 Hello, good evening everyone, can you hear me? Okay, so like last class, I think we had discussed the various molecular speeds, right? So molecular speeds we have discussed. Next, before going into the real gas, we have to start with Maxwell distribution curve. So heading right down, it is a Maxwell distribution of, distribution of molecular speed. Molecular speed, if you see, we had discussed last class. There are three types of molecular speeds we have. And that is given as root mean square speed, that is V rms. And the formula is root under 3rt by m, m is the molecular mass of the gas. And then we have average speed. We have average speed, that is V average. And it is given by 8rt by, 8rt by pi m. And the last one is, that is most probable speed, most probable speed, that is V mp, is equals to 2rt by m. And we had also discussed, V rms is maximum, then V average and then V mp, that we had discussed. Now, the law of probability means, what molecules will have what speed or how many molecules, fraction of molecules will have what kind of speed, okay? Maxwell has given, yes, that we are not doing, I'm just giving you the formula here, because we'll take the reference of it, Pranam, okay? So Maxwell has given a distribution curve for the molecular speeds. That we call it as Maxwell distribution curve, okay? And the graph that we have here, it is drawn between the graph, that is Maxwell distribution graph, of speed of the gas. Maxwell distribution graph of speed of molecular speeds, it is drawn between the x axis is the fraction of molecules, x axis is the speed we have, speed of the molecules and y axis we have, fraction of molecules, fraction of molecules. That is dn divided by n, okay? Fraction of molecules having one type of speed. Okay? So this graph gives you the distribution of molecules having one type of molecular speeds, okay? How many molecules will have one type of molecular speeds that we are going to understand here? Okay? So the Maxwell distribution curve, it is a bell shaped graph, okay? It is a bell shaped graph. Actually have we discussed, one more thing I wanted to ask, have we discussed Graham's law of diffusion, class plus? Graham's law of diffusion and diffusion, one second? No, okay. After this we'll discuss that also, okay? So the graph here, it goes like this. Molecules will have certain speed and in the bell shaped graph, so it goes up to a maxima and then comes down and goes like this, right? This is the graph of the Maxwell. It is given by Maxwell distribution curve, okay? Now here, this point you see the peak of this graph, it contains the maximum fraction corresponding to this point you see, we have the maximum fraction, okay? Corresponding to this point, we have the maximum fraction, maximum number of molecules. So this speed corresponding to this point is by definition, if you see, this becomes the most probable speed, VMT, because it's a speed of the maximum molecules, right? Maximum fraction means maximum number of molecules, okay? VRMS we know it is the maximum value we have, right? So VRMS will be somewhere here. It is maximum somewhere here, okay? And in between these two, somewhere here we have V average, okay? So this point here, I'll draw a horizontal line also, this fraction, so all these things represents the fraction of molecules having different, different speed. Here we have VRMS for this one and this is VMP, okay? We know VRMS is maximum, that's why you see the maximum value of VRMS, then VMP and then this. This means what? The molecules having VRMS means the maximum speed, the fraction of molecule is very less, means in general, what happens, the gaseous molecules with higher speed does not exist, okay? As the speed increases, okay? Means we can say one thing here that molecules with higher speed, the fraction of the molecules with higher speed will be lesser than the other two types of the speed that we have, this is VMP, this is V average, by mistake I have written VMP here, this is V average. This is the three speed. So there are two, three features for this graph. The first point is, and obviously this graph is drawn at a certain temperature, right? When we change the temperature and what happens, we'll see that, okay? The features of this graph you write down, it is based on, this graph is based on the law of probability, law of probability. The second point for this graph it, it is a bell shaped graph, bell shaped graph, the fraction of molecules, the fraction of molecule having zero speed is very less, having zero speed is very less, as well as the fraction of molecule having high speed is also very less. Give me one second. Yeah, so fraction of molecules having zero speed is very less, the graph is coming down here, molecules having high speed is also very less, the graph is also coming down here and hence the graph is a bell shaped graph, okay? The fourth point here is the maximum fraction of molecule, fraction of molecule will have one velocity or a common velocity, which is, which is VMP, the most probable velocity and this is corresponding to the peak of the curve. Copy this down, done. You can take here also stairs, that's not a problem. Randomly I have placed this point, okay? You can take this here also, the point is simply that VRMS is maximum and VMP is minimum and VMP by definition, we can easily say this point gives you VMP, but VRMS could be anything, but on the extreme right side, that's first point. Second point we can take from this is what? That the molecules having highest speed, they have very less fraction, that's what the point is. Yes, V average will also vary. See, I'm not giving you one, it's not like the average is this only, I'm just giving you one general point here, the point I'm trying to make, that in between VRMS and VMP, we have the average, right? It's just a general thing I have written over here because I haven't mentioned the temperature, I haven't mentioned the gas here. So for any gas, the nature of the graph would be like this and VRMS V average VMP will be placed like this, okay, in this graph. Yes? Okay, now you see, if you talk about the VMP here, right? VMP, we know the formula for most probability speed and that is VMP is equals to root under two RT by M, molecular speed, molecular mass of the graph of the gas, right? As we increase the temperature, then what happens? VMP also increases, right? So VMP, we can write, it is directly proportional to the square root of temperature, which means as temperature increases, VMP increases, and in fact, VRMS increases, and V average also increases for a given gas, yes or no? With temperature, all these three relations are correct? With the formula, we can say that. Now, so here you see, if you draw a graph of a given gas at different, different temperature, so the y-axis is again the fraction of molecule and x-axis is the speed, right? So the graph goes like this, this is a temperature T1, right? Like this, the graph goes, correct? The point I'm trying to make is, suppose this is at T1, this graph is at T2 and this graph is at T3. So could you tell me what is the relation of T1, T2 and T3 here? T1, is it T1 is maximum? T3 is maximum, right? So we can say from the relation of VRMS we have, we can say that corresponding to this point, we have VMP, right? This is a VMP at temperature T1, right? This is a VMP at temperature T2, this is a VMP at temperature T3, okay? T1, T2 and T3. So corresponding to this point, this is the most prevalent speed at one temperature, most prevalent speed at another temperature and this is the most prevalent speed here we have. So VMP1, this is VMP2 and this is VMP3. So since we write here that we can clearly say, see that VMP is maximum here, right? So we can easily write since VMP3 is greater than VMP2, is greater than VMP1, which means T3 is greater than T2, is greater than T1, that's the one point, right? This means what, what we can conclude from this as we increase the temperature, right? The curve becomes flat, right? It becomes flatter. You keep on increasing the temperature, the peak of the curve of the graph will come down and it becomes flatter, right? And the graph shifts towards the right, right? And the VMP, since the VMP is increasing, right? So next point to write down as temperature increases, as temperature increases, once again, as temperature increases, the curve becomes flatter. It means the graph shifts towards the right and the graph shifts towards the right, okay? One more thing is what at a given temperature, the fraction of molecules that we have here, that is an area under the curve, this molecules be considered to be as unity, right? Total area under the curve at temperature T1, that will be same T2 and that T3. Means the area occupied by this curve is same as the area occupied by this curve, which is same as the area occupied by this curve. This means what this side you see, the change that we have here, here the difference, this area, you know, since the graph is shifting towards the right, this area is exactly equals to this area, which is covered when the graph shift towards the right. And this is the common point we have. Means at a given temperature, the area under the curve is always same, right? That is assumed to be unity because the fraction of speed we are assuming here. So area won't change, but the graph shifts towards the right, okay? This is must figure out. Sometimes what happens, they also give you, suppose I give you three different gases, O2, H2 and N2, O2, H2 and N2, okay? For a given temperature, then which graph represent O2, which graph represent H2, which graph represent N2? And for this question I am assuming, T1 is equals to assume here, T1 is equals to T2 is equals to T3. If you assume all the three temperatures same, then which graph that is one, two and three represents O2, H2 and N2. And how do we do this? Since the temperature is same, so we can compare VMP, the molecular speed, is equals to two RT by M root under, means as the molecular mass is more, VMP would be less, okay? Because VMP is inversely proportional to the square root of molecular mass, okay? So the gas which has the least molecular mass will have the maximum molecular speed, right? So least molecular mass we have for H2, then N2 and then O2. So for the given set of data, the answer for this question is this graph represents H2, this graph represents N2 and this graph represents O2. Is it clear? Yeah, no doubt in this. So this you must take care of for Maxwell distribution graph, okay? All these information regarding the graph, okay, based on that. No, for a given gas, it will be constant. It's not for all gas, Aditya. For a given gas, this graph will be constant. Means for a given gas, if you draw the graph for different, different temperatures, then the area under the curve for different, different temperature for a gas will be same, understood? Right? Similarly, for other gas, we can have some different area, right? But that area would also be same at different, different temperature for that particular gas. Usually we consider this as unity because the fraction of molecules we have. So area under the curve is nothing but the number of molecules, which we consider as unity. One, yeah. Next, we are going to see diffusion and diffusion. One second, write down.