 Welcome back. We have been in the last few lectures looking at the effect of chemical reaction on mass transfer and we first did this within the framework of the film theory and we found that the effect of the reaction can be studied in order of increasing severity of reaction and the various possibilities fall into several regimes. So, the reaction can be very slow relative to mass transfer in which case we have the slow reaction regime in which the reaction takes place essentially within the bulk of the liquid and the film has the role of simply conducting the solute from the gas liquid interface to the bulk where it reacts. So, in that sense the film and the bulk operate in series here the film has diffusion and the bulk has reaction and the diffusion and reaction operate in series. Now, as often happens when rate processes take place in series it is the slowest rate process that dictates the overall course of events. In other words the overall driving force that is there is partitioned in such a way that the greatest share of the driving force is available to that process which has the highest specific resistance. So, this using this principle the slow reaction regime gets further subdivided into a kinetic sub regime and a diffusional sub regime. The kinetic sub regime being one in which the reaction is by far the much higher resistance as compared to the mass transfer and in the diffusional sub regime the diffusional resistance of the mass transfer step is the one which provides the greatest resistance and therefore, is the controlling resistance. So, that is the slow reaction regime and as the reactions become faster and faster the occurrence of the reaction within the film that is simultaneously with diffusion can no longer be ignored and we have the transition to what we call as the fast reaction regime which itself is a situation in which reaction and diffusion are taking place completely in parallel there is nothing happening in the bulk. The bulk just acts as an inventory of the liquid phase reactant and serves to store the reaction product, but otherwise everything that is of concern to us is actually happening within the diffusion film. And as the reaction becomes faster and faster the concentration profile of B starts to develop inside the diffusion film. In other words variation in concentration of B within the diffusion film can no longer be neglected and the reaction transits towards what is called as the instantaneous reaction and the instantaneous reaction is an extreme situation in which there is no region in the liquid which contains both A and B because the two are so prone to react that the moment A and B are present in the same place they annihilate each other and produce the reaction product. So, obviously the reaction can take place only at a plane to which from the interface the mass transfer process supplies A and from the bulk the mass transfer process supplies B. And the supply of A and B is coordinated in such a way that they arrive at the reaction plane in the ratio in which they are required by the reaction stoichiometry. And so this requirement fixes the position of the reaction plane and the position of the reaction plane in turn fixes the magnitude of the enhancement factor that is possible. And this of course is the maximum enhancement factor that is available to a system given the concentrations of B and the concentration of A and so on. So, this is the sequence of regimes that reaction goes through as the reaction velocity becomes higher and higher within the framework of the film theory. And in the last lecture we started looking at the effect of reaction within the framework of the surface renewal theories. And in order to keep the discussion simple and uniform we decided to ignore the differences between the Higbee's version of the surface renewal theory and the Danckwert's version of the surface renewal theory by defining a characteristic time tau which has two different connotations slightly different connotations in the Higbee and the Danckwert's version of the surface renewal theories. So, this characteristic time was used to non-dimensionalize the time in the diffusion reaction equations. And we saw that the there is a place for something like the slow reaction within the framework of the surface renewal theories as well because it is a situation in which once again the reaction is so slow that within the lifetime of the surface element there is not too much reaction that occurs. And therefore, the concentration profiles develop just as if the reaction were not there. So, the of course, the physical mass transfer coefficient does not change because it is governed by exactly the same equations as in the case of the physical mass transfer. And so, we arrive at a situation where the surface process is exactly the same as in the case of physical mass transfer. As a result of the surface process by the time a surface element leaves the interface it has a certain amount of A inside it this A is delivered into the bulk where it now reacts. So, in order to understand the effect of reaction we go back to having to make a balance for the solute A within the liquid bulk. And of course, this works exactly the same way as it does in the film theory because what happens in the bulk is not subject to the film theory or the surface renewal theory it is outside the ambit of both of these theories. It is a simple balance which says that whatever is delivered by the mass transfer is consumed by the chemical reaction if the bulk can be assumed to be in a kind of quasi steady state. So, everything that we said for slow reactions in the ambit of the film theory applies verbatim to the case of the surface renewal theories. And therefore, we again can define a diffusional sub regime and a kinetic sub regime. Therefore, there is nothing different about the slow reaction regime whether you talk about the slow reaction regime from film theory standpoint or from a surface renewal theory standpoint. Of course, we must remember in all this that we are considering those reactions which can be while being second order can be considered to be in a pseudo first order kind of situation because the rate of supply of B to the places where the reaction occurs is so much larger than the rate at which B needs to be supplied in order to account for the reaction in proportion to in stoichiometric proportion to the rate of supply of A. And therefore, the concentration of profile of concentration profile of B is essentially flat right up to the interface throughout the lifetime of the surface element. So, we will continue our discussions and go to situations where the reaction is now of such a velocity that it does make a difference to the diffusion process within the film. In other words what we are saying is we are looking at let us say faster reactions faster in the sense of these reactions do take place to an appreciable extent within the lifetime of the element and therefore, they can no longer be ignored. So, the kind of situation we are considering is m far less than q which ensures a pseudo first order regime and m is of the order of 1. So, under these situations we have to solve in non-dimensional terms this equation that we have written earlier of course, there is a B there which becomes equal to 1 and the relevant initial and boundary conditions are at theta equal to 0 A is A B and this is equal to 0 in most practical situations for reasons that we have elaborated earlier. And we shall make this assumption we will not solve this equation for the general case of non-zero A B and for zeta equal to 0 A is 1 and as zeta tends to infinity A is A B which is once again 0. Now, this is a much nastier equation than the equation that we solved in the case of film theory for a similar situation because that was a steady state theory. This is a partial differential equation, but there is a way of solving this which was introduced by Danckwerts and we shall simply indicate what this method is and leave the details to be worked out by you. So, this essentially constructs the solution from the case of pure diffusion that is what Danckwerts says is that if A 1 of zeta and theta solves in other words if A 1 is a solution to the pure diffusion problem which is dA 1 by d theta partial equals d squared A 1 by d zeta 1 squared then. So, this is subject to the usual I C B C then you can construct this quantity A in the following manner m integral of integral from 0 to theta sorry A 1 e to the power minus m t dash where t dash is a dummy integration variable d t dash plus A 1 e to the power minus m theta. So, this solves the diffusion reaction equation with the same I C and B C's. So, in other words if you know the solution to this equation with the initial boundary conditions that we have set out earlier then we can construct the solution to the diffusion reaction equation by transforming the A 1 in this manner the solution to the pure diffusion equation in that manner the same initial and boundary conditions apply. And as we have seen the initial and boundary conditions do not change whether there is reaction or there is no reaction. And we already know the solution of this one which is the pure diffusion equation it is the error function complementary solution that we have written earlier. And therefore, we can use this method in order to calculate the solution of the diffusion reaction equation. So, when we do that so I leave the details of that integration to you, but the essential result is that the concentration profile of A that is assuming that A B equal to 0 concentration profile of A is half e to the power minus square root of m zeta error function complement of eta minus square root of m theta plus half e to the power positive root m zeta error function complement eta plus square root of m theta where this eta is the same combination variable that we have defined earlier and it is this quantity here. So, we know the concentration profile and from the concentration profile we can calculate the instantaneous mass transfer flux as in dimensional terms it is d C A divided by d x equal to evaluated at x equal to 0. So, we can do this and so once we do that then we have two options. We can choose I of t for the Higbie or we can choose I of t for Dankwurz and we arrive at the average N A. So, this is the process. So, if we did this we arrive at an equation which is compared with the film theory equation in this slide here. So, we have already seen that the film theory gives you enhancement factor is equal to square root of m divided by tan h square root of m. If we did the the if we go through the process that I described on the previous slide then we and use the Higbie's I of t function in order to average out the instantaneous rates and compare that with the physical mass transfer rate. We find that the enhancement factor is given by this fairly complex expression as compared to what we see here. But on the other hand the Dankwurz surface renewal theory gives a particularly simple expression for the enhancement factor that is square root of 1 plus square of the hat number 1 plus m. Now, while these expressions look very very different it turns out that if you actually put in the values of root m and calculate the numbers they are not all that different. So, we illustrate that by making these calculations and comparing them for the slow to fast transition regime in this plot here, where the hat number is varied in the range of importance to us that is from root m equal to 1 to root m of about 5 or so and the enhancement factors are plotted and this scale goes from 1 to about 5.5 and we have the Higbie theory and the Dankwurz theory and the film theory compared here and Higbie and the Dankwurz theory virtually fall on top of each other there is that is the upper curve here and the lower curve is the film theory. So, the film theory predicts slightly lower enhancements as compared to the surface renewal theories and the surface renewal theories pretty much agree in as far as the magnitude of the enhancement factor is concerned. And the difference is actually very very small it is about a few percent you know it is about 7 to 8 percent at the maximum and you know at both ends the differences tend to narrow out and as you proceed in this direction you are going to the slow reaction regime and therefore, there the enhancement factor is 1 and as you go towards the higher reaches of this curve that is as enhanced as Hata number becomes faster and faster once again the curves approach each other and there is a particular reason why the curves become more and more identical as enhancement factor increases and as we know for as the Hata number increases and as we know for Hata number greater than 3 we have the fast reaction regime within the framework of the film theory and we shall show in a moment that the case of the surface renewal theories also predicts the same kind of situation and in fact, it agrees identically in as far as it is quantitative prediction of the enhancement factor for that situation is concerned. But before we do that let us try to understand from a point of view of a physical understanding as to how the concentration profiles are developing in the case of the slow to fast regime transition. In other words when we when our reaction is kind of fast enough to be taken into consideration while solving the processes in the surface element. So, let us look at those concentration profiles again. So, this is our zeta and this is our concentration profile of A and the concentration profile of B throughout is 1 that is because it is the pseudo first order. Now, let us look at these concentration profiles. If you look at a short time then the concentration profile of A will be something like this a slightly longer times it will be like this at even longer times it will be like this. Now, let us look at what is the effect of chemical reaction. So, this is what I have drawn here these black curves are physical mass transfer that is there is no reaction taking place. Now, what is the effect of chemical reaction? At every distance at the same time we would expect the concentration to be lower than in the case of physical mass transfer because some of the A is being consumed by chemical reaction. In other words for this time the concentration profile might look like this. For this time the concentration profile in the presence of chemical reaction may look like this and for this time the concentration profile might look at this. So, what happens is the longer the element spends at the surface the greater is the difference between the red curve and the black curve. So, the red curve is the case of mass transfer with first order chemical reaction. So, there is another way of visualizing this situation. So, what I just said was that the longer the element spends at the interface the greater is the difference at a given distance of the concentration it would have as compared to the concentration it would have in the case of physical mass transfer because this is a cumulative effect the longer it spends greater is the amount that gets consumed. So, we can visualize this situation in another manner. The mass transfer process is trying to push these concentration profiles is getting to getting A to penetrate deeper and deeper into the surface element. On the other hand reaction is trying to consume it and therefore, it tries to push it more and more towards the gas liquid interface itself. In other words mass transfer tends to push the gas into the liquid the reaction tends to consume it and prevent it from going further into the surface element. So, the net result of these two processes which are at loggerheads with each other you can imagine is that the concentration profile does not proceed does not get to proceed at the same rate as it would in the case of physical mass transfer. In other words it lags behind. So, a limiting situation can be imagined in which the tendency of the mass transfer to push the rate into the push the concentration profile into the surface element is exactly balanced by the tendency of the reaction to keep it from going forward at some stage. And therefore, the concentration profiles achieve some kind of a steady state. In other words there are two opposing tendancies one of which tends to get the gas to penetrate deeper into the system the other of which tends to prevent this from happening. The two tendencies will operate at equal rates at some stage and at some point during the surface during the life of the surface element the profile will stop from moving any further. In other words. So, it will move like this and at a later time it will not be much different from this and so on right. Now, the time at which this kind of a steady state occurs depends on the value of the value the diffusion reaction parameter or the Hata number. So, we can say the larger we can make the statement the larger the value of root m the sooner does a steady state occur. Now, this is a statement of some consequence because you can imagine a situation in which the root m value is so large that the steady state happens very early in the life of the surface element. If that happens the surface element is going to spend a certain amount of time at the gas liquid interface much of the time is being spent in this steady state that we are talking about. And therefore, in the when it comes to averaging the local rate or the instantaneous rate with respect to time you are essentially averaging a time independent instantaneous rate the steady state instantaneous rate. So, if this is the situation that if the this is the limiting situation we are interested in that situation is essentially governed by this equation. If root m is sufficiently large then we can say that we are essentially solving this equation which is the steady state equation with there is no initial condition required for this one zeta equal to 0 a equal to 1 and zeta tending to infinity a equal to 0. Now, if you look at this this is exactly the same as the film theory equation first reaction. So, in order to just recollect that one we will just spend a minute trying to see what the first reaction was for the case of film theory. So, if you recall this was the in the film theory. So, this was zeta equal to 1 that is the end of the film this is zeta equal to 0. And in general the concentration profile of within the film theory looks something like this. And as far as this diffusing solute is concerned because it is not going beyond a certain distance which lies within the film the effective thickness of the film is infinite as far as the diffusing solute is concerned. In other words, so this situation is we are written the equation with the far boundary condition as zeta tending to infinity a equal to 0 and d a by d zeta equal to 0. The concentration and the flux both go to 0 as the distance increases. And for all practical purposes the end of the film could be it does not matter if it is there or there or there because the concentration profile does not know where the end of the film is. So, we have an interesting situation that we have these two theories. One is a steady state theory and it works in a finite field between x is equal to 0 and x is equal to delta. And then there is the other theory is surface renewal variety of theories which are unsteady state theories and they work in a semi infinite field. The diffusion occurs in a semi infinite field bounded by the gas liquid interface at x is equal to 0, but really stretching to infinity on the other side. Now, it turns out that this is a situation the fast reaction situation is one in which the governing equations for the two theories become identical. So, this should not be surprising because for two reasons one is if you recall the resulting equations for the film theory treatment we found that the results are independent of delta that is because of course, what I said just a moment ago that delta does not matter for the diffusing solute under these situations. Once it is independent of delta which is a construct of the theory the resulting equations are essentially independent of hydrodynamics. And hydrodynamics or the effect of hydrodynamics is the precise point on which the film theory and the surface renewal theory have very different types of perceptions of the gas liquid mass transfer process. Put another way what is happening here is that we have the film theory operating in the finite field that is between the diffusion takes place between x is equal to 0 and x is equal to delta. In the fast reaction regime for all practical purposes the x is equal to delta might as well lie at infinity. Therefore, the finiteness of the diffusion field does not matter for the diffusion process. On the other hand, so in that respect the film theory becomes something similar to the surface renewal theories at least in so far as the mathematical formulation is concerned. On the other hand the surface renewal theory which is an unsteady state theory becomes essentially a steady state theory because the fast reaction is such a situation in which the reaction profile is arrested from proceeding in a time dependent manner into the liquid very early in the lifetime of the surface element. And therefore, for substantial part of the life at the gas liquid interface the gas liquid the surface element is absorbing the gas as if it were a steady state process. So, it is a steady state process taking place in a semi infinite field as far as the surface renewal theory is concerned it is a steady state process taking place in a semi infinite field as far as the film theory is concerned. So, the two processes become identical. So, needless to say since we are solving the same equations we expect the same results and so for fast reaction. So, this is what is called as the fast reaction within the surface renewal theory. So, the definition of the fast reaction in the surface renewal theory is that it is a situation in which the surface element essentially absorbs in a steady state manner for a majority of its stay at the gas liquid interface. We expect no difference to be made by the choice of the IFT function because whether it is surface renewal theory of the Hibbe variety or surface renewal theory of the Danckwertz's variety. We are averaging essentially a time independent function with respect to the awaiting function. And the waiting function itself is normalized therefore, in other words what I am saying is if Na i of t is constant independent of Na i then your average rate is Na i of t i of t d t 0 to infinity this is Na i 0 to infinity i of t d t and this is nothing but Na i of t rates. The average rate and the instantaneous rate agree because it is a steady state process there is nothing to be averaged. So, we can now come back to the question that I raised very early in our consideration of the effect of chemical reactions on the mass transfer process. And I said that while the comparison of experimental data with the theories seems to suggest that the film theory is less accurate than the surface renewal theory in terms of the way it pictures the processes at the gas liquid interface. We have seen in so far as the prediction of the effect of chemical reaction on the mass transfer coefficient or in other words in so far as the prediction of the enhancement factor is concerned there is a slow reaction regime in which the two theories behave identically and there is a fast reaction regime in which once again the two theories behave identically. So, in order to picture this on the enhancement factor Hata number diagram we have a Hata number of 1 and out here is the slow reaction regime and around about 1 the curve starts to lift off and we have about 3. Incidentally, if you look at the way in which the equations work for the surface renewal theories it is once again a value of 3 or 4 that around which the equations become independent of time. The instantaneous absorption rate becomes essentially independent of time very early in the surface elements age and therefore, the precepts of the fast reaction regime start applying from a value of about root m equal to 3. So, that is clear from a numerical calculation that becomes clear from a numerical calculation of the enhancement factor from the transition regime equations. So, we have we have E equal to root m here and from whatever we said earlier we have the surface renewal theories there and the film theory here. In other words this is the film theory and that is the surface renewal theory. In other words the theories are identical in this region the theories are identical in that region in between you can always expect that the theories are not going to be very different because it is only a small region that we are considering and given that you are holding the two curves two definite asymptotes at the two ends. There is only so much they can do in between and if you actually work out the numbers as we saw in this earlier curve as we saw in this plot here the difference between them is rather small and therefore, we can continue to use either expression for the enhancement factor in the in this region in the. So, this region is the transition region transition slow to fast. So, then why do we need to so we now take up the opposite question of. So, we were asking why are we looking at film theory at all if film theory is not accurate and we know surface renewal theories to be the better theories and the answer is now clear that the film theory gives the same results as the surface renewal theory over a large part of the Hata number range and we shall see what happens as we go to even larger Hata numbers in a moment and it is a much simpler theory to conceptualize and solve and therefore, we use the film theory. So, the alternative question is why do we then worry about the surface renewal theories at all at least in so far as what we have seen so far goes the answer to that is you have seen that the equations that result from the three theories are of different complexities and this and the equations all look different and we go back to these three expressions that we showed there is the film theory expression there is the Higby expression and there is the Danckwerts expression and this is sometimes of advantage because you can use the equation that suits your purpose from the point of view of mathematical convenience in order to design experiments and in order to calculate quantities of interest. To take an example the Danckwerts theory leads to the use of what is called as Danckwerts plot for determining k l and a separately. Now, this is usually a tough ask because you either have situations in which you can only determine k l a as a product or you can you have situations in which you can determine only interfacial area that is the fast reaction regime. But if you think about it in the diffusional sub regime of the slow reaction regime which is the one that happens before the reaction becomes fast enough to occur in the film you have a rate that depends on k l a the product of the mass transfer coefficient and the interfacial area and on the other extreme when it goes completely into the fast reaction you have a rate that depends only on the interfacial area that does not depend on the mass transfer coefficient at all. So, it stands to reason to expect that in between that is in the transition from slow to fast reaction regime the rate of reaction should depend on k l a as a product as well as on the interfacial area. So, in other words both k l and a independently influence the rate in the transition regime. Now, this is while this can be anticipated the expression that we use that we have derived for this regime is not very convenient to extract k l and a independently if you consider the film theory expression or the Higby's expression. On the other hand the Danckwert's expression allows you to do this and that is the matter that I want to just spend a couple of minutes on. So, if you write the rate expression for Danckwert's theory it is of course, k l a C a star multiplied by the enhancement factor which is the Danckwert's theory 1 by m. So, if I take r a by C a and square it then what I have is square of k l a plus square of k l a times m. Now, so this is square of k l a plus a square k l square will cancel with the k l square in the denominator in m and therefore, we are left with d a k C b b the pseudo first order rate constant. So, what this allows us to do is if you can vary the pseudo first order rate constant either by varying C b b or by making use of a catalyst in different amounts which causes a variation in k itself and this is this is possible for very many reaction systems. Then you can plot for different values of the pseudo first order rate constant you can plot r a by C a star square versus k C b b. And if you do this then you have a slope from which you can get interfacial area and you have an intercept from which you can get k l a. So, this is the advantage of the Danckwert's expression and given that it gives you results which are numerically almost the same as any other theory this can be used to advantage in the characterization of mass transfer equipment. So, we shall leave the first reaction regime at that point and look at the situation that arises when the reactions that we are considering are even faster than what we have considered so far. So, the only assumption that has remained with us so far is the fact that the Hata number is much less than q or m is less than much less than q. So, if this as m becomes larger and larger as the reaction becomes faster and faster this assumption has to be violated at some point. So, now we consider situations in which situations for which it is not true that m is far less than q. So, this assumption is violated. So, let us it is clear that you know the equations have to be solved without any simplification at this for this case in which by which what I mean is you have got the second order partial differential equation for A you have the second order partial differential equation for B these are coupled because the equation for A contains the concentration of B and the equation for B contains the concentration of A there is a certain set of initial and boundary conditions. So, this coupled set of non-linear partial differential equations has to be solved for the case of m not equal m not much less than q. This is of course, not a easy task it is not possible to do this by analytical means and you have to solve this by numerical techniques. But let us consider physically what is happening much as we did in the case of the film theory. So, that we will see for are there any extreme situations that we can identify in which we can derive solutions for a much simpler set of equations. So, we have these you know we already are in the fast reaction regime and this is the concentration profile. And if this is the situation if m is comparable to q then what happens is that the concentration profile of q is no longer flat I mean concentration profile for B is no longer flat and it decreases into the film. So, now the unsteady state nature of the process comes into the fore once again because the concentration profile of A has developed to an extent. But because of that B is consumed to a little extent near the interface because of which the reaction is slowed down the concentration profile can push a little further the concentration profile becomes deeper and so on. So, as time goes on you have situations of the profile developing in this manner. Now, what happens is at some stage during the life of the surface element this concentration profile of B can hit the interface at B equal to 0. In other words B is completely consumed at the interface and later on what happens is that there is a region that develops close to the interface which contains no B much as we saw in the case of the film theory except that in this case the events are happening as a function of time within the same surface element. Now, all this can happen pretty much pretty early in the life of a surface element if m is much larger than q. So, in that situation so if we take an extreme situation then what happens is there is a region in which A is present there is a region in which B is present they do not overlap because the reaction is so fast that the two meet and annihilate each other consume each other at a plane and this plane moves forward into the film as the surface element ages more and more. In other words you may have a situation like this at short time and a situation like this at long time. So, this is the so this is zeta zeta so this is theta increasing this is A and this is B. So, we have a situation in which virtually throughout the life of the element in other words the surface element comes to the liquid element comes to the surface at that instant the B at the interface is completely consumed. So, the concentration of B falls to 0 instantaneously the moment B reaches the interface because there is no B now A can proceed a little into the film and so the point at which B becomes 0 moves into the film. So, B is consumed at that point now A can push a little further and so on that is how you get this progression of profiles. So, now what is the if you take this central curve for example, let us look at this now this is a region which contains A this is a region that contains B and the two regions are separated by the reaction plane which is not stationary as in the case of film theory, but the position reaction plane is at a position zeta 1 which changes with theta because this is an unsteady state theory. So, the governing differential equations for this case look as follows in region A we have d A upon d theta equals d square A upon d zeta square pure diffusion equation because there is no reaction that can take place there is no B and so this applies between the interface and the reaction plane in region B. So, this region A only exists for times greater than 0 and for region B you have a similar equation except that you have the ratio d A by d B and this applies between the reaction plane and infinity. So, these are the two equations and the initial and boundary conditions were the same way except that at the reaction plane at zeta equal to zeta 1 of theta we have A equal to 0 and B equal to 0 and the fluxes have to be matched. So, the condition that determines at what position the reaction occurs is this condition that nu times the flux of A equals the flux of B in the opposite direction. This is the same condition that we applied in the case of the film theory, but since now the partial derivatives of the concentration with respect to x are time dependent this gives you a solution for zeta 1 that depends on time. Now, this is what is known in mathematics as a moving boundary problem in the partial differential equation theory as a moving boundary problem and it is not easy to solve, but it can be solved analytically because it is a much simpler equation although it looks complicated it is much simpler as compared to the situation in which you had the reaction term in both the equations and that brings about a nonlinearity. So, these equations are linear and if you look at the equations you would suspect that they would have error function type solutions and so you can formulate an error function solution for this with two constants an error function solution for this with two additional constants and these four constants have to be evaluated by the application of the initial and boundary conditions and this condition this pair of conditions which fixes the value of zeta 1 of theta. So, this solution is of course, the mathematical details of this solution we will not go through it is available in books such as Dan Quartz's book on gas liquid reactions and or even Burr Stewart Lightfoot the book on transport phenomena, but what we will do when we come back is we shall look at the final results of this treatment the solution what it leads to and what are the implications of that and how those situations compare with the situation for. So, this is what we have formulated is the case of the instantaneous reaction within the framework of the surface renewal theories. So, we have the instantaneous reaction within the framework of the film theory and we shall compare the results of these two theories when we come back.