 Right down. So there is a presence of nuclear force, which is very important when we discuss about nucleus. Okay. Let us talk about the nuclear force exclusively. Okay. So several experiments have been conducted between 1930 and 1950. So there is an entire decade of experimentation that is done on the stability of nucleus and there is a graph that has come out between, distance between the two nucleons. Nucleons is proton or neutron. So you are taking two nucleons, you take two protons or two neutrons or take one neutron, one proton. Okay. And you just vary their distances. Alright. If you increase or decrease their distances and you vary it in such a way that their distances go from 1, 2, 3. These are, you know, multiplication factor is 10 is for minus 15 meters. Okay. So this is perimeter. 10 is for minus 15 meter. So when you say 1, 1 is 1 into times or minus 15, which is 1000 times less than the size of item. Okay. So remember that. Now this on the Y axis, you have potential energy. This is potential energy. M E V. Okay. So we have minus 100 over here. We have zero here and we have let us say 100 over here. Okay. Now when this graph is plotted, it is observed that this graph goes like this. Draw this all of you. Now, do you guys remember that the relation between the force and the potential energy is minus the derivative of potential energy like this. Do you guys remember this? Right. We define potential energy like this only negative of the work done by the field, which is minus f dot dr. So if you are talking about the displacement in the direction of force only, so the full derivative becomes partial derivative. So it will be minus of du by dr becomes force. So this is the relation that is there. Now you can see here that the slope of this line, when you are less than r naught distance. Okay. When you are less than r naught distance, slope is what? Positive or negative? Negative. Negative. slope is negative and it is very, very high because the x axis is, you know, this scale is literally stretched. You know, the distance from here to there, you know, from here to there is 10 raised to power minus 15 into 3, which is almost nothing. Okay. So we are stretching it and then seeing it. So this is having like infinite slope. Are you getting it? This is like literally a straight line. Okay. So the force of repulsion will be, sorry, the force will be positive. So there will be a repulsion force before r naught. And then if you go slightly away from r naught, then you will see that this is, this is the positive slope. So it will be attractive force from r naught distance. But then if you move to this distance from r naught, if you travel this much distance, then the slope will rapidly decrease. Here it is near infinity and here it is almost flat, which is zero. So here du by dr becomes zero. So as soon as you reach about 1.2 into 10 raised to power minus 15 meters, the nuclear force goes to zero. Fine. That is why we call it short range force. It cannot be felt beyond a certain point. It suddenly drops to zero. Okay. So just write down a few properties of the nuclear force. Write down the first one is nuclear force is much stronger. You can see the graph. The slope is almost infinity whenever it is applied much stronger than Coulomb force. Okay. Second. So that is why you know two protons, even though they are very close to each other, they should feel electrostatic repulsion. But nuclear force being much larger still holds them together. Okay. So nuclear force is an attractive force inside the nucleus between the two protons. Write down the nuclear force, nuclear force falls rapidly to zero for distance greater than few centimeters. It is very or femto. You guys have any idea? No idea. Okay. Anyways. So what is that for distance greater than something you've written? For distance greater than few centimeters. It is femto. Femtometers. Okay. Fine. Write down the third point. Nuclear force between neutron, neutron, neutron proton and proton proton is approximately same. It does not depend on charge. Okay. So these are the few points about the nuclear forces. Fine. So we have done with the major portion of the theoretical part of the chapter. So after the break, we will start with something called radioactivity. There you will see a lot of numericals. Okay. We'll take 10 minutes break. Right now it is 7. Fine. So next we will discuss about radioactivity. Write down. Ramchar, you're there, right? Yeah, sir. Yeah, I'm just making sure everybody came back. Fine. So we are talking about radioactivity and radioactivity is nothing but it's a spontaneous reaction. Okay. It is spontaneous. Write down spontaneous disintegration of heavy nucleus into daughter nucleases. Fine. So it's a spontaneous reaction that happens. You don't have to do anything like several spontaneous chemical reaction you might have seen, right? That happens on its own. Similarly, this is also a spontaneous reaction. So when a heavy nucleus disintegrate, you can see that, you know, entropically it is favorable because entropy is increasing, number of nucleases are increasing. And also the binding energy per nucleon goes up because for a heavy nucleus, the binding energy per nucleon is lesser. So it is favorable both ways. Thermodynamically as well as, you know, entropically. So that is why this is a spontaneous reaction that happens. And it is disintegration. Okay. So there is just one nucleus that is there. It's not that two or three nucleases are reacting. One nucleus is disintegrating into the daughter nucleus. Maybe one will give three nucleus. Are you getting it? Now, when this disintegration happens, it is a good idea to find out the rate of disintegration. That will actually give us an idea about how much time it will take to totally consume the entire nucleus. For example, you have n nucleases of uranium 238. And it is disintegrating into several atoms. Let us say this is atom one, atom two, atom three. Okay, this has three different nucleases. So this many will disintegrate into these three. And some amount of maybe energy is also released. Okay, it's a good idea to find out how fast this disintegration is happening so that you will know that after particular time, what will be the remaining uranium 235 and how much energy will be liberated or how many daughter nucleases will be there. Okay, so in a way, what is happening is that the uranium 238 is getting consumed. Okay, right now let us say 50 nucleases there after some point in time, maybe 40 will be there, then 30 will be there. So like this, the number of nucleus of uranium 238, they are decreasing. Okay, and you can relate this scenario to let us say a city which has, suppose this city has a population of 1000 people. Okay, and this city has a population of 10,000 people. Fine, demography is similar as in the percentage of people who are to 20 years age, percentage of people who are 60 years age, like that it is similar, male number of percentage of male, percentage of female, everything is similar. So if I say that here, the number of deaths that are happening here is five deaths per day. So how many will be there here in this city, 50, right? So this number of people who are dying here can be related to the number of nucleases that are decreasing. Are you getting it? So you can say that the number of nucleus that are decreasing at any moment is proportional to the number of nucleus that are present at that moment, isn't it? If number of nucleases are more, then more number of nucleus per unit time will be decreasing. Are you getting it all of you? Yes, sir. Okay, and since number of nucleus, they are going down, so dn by dt will be a positive or a negative quantity? Negative. This will be less than zero. So if I have to bring a correlation between dn by dt and number of nucleus, number of nucleus will be a positive quantity, right? So I have to write down like this, dn by dt will be equal to minus of some positive constant multiplied by number of nucleus at that point in time, okay? So this is the relation which we are going to use to derive several things, all right? So this is purely probabilistic in nature and you can solve also this particular equation. So if you say that if we consider at t equal to zero, when time is zero, number of nucleus that are there, they are n naught, then how many number of nucleus? So is it equal to n naught into e to the power minus lambda t? All of you getting that? Yes, sir. Yes, sir. This is a simple linear differential equation. You can just take n in the denominator of the left hand side and this is equal to minus of lambda into dt, okay? Then lambda is constant. So when you integrate from zero to t time, number of nucleus goes from n naught to let us say n. So you will get log of, natural log of n from n naught to n. This will be equal to minus of lambda times t. So when you put the limits, it becomes ln n minus ln n naught. So it will be ln n divided by n naught. This is equal to minus lambda t. So n divided by n naught will come out to be e to the power minus lambda t, right? So n will be equal to n naught e to the power minus of lambda into t, okay? Now you can look at this expression and make out that when t is equal to zero, n is n naught only and ideally when t tends to infinity, what should be the number of nucleus present? It should tend to zero, isn't it? After infinite times, the number of nucleus should just vanish. Are you getting it? Yes, sir. Okay. So now see, this is rate of disintegration as a function of how many number of nucleus are there at that point in time, okay? You can also write down rate of disintegration with respect to time. So this will be lambda n naught e to the power minus lambda t, okay? This is the second equation. The first equation gives you rate of disintegration as a function of n and second will give you rate of disintegration as a function of time, okay?