 Friends, I am Ganesh Viswanathan from Department of Chemical Engineering Indian Institute of Technology Bombay. I am going to give several lectures in the second course of chemical reaction engineering and today I am going to start with heterogeneous reactor analysis. So we will start with looking at heterogeneous reactor data and the reactor design. So we will look at the heterogeneous analysis for reactor design. So the general algorithm for designing a reactor is as follows. So the first step is to deduce a rate law, the second step is to find the mechanism behind the reaction and the third is to find the rate parameters and the fourth step is to design a reactor using all the information that has been found from the previous three steps. So that is what we are going to start today. So today we are going to look at the rate law and we are going to look at the mechanism and how to estimate rate parameters and then we will go and march forward to the design of the reactor. So as an example, let us take the hydro demethylation of toluene whose reaction essentially follows this scheme C6H5CH3 which here after I will call as T, T stands for toluene and plus hydrogen which will here after be called as H leads to benzene which will be hence forth called as B and methane which will be hence forth called as M. So the objective is to design a packed bed reactor which will be hence forth referred to as PBR and in order to do this we need to find a rate law that is we need to find the rate of generation of toluene per unit gram of cattle or per weight of the catalyst that is used in the packed bed reactor. So in order to achieve this objective let us start with the experimental data. So there has been experiments that have been performed under different conditions and different partial pressures of these different species toluene, hydrogen, methane and benzene and different combinations various experiments have been performed and the rate in terms of gram mole of toluene per gram catalyst per unit time have been measured. So there is 16 experiments and you could actually classify them into four different sets the first one and the second one by changing the partial pressure of methane and then the second one by changing the partial pressure of benzene and then the third set by varying the partial pressure of hydrogen and the fourth set by varying the partial pressures of toluene. Now remember that partial pressure is a reflection of the mole fraction of each of these species. So we can now deduce a rate law by looking at this experimental data and so as a first step let us try to see how the rate law depends on. So dependence on let us say methane. So start with let us look at the dependence of methane the concentration of methane on the rate of the reaction. So if we look at the data here so if you look at the first two data sets we will see that as the partial pressure of methane is increased by 5 times one can observe clearly from the data that the rate is hardly changing which means that the methane has little or almost no effect on the rate of the reaction. So clearly we can observe that the methane M is weakly or weakly adsorbed or goes directly into the gas stream into the gas phase. So that is an important reduction which means that methane is hardly contributing to the rate of the reaction. Methane does not affect the rate of the reaction. So that is an important reduction that one can observe from the experimental data. So next let us look at what is the dependence of benzene on the reaction rate. If you look at the second set that is runs 3, 4, 5 and 6 and you can see that there is a significant effect of benzene on the reaction rate particularly if the partial pressure of benzene is increased then there is a decrease in the reaction rate overall reaction rate. So which suggests that the concentration of the benzene has to appear in the denominator of the reaction rate. So which means that so if I look at the dependence if I look at the dependence so runs 3 and 4 they suggest that the rate decreases with increase in the benzene concentration or the partial pressure. So this suggests that the reaction rate must somehow be proportional to the partial pressure of benzene which appears in the denominator of the rate law. This is because the rate now decreases with increase in the concentration of the benzene that is the partial pressure of benzene. So therefore the rate of generation of toluene which is RT prime which now is given by 1 by should be approximately be proportional to 1 by 1 plus kB into Pb plus other components where kB is basically the corresponding equilibrium constant. So next if we look at the dataset again and we want to know what is the dependence of toluene on the reaction rate. So let us look at the dataset 11 and 12 10 and 11 excuse me and it suggests that 10, 11 and 12 suggest that as the toluene partial pressure is increased there is an increase in the overall reaction rate which means that at lower concentration of toluene there is an increase in the reaction rate. However if we go to a much higher concentration of toluene for example look at the run 13, 14 and 15 that a significant increase in the concentration of toluene does not have a any effect or marginal that has only marginal effect on the reaction rate. Therefore this suggests that the dependence of toluene must appear both in the numerator and the denominator of the reaction rate. So let us summarize this here. So dependence of T so runs 10 and 11 it suggests that at low concentrations at low concentration of toluene the rate increases with increase in concentration of T. And similarly runs 14 and 15 it suggests that at high concentrations of T rate remains constant this almost constant. So that is the reduction that we can get from the experimental data on the dependence of toluene concentration of toluene on the reaction rate. So the only other component which is left is basically the hydrogen and so let us look at the experimental data again. So the partial pressure of hydrogen if it is increased by 2 fold 1, 2 and 3 it appears that there is a linear increase in the reaction rate which means that when the partial pressure goes from 1 to 2 the reaction rate is almost double and from 2 to 3 it is almost double of that. So this suggests that the reaction rate perhaps depends linearly on the concentration of hydrogen. So if we look at the dependence of of hydrogen on the reaction rate so runs 7 and 8 and 9 suggests that the rate increases linearly with the concentration of hydrogen. So this suggests this means that the hydrogen is perhaps either not adsorbed on the surface or it is immediately goes into the it is not adsorbed on the surface or the surface coverage of hydrogen on the catalyst site is insignificant. So therefore now we can combine all of these different reactions different observations from experimental data. So combining the observations one can deduce that the reaction rate perhaps must have a form which looks like this. So it has to depend linearly on the concentration or partial pressures of hydrogen and it has to depend it has to increase with the concentration of toluene when the concentration is lower. However it has to remind constant when the concentration of toluene is larger. So therefore it has to appear both in the numerator and the denominator and similarly the benzene has a reverse effect benzene as the concentration of benzene increases the rate decreases and therefore it has to appear in the denominator. So now we can convert this proportionality into an equivalence by putting a rate constant in front. So if I stare at this equation you can see that the rate now is directly proportional to the partial pressure of hydrogen and it appears both in the numerator and the denominator of the reaction rate for with respect to the partial pressure of toluene and then it appears in the partial pressure of benzene appears in the denominator of the rate expression. So this actually provides a method by which you one could actually deduce what is the possible reaction rate law based on the experimental observations. This data is only an example and in general data may not be available in such a form and it may not always be possible to deduce rate law by inspection. So the next step towards finding the rate law if we go back to the set of different steps which is involved so we have now found the rate law. So we need to next go ahead and find the mechanism that governs this particular set of heterogeneous reaction. So the next step is towards finding a mechanism next step is finding a mechanism. So now we have to make certain assumptions here. So if we assume that the toluene is adsorbed on the surface. So remember that from the experimental data we observed that at low concentrations of toluene the reaction rate actually increases with the partial pressure of toluene. However at high concentrations or high partial pressures of toluene the reaction rate almost remains constant which suggests that the toluene must actually be adsorbed on to the surface and so based on that observation we will make an assumption that the toluene actually is adsorbed on the surface of the catalyst at which the reaction is occurring. Then the second important assumption will make is that the toluene which is adsorbed on the surface it reacts with the hydrogen which is present and when it reacts with the hydrogen which is present because the partial pressure of hydrogen affects linearly the reaction rate we will assume that the reaction rate reaction actually occurs between the toluene which is adsorbed on the surface and the hydrogen which is present in the gas phase. Now after the reaction is completed the products which are formed are basically benzene and methane. Now we observed that the methane hardly has any effect on the reaction rate which perhaps suggests that the methane must actually directly immediately go into the gas phase and benzene must it affects as the as the partial pressure of the benzene increases we can see that there is a reduction in the reaction rate which suggests that the benzene actually has to stay adsorbed on the surface and then later get released into the gas phase as a product. So, this can be captured in this statement here. So, the reaction the toluene which is absorbed reacts with hydrogen in gas phase in gas phase and it leads to the production of benzene which is which reminds adsorbed on the surface and methane it goes into the gas phase. Then as a third assumption after the benzene gets adsorbed on the surface the the product has to come out of the reactor. So, therefore the benzene actually gets desorbed from the surface and then it goes into the gas phase and then leaves the reactor. So, this can be stated as benzene is desorbed from gas phase and then so you must have learnt in the previous lectures that there has to be a rate limiting step. So, we will assume that the heterogeneous reaction is actually a surface reaction limited step and this is in fact not not a bad assumption because 7 to the percent of the heterogeneous reactions are actually limited by the surface reaction. So, therefore, these let us assume that it is a a surface reaction limited its surface reaction limited. So, let us take each of these steps one by one and then try to capture write a simple rate law for each of the steps. And so, there are three key steps which we have actually deducted from the experimental data and the form of the rate law. So, the first one is the adsorption. So, this is the toluene which gets adsorbed onto the catalyst surface and so this can be captured by the following reaction. This can be depicted by the following reaction. So, it is the toluene in the gas phase which actually goes and occupies a vacant site S, S represents the catalyst site at which the toluene gets adsorbed and that leads to the formation of this complex which is basically the toluene and the site which is basically means that the toluene is now attached or resides on the location of the catalyst site. So, we can now also represent saying that there is a specific constant which corresponds to the formation of or adsorption of toluene on the catalyst site and also it is an equilibrium process and so therefore, there can also be a simultaneous desorption. So, if Ka corresponds to the specific constant which captures the adsorption of toluene onto the site and Ka-a corresponds to the desorption of the toluene from the catalyst site into the gas stream. And then the second step is the reaction. So, the hydrogen which is present in the gas phase that now reacts with toluene which is residing on the catalyst site to give the products where this will be benzene which is adsorbed which continues to be adsorbed on the surface plus methane which goes into the gas phase. So, this is the second step where hydrogen in the gas phase reacts with the toluene which is now adsorbed onto the surface of the catalyst site and that leads to the product formation where the benzene which is one of the product it remains adsorbed on the catalyst site and then methane is formed along with it which immediately goes into the gas phase. And the reaction rate corresponding to that can actually be here it is depicted as KS and K-S for forward and the backward reaction. The third step is the desorption of benzene. Third step is the desorption of benzene where the benzene which is adsorbed onto the surface of the catalyst is now desorbed into the gas phase where benzene goes into the gas phase and leaves the and the catalyst site is emptied. And so this reaction the specific constants can be represented as KB and K-B. So, these are the three steps that we have identified based on the experimental data and also based on the some of these observations that we got from the experimental data. So, now what we are going to do is we are going to take each of these steps each of these individual steps and then try to find out what is the reaction rate law and then we are going to we have identified what is the we have assumed what is the limiting step in each of these three which one which one of these three is a limiting step and then based on that we are now going to find out a rate law for this particular heterogeneous reaction. So, let us now go into the first step of adsorption. So, let us look into the let us go a little bit deeper into the adsorption process. So, toluene in the gas phase it gets adsorbed onto the vacant site S on the catalyst and leading to the TS which is the site which is adsorbed with toluene and Ka and K-A are the corresponding specific constants. As toluene adsorbs to the catalyst site and the catalyst and the adsorbed species reacts with hydrogen in gas phase we assume a single site mechanism. So, the rate of adsorption can now be written as Ka multiplied by the concentration of the vacant site CV multiplied by the partial pressure of toluene that is basically captures the rate at which the forward reaction is going to happen in order for toluene from the gas phase to get adsorbed onto the catalyst site minus the concentration of the or the number of concentration of the sites in which the toluene is already adsorbed divided by the corresponding adsorption equilibrium constant. So, Ka is the specific constant adsorption rate adsorption constant adsorption and CV stands for concentration of the vacant site vacant site on the catalyst and Pt is the partial pressure of toluene Pt is the partial pressure of toluene and then Cts is the occupied site concentration Cts is the occupied site concentration and Kt is the adsorption equilibrium constant and that is typically given by Ka divided by K minus A. So, that sort of captures the rate at which the toluene in the gas phase is gets adsorbed onto the catalyst site. So, let us take a look at the let us go into the details of the next step that we outlined that is the surface reaction part. So, let us look at the surface reaction let us look at the surface reaction. So, let us assume that it is a single site reaction where only toluene molecule which is adsorbed onto one site is what is involved in the reaction in the catalytic reaction and so we have hydrogen which is in the gas phase plus the toluene which is now adsorbed onto the surface reacts with each other and then it leads to the formation of benzene and methane in the gas phase. So, now we can now capture the rate at which this particular reaction occurs. So, the surface reaction rate is now given by Ks multiplied by partial pressure of hydrogen into the concentration of the sites in which the toluene is actually adsorbed minus the concentration of the number of sites concentration of the sites in which the benzene is adsorbed which is a product multiplied by the partial pressure of methane divided by Ks. So, the first term here corresponds to the rate of the forward reaction and the second term here corresponds to the rate of the reverse reaction. So, here Ks is basically the the specific constant for forward reaction forward reaction and Th2 corresponds to the partial pressure of hydrogen in the gas phase and Cts corresponds to the concentration of toluene number of sites in which the toluene is adsorbed and Cbs corresponds to the concentration of the number of sites in which benzene is adsorbed the product benzene is adsorbed and Tm corresponds to the partial pressure of methane in the gas phase and Ks is basically the corresponding equilibrium constant which is given by Ks divided by K minus S. So, next we look at what look at the desorption process. So, that is the third step which is the desorption process. Now, here the benzene which is adsorbed onto the surface gets desorbed to give benzene in the gas phase and an empty site or a vacant site and so if the corresponding specific constants desorption constants are Kb and K minus b then the rate can be given as rate of desorption is given by Kb into Cb dot S minus Kb into Pb into Cb where the Kb corresponds to the benzene adsorption equilibrium constant and Cb corresponds to the concentration of the vacant site. Remember that the reverse reaction is basically where the benzene in the gas phase can actually go and adsorb onto the vacant catalyst site and therefore Kb Pb into Cb tells you what is the rate at which the free benzene which is available in the gas phase gets adsorbed onto the catalyst surface. So, next we have assumed that the surface reaction is the limiting reaction. So, therefore all the other reaction rates really do not contribute. So, the key reaction which key rate which contributes to the overall reaction rate of this heterogeneous reaction is basically the surface reaction. So, we said that it is a surface reaction limiting. So, therefore the rate of generation of toluene should actually be equal to the rate of the surface reaction, the rate at which the surface reaction is occurring because it is the limiting step and so that will be given by Ks into Ph2 partial pressure of hydrogen multiplied by the concentration of the sites in which the toluene is adsorbed minus the concentration of benzene that is adsorbed to the catalyst surface and then multiplied by the partial pressure of methane divided by the corresponding equilibrium constant. Now, this also means that the other reaction rates actually have to be 0. So, that also means that the R adsorption divided by the corresponding Ks is approximately equal to 0 and also R desorption divided by the corresponding constant is also approximately 0. Now, from this we can actually deduce that the concentration of the benzene which is adsorbed to the vacant sites is actually equal to Kb Pb into Cb. Similarly, the concentration of the vacant sites, I mean concentration of the sites on which toluene is adsorbed is actually given by Kt Pt and Cb. So, this is actually obtained by setting R adsorption divided by Ka equal to 0. This is obtained by setting up this particular equivalence by setting up R adsorption divided by Ka to be approximately 0. Now, in addition to this, the total number of sites in a given catalyst is approximately constant. So, we can now depict that by saying if Ct is the total number of sites in the catalyst that should be equal to the total number of vacant sites plus the sites on which toluene is adsorbed plus the sites on which the benzene is adsorbed. Because these are the two possible ones that can in principle be adsorbed onto the surface because we assume that hydrogen is primarily in the gas phase and so is methane. So, therefore, Ct total should be equal to vacant sites plus the concentration of the sites in which the toluene is adsorbed plus the concentration of the sites on which benzene is adsorbed. So, by using this conservation property and also the expressions for the concentration of toluene adsorbed onto the catalyst site and the concentration of benzene adsorbed onto the catalyst site, we can find that the concentration of the vacant site is given by Ct divided by 1 plus Kb partial pressure of benzene plus Kt into partial pressure of toluene. So, if we know what is the rate limiting step and if adsorption, desorption processes in this particular case are not the rate limiting step, then we will be able to estimate the amount or concentration of the vacant site in terms of the observable quantities that is the partial pressure of benzene, partial pressure of toluene or the measurable quantities. So, now, so once we know the concentration of the vacant site, we can now go back and try to estimate what is the overall rate at which the heterogeneous reaction is occurring. So, because surface reaction is the limiting reaction, so the reaction rate is now given by Ks into C total divided by 1 plus Kt partial pressure of toluene plus Kb Pb multiplied by partial pressure of hydrogen Kt Pt minus Kb Pb into Pm divided by Ks. So, this can be obtained simply by plugging in the expression for the vacant site catalyst concentration into the expression for the reaction rate and we can obtain this particular expression. So, from here, we can rewrite this as minus RT prime equal to Ks Ct. So, pull out Kt from the expression divided by 1 plus Kt Pt plus Kb into Pb that multiplied by partial pressure of H2 partial pressure of toluene minus partial pressure of benzene partial pressure of methane divided by another constant Kp. So, this Kp is nothing but the equilibrium constant of the surface reaction multiplied by the corresponding constants for the adsorption of toluene and the adsorption of benzene. Note that Kp can be determined from the thermodynamic data of the overall reaction. Now, suppose if we neglect the reverse reaction, if we suppose we say that the total amount of the reaction primarily goes in the forward direction. So, if we neglect the reverse reaction, neglect reverse reaction, then we will see that the rate can actually be given by Ks which is the corresponding rate constant for the surface reaction multiplied by Ct which is the total number of catalyst sites which is available in the catalyst and Kt which is actually the corresponding constant adsorption constant for toluene and partial pressure of hydrogen partial pressure of toluene divided by 1 plus Kb Pb plus Kt into Pt. So, by clubbing in the 3, the Ks, Ct and Kt into 1 constant, we can write this as K into partial pressure of H2 partial pressure of toluene divided by 1 plus Kb Pb plus Kt into Pt. So, that can be written as this expression here. And so, now we can actually rearrange this expression and we can rewrite this as rearrange the expression as partial pressure of hydrogen partial pressure of toluene divided by the corresponding rate should be equal to 1 by K plus Kb Pb divided by K plus Kt Pt divided by K. The final form of the rate expression is actually a linear equation. So, with this, we have actually found the second step. So, remember that there are 4 steps here. First, we deduce the rate law and then we find the mechanism. So, what we have done is we have found the mechanism and after we found the mechanism, we now need to go ahead and estimate the rate parameters. So, in order to estimate the rate parameters, it is useful to write in this, it is useful to write in the form of this rearranged expression and you can see that there are 3 constants which are present. One is K, the other one is Kb, another one is Kt. So, now these 3 constants need to be estimated and we will have to use the experimental data in order to estimate these constants. So, when we write the expression in this form here, then we will be able to use the experimental data and we will be able to perform certain regression analysis in order to estimate these parameters. So, the rearranged equation can actually be written as a plus b into partial pressure of benzene plus c into partial pressure of toluene minus z equal to 0, where z is given by partial pressure of H2, partial pressure of toluene divided by the rate which is measured experimentally and a is given by 1 by K and b is given by Kb by K and c is given by Kt by K. So, once we have experimental data which is presented in the experiments that have been measured, as can be seen from these experimental data. So, the partial pressure can be measured, the partial pressure of toluene, hydrogen, methane, benzene have all been measured and different reaction rates have been measured under these conditions. So, using this data one can perform a regression analysis and using the regression analysis one can find these constants A, B and C. So, use experimental data and one can perform a linear regression analysis and using the linear regression analysis we can now find these constants A, B and C. So, once we know these three constants then we will be able to estimate what is the value of K, we will be able to estimate the value of Kb and we will be able to estimate the value of Kt and then we can find out by using the experimental data which was presented, the constants have been found and from the data K will be equal to for the data that is shown to you, the K will be equal to 6.18 into 10 power minus 4 moles per atmosphere square multiplied by kilogram catalyst per minute and Kb which is the corresponding equilibrium adsorption, desorption constant for benzene is comes out to be 3.5760 atmosphere minus 1 and Kt turns out to be about 1.48 atmosphere minus 1. So, these are the constants that are being estimated from the experimental data using the rate law mechanism that we have just found out and by performing a linear regression analysis on the experimental data. Now, what do we do with this data? So, what we can do is we can estimate some other information, we can deduce some other information in addition to these. For example, we can find out what is the ratio of the ratio of the sites on which the toluene is adsorbed to the sites on which benzene is adsorbed. So, how do we do this? Because we have assumed that the surface reaction is a limiting step, so the concentration of the adsorbed sites on which toluene is adsorbed is given by Cv multiplied by the corresponding equilibrium constant adsorption, desorption constant k toluene multiplied by the corresponding partial pressure. So, using that expression, we can rewrite this as by substituting those expressions, we can rewrite this as Cv kt into Pt divided by Cv kb into Pb. So, that is given by, so therefore cancelling the vacant sites, we will see that this is equal to kt into, if the total partial pressure at the inlet of the reactor is given by Pt0, then Pt is given by Pt0 multiplied by 1 minus x, where x stands for the conversion of that particular reaction divided by kb, which is the corresponding adsorption, desorption constant multiplied by Pt0 into x. And this is equal to kt into 1 minus x divided by kb into x. Suppose if the conversion x is about 0.25, that is it is a 25% conversion, then we can plug in these numbers and we will see that this is equal to 1.48 divided by 3.57 multiplied by 0.75 divided by 0.25. So, this is approximately equal to 1.24. So, what this suggests is that the number of sites on which the toluene is adsorbed is actually about 25% more than the number of sites on which benzene is adsorbed for the given set of experimental conditions. So, that is an important piece of information that one can actually deduce. So, if one needs to have a 25% conversion or x to be 0.25, then the 25% of this sites has to be greater than, excuse me, the number of sites on which the toluene is adsorbed has to be greater than 25% greater than that of the number of sites on which the benzene is adsorbed. So, let us get back to the algorithm of designing a reactor. So, we looked at rate law, we looked at the mechanism and by using the regression analysis, we found out what is the rate parameters, that is the parameters that is involved in the rate law. So, the next step is actually to design a reactor. So, let us consider a tubular reactor filled with catalyst. So, we get into the reactor design. So, let us consider a tube which is filled with catalyst and if Ft0 is the molar flow rate at which toluene is actually entering this particular reactor and let us assume that Ft is the molar flow rate at which the toluene leaves the reactor. Now, if I take a small element and if the total weight of catalyst which is packed till that location is W and the weight of catalyst packed at the other end of the element is W plus delta W, then we can write a simple mass balance in order to account for what is happening inside the reactor. So, the mass balance is what enters the element, what enters the element minus what leaves this element plus whatever is generated in this particular element in this element, whatever is generated in this element should be equal to the amount of material that is being accumulated in that element. So, what enters that element? It is the Ft, that is the molar flow rate of T at W where the weight of the catalyst up to that point is W minus the molar flow rate Ft of toluene at W plus delta W plus if the reaction rate is minus RT then RT prime multiplied by delta W equal to 0. So, that is the rate law. So, where RT prime is the generation of the rate at which the toluene is being generated and delta W is basically the amount of catalyst which is packed in that particular element. So, we can simply rewrite this as Ft W plus delta W minus Ft W equal to RT prime into delta W and this can be rewritten as DFT divided by DW equal to RT prime. Now, Ft can be written in terms of the conversion that is equal to Ft naught into 1 minus x. So, substituting the expression for the relationship between the molar flow rate of toluene and conversion, we can rewrite this as minus Ft naught into dx by DW equal to RT prime. So, therefore dx by DW equal to minus RT prime divided by Ft naught. So, that is the performance equation by assuming that it is a plug flow reactor by assuming that the reactor is a plug flow reactor. Now, moment we find the moment we write the model equation for the reactor, we need to now plug in the rate law. So, we know that the rate law is given by k into the partial pressure of hydrogen partial pressure of toluene divided by 1 plus kB into partial pressure of benzene plus kT into partial pressure of toluene. So, we can now plug in this rate law into the model equation and then we will be able to estimate what is the how the conversion changes, how much toluene is actually being consumed inside the reactor. Now, in order to do that we need to now express the partial pressures of toluene in terms of the conversion. So, we can do that. So, partial pressure of toluene is given by the concentration of toluene multiplied by R into T using an ideal gas law and from stoichiometric relationship this can actually be expressed as Ct naught into 1 minus x divided by 1 plus epsilon x into RT multiplied by P divided by T naught into T divided by T naught where P naught is the total inlet pressure and epsilon is basically given by yT naught into delta where yT naught is the partial pressure of toluene at the inlet divided by the total pressure at the inlet and delta is the change in the number of moles and so because in this particular reaction the number of moles is actually 0. So, therefore this epsilon is equal to 0 which accounts for the change in the volume and if you assume that it is isothermal, if we assume that the reaction is conducted on isothermal conditions then T by T naught is equal to 1. So, now using these assumptions we can now rewrite the we can now express the pressure partial pressure of toluene in terms of the conversion as P T naught multiplied by 1 minus x into P by P naught. Now P is essentially the pressure at that particular location and so if I call that as y then we can write this as P T naught into 1 minus x into y which is the ratio of the pressure at that location divided by the total inlet pressure. Now similarly we can write the partial pressure of hydrogen as P T naught multiplied by theta H 2 minus x into y and partial pressure of benzene can be written as P T naught into x where theta H 2 is the ratio of the concentration feed ratio of the amount of hydrogen which is present in the feed with respect to the amount of toluene which is present in the feed. So, in order to find the conversion profile as a function of the catalyst weight we need to estimate this quantity y which is the ratio of the local pressure with respect to the total pressure at the inlet. So, how do we find this? We can actually use the Ergen equation we can use the Ergen equation in order to find this ratio find this quantity y. So, the Ergen equation can actually be written as dy by dw equal to minus alpha by 2y into 1 plus epsilon x into T by T naught. So, what we have seen so far in today's in this lecture is essentially how to use the experimental data in order to find out what is the deduce the rate law which corresponds to the particular heterogeneous reaction and then using that rate law and to find using the observations from the experiment to understand and deduce a mechanism by which that particular heterogeneous reaction perhaps occurs and what is the rate limiting step in that particular series of in that particular heterogeneous reaction. Once we find this rate limiting step then we can actually using the rate law we can write the rate expressions for each of these steps and by identifying the rate limiting step we can find out what is the rate of reaction in terms of the observable or measurable quantities and from that we will be able to use that rate law expression find the rate law expression that we have derived in terms of the measurable quantities and use the actual measurements and the reaction rate and plug it in and using regression analysis we can find out the rate constants and the other constants which is involved in the reaction rate law. Thank you.