface. As we know that it is one of the very critical parameters in terms of deciding the kinetics of a particular process, because depending on how many ions or molecules they get adsorbed on a surface, they tend to react or they tend to form some species or they tend to precipitate on the surface to result variety of properties. But again, it is different from adsorption, because in adsorption, volume of material, it holds the ions and molecules and not the surface. So, in this particular process, we have adsorption in which the bulk of a material takes part in terms of attracting a species. Whereas, in adsorption, the particular species, it can be ion, it can be molecule, it stays on the surface of a particular substrate or adsorbent. So, in this particular manner, because once we have any surface being exposed to any environment, it is a surface which comes in contact first. And once surface reaction has occurred, then only it can go on further to the bulk or basically volume of that particular material can take part in terms of dictating the subsequent phenomena. Again, in this case, in adsorption, atoms on the surface are not in contact with the adsorbent. So, this is what, that we have a free surface. So, free surface, once we see the arrangement of particular atoms and then we can see that the atoms which are not on the bulk, that means the atoms which are on the surface, these atoms, they now have free bonds and hence they can attract the adsorbs to their sides. This is why it happens is because the bulk material, the bulk material which has all the bonds satisfied. So, bulk, we have all the bonds which are satisfied, whereas, in case of surface, what we have? We have unsatisfied bonds. So, this becomes a high energy area. So, since it is a high energy area, definitely once we have any secondary species, they will tend to come and attract on the surface of this particular entity to reduce the overall surface area or the surface energy of this particular entity. Again, the adsorption can be defined as a physioption or a chemisorption. In case of physioption, we have particle which is trapped on to the surface and once it is trapped on the surface, it is more like a potential well. So, I have a particular entity which is entering the well by some means and because of elastic collisions, the species can go back to the gas. So, we have particular entity, particular ion or atom or molecule. It tried to go, but it could not because of elastic collision. It might try to come back, but it might also happen that upon collision, it can lose all its energy. So, it has lost so much energy that its energy basically comes down to a very lower value and then it can just stay here without bouncing back. So, in first case, once we have elastic collision, it goes, strikes a molecule and then it comes back or goes to some other location without sitting along this particular site, but in this case, but in case once it is losing all its momentum because of elastic collision, it will remain stuck to the surface by some weak forces. Those can be van der Waals forces or London attractions. So, this is being called as a physioption. So, in this particular case, we have physioption and physioption, we have particle which is not trapped on to the surface. So, in this particular species, we can see that we have a zero potential and then it basically now stays in the potential well for some time. So, this is our, so keeping it as a potential and initially, we have a zero potential and then this is the surface of the adsorbent and then we see the particular item or molecular ion. It comes and sits and it basically gets entrapped into the potential well. So, now it requires certain energy to get desorbed from the surface and this may not happen in very quick succession. So, the iron or the molecule can stay there for a little longer time and this happens because we have elastic collisions happening with the particular molecule. So, in case the molecule starts losing the energy, in elastic collisions, it might get entrapped into the potential well, which is now the surface of a particular material and that second ion or molecule which is coming and striking the surface. Now, it is reducing the energy of the surface and that is why it is harder for it to go back because it has lost all its energy in satisfying the elastic collisions. So, it remains stuck to the surface by some weak forces, which results in the physical option. So, ideally this particular process has no barrier because it is more like resting yourself on a particular wall, so that you can lower your energy. So, it does not have any barrier, it is more like a physical attraction or the physical movement of a particular item or a molecule to a particular surface side and it is very fast because in this case it wants to reduce, the surface wants to reduce its energy. So, whatever molecular item is coming on there, it is basically trying to get bonded with that particular species. So, in terms of making it stuck on its surface, again these interactions are very very, they have very low energy, it is less than 0.4 electron volt. So, that is what is happening out here and they are always atomic or molecular in nature. And since there is no chemical bond which is forming out here, they are always reversible. At the same time, they are surface symmetry in sensitive because it is more like a physical movement of a particular item to a particular location. So, they remain in sensitive to a surface symmetry. In case surfaces are some symmetry, they do not basically pick a particular symmetry in that particular sense, it may not want to go to a particular site because they are not sensitive on the, sensitive to the particular molecule which is sitting on the particular surface. And they can also lead to go on forming multilayers because if I have particular species and secondary species come and sits on the surface, now that particular adsorbed species species, now again it has some free surface available to it. So, it can again keep forming some multilayers and the surface temperature has to be lesser than the condensation temperature. So, that the particular species can lose its energy and it can stay on the surface for a pretty long time. So, this is the overall deal with the physical option that they do not have a barrier. The process is very fast because surface wants to somehow reduce its energy, surface energy and these are weak interactions of less than 0.4 electron volt. And they are always atomic or molecular in nature because this has to happen by atom to atom or by molecule to molecule. And since these are only physical entities which are basically moving and going to rest on a particular surface being held by some weak or weak or vulnerable forces. So, this is again reversible process and they are insensitive to the surface symmetry. And since again this newly adsorbed species on the surface, again it will have some free surface available to it. So, it can again keep forming multilayers and again for this to all to occur, the surface temperature has to be low enough. So, that the particular ion or species it can lose its energy or it can lose its movement. So, the all the vibrational movements can be reduced by lowering the temperature. So, surface temperature has to be lower than the then that of a condensation temperature in order for the physical option to be much stable for a little longer time. And going to the chemisoption, chemisoption is defined as a strong interaction. In the earlier case, in the physical physical option, we had weak interaction. But in this case, we have a strong interaction between the adsorbent and the adsorbent surface. Adsorbent is where the overall substrate is, adsorbent is the new species which is forming and sitting on the adsorbent. So, in this particular case, we can define it like this, that we this is a zero potential and then we have some sort of a physical option which can occur. At the same time, we can also see the chemisoption is much this is for the physical option and this for the chemisoption depending on the potential and this is depending on the distance on the solute solvent, solvent surface, atom surface, it can be atom surface. So, what is happening out here is that the closer it is going initially with time, this option will occur because it has now much more probable that because of its lower energy, particular entity can go and sit and sit on the surface. But eventually for chemisoption, it is lowering down its energy by a much larger extent. So, eventually chemisoption is much more preferred once it is, once it can lower its energy by a certain value. So, in this particular case, we have very strong, we have very strong in the very strong bond which can form out here. So, it has strength of greater than 0.4 electron volt, it can again be covalent bond, metallic bond or an ionic bond and once a particular entity is formed because it is a chemical reaction, it tends to be irreversible in nature. Again depending on that, it might have a barrier as well. So, in that, in case of a barrier, it has to jump from physical option to chemisoption. So, we will see that how this particular barrier can affect. Again, it can also have variability in terms of kinetics depending on what is the overall barrier, it can, it can detect the overall kinetics of the particular process. Again, it can be again dissociative, what is that, we can come back to it. But, it is very highly sensitive to the surface symmetry because it highly depends on, if a particular atom or a species is located on the surface and there is a particular tendency to attract a particular ion or a species, then that particular ion or species will tend to go exactly at the site where it is finding much more comfortable with. So, in case of chemisoption, it forms a very strong bond. So, it has, it becomes much more sensitive to the surface. So, the species which is getting chemisoption on the surface, it has a, it becomes very sensitive to the surface symmetry and at the same time, it becomes limited to mono layers because it is so sensitive to the symmetry part or the species part which is attracting it, it becomes limited only when that particular species is available on the surface. Once it gets chemisoption on the surface, there is no other ion which is available for it to deposit further. So, once it forms a mono layer, there are no free substrate species available to again trap those adsorbing species. So, that is the reason it is basically limited to the mono layers, but more than that it can occur for a wider range of a surface temperature. So, when the surface temperature is pretty high or pretty low, it can still occur and physioption can be a first means of leading to the chemisoption as well. So, in chemisoption, we can find that we can have much more lowering of the energy of the surface which can occur via chemisoption because it tends to form a very stronger chemical bond. The strength of the bond is pretty high, it can be either covalent ionic or metallic and once it is allowing the particular species to form on its surface, it is because of chemical interaction, the chemical reaction between the two species, it is limited to the surface only, but at the same time, it tends to form only mono layers because now once mono layers form, there is no freely available surface to react with the adsorbate further. So, in that particular manner, it tends to form a mono layer only, but it can now occur in a variety or at variety of temperatures because even physioption can lead to chemisoption and even at higher temperature, the kinetics factor comes into play and then again that can again lead to some formation of chemical bond and then it can release its energy to make the system much more stable. So, we can see that it is sensitive to the it is sensitive to the initial orientation of the molecule because once you have some species out there, some chemical species to attract adsorbate, so it will depend on where the particular species is. If I have a surface and if I have a particular attracting species for the adsorbate out here, then the adsorbate instead of sitting here, it will tend to form go and see these locations and form a chemical bond. So, we can see specifically that this particular atom will tend to go and form sit out here and form a chemical bond and release its energy, so thereby lowering the overall energy of the system. So, second atom it will try to go here now and form a chemical bond with the targeted species as well. So, again it can also lead to changes in the internal bonding because of that it might tend to destabilize it and it might tend to lower its energy and again because of this, it is also sensitive to the point of approach. So, that is what we just discussed that because this particular species is now much more comfortable or much more attractive to this particular approaching species, this becomes highly sensitive to the point of approach or the adsorption side. So, that is what happening with the chemisorption part. So, this is how it becomes that it becomes sensitive to the initial orientation of the molecule. It might also be the case that particular molecule is coming like this and my site is out here and which can accept this particular molecule. Then ideally this part has to come down here and sit more like this. So, that is the reason this one becomes much more sensitive to also to the orientation. It is not only specific to site, but it is also sensitive to the orientation of the particular molecule and again the type of bonding which generates out of it, it again is now much more sensitive to that particular thing and again we can also see dissociative chemisorption and generally the direct chemisorption is generally not observed because it is basically being followed by the fizzy option. So, fizzy option is the sort of transient phase and which mediates the adsorption. So, it serves as a precursor to mediate the adsorption for chemisorption. So, in this particular case what we can see is once we have a particular potential out here and then we try to have a fizzy option. Then we did see that after some time depending on how the potential curve is, we might have some certain barrier to it. This is for the chemisorption and this is for the fizzy option. So, we can see that this is a potential and it might happen that our surface is much more activated. So, what we can see that the initial potential is now much higher at much higher level, but by lowering its energy it can lower down its energy by so much extent. So, this option will definitely try to go and jump into this particular well. That is what we can see here that once I am here, that once a particular species has come here and it wants to lower its energy, it has to cross this particular barrier and then jump on to the lower energy well. So, it can lower its energy by this particular extent, but to overcome this particular to basically attain this particular much of stability, it has to overcome this much of a barrier. So, that can happen that also is required to happen that this barrier height will affect the overall adsorption characteristics because depending on the height what is the overall probability, it will define how is the lifetime of the transient because it has to it is a transient state and it has to jump to the lower energy state. So, this is nothing but the transient time and also the desorption kinetics, it might happen that this jump from here to this from point A to point B is such that it is long enough. So, what can happen it can basically get desorbed on to the surface because this much energy it can take and it can basically go back or it can get desorbed from the surface itself. So, the overall transient time is very important, the desorption kinetics is also very important and the overall lifetime also of the transient is also very important for the transition from physical option to chemist option. So, generally we see that direct chemist option does not occur and then this is the potential out here, this is the overall distance of the molecular surface interface, this is the overall distance and that is what we are seeing here. So, in this particular case we can see that we need to first of all the molecule has to go under physical option and then basically it has to cross this barrier at this particular junction and then come back to the chemist of chemist state to in order to lower its energy. So, physical option basically serves as a precursor to mediate the adsorption, the mediate the chemical chemist option and then again the barrier heights are basic factors in terms of dictating the adsorption kinetics or defining the lifetime of the transient or the desorption kinetics, it is being dictated by this particular barrier height and sometimes it might also happen that physical option to chemist option may not be feasible at all, let me go back to it. So, this is the chemist option it can happen like that that we have a particular entity which now can go and serve go and basically sit on the substrate, but later on what can happen is that it can now start chemically reacting with itself and it can leave the surface as a molecule. So, in initially atoms they come they go and sit on the surface and they basically get dissociated by forming a molecule that is what is happening this is what is called a dissociative chemist option that we have species arriving arriving as an atom and then it stays on the surface and then basically it leaves out depending on their chemical stability. This can happen is that in that hydrogen atom it can basically sit on a gold surface, but once it is desorbing it desorbs it the hydrogen atom is basically coming and sitting on the gold surface, but it goes out as a hydrogen molecule. So, hydrogen atom is coming, but it is leaving as a hydrogen molecule that is what basically can happen, but in some cases like it can it can it can happen that once hydrogen is absorbed fizzling on the gold surface chemist option is exothermic. So, again that may not assist the formation of hydrogen fizzling option on the gold surface at all. So, in that particular case we can see that in certain cases the fizzling option to chemist option transition may not be feasible in certain cases because in certain cases like if hydrogen wants to hydrogen molecule wants to absorb on a gold surface chemist option becomes exothermic and again it is basically it is not becoming feasible because it requires very high energy to get fizzled on the gold surface. So, in that particular case we can see the transition will be more like this that we have fizzle option which can occur like this. Then the chemist option, but in chemist option we have more like this kind of pattern. So, we have certain energy which is much higher. So, we have this barrier height which is again pretty high. So, we have this particular desorption energy is from here to here. So, chemist option to desorption will basically happen here this for the desorption, but whereas for the adsorption we require such and such a high energy delta E adsorption, but what my molecule is already sitting here or the item is directly sitting here then it may not be feasible for it to basically go from this state the lower energy state like A to B. So, we can see that in certain cases once I do not have the dissociative thing activated adsorption, my delta E desorption is not equal to the delta E adsorption. See I can expect that this my once it is at point B when chemist option is occurring once it is lowering the energy then it makes sense for the jump of atom for the particular point for the particular species to go from point A to point B because it is lowering its energy. But in certain cases when this particular desorption thing is pretty high enough and again my delta E adsorption is even higher. So, in that particular case it will remain adsorbed because it is much lowering of the energy like here. So, we have our delta E adsorption and the barrier height is pretty high in this particular case we can see the delta E adsorption is pretty high and for desorption it is again lower, but it will be it will the barrier height itself is so high that the point A may not want to go to point B and in that case the physical option to chemist option transition may not be possible at all. So, to define that how this how this particular thing how the adsorption is occurring we can define them in terms of how what is the amount of adsorbate on the adsorbate as a function of constant temperature that is how we get the term isotherm. So, we are defining this particular relationship at a constant temperature to define how much amount of adsorbate is sitting on the adsorbate and that can be defined by the pressure or the concentration of the species on a particular adsorbate. So, if it is a gas we define it by pressure if it is a liquid we define it by its concentration. So, we can define it by the quantity which is being adsorbed on the particular mass of adsorbate. So, to basically we can normalize it and that thing is being defined by the pressure of the adsorbate the and K and N being the constant for the adsorbate adsorbate pair which is being specific for a particular adsorbate adsorbate pair. So, in this particular case we can see that higher the quantity being adsorbed I can get higher amount of pressure which can be which can for the pressure of the adsorbate which can be rendered. So, higher the quantity of adsorbate my pressure is now much more higher. So, that is how it basically basically this particular relationship comes about, but again in this particular case we can also define it using a Langmuir isotherm that gas molecule will find a side and it will stay there because there are certain assumptions which are required and again once we have theta as the surface coverage in the fraction of adsorbate which is now cited it basically is now occupied at the equilibrium. So, that is that is what we can see here the overall equilibrium equilibrium fraction it can be basically being defined by the species of the with the which has the direct rate constant and with the inverse rate constant. And so that it eventually comes out to be the overall coverage this is the overall coverage. So, this is overall free space which is available to the particular species and this is overall partial pressure of the gas or molar concentration of a solution. So, this from this we can see what is the overall fraction of the adsorbed side which is now occupied at particular equilibrium. So, that is being given by k. So, again we can see. So, this k can be defined as the equilibrium number of fraction of the side which are now occupied. So, this is defined by k. So, we have particular species of a which is gas and is going and sitting on a particular surface s to form a s already occupying a particular surface. And at equilibrium we can define the overall fraction of the adsorbed side by the overall surface coverage and the overall partial pressure which is being generated. So, from this we can directly see is from this we can directly see is for very low pressures when I have p very very low I get overall theta which is becomes equal to the k p. Because once my this term is very very low once it is very very low and this becomes k p by 1 and this is again nothing but equal to k p. But once for very high pressures once I have pressure very very high then what happens is my this term become this starts dominating and then k p by 1 plus k p it becomes eventually to be 1. So, my theta comes becomes approximately 1 for very high pressures. So, this is what the this is how it is being defined by the Langmuir isotherm that we are finding a fraction of adsorbed side which are being occupied at equilibrium that is being defined by the overall surface coverage and the overall pressure of the gas which is now being generated or the concentration of the solution. And from that we can define the overall coverage and for a very low pressures my theta is equal to k p for high pressures my theta is equal to approximately 1. And Langmuir isotherm is now based on certain assumptions and this assumptions are very like they required to be straighted out here. Because this is this forms a modeling for doing the particular for defining a particular isotherm in terms of deciding its coverage. So, assumptions are the surface of the adsorbent is uniform it means that all sides which are required for adsorption they are equivalent. So, if a particular surface if my particular species coming here it is it is equal probability of going and setting on any of the sides. So, this is what is being defined that the surface of the adsorbent is uniform. So, this surface of the adsorbent is uniform and all adsorption sides are equivalent. Secondly it says that adsorbed molecules is not interact. So, once I have a particular species which is now adsorbed on the surface it does not react or interact among one another. So, adsorbed molecules among themselves they are not interacting and again all adsorption is occurring through the same mechanism. So, that is one more feature of it that the way this particular a particular species will go and sit on this particular side. The similar manner another species will go and sit on some other side. So, they are being dictated by the same mechanism for this particular adsorption and at the maximum adsorption only a monolayer is forming, but that again becomes a limitation for this Langmuir isotherm because generally we define for fissure option more than one layer can practically form, but for in this particular modeling Langmuir isotherm model we can we can take it only as a single monolayer for keeping the simplicity because the way we are defining the coverage that is again dependent on the kind of pressure which is now being generated by the surface. So, that becomes a limitation for defining a Langmuir isotherm. So, molecules of adsorbed they do not deposit on the already adsorbed molecules of adsorbed and they deposit only on the free surface of the adsorbed. So, if I have particular surface where I have see I see the deposition of already certain molecules or ions or atoms out here. Second atom it may not see and may not sit here it might go to some it might go to some available site and then it can follow and sit here. So, that is the that is the deal about the adsorption of a particular molecule on a free surface. So, this we can see that it has it poses certain limitation that we are assuming that the surface of adsorbed is uniform and adsorbed molecules do not interact whereas practically these adsorbed molecules can again start interacting with one another. So, that again is a possibility, but again we are saying that adsorption is occurring to the same mechanism and it forms only a monolayer and a newly arriving species can sit only at a free surface. So, to define that we have a method of seeing how much is the how much how much coverage is being now done by a particular species. So, we can define that by a BET technique that is called browner, emet and teller technique and this is named on the three scientists in who basically define this particular isotherm. In 1938 those are Stephen Browner, Paul Emmett and Edward Teller who are basically developed a model for isotherm. So, as we see in the previous case we are assuming only a single monolayer to be getting adsorbed as a monolayer, but in this case we can see we can basically classify as a network or a sub routine of it that we can see that first we are forming a monolayer and then we are forming a bilayer then we are forming a trial layer. So, we can see is that if molecule is already being adsorbed on a surface then we cannot extend the Langmuir isotherm once the molecule is already now adsorbed. So, BET technique now comes into picture that it can define them as a multi layer we can see that there are multiple levels of adsorbents on the surface forms a monolayer then that monolayer comes and sits and the next species it sits on the surface to form bilayer and then keeps on going forming trial layer and so on. And that is what is more practical because it might be it might happen that second species they can come and find this surface to be more feasible and start sitting on the surface of a already adsorbed species. So, that is how it is forming bilayers and multiple surface layers. So, basically we assume that gas molecules they are getting physically adsorbed on the surface and forming infinite layers. So, depending on how many gas molecules come they can there is no limitation that they be restricted to certain monolayer or a bilayer. So, in this particular assumption we are making a practical case that gas molecules can come and sit on a already adsorbed surface as well and it can form infinite layers. At the same time there is one more limitation that we are also assuming that there is no interaction between the heat adsorbed layer and again we can assume we are also assuming that Langmuir theory can again be applied to each layer. So, this is again a limitation of that, but we are limiting we are eliminating one limitation by making it making it a multiple layer or the infinite layer adsorption on the surface, but we are assuming that these all layers they are independent and they do not interact with one another and the same time we can apply this Langmuir theory to each adsorbed layer. So, that is the advantage of this particular B E T theory and it basically comes out to be a formula like that that our adsorbed gas quantity it has some relation with the equilibrium pressure and the saturation pressure this is the saturation pressure and the equilibrium pressure with respect to the monolayer which is now being adsorbed on the surface. So, that is how it basically comes and again this seek is nothing but a constant which is which arrives from the liquefaction and it is given by the exponential of E 1 minus E L that is nothing but the heat of adsorption which is now emitting from the first layer in order to lower the energy and E L is for the second and higher layers. So, how much energy is now being reduced by the adsorption of all these species. So, this is more like a reduction in the energy of a particular surface by the adsorption that is being accounted by the term C and that is how it is now associated with the how the monolayer adsorption will occur with respect to the pressure it is forming that is now being given by the equilibrium pressure to the saturation pressure of the adsorbed. Saturation pressure of the adsorbed is basically the maximum pressure it can exert and equilibrium pressure is the pressure which is now being exerted at the equilibrium under the equilibrium conditions. So, this is how which can be utilized for defining the BET isotherm. So, in this case we are doing it like that we are defining a dependence of this monolayer with respect to multiple layers and be able to give the overall a saturation pressure with more and more number of atomic species. So, for a particular species we are getting certain equilibrium pressure and as we have more and more number of adsorbed species it will start increasing the overall saturation pressure. So, in that particular manner we can eventually find out what is the overall composition or the overall concentration of a or the overall coverage of a particular species. So, basically we can so we can basically see is from the basically from the adsorbed quantity adsorbed quantity of the adsorbed quantity of the of the of the gas on the particular substrate this is the defined quantity the kind of pressure it is generating the saturation pressure and the equilibrium pressure that part is also known. So, relationship between between them can provide us a particular slope and a particular intercept and from the intercept and the particular slope we can always calculate what is the overall gas quantity which is now adsorbed as a monolayer and again the concentration which is now basically being arrived from the initial part of the even an EL that is the heat or adsorption for the first layer and from for the higher layers. So, from that we can always calculate the overall reduction in energy and also the overall gas quantity which is now being adsorbed on a particular surface and again from this intercepts we can basically calculate the overall overall surface area and we can also calculate the specific surface area from this particular technique, but the idea is that now we can from so total surface area can be a very useful because it will tell us what is the overall vacant site available in a particular volume. So, that is that is being defined by the BET isotherm and from that we can eventually calculate what is the overall porosity of a particular material. So, once we know what is the overall free volume available in a particular material through this particular technique we can using a particular equation of the total surface area by the monolayer which is now being adsorbed on it, we is the overall volume of the adsorb and gas and S is the Avogadro number and S is the adsorption cross section. So, from that we can always calculate what is the total surface area and if we divided by the overall molar weight we can also get the specific surface area for a particular species as well. So, we can define the overall what is the overall specific surface area which is now available for a particular material. So, BET comes a very handy tool in terms of defining the overall porosity in terms of its volume as well as. So, in this particular case we can see BET plot is very critical tool in terms of being able to define a particular total surface area as well as the specific surface area to be able to calculate the overall porosity of a particular material or define what is the overall area available for a particular entity or a gas species to react with a particular surface. It is very much critical in terms of defining catalysis because we want a particular gas or any entity to come and get adsorb on a particular surface and higher surface area higher will be the adsorption and that will be defined by the available porosity volume or the overall surface area available for the reaction. So, coming to the surface adsorption many atoms basically they can from ordered adsorb layers on a basically lower index planes. It can again have a very defined relationship that is commensurate or it can have just random structure which is independent of the substrate. So, it can form incommensurate structure as well again they might have dependence on the temperature it might have dependence on the chemical species on the which is available on the surface. So, it has certain criteria which is available which can happen or in terms of defining the surface adsorption. So, there are certain notations which are now being developed. So, one is called the woods notation. So, if you take a particular crystal of a simple cubic just showing a 1 0 0. So, we have a particular unit cell of 1 0 0. So, we can see a simple cubic has certainly 8 atoms on the corners of a particular particular crystal. So, we can see a one of its surface if you take 1 0 0 and we define it by x y and z. We can see that one of its species will look more like this that little atoms along x and y more like this. So, it can again be defined like that, but once we have once we start seeing some adsorbates on its surface they might have certain symmetry to it. So, this can again be defined using the woods notation that in this particular case we are taking this one as the unit cell or it can also have we can take this particular entity as a unit cell for the adsorbate. This is what we require for the substrate this grayish portion. This is what we require for defining a particular substrate unit cell for a particular substrate, but for defining the orientation of the adsorbate we have to take a bigger unit cell and from that we can see that it is defined by a simple cubic it is more like a simple cubic structure with a 1 0 0 surface and it is a primitive lattice, which is now 2 by 2 of the original of the original lattice length. We can see it is starting from here it is going to 1 and 2 lattice points 2 atomic positions to sit out here. So, we have this atom which is now 2 atoms apart. So, it is forming 2 by 2 it is defined by simple cubic which is now sitting at the 1 0 0 plane of the initial substrate. It is forming a primitive lattice which has a length of 2 by 2 it is going 2 in the x side and 2 in the y side. So, it is going 2 unit vectors in the x side and 2 unit vectors in the y direction. So, that is why it is forming simple cubic 1 0 0 and it we can just eliminate the primitive word p and we can define it by simple cubic 1 0 0 and into 2 by 2. In case we had certain lattice like that we could have also have something like this. We have 1 atom, 3rd atom, 4th atom, 5th atom like this and then we can also see that the we can also see that if the if our adsorbates they are sitting more like this. Then in this particular case we have to take this one as a unit cell and that will be defined by a simple cubic sitting on a 1 0 0 surface and now it is 2 unit vectors in x direction and 1 unit vector in the y direction. So, this is how it has to be defined like that. In some other case if you can go back and say that if my atom is like this for a simple cubic 1 0 0 and say if I had something like this, if I had something like this 1 atom adsorbate adsorbate out here, 1 out here, 1 out here, 1 out here. I can take either this is a primitive lattice or I can also take this as a primitive lattice. I will definitely have to take this as a primitive lattice which is not dotted because this one is forming my now C centered and this one is now my primitive. So, I can define it in two manner. I can say it is a simple cubic, first I am taking only primitive case. So, it is a simple cubic now it is it has to travel root 2 distance. This is to travel root 2 by root 2 at 45 degree. So, I am defining it as again let me define as a for the primitive I take it as simple cubic sitting on the 1 0 0 surface. Now, it is a root 2 by root 2 because I am traversing root 2 on this side on the x side and root 2 on the y side to reach the second adsorbate species or I can also define it as a non primitive simple cubic 1 0 0 substrate surface. Now, what I am doing I am traversing now from here I am going to this particular side. So, if what I am doing I am traversing from here to here and then from here to here it is more like the previous case. So, I am traversing 2 by 2, but now this particular species has become it has one more adsorbate at the center. So, I make it C centered. So, I can define it by simple cubic 1 0 0 just C centered and I am traversing 2 by 2 or it is a simple cubic 1 0 0 plane and I have to traverse root 2 by root 2 in x direction and in the y direction. So, I can define it by vectors of root 2 by root 2 at angle of 45 degree that is it that is how I am defining my adsorbates using the roots notation. I can also define them using a matrix notation. So, I have a similar arrangement of them. So, I can take a simple cubic 1 0 0 and in this particular case I can have a 1 equal to a 2. So, in this particular case I can have a 1 equal to a 2 more like that. So, my a 1 direction and a 2 direction they are basically equal as well as they are perpendicular to 1 and 1 to each other. So, in that particular case I can define them using like that or alternately I can also some other surface like in FCC I can take a 1 0 0 surface. So, my atoms in the x direction are much further apart as compared to my y direction. So, I can see more like this. So, in this case I will see that my a 1 is much smaller as compared to the a 2 direction. So, I have my a 2 like this a 1 like this. So, I am seeing this is my a 2 this is my a 1. So, I am seeing that my for my FCC 1 0 0 lattice my a 1 is much smaller than a 2 at the same time they are perpendicular. But, there can be again one more case that in some other location I can have some different arrangement of FCC I can have a b kind of a structure. So, I can see more like this type of structure where my a 1 is like here my a 2 is out here and now my a 1 and a 2 they are not perpendicular but they are equal. So, a 1 is equal to a 2 but they are not perpendicular. So, again I have to define the a 1 and a 2 based on that. So, if I take a particular other system I can define I can correlate basically the what is notation and the matrix notation more like this. So, I am just defining the initial my substrate will my substrate will my substrate that will be simple cubic 1 0 0 surface. And if I go back and take my take my adsorbate species more like this out here. So, I define it like that. So, in this particular case I know that it is forming a simple cubic 1 0 0 surface and I am defining my primitive lattice of 2 by 2. So, similarly what I am doing here I am traversing if I define them as direction b 1 and direction b 2 and the initial direction b a 1 and a 2 of the initial crystal surface of a substrate. So, in this case what I am doing I know that my b 1 and b 2 they are dependent on the a 1 and a 2 by 2 0 0 2 of a 1 a 2. For achieving b 2 b 1 I am traversing 2 distance in a 1 and I am achieving my b 2 I am traversing 2 distance in a 2. So, that is how it is equivalent to this this is the matrix notation this is a Witt's notation. Had I had some other some other geometry. So, in case I have some other geometry. So, I had I had particular different geometry. So, let me take it more like this. So, say my adsorbate atoms are out here out here out here regularly placed like that. So, how I can define them I can define them by b 1 b 2, but in this case my a 1 and a 2 are again in the same direction this is my a 1 this is my a 2 direction. So, this would this I would have defined by utilizing primitive lattice of defining them as simple cubic 1 0 0 and again it was forming my primitive root 2 by root 2 R 45. Now, I can I can define that through my matrix notation more like that that in case my b 1 and b 2. Now, what I am doing I am traversing 1 distance in the x 1 distance in a 1 and 1 distance in the a 2 as well. So, I am defining it by 1 1 a 1 a 2 similarly, for the but for the for for reaching here I am traversing minus 1 in the a 1. So, minus 1 in the a 1 and 1 in the a 2. So, this is what is the matrix notation for this particular system and this is the Witt's notation for this particular system out here some or the matrix notation for defining the way adsorbed adsorbed is sitting on the adsorbent and for that we can use a very simple Witt's notation or a matrix notation. So, we can we can play around and we can see how it is to be defined and. So, just to give you a equivalent if I am taking a nickel substrate taking a plane of 1 1 0 it can be C centered and the defining the which how much distance I have to travel. So, 2 in a 1 and 2 in b 2 and we can also define adsorbent which is sitting on the substrate. So, again just to give you a equivalence I can I can traverse more like that that I am defining a platinum surface of plane 1 0 0 I would traverse 2 root 2 in x and root 2 in y at angle of r 45 and the adsorbent is again oxygen. That means equally I am in matrix notation I have a similar plane, but now I define this particular part as traversing 2 distance in x 2 distance in y and minus 1 in x minus 1 in y. So, these terms become equivalent, but one very important criteria which separates which basically is not at all available and we tend to think that it is that neither the Witt's notation or not the matrix notation they will tell us anything about where the actual adsorbent is. Let us tell us how this adsorbent is now oriented had I had adsorbent like here or may be like here I would know how what is the difference between these two it will define both the Witt's notation as well as the matrix notation will define them as the same it would not differentiate whether it is on a particular or exact location. So, again it depends on the nature of the bonding to the surface it might happen that the non directional bonds they may result higher symmetry because they are not at all they do not have any certain direction. So, atom can go and sit anywhere and because of certain repulsion among itself it will tend to achieve a very high symmetry, but in case of directional bonding it might need to go and sit exactly at a particular location. So, it will limit basically this some restriction. So, that is what it might create a low symmetry bridges on the top side. So, that is how I will deal with the define definition of the adsorption on a particular substrate. So, in this particular lecture we did see that how we define a isotherm that we keep we initially we assume it to be a mono layer, but then we can go on defining it in terms of bilayers and trial layers. So, we extended our initial isotherm to making it multilayers and then assuming that these layers which are now multilayers they do not interact with each other and they have basically they basically form uniform layers and then they go on to forming multilayers and from that we can also we also see that how physioption and chemisoption tends to basically affect how this particular species is now getting dissociative or how it is getting attach to a particular surface. And then we also learnt about how we can utilize this matrix notation and Woods notation to basically say how where the particular adsorbent is adsorbent is sitting on a particular adsorbent surface and how we can play around with it, but these techniques do not tell us exactly where the particular adsorbent is sitting it only tells us how they are basically sitting. So, I end my lecture here. Thanks a lot.