I'm Greg Lee. Thanks for joining us for another edition of Made in Riverside. But tonight, they're here rehearsing for the Festival of Likes, such as the West Nile virus. And so recently, City Mosquito Control was undertaken by the City of Columbia. I fell in love with Greg's personality. He was always hosting events. He was always involved in the community. He just talks to everybody and knows everybody. I'm in the business of talking to people. They said I could do it with these. They said he came natural. I think it's a gift. He really brings people together. Greg and I became engaged in 2014 in Oahu, Hawaii. We were so much in love. In September of 2015, Greg started to develop flu-like symptoms, and I ended up taking him to the emergency room. Greg presented with nonspecific symptoms. He had some nausea, some vomiting. He then developed a condition known as hydrocephalus and that swelling of the brain, and that can quickly cause damage to the brain and permanent injury. They ended up inducing him into a coma. When Greg arrived, he had been diagnosed with West Nile virus, and we confirmed that diagnosis. Our team quickly realized that he was going to need the expertise of many different areas. He was soon seen by our intensive care physicians, our infectious disease specialist, our cardiologist, our neurologist. He's the most severe West Nile encephalitis patient that we've seen. He then developed cardiac arrest and respiratory failure. I remember the care team at Kaiser Permanente being very aggressive with their care, being very open to letting us know what was going on. Knowing that really takes a burden off of you. Our entire team came together to surround Greg with all of the specialty knowledge that was needed to get him through this hospitalization. All of those specialists communicating with one another seamlessly depended on our integrated system of care. Dr. Dowd started to build a sense of clarity. He knew that there was potential to move forward. I woke up like, where am I? Why am I like this? My legs couldn't move. It couldn't move. The only thing that was moving was his eye, and I can see that little spark in his eye that something there is something that we're fighting for. He's just one of a kind. There's no one like Greg. Greg basically had to start from scratch. He had to relearn how to eat, how to talk, and now walk. He started regaining his upper body strength. He needs his bicep and tricep. I have a great care team around me. Dr. Dowd has always been there for me. Guys are permanent. It's always been there for me. They brought me back to life. Dr. Dowd is one of our guardian angels throughout this whole process. He just genuinely cares about Greg and his recovery. Hard work, patience, and faith that this will end. To see him continue with that attitude of determination is something that I can't describe. It just, it makes me very proud. Greg's taken the initiative to start planning our wedding again. Walking Tina down the aisle. Look forward for that moment. This whole deep breath has brought to our family, but that keep going. That's the only way to keep going. There's two steps.


Good morning mind explorers. I just told Julia a joke that's why she's laughing. Here we are hiking up to a large orbin with looks like three or three shoots coming out of it. I believe there's an open portal behind it. Looks like a talc mine. Makes sense because we're in talc country. So let's go a little closer see what we have up there. That's a pretty large orbin in it. I see two shoots that are still there. Looks like there was another one to the right. Oh the other one fell off it's down there. Think of the lumber that's involved in making one of these things. I see a birch nest up there. Yeah it is. Looks like a hawk's nest where they make use sticks like that. That thing's about 25 feet tall to the top of that post up there because in my surveying days that's how long my rod was and that's about how long it seemed like it was. I land surveyed for eight years when I was younger. Obviously I want to go over here. Feel the easy way. Interesting looking wash coming out of these mountains here. Wonder what's back in there. I want to look at the view out here to the dunes. We're looking at Death Valley over that way. Here's one on top of the orbin. It's a big one. Probably 15 by 20 something like that. Even at the sideboards on there they got three sideboards. Two by 12. Three or four even. Just looking down. Just looking back towards the portal. Okay that's the cavernous portal that we have just entered through into this talc mine. And here we are. Looks like something's been burned in here at one point. I see some timbering right here. What's this all about? Big stove. Cavernous added here. Another stove going up that way. Orays. Yeah some big stoves up there. Big blocky. Well there's a little cone all right. Not quite a few. Look at them all. Where are they loading this from? Look at all the load counts. Miners did that in there with their headlamp. Wow that is huge. You don't realize how big that is but that is that is really big. So that's why it opened there. Looks like a big timber here. More load counts. Quite of a normal ladder it looks like. Quite a lot of these raises that go up into a real blocky type stove. Left a big pillar in the middle of that room. Here come some stoves. At about a very sharp angle. 45 or more. Some big lagging. So that's all they've done. It's like rumen pillar type thing. They've come up and there's another shaft up there. You can see the daylight. Look at the size of those timbers. Those are at least eight by eights and pretty heavy duty. I guess you don't want this stuff coming down do you? Looks like a cable has been cutting through that. That must have been used for something else at some point. Kind of a different talc mine isn't it? Yeah. It's such big blocky stoves that water has been coming down in through here. There's another opening to the outside. I'm guessing some of those are gated up there just because. I wonder why they've done it like that. I suppose they know what to do. This seems like it's missing a support here somewhere. This thing. Yeah there's supposed to be four shafts. This whole thing is on about a 45 degree angle. You can see the angle of the van apparently.  I'm guessing this isn't going to be much longer. There's a little track in here that I can see. It's winding around. We're still rolling. Looks like somebody came sliding down there. There's more than four shafts up there. I can tell you that. I've seen about six already. I still feel air moving through here though. Another big area up here. There's some ties anyway. There's another set of timbers. Yeah that one's coming apart. That's a lot of times when they made them angled in a little bit. So they wouldn't come out like that so easily I guess. Oh yeah. Yeah you can see the the cable marks on this on the back of this stope slusher was running up into there. I still feel air moving through here so it must go a little more. There's a fence post. Some fumes it looks like. Well look at the drill marks. They're 20 feet above the ground. What do you see? So that's just some bracing they've drilled in to the back of this area just to keep it from collapsing. This goes down there. I'm gonna go this way since I'm already over here. It's like a huge cavern. All right continuing on. Here's another gnarled up slusher cable. I guess those things really take a beating. Of course you can see when they're rubbing against rocks and everything. I just can't get over the cavernous nature of this. It's just that's got to go up about where they've got some some wire up here for reinforcement. But this goes up probably it's got to be from right here to the top is about 20 feet up there and it goes up at I don't know it must be a hundred feet up in there. The others will come out here. Big huge slabs. The hanging wall that they take. Took some out. This is one of the most open tucked mines I've ever seen. Here's another one that just goes up. We're timbering. Wait look at all the stuff they got jammed in there.  Part of a cable. I'm not sure where that goes. Look at this thing wow That's crazy. All right moving on here. This piece of drift was heavily lagged and reinforced. They chucked everything in there that they had. One of the same. I thought that would be the end. Actually it says it keeps going. They had their ventilation and everything hung on here with the wooden stakes and wire as usual. Now I don't feel any air anymore. Yep so we're probably getting close to the end. Who knows. Look at all the hangers. Apparently there was a lot of material taken out of this mine touching from the size of these drifts and the stoves up there. It's huge. Oh boys are getting out now. Wow it's like we just walked into a sauna. Are you squeaking? There's a squeak. Looks like there's some guano there maybe. There's a short cross cut to the left of the main drift. Here's a big timbered hallway. Oh that's my pack squeaking. Unless there's a rat in it. It is hot. I'm not complaining I'm just amazed how much it changed there. Yeah it was nice and cool before. There must be some bats in here. I haven't seen any. That's guano though. I haven't seen any bats. Another big stove. It's amazing isn't it? Surprised we haven't seen any bats. I know because you're sure what it opens up. Yeah there is. Can you stop for a second and see what's squeaking in my pack. Another big stove. There's a huge pillar in the middle up there. Cable around it. Wow this is a big mine isn't it? Most health mines are much more heavily timbered than this and not quite some big openings. What's that doing there? So this was a ramp they put over here. This was a ramp they put over so they could drop stuff down and have it miss this little lower area. Did you see the end of this? No. There's a cam top there too. What's that? It's got something on it. Golden gate sold on merit. One pound net. Huh golden gate. What's that? It's a cam top for some kind of food. Well I think that is about it. It's good because my light's getting hot. It's hot. Yeah it's hot in here no doubt. So that's it they went from from white to red and they quit. The vein ran out. I guess she's leaving. She's out of here. You know there's a rubber cap or something. I don't know what that is.  Keep it from banging around. There it is. All right we're back to this spot. So Tom can't resist. He's got a little bit of ladder and he's curious. People all frayed up. Real ham shredder. On the table yeah. I said shred your hands if you didn't have gloves on. Do you think that they use this ladder again? No. He's curious as to what's above. Talc is slippery. For those of you who have not walked on it. It's just up into a big stove. Big stove up there. Have you thought? But now you know. I just climbed up that cable. Look at the size of these stoves. My god it's insane. And then Julie's down here. You can see the scope of these things though. It's just crazy. There's some timbering up here. She's down here. Goes this way a little ways. I might as well walk over there as long as I'm up here. They sure do shred the cable here I gotta say. These slushes were hard on cable. Here's some townhouse Bartlett pairs. Miners like their canned fruits I think. See that quite often. Okay I'm kind of walking along the top of these stoves. Which is a little strange. Here's more of that box whatever it was. Bandit buckles. Box of bandit buckles. Say that fast. This is up into the stove again. Wow. Crazy. I don't know. I'm not sure if it's worth it. Look at all the reinforcement. This is crazy. So yeah I'm not going to go any further that way because I just got to come back this way to get out of here. And this bridge doesn't look that fun. So that's the end of my road. I'll go back this way. Here's some of this anchoring that has failed as you can see. But you don't want it to fail when you're standing up in here. This little bridge isn't so bad although you wouldn't want to fall down this thing. It's a good 40 feet down there. This is back to where I came up that cable. And there's Julie. All right Tom's gonna come down hopefully not the fast way. Really steep. Yeah we have been on worse. The bottom two rungs are missing. You betcha. Wow this cable is totally free.  We appreciate you guys watching the video. Hope you enjoyed it and we'll see you on our next adventure. This sand is also uniform. It kind of gives you a vertigo doesn't it? How do you feel about being out here in the Sahara desert? I think it's great. Great. It also makes you want to have a margarita. Are you Irish? You're speaking Irish now you want to have a Mexican drink. That's because I've been in the desert for too long. Your brain's drying out. Your brain's drying out. Look at the dune. You want to climb the dune? Sure. It'll be great. It really is kind of disorienting here. There's nothing to, you know, nothing to uh. I'm going for the big pool ahead. Judge the big pool. All right I'll see you in the pool. It might be a mirage. Going up a wall of sand. It's like an old pack mill. Tom's walking off the no man's land. He's been in the desert so long. There aren't any ladders up that way.


the batch reactor which we have discussed yesterday all the ingredients are there in this problem okay what you have to find out is the reaction time okay and after calculating reaction time then you have to add 30 minutes that is the last line you have 30 minutes required for discharging cleaning recharging and all that then you have to calculate how many batches and then you have to calculate what is the batch volume yeah it is a reversible reaction and equations are already there derived and all that in 11 spiel book also right so you can if you derive then you will be more comfortable if you just copy from the textbook if I give the same thing in the examination I do not give you textbook okay so then you will have problem and I also would not give marks if you are putting from memory okay memory means only computers okay so that is why if you recall from memory and then just write there I do not give you have to derive good and the bracket A B R S are the like A plus B going to R plus S all liquid phase reaction also constant density system everything is there good okay anything okay good so that is the problem now we will start logically plug flow reactor why logically plug flow reactor why did I say logically plug flow reactor after batch plug flow and batch will be same under some conditions performance yeah I mean both are same and if they are same why they are same logically I said because the same equation under certain conditions is exactly same as batch reactor equation for constant density systems why they are same and all that now you have to get that knowledge okay why and that with clarity not simply you know like parrot because everyone says that they are same or they may be different then you are also repeating they are same they are different right but if I ask you why they are same why they are different not able to tell so in this course what you have to get is that clarity all the cobwebs removed from the mind clean house correct no so that is what I think you know definitely when you are going to choose continuous system when you are going to choose batch system that clarity has come okay with examples and also with reasons and all that so now in batch sorry in batch reactor again how do you calculate volume even though volume is not directly present in the design expression that also you know how to calculate so any batch reactor the time is same even if you want to produce 5 kgs 50 kgs or 5 tons 500 tons batch time is same but depending on the production rate you will have different volumes okay sometimes you may use 1 meter cube production may be less sometimes you use 5 meter cube because production is more so to calculate all that only this problem is given okay so plug flow reactor diagram what we normally draw is something like this it is a tubular reactor it is a tube right and then something enters something comes out and we also show whenever we have plug flow reactor that we have like this correct no yeah so like that what is the meaning of that constant velocity and this constant velocity when do you get throughout the tube fluid mechanics wise hi how hi because only at infinite velocity you will have the velocity near the wall also same as otherwise as long as walls are there is no real plug flow reactor existing okay without walls how do you conduct the reaction you can conduct in some heterogeneous system beautifully you can conduct the true plug flow is there actually you may not know that but anyway I will tell later that one also yeah so this is how we have to show we will show the plug flow reactor then normally we will say that we have f a not entering and this one is final f a f and here also you may have c a not v not and isothermal we are talking now isothermal isothermal plug flow reactor and temperature is same that I do not have to write throughout because isothermal then what else is missing here of course x a we can write which is 0 there okay x a equal to 0 then correspondingly here I have c a f v f which may be final and also x a f which has some value if that is also 0 then beautiful design no reactor because where is the reactor there because x a entering 0 and leaving also 0 therefore that is a wonderful design because nothing will happen there sometimes you know not doing anything is safe good okay so this is what and I do not know whether you remember any other names than plug flow yeah we call piston flow any other name tubular flow but rod flow means you have to slug flow means it is only two phase system we are talking about homogenous okay we have sufficient names because do not invent slug flow and all that that will not come slug flow will never be plug flow okay so slug flow we will discuss later when you come to heterogeneous systems we are talking about homogenous yeah normally the names what you have already told we may call it as piston flow okay and other name you said tubular flow but when you say tubular flow normally tubular flow tube associates with flow where definitely at the walls you will have velocity equal to 0 so that is why when you say tubular reactor it does not mean that it is ideal plug flow reactor if you want to specifically say ideal tubular reactor okay so that means it is plug flow what is the ideality in tube our assumption is that we have flat velocity profile okay that only gives me that plug flow condition so piston flow then ideal tubular flow reactor and rod flow also is rarely used it is right what you said is right but rod flow rod flow also very rarely used it is that means it is like a piston piston is a rod correct no see if you look at the piston when it is moving through a cylinder how it moves exactly you have the cut there okay the piston the piston diameter is cut like this exactly so then this moves like this okay so that is why we say it is a piston flow because we expect in our tubular flow that kind of flow like a piston flow okay so why should we assume that we have that kind of piston so the ideal definition for this is that all particles must spend exactly same time inside the tube okay inside the reactor each and every particle okay how do I imagine that okay I will give you a very simple example most of you would have seen that okay before that I will also write time times okay time spent by each and every molecule is same in a plug flow reactor yeah plug flow is always ideal we do not have to again say that ideal plug flow okay plug flow itself is ideal okay so then why we should have this name you know plug flow the reason is that when you assume this time spent by each and every molecule yeah time spent by each and every molecule is exactly same means that will be possible only when you have flat velocity profile or axial mixing equal to zero axial mixing means this is the axial direction this is radial direction or lateral direction okay so in this direction there is no mixing what do you mean by mixing a molecule coming here and then again coming back back and forth okay so that kind of thing is not allowed why if a molecule comes here there is a flow continuous flow you are sending continuously volumetric flow rate wise v not if you divide by cross sectional area then it will be velocity so volumetric flow rate wise it is going with constant velocity okay for example first time we discuss constant velocity afterwards we can extend that to what will happen if you have variable velocity also okay so constant velocity and then the molecule comes here and by some reason that will also may come back a little bit backward so that means this molecule which has which has been moving with some other molecules forward now coming back and joining the molecules which have which have come later than this molecule okay I think okay may be explanation is not probably you are not understanding so this molecule entered five seconds back at this point okay at this point if I take that it is time it is five seconds now there is another molecule which entered three seconds back right when you say axial mixing equal to zero means this molecule this molecule will never mix in reality what will happen this molecule because of some disturbance will come back and then mix with this molecule that is not allowed in plug flow simple example is conveyor belt but that velocity is with respect to what velocity of the conveyor belt is with respect to what there must be some reference you know ground with respect to ground may be the velocity of this conveyor belt is five kilometers per hour okay that is normal walking speed that is why and all of you are standing there just and behind you also there is another row three people behind you also there is another row I am talking I am imagining you as molecules so three molecules three molecules three molecules in reality there will be millions of molecules but only just for imagination we are telling only these three molecules just standing there and then just you are pushed by the conveyor belt where that is equivalent to for us here convection the flow so this flow is simply pushing those molecules without any disturbance simply like that okay so then in first minute your color changes to green your normal color this color okay so then after another minute then it becomes red after another I think it becomes green sorry orange so like that by the time end if you go so you may get some purple color or something that is equal to 100 percent conversion right easy imagination now at any time when I look into this when I look into this people standing here will have red color all of them are uniformly have red color okay what is the meaning of that that means the concentration if you if I imagine that color as concentration all that color is same for each and every person and the concentration is same yeah at that point at this point so that means what is the assumption we are giving without knowing also that the radial mixing must be infinity we say 100 percent or infinity we say what is the meaning of that yeah there is no change in the concentration and that is happening because all the molecules entering in that row just moving are uniformly converted not for 100 percent 100 percent conversion is not occurring at this point it may be 30 percent but uniform at this point it may be 60 percent still uniform at this point it may be 90 percent still uniform this is 100 percent if you get 100 percent conversion so that means whenever I look at the reactor across the cross section lateral direction we say that we have uniform concentration when do you get uniform concentration in a mixture yeah when you yeah exactly when you have perfect mixing then you say that I have mixing throughout the reactor same so that is the reason why even here at any cross section we say that we have perfect mixing and in other words we will also say because we are saying zero there the axial mixing this direction okay axial mixing is something like this right some molecule is here again it will come back some molecule is here another again it may come back molecule here again it may come back same molecule may be coming backward and forward so that means this is mixing with molecule which is which has entered a little bit late and again going forward and again coming and mixing with some other molecules which have just entered there so all that mixing is called axial mixing axial mixing but in plug flow we say that that is not allowed ideal condition so now what we say is we have this axial mixing axial mixing equal to zero in fact which means that radial must radial mixing must be infinity we do not have to say again but we say axial mixing is zero radial mixing equal to zero and then what are the other things you said a flat velocity profile unless you have the flat velocity profile all three conditions will not be satisfied but three times we reiterate okay axial mixing equal to zero same thing as radial mixing equal to infinity these two can happen only when you have flat velocity profile all three can happen only when you have time for each individual particle and exactly same so that is why the basic definition what we would like to remember is the time of each and every particle okay why they are same means without any disturbance in the sense that all these molecules okay now we will expand our imagination slightly more now I will say that I am now feeding the molecules continuously but these molecules I am sending in the form of pockets each pocket may have some molecules where there is some concentration and all that so that means at time t equal to zero I imagine these are the molecules just entered okay these are the molecules just entered so now if we accept these definition like axial mixing equal to zero and radial mixing equal to infinity and of course the other one what is that the flat velocity profile all that if I assume then the same pockets will move from here to here each pocket now you can imagine as a very small batch reactor inside that batch I may have good mixing and all that okay so our imagination we are making smaller and smaller and smaller till individual molecules but first let us I started with molecules but let us imagine that we have small pockets and in that pocket the reaction is going on and this we are not allowing them to move forward and backward and that means axial mixing equal to zero which also means that radial mixing equal to infinity then at this point what is the conversion in each and every small reactor that small batch reactor conversion it is same and that conversion depends on how much time it has spent inside the reactor if it is 1 minute some conversion 5 minutes some conversion now let us say total time is 10 minutes required for 10 minutes you may get 99 percent conversion so at the end if I see oh they are not become bigger same yeah they may become bigger under some conditions also okay same size yeah at this point if I look each and every pocket also will you have exactly okay now slightly relax our I think this is clear now yeah the turbulence and all that when we say when you know you may have that very high Reynolds number you have turbulence what do you mean by turbulence there it is not that turbulence what we are talking when a molecule here can suddenly jump here a molecule here can suddenly jump here that will happen if I have a tank reactor if I have tank like batch reactor we have studied so batch reactor you take and then you put the all the reactants and then start studying can you identify its location you cannot because one molecule may be momentarily at the bottom of the reactor because of mixing suddenly it may come out or again because of mixing it may go to one corner and again you know another corner like that at any time any point of yeah at any time inside the reactor at any point it may be there in fact that is the definition of mixing you cannot say that yeah you cannot say that I have mixing but molecules are not moving okay so then when we say that we have perfect mixing the possibility for the molecules to move from any water anywhere is possible for I mean for for the because of the mixing yeah Abdul you know that the reaction happens by collision of molecules and in plug flow it is highly limited there is there is lesser conversion there is lesser collision of molecules how do we decide the lesser collision of molecules if we let it devide from plug flow and let it come out let there be actual actual mixing no but why do you say lesser collision I think you say that the collisions are there when only when you have tank reactor there will be some lateral mixing but yeah but even here when the molecule molecule level what is the size of the molecule how many molecules you can put in a in one inch pipe diameter so there yeah so there is again you know all those collisions and all that is there definitely but now that is not this way but only this way why only this way only why this way because if the flow is so good that it is not allowing anything to come back any molecule this is not allowed in this direction because of the convection velocity push that is bad actually because if I have a yeah I think that is a good question also because so that others also will learn that if I am allowing a molecule which is already converted with the molecule which is not converted what is the concentration now we cannot say one molecule but few molecules here they are converted 50 percent now suddenly they come and join here mix right so then they are coming and mixing with these molecules these are 25 percent or 30 percent converted what will happen to the concentration now it will go down and what will happen to rate of reaction it will be down because I think rate is a function of concentration see if you first order if you take very simple imagination directly proportional so that is why the concentration falls this is the reason why the mixed flow reactor is inefficient correct no we know that if I take one meter cube mixed flow reactor one meter cube plug flow reactor one meter cube mixed flow reactor will give me low conversion because only means fixed and one meter cube plug flow reactor will give me high conversion you know why now that is the reason the reason is because of that mixing so that is why here that same collisions are taking place but that is only in the restricted region theoretically speaking it is very very thin infinitely small region not at molecular level we are talking molecules levels are very very small okay so that is the reason why we say that yeah the plug flow reactor is more efficient than mixed flow reactor because mixing always you know this is what many students will have this imagination that mixing is better for reaction no mixing is bad for reaction okay perfect mixing is bad for reaction but you are allowing the you know again do not get confused so here even in this cross section I have perfect mixing so that means at this cross section I have the same concentration for each and every pocket because molecules we cannot say and then say concentration molecules are so many moles per unit volume only that is what is the concentration that is why this concentration is not diluted here in a plug flow you already told me no sir the concentration slowly changes okay very good that is the definition of also plug flow slowly means how slowly at only at each and every cross section only that concentration is changing slowly not across okay but definitely across means at this point I have different concentration at this point I have different concentration so that is why when I plot this graph concentration versus length this is the length then we will have this is C A not it slowly decreases and you also know that I mean as I already discussed that mixed flow reactor is better than sorry mixed flow reactor is less efficient than plug flow reactor we discussed that the reason is this concentration it is slowly decreasing depending on it is time here but across one cross section so now next we have another cross section okay another concentration next one another concentration that means at every point this rate rate is maximum here because assuming ideal I mean non isothermal reactor I have first order reaction K into C A not is the rate or C A is the rate at time t equal to 0 just when it entered what is C A not highest so that rate is more okay I am talking about simply A going to R first order reaction then slightly inside now it is K into C A but C this C A is highest okay highest means it is not highest not more than C A not but next to next one it is highest so like that slowly it is the concentration is decreasing when you have slowly concentration is decreasing the rate of reaction you know then that means I have one rate here one rate here one rate here one rate here one rate one rate one rate so many rates right so all those rates when you average them that only gives you the overall conversion okay but that is why the concentration of each and every I mean at each and every point very optimally used in a plug flow reactor whereas mixing flow reactor you are taking a I mean one tank and then continuously putting water I mean reactants continuously you are taking out stirring is there perfect mixer so the moment you put C A not then yeah then it is suddenly mixed instantaneous mixing because you are perfect mixing what is the meaning in everything is instantaneously mixed now when you see the instantaneous mixing do you see C A not at all then you see slightly less than C A not yes sir yeah how much slightly want concentration you see in the reactor yeah it is the outlet concentration and outlet concentration is how much 90 percent so your C A not is suddenly decreased to 90 percent concentration which is equivalent to your outlet concentration right so that means all that concentration history gone like here I have concentration history at this point okay if this is 10 moles per C A per liter 10 moles per liter this will be 9 okay this will be 8 7 so at this end I will have 1 mole per liter what is the conversion 10 moles 9 conversion equal to 90 percent right so now yeah but that 90 percent occurring here through so many rates correct no because concentration here different concentration here different so that will have a rate that will have a rate what you see is the overall effect of that concentration at the end whereas if you go to a mixture flow reactor where you take the tank and then just put let me also because for discussion I think I have to give this yeah continuously this is entering continuously you are taking here so now here I have C A not here I have C A F and by perfect mixing assumption which is simple assumption this is also C A F and now you see there is no temperature history at all yeah what is the plot here concentration versus C A not versus okay length there is no length actually but I think you know space coordinate so what is that much now like this and then you have C A C A F so all temperature history totally gone there that is why to maintain this conversions because given a volume given volume this conversion will be less this will be more right but to compensate let us say that you want to also get the same conversion like this where concentration come maintained concentration maintained every rate all rates are average and then you are not averaging system averages and finally it gives you so much conversion there if you want to compensate what you have to do oh no that is different same one reactor you should go for larger volume bigger volume so that is why you see you do not need any mathematics simple logic we know that you know the volume of the mixture flow reactor will be definitely larger than the volume of plug flow reactor for a given conversion that you have to always say for a given conversion for given volume the conversion in plug flow reactor will be more than C S T R okay good so that is what is the plug flow and that will happen only when each and every particle exactly moves like that and now I will give you another example as I told you that you have you know heterogeneous system I told you you know the coal is burnt using moving grades I do not know whether you heard or not coal coal for combustion so that will be something like this you have a wheel here which will be rotating another wheel here which will be rotating over this you have the conveyor belt so this moves like this comes out and then moving so now here I will feed coal continuously I am feeding and I will now calculate what is the time required for each and every part let us imagine that every particle is one centimeter easy to imagine okay one centimeter uniform size right so all the particles are coming there and then by this time it comes here if it is pure coal you will not have anything you do not have even ash ash also you will not have ash you will have whenever you have some silica and minerals and all that okay then only you will get ash otherwise no so that means each and every particle is entering coming here by this time at this point the particle completely disappears right because you have now fixed the space the not space the speed of the conveyor belt okay at 50 percent of the time that means you know if the total is let us say 5 minutes at 2.5 minutes at this point each and every molecule each and every particle would be uniformly converted right uniformly converted then when I look into this direction that means the yeah so this third dimension that is this flat plate this is moving like this okay this is moving like this this is what the distance I told so now at the middle if I just see here I will have the conversion of this carbon particles uniform throughout so at this point if I draw a line and see the concentration what do you say about that concentration same that means radially you have same concentration uniform concentration that happens because of infinite mixing see that is what is the meaning of that infinite mixing there because you know it is only uniform concentration normally when do we imagine that we have uniform concentration when you have mixing so what we say for axial direction is the mixing equal to 0 right and then in the radial direction mixing equal to infinity because you said 0 there you are putting here infinity right but in in in words it is that the uniform mixing means infinite mixing and uniform concentration means that will only happen if you have infinite mixing the infinite mixing means good mixing how can you provide infinite mixing there are many words which we do not mean anything including love and when you say I love you it would mean anything but as I told you know now explain to me what do you mean by I love you you cannot tell that is a feeling infinite mixing means it is a good mixing that is all where you have uniform concentration that is why those words infinite mixing and axial mixing equal to 0 axial mixing equal to 0 I will explain that may be true but infinity we can never see in the moment you see infinity you cannot come back again gone you are gone to infinity so that is the problem you know with these words but still we use very frequently infinite mixing for radial direction 0 mixing in axial direction 0 mixing in axial direction is explained by this you know very clearly that this molecule this pocket is not allowed to mix with this molecule okay and this kind of mixing when you have you will not have this condition what is that condition time spent by you cannot you know have that condition the reason I will tell you the same thing same mixing is happening everywhere axial mixing there is axial mixing so by the time you come here you will also see same thing same thing I am just pushing it there so a molecule comes here and a molecule coming here so now each and every molecule is not going and mixing with that okay so one molecule which is coming here may come out other molecule which is about to leave may may come back so now what is the residence time of this molecule which has gone out and what is the residence time of this molecule which has come in and the average of that is 10 minutes the average of all those timings if I take that will be 10 minutes correct no you know you have seen recently olympics 10 people are running so few people will come faster some people will come you know just behind and you add all those things and take average so then we know that average run time for that event we are not yeah that is what is the mean residence time here okay but some people would have come early some people would have come late right but now when I have 10 minutes there may be a molecule which has come in the 9th minute itself and there may be one molecule which is coming at 12th minute itself here right so that is not plug flow why every molecule or every particle should come exactly at 10th minute that is what is plug flow if we have plug flow in olympics you have to give medals to everyone correct no all of them are coming exactly at same time but in chemical engineering no we do not allow that all of you should exactly come out of the run with the same time that is what what we tell there and if you want to have a feeling for this I will tell same example when you are walking when you are walking some people may be more enthusiastic people they may move faster they will push you aside and then cross you correct no so that way may be going another 5 minutes may be 2 minutes before then you and there may be very slow fellow who is who is having diabetes heart attack and everything so those people cannot move so he is going behind only not going forward so those molecules may spend more time so that is not allowed in plug flow that is what is axial dispersion model when you really allow that because we can allow we have to allow that because in reality you will never have plug flow plug flow is only in our minds as a definition but plug flow for heterogeneous system this is the excellent example you know this conveyor belt coal conveyor belt perfect example is only this that means you are designing this and also you know perfect means I cannot really say perfect because when you are designing this you should design that without any vibrations right and without any sagging you know when you have this side one wheel this side another wheel when you put a belt it is definitely slag a little bit right so without that that slagging also you should sagging also you should design no vibrations so what will happen if I have vibrations and allow this a little bit articles will be coming forward and backward okay that is what is axial mixing right so that is why and that kind of perfect design even for conveyor belts I doubt whether there is any that is why I told plug flow is only in our mind but still that will help us in design you know very beautifully so that our volumes will be not exact but you think it will give one extreme that okay if it is ideal condition this is the volume if it is the real condition like I have axial mixing you need definitely more volume right because because of axial mixing the concentration is decreasing so instead of 1 meter cube you may have to use 1.2 meter cube so when you are moving in that cube if everyone is exactly moving with the same speed without pushing without pulling okay correct no that cannot happen anyway those who entered there that cannot happen without pushing without pulling then that is ideal plug flow everyone will see the god exactly after half an hour if it is continuously allowed okay but most of the time it is not continuously allowed they will put you in a place called cell okay cell and then lock you yeah that is a batch reactor that is what what you what you are doing in batch reactor you are not allowing any flow so intermittently they do that but I think after last cell you are allowed to move so when you are moving that when you are moving in that cube sorry okay batch PFR okay so then after we are talking about only plug flow section when they are moving so when each and every one is exactly moving with the same speed no push no pull by the time you lose patience okay because last cell will really kill your patience by the time you should have spent 10 hours or 15 hours you know the entered the entering time so that is why and then you start pulling and pushing okay so when you start pulling the particle will go forward correct no you push them so so in that cube there may be five six people or ten people you know so if you are so strong you will push them out and move or maybe one person will move forward so that fellow will be very angry you will say like this so he is pushing them back so that is why he said push and pull you know yeah so when they are when this fellow is pushing them backward then that molecule will come here okay so now is a weak person may be moving always from here to here here to here that is what is the example of axial mixing so next time when you enter the cube remember plug flow and axial mix okay yeah that is what an ideal plug flow that is not and of course even with conveyor belt we are so impatient we will be pushing and pulling there also okay so conveyor belt is to make them uniform time but that I think proposal was not successful because there is not much space all that I think so that is why maybe one day it may come right so that is another example for this axial mixing but in reality you will have that kind of pushing and pulling that is the same thing in that is happening when you have ideal I mean tubular reactor not ideal tubular reactor in tubular reactor you have now I am coming to your question you asked me know sir there is turbulence in the packet bed yes there is turbulence but that turbulence in packet back sorry in plug flow is only limited like this you can show that you are showing how are you showing turbulence in in okay in turbulent flow how are you showing in pictorially yeah but what is that what do you call that and all those things are velocity fluctuation we call them as velocity fluctuation in ideal plug flow what you are mixing is that I mean what you are imagining is that fluctuations are not allowed sorry they should have when we imagine plug flow ideally then they should have that kind of no velocity fluctuations are allowed in plug flow okay so that is why the turbulence is happening because of the velocity fluctuations because of the velocity fluctuations why the velocity fluctuations are there molecules are coming backward and forward and when they are coming backward and forward they are losing concentration because they are converted to slightly higher conversion but they are coming back and again mixing with lower conversion I mean higher conversion making that entire region is lower conversion average conversion right you know average conversion is between these two conversions this నిఘి నియికినెతా ఒారిమిచుందా ని హిమా ఆరినినిందూ కారినా and you are spoiling this fellow, this fellow has higher concentration, right, back, somewhere here, okay. This is higher concentration, this is lower concentration, we are talking about only reactants. So then you are, in this region we are diluting the concentrations. So when you dilute the concentrations, rate of reaction will be lower. When you have rate of reaction lower, then naturally your reactor volume should be higher. Why? In all reactor designs, I mean how many, we have only three reactors, batch reactor, plug flow reactor and mixed flow, not so many. So in all these reactor expressions if you remember, if you recall your memory, in the denominator you have rate, okay. So when denominator is smaller, then volume will be higher, okay. I think you know the derivation of this equation is very very simple if you understand this, okay. Ideal plug flow is the, sorry, plug flow itself is ideal. Yeah, the plug flow what we are imagining in our mind to be in our reactor is like conveyor belt where no one is moving, like coal particles, perfect conveyor belt without any vibrations, without any sagging and all that. So that is what, then each and every particle will spend exactly same time. So when I calculate, I mean when I, when I measure the concentration at any point across, then I will have uniform concentration. That means I have, I will say now that I have infinite mixing radially. No molecule is going forward, no molecule is coming backward. So I will say axial mixing equal to zero, right. And then the other one, this one, time spent by each and every molecule is exactly same. So all these things mean only one definition that is time for each and every particle is same, okay.


Welcome to our Spotlight Conversation Program podcast thing. Write in the comments if, while you're watching this, you come up with a great name for what we're doing. Yes, perfect. I am Adam Novus. I am here with the wonderful Liz Wade. Liz, hello. Oh, thank you, Adam. Hello. Hi. It's good to be here. Good. Liz, if you are new to this program, we are talking about our week's featured program, which is Equadorian, I'm going to say it wrong, Finesca, Equadorian Finesca Soup. We will put a link to that program in the description and we'll try to link to it a couple of different places. You can listen to it. You can decide, Adam, who needs to point up in the corner. Probably you. Probably you. This way? Other way. Okay. We'll put the link right here. Yeah. We think. We're doing that thing where you point to the part of the screen and down, like that. Anyway, so you can follow along with that program on our website. You can find it there. But we really encourage you to listen to that program. It's at a slower speed than we're talking and then come back and listen to us talk about it. Yeah. We're catch the, yeah. Yep. Usually there's an advanced version, but there isn't an advanced version. So I'm going to keep my mouth shut. So Liz, how are you today? We are going up. We are a week out towards the holiday of Easter and that is a, I mentioned that because Happy Easter, but also because that plays an important part in this program. Well, as you know, yesterday was Palm Sunday. So that's when Christians celebrate that Jesus came and people waved palms. So now this is when, yeah, the holy week where we lead up to Jesus's death and then resurrection on Easter Sunday. So this is a huge holiday celebrated in the United States and all around the world. Yeah. Yeah. All around the world. Sorry. I should have said that. Yes. Definitely all around the world. And as we will see in this program or you already know, if you've listened to this program, it's definitely celebrated in Ecuador. So and then, yeah, this is about. If you don't know where Ecuador is, as an American in who is very poor at geography, give me a sense of the continent. Where would I find Ecuador if I'm going to look for it? Ecuador is in South America on the Northwest corner. And the equator runs through Ecuador. About Ecuador. Yeah. Well, it's the equator, which goes through the middle of the earth and imaginary line, which separates the earth in half. I think Ecuador is named after the equator, right? It's right on the equator. Yeah. Or whoever named the equator also named Ecuador. I don't know. And I will say I have been to Ecuador and it is a beautiful country with probably the best weather I have ever experienced. Because it does lie on the equator. So yeah, I would like to go there origin there. I would like to go there. I've never been. Yeah. Maybe we need to plan a listener trip where we where we visit listeners everywhere around the world. Wouldn't that be so once this pandemic is over where we'll be all about it. A world tour. I feel like once this once this pandemic is over, I'm just going everywhere. Yeah. I'm never coming home again. I'm just going to live on the road. Exactly. So this week we're preparing for Holy Week and this program is about a food that is prepared in Ecuador during Holy Week. So yeah, and eating this food is a family tradition. It's yeah, it's a special food that people prepare and eat. It's got special meanings for the Easter holiday. Yeah. There are actions. It's not only a food that you eat to sustain yourself, right, but a food that you can eat and remember things through. Right. There is food that has symbolism and there are a lot of traditions around the world where people use food to symbolize whether it's spices or a certain meat or not meat sometimes. Right. There's people eat fish around this time rather than meat and because they're giving up something for the season leading up to Easter. Liz, do you have any food traditions? I'm trying to think of. No. We don't really have a big tradition, well, I don't know. It depends. Usually my family does get together for Easter and we usually travel to be with my husband's family and so we're down there. But here's a thing that you might not know about me, Adam, or you might already know this. Okay. I am a terrible cook. What?  No, I wouldn't have put you as terrible. Oh, yeah. I do not enjoy cooking. I enjoy eating, but I don't do very much of the cooking in my house at all. Even with recipes, like my family just tells stories about how I mess up recipes. Really? Yeah. Oh, it's bad. It's very bad. I'm a good baker. I can bake things like bread or cinnamon rolls. So let's dive into this a second, if we can. Was your, were your parents, did they enjoy cooking or bake? I'm trying to think, we're talking about traditions and traditions usually get passed on from grandparents to parents. My mom would often make a very big meal. It was simple food. So it might be like a turkey or a ham or something like that.  And she did like to try new things every once in a while. But the one thing I do remember from her cooking is that she would invite anyone who didn't have anywhere to go. And we would always have someone extra at our holiday table, whether it was Easter or Christmas or Thanksgiving. We would always be room for people to join if they didn't have somewhere to go. That is, that is very sweet. She has a very big sense of hospitality and an open door, as we said. Yes. Yep. Exactly. And there was always too much food, you know. So the joke was that nobody could leave until all the food was finished. And then everyone would laugh and laugh because there was so much food, no one could finish it. No one could ever leave. Yeah. So, yeah, so I don't really, I don't really do any of the baking. Or sorry, I don't really do any of the cooking for a holiday meal. Sometimes my mother-in-law will ask me to bring something. She'll say, you know, can you make the mashed potatoes or can you make the rolls or whatever? But she tries to ask me to bring very simple things. Yeah. Because she knows I'm not good at it. Yeah. But she does make very elaborate meals. She likes to try very fancy recipes and do special foods. But that's fine for me. Yeah. I like to go and eat it and spend time with family and do that. Well, you would do well with my mother who loves to cook. Oh, yeah. She will say, what can we bring to the holiday meal?  Yeah. And then she'll add, well, I also will have a pie. So this will be the second. I was just going to ask, also make pies as well. Yes. Yes. Usually there may be seven of us and we'll have three pies, different flavors. Or again, it's too much food. And I think there has to be something about how a part of the holiday is having more than you need. Right. Like that's the celebration where you don't feel a lack, where you don't feel like you're missing on anything. So I think that's probably something that happens. You know, I think that's got to be a sort of a worldwide thing. I would love to hear what people have to say about that in the comments below. Like, is that normal? I mean, of course, there are times when you don't maybe have enough money to feed as many people as you want. Or, you know, to get the ingredients that you want or whatever case. But I think even in those times, people have the want to have there be a celebration and to put all of your resources there and to be together with family. I'm curious that that's something that people also experience. I would love to hear about it in the comments. Or I think if your family, let us know if your family was also like Liz's mother, very hospitable. If you invited people into your home because it's the holiday or at any time, let us know, tell us a story about when that happened in your life. And if that's a tradition that you've continued. Yeah, I wonder if that's culturally that is the same. You know, inviting a person into your home rather than going out or whatever. So, yeah, I would love to know that. Yeah, speaking of baking, Adam, while we were speaking of baking a few minutes ago, we do also have a video in our video extras for holiday videos about making a baked treat for in the UK. Yes. So obviously, Adam and I are not from the UK. We're from the US, but we do have people who make spotlight who are from the UK. And they made a video about a very popular baked good there called Hot Cross Buns, which, you know, I know a little song about. I'm sure you do, too. It is often a song you learn on the piano or the recorder. Yeah, or the recorder. I wish I had my recorder here. It's probably it's probably nearby. I could play it for you, but there we go. Should I do a little concert? No. But anyway, it's about Hot Cross Buns. And then I have actually never had a Hot Cross Bun. Oh, have you? No. Yeah, I mean, I always I always know about them because, you know, they're in the song and I play the recorder. But yeah, as far as eating one, I have never did you know that they have like dried fruit in them? Only because of this video. Yeah, see, I didn't know that either. And then also they have to be there's something about the dough where it has to like sit for a really long time or something like that. And it sounds like you and I need to go watch this video. I know, right? Yeah, I'll link that over here. Ding! Ding! Yeah. Goodness, well, we we are so glad that you have stuck with us as we've rambled on. But oh, before, OK, I just wanted to say this, this program is about finesca soup in general. But there are more programs that we have about Easter foods around the world. So if you have a food that you commonly eat around Easter, you can check out this program and see if I see if we mention your food and if not, tell us about it. Excellent. Yeah, so we hope you've really enjoyed this conversation.  And I feel thrilled and privileged. We always say that spotlight has the best audience. You guys are wonderful and kind and sweet. And we hope that this has helped you on your English journey. Practice listening to two native English speakers. Make sure that you like and subscribe to this video on YouTube. But also check out our Facebook and Twitter and our website and our podcast. Wherever you're looking for us, we try to be there. Yeah. So we hope you have a good day and happy Easter, Adam. Happy Easter, Liz.


All right. So let's start the session on a relation and functions. It's a third session for you And I'm assuming that all of you were present in the previous two sessions. If you're not I would request you all to watch the videos already posted on the group So let's resume with today's session Dear students if I recall correctly last class I talked about one one function. So we are doing types of functions We were doing types of functions and in the type of functions. We started talking about one one function Okay, so types of functions and The first one under that was a one one function a one one function Okay, one one function. What was the definition of it? I don't worry. This is just a quick recap So if anybody is joining in late, okay, don't panic. I have not started with the session So this is just a quick recap of whatever we ended in the last class Last class. I remember correctly that I did equivalence relations with you we did a lot of questions on them and We started with types of functions and under that we were doing one one function. Yeah, so what's the one one function? Can anybody define me one one function? Just quickly unmute yourself Plain and simple language plain and simple language. What's a one one function a one one function is where your different different different images come from come from different pre-emages Come from different pre-emages just plain and simple. You don't have to break your head in understanding anything else So if an example to give an example of a one one relation, so let's say there is a function from set a to set b Okay, set a has let's say a bc Set a b has let's say one two three four. So this type of mapping Okay, I'm just giving you a simple example. This type of a mapping is called a one one function Okay, you also call it as an injective function Injective function some books will call it as injections Okay, it's a combination of injective and functions Okay, so you can see different images So one three and four are images over here. They come from different pre-emages Right. However, you would recall in the definition of a function. We had learned that two pre-emages can map to the same image Right, that is also a part of a function But in one one function, we do not entertain such cases where two pre-emages are pointing towards the same image Okay, so definition wise it is clear. No doubt about what's a one one function And if I recall correctly, I had also done how many one one functions are possible Number of one one functions So let's say if a set a contains N elements and if a set b contains m elements How many one one functions were possible? How many one one functions were possible? So we took various conditions first we took the case where m is less than n If m is less than n that means the number of elements in set b is lesser than the number of elements in set a Could we have any one one function possible? Yes or no. No, no There was zero one one function in those cases Where m is greater than equal to n the number is m p n p stands for permutation here I'm sure all of you know the formula of n pr just a quick recall from your last year chapter PNC chapter n pr formula is what n factorial by n minus r factorial Okay, so this is what we had done the last class So just quick five minutes. I took to recap it Any questions on this anybody? Okay So good morning to everybody who has joined in Hope you're doing well. Hope you are safe at home studying. Yeah Okay, so now we'll we'll go into more details of this one one function The first question that arises in your mind is how do we identify a one one function? If some function is given to me, how do I know that it's a one one function? So let's do an identification of one one function. Okay, so how do we identify a one one function? identify or prove Where there are function is one one function Identify an injective function Okay Now all of you, please Pay attention over here because this is a part which is going to be asked in your board exams also Okay, so just pay attention to this what I have to say if possible put your pen down I'll give you ample time to know no down things Now just try to understand these you know steps, which I'm going to tell you Now when you're going to identify a Function to be injective function. Basically, we can do it in three ways Right, there are three ways to identify Okay The first way is By drawing the graph of the function Okay, or you can say graphical method, but let me tell you this method is just a optional method Okay. Hi, Vinay. How are you? Vinay had accidentally on his video cam. No worries So graphical method is basically a method by which you can identify By plotting the graph or sketching the graph of the function But let me tell you this method is only a supplementary method Right in the board exams probably your examiner will not entertain Just the graphical method of proving a function is one one Another drawback with this method is many of us may not be very good in sketching graphs and Many at times the functions may be too complicated for us to sketch the graph Okay So nevertheless, I will tell you this method. What is this method so that when you are you know applying it to? Verify your result you can do it. So what does graphical method say it says that a function is one one a Function is one one if a horizontal line drawn a Horizontal line drawn What is the horizontal line mean horizontal line means a line parallel to the x-axis a Line parallel to the x-axis Okay, so if you draw a horizontal line It must not cut the function at more than one point Okay, a horizontal line Must not cut Must not cut Must not cut the function graph The functions graph at more than one point at more than one point Okay, now, let us understand why does this you know horizontal we call this as a horizontal line test We call this as a horizontal line test horizontal line test line test Now don't confuse it and don't confuse this with the vertical line test vertical line test is used to You know in and establish whether that given relation is a function or not Right if you recall the very definition of the function was Every image or every pre-image must have a unique image, right? This was the definition of a function, right? In order to establish whether a relation is a function or not. We used to have vertical line test. I Think you would have done this in class 11 Right everybody's clear about vertical line test I'm not talking about vertical line test here. I'm talking about horizontal line test Okay, so if you have been given any function, I'll just a sketch a you know Just rough figure. Okay, let's say my function is like this Okay, so this is my function graph Some arbitrary, you know function I have sketched if you draw a horizontal line If you draw a horizontal line, let's say this yellow line is a horizontal line You will see that this function will be cut only at one point by this horizontal lines only let's say at point a In that case this function will become a one-one function. So this becomes a one-one function, okay? If it is cut at more than one point Let me give you a typical example of that. Let's say I have a function like this. I have a function like this Okay, and if I draw a horizontal line like this, okay What is going to happen in this case? It is cutting at one two Three okay, so there are three points at which this function has been cut by this horizontal line in this case This function will be not a one-one function. It will be a not a one-one function Now the question arises why? Can anybody explain me why it happens? Why if it is cut at more than one point we don't call it as a one-one function any any answer See, it's very simple. It's very simple If you look at the second case, okay, what is happening? Let's say this is an output output is B or you can say output is Y Okay, this output is coming from three different inputs. Let's say I call it as a B and C Right, so what is happening is that okay? Let me name it as D Let me name it and let me name this output as D or O. Okay, O will be a good idea So this output O is coming from three separate inputs a b and c Okay That means if you're going to sketch an arrow diagram for such a scenario if you're going to sketch an arrow diagram For such a scenario what is going to happen is that? a, b, c will all point towards O correct So you can clearly see it's not a one-one scenario Okay, in one-one scenario every image should come from a different pre-image But this image O has come from three different pre-images right, so same Image has come from three different pre-images. That is not a one-one function by the very definition of a one-one function Getting my point, but what is happening in the first scenario? Let me name it as case one and case two But what is happening in the case one? Let's say this output. Let me name is as O dash is coming only from one input Which is a dash Okay, so let me call this function as g function So g a dash only is giving you O dash and no other pre-image is giving you O dash Getting my point. So this is this is accepted as a one-one function. This is not a one-one function Is that fine any question regarding horizontal line test again? I'm repeating it do not get confused with vertical line test Vertical line test is used to a certain or used to establish whether a given relation is a function or not Okay, but horizontal line test is used to a certain whether a given function is one one or not So there's a difference between these two Is this clear please type clear on your chat box so that I can proceed at least I should see 15-20 clears on my chat box Thank you so much my dear students for confirming this Now we'll move on to the second method now. We'll move on to the second method Which is the non-graphical method? Very good. Thank you Okay, second method that we are going to talk about is basically a non-graphical method Which we call as the uniqueness test which we call as the uniqueness test This is very important and Majorly, we are going to prove the one-one nature of a function All right only by these this test graph is supplementary. You can draw the graph to support your answer Okay, don't don't draw the graph and say okay see in the graph. That's a one more thing. I forgot to tell you When you are trying to prove the one-one nature of a function or trying to prove whether a function is one one Don't draw such a horizontal line, which will save it from cutting at any point Okay, you don't have to save the function for example, let's say There's a graph like this Okay, right is this function one one or not you don't have to draw a horizontal line here and Say sir see this horizontal line is not cutting the function at any place So it is a one-one function. No my dear you don't have to save the function You have to scrutinize the function Right, so you'll draw at such you'll draw at any such place where it is cutting it at more than one point Okay, if it does it will not be a one-one function Okay, so don't try to save the function Try to scrutinize the function. Okay, so don't do these kind of activities Okay So I was telling you that uniqueness test is the major test that we normally follow for proving a function is one one Graph is only there to support it Right, so what is uniqueness test now all of you please listen to this very very carefully See whenever a function is given to you, they will mention you the domain of the function So basically they will mention you the domain of the function And they will mention you the co-domain of the function, right? DF and CDF means domain and co-domain of the function correct Normally even if co-domain is not mentioned domain is definitely mentioned Let's say both are not mentioned Okay, then you can find your own exhaustive domain all of you know how to find domain last year You would have done that in the function. Okay in class 11 Function chapter only comprises of finding domain and range So many a times when they will give you a function I would say 99% of the time They will mention you the domain of the function at least if not the co-domain. Okay, mostly what do we do? We mentioned both of them to the students. Okay, so the function would be now given to you in this form This is the this is how the function is, you know written in the exam. Okay Now when you're checking whether this function is a one-one function or not, what do we do is we say that? Let's say there are two different Inputs x1 and x2 Coming from the domain of the function Okay, so what do we do? We we choose two inputs x1 and x2 from where? From the domain of the function Okay, such that such that F of x1 is equal to f of x2 f of x1 is equal to f of x2 Now try to understand what is happening You are assuming that there are two inputs x1 and x2 which give you the same output When you're writing this Basically, what are you assuming you're assuming that there exists? two inputs x1 and x2 Okay, which give you the same output which result into the same output Okay, you can say same image So these are two pre-images which result into the same image This is what you are assuming in the beginning of this proof Okay, and by using this I'm putting dot dot dot because it all depends upon what function is given to you and if by using this you end up showing that x1 equal to x2 is the only possibility Please note this word only possibility only possibility Well, let me explain this to you because this is Confusing at times if you don't listen properly you will get confused So request you to put down your pens. Okay, if possible do this to your ears No, I'm just joking and just listen to this carefully See what I'm trying to say is you start with two inputs x1 and x2 from the domain of the function Whatever is the domain given to in the question Such that the two inputs or what we call two pre-images Give me the same output or map to the same image and By assuming that you later on you know end up realizing that If such a thing is to happen it can only happen when the inputs are also the same What does it mean that means two different inputs cannot give me the same output Are you getting my point? Why only possibility only possibility is because when you are solving this kind of equation x1 equal to x2 will always come as a solution Trust me. We'll take some I know questions in today's session x1 equal to x2 will always come as a solution Right, so that doesn't mean every function in this world has become a one-one function Right apart from x1 equal to x2. There may be other relationship between x1 and x2 also Okay, if you are able to negate that or if you are able to prove that the other relation is not possible Only possibility is this one x1 equal to x2 Then only you have proved that the function is one one Let me give you an example one example is you know better than explaining it hundred times. So let's say Let us say Let us say I have a function From real numbers to real numbers You all of you understand the meaning of R with a double-danda double-danda means it's a it's a symbol that we use for real numbers Okay, and the function is defined like this The function is defined like this x2. Okay, normally this thing will be given to you by the examiner This will be given to you by the question set it Now, how will I proceed see the steps? I will say let x1 and x2 be two inputs taken from the domain of the function now This is the domain of the function Okay, such that such that F of x1 is equal to f of x2 So this is my assumption Right, so I have taken two such inputs, which will give me the same output Now if I say f of x1 equal to f of x2 indirectly I'm saying x1 square is equal to x2 square because my function itself is x square So if you put x1 there, it will give you x1 square if you put x2 there, it'll give you x2 square Now if you simplify this now now once one important step over here guys I'm just making a request to you. Please do not cancel out these powers from both the sides Please stop doing that if you have been doing it before Okay, I have seen student x1 square equal to x2 square square gone x1 equal to x2 Are you that doesn't work like that? It doesn't work like that. Let me tell you if you're doing that you are losing out on the roots Never cancel factors never cancel powers on both the sides Right. This is not a way to deal with this. Please Don't do this. Don't do this since your requests. Okay So how to deal with this The approach to deal with this is bring it to one side. So let it let x2 square come to one side Okay Since I don't have space. I'm just scrolling my screen to the right The factor is this like this. Oh