 In this video, we will talk about nuclear size, particularly nuclear radius and its density. From Rutherford's Alpha scattering experiment, where he took a thin foil of gold and we can see one gold atom over here with its nucleus right at the center. This thin foil was bombarded with alpha particles. Alpha particles being the nucleus of a helium atom. It has two protons, two neutrons, positively charged and and the thin foil of gold was bombarded or being shot at with alpha particles. So because alpha particles are also positively charged, as they started interacting with the gold atom, you had some beams that were farther away from the nucleus and they underwent some deflection. Then you had some alpha beams comprising of many many alpha particles which were much closer to the nucleus and because of the positive positive repulsion between the nucleus and the alpha particle, this one underwent greater deflection. And if it is closer to the nucleus, it would then undergo even more even more deflection. And Rutherford, Rutherford was especially interested in one case when the alpha particle was directed at the center of the nucleus. In this case, because of high energy, high kinetic energy of the alpha particle, it came closer to the nucleus and nucleus being positively charged, there was a huge repulsion that was faced with the alpha particle and it momentarily stopped and then started going back. So this was a huge deflection, almost 180 degrees of deflection. Let me let me put in the arrows here to show where they are moving. Now this case, when the alpha particle went completely back to where it came from 180 degree turn, we see that all the all the kinetic energy that the alpha particle had at this point, it came momentarily to rest. So all of this kinetic energy was changed to potential energy of the nucleus and the alpha particle of the system of nucleus and alpha particle. And using the idea of conservation of energy, doing some calculations, we were able to figure out this distance, this distance which was called the distance of closest approach. How close the alpha particle of some known kinetic energy can get to the nucleus. It turned out from many experiments that this distance, this came out to be equal to approximately 3.2 into 10 to the power minus 14 meters and this number gave some idea to the size of the gold nucleus. Because the size of the gold nucleus, it cannot be more than this number and when we say size, we can think about radius, the radius of this gold nucleus, it can in no way be bigger than this number. So this number does not give the actual size or the actual radius of the nucleus, but it does give some idea, some estimate into the size or the radius of the nucleus and turned out there were many more experiments that were carried out. There was electron scattering, there was neutron scattering and with all of these experiments one common observation that was noted was that the volume of the nucleus was directly proportional to the number of protons and neutrons inside the nucleus. So you have the volume, the volume of the nucleus. This is directly proportional to the number of protons and neutrons. And the total number of protons and neutrons inside a nucleus, that's just the mass number. That's the mass number which is denoted by A. So we can write this as V that is directly proportional to the mass number. And this also makes sense because if we think about it, if we look at the helium nucleus which has two protons and two neutrons, we have two protons and we have two neutrons. So the mass number, the mass number here is four and this could be the size of the nucleus. If we take a bigger nucleus, if we take a lithium nucleus, this one has three protons and it has one, two, three, four, four neutrons. So now the mass number becomes, now the mass number becomes seven. So this one has A, that's the mass number, seven. And if you fit in more protons, more neutrons, they will take up more space. So even the size of your nucleus gets bigger and bigger. If you take a bigger nucleus of, let's say a sodium, this one has, it has, it has, let's say it has 11, 11 protons and 12 neutrons. So the mass number, mass number here is 12 plus 11, 23. So now you have more protons, you have more neutrons, so you need a bigger space to fit all of those, all of those small particles, those subatomic particles. Now another thing that was, that was noted experimentally that the shape of the nucleus, it's mostly a spherical shape. Some of the nuclei do have different types of shapes. Even the ones that do have a spherical shape, it's not entirely or purely spherical. We can say, we can, we can say that it's almost a sphere. So while doing the mathematics, we can approximate it to be, to be, to behave just like a sphere. So when we think about the volume, we can write this as four by three. This is four by three pi r cube. r here is the radius of the nucleus, not the atom, just the nucleus. And this is proportional to the mass number a. We can, we, now four by three pi, four by three pi is a constant. So that doesn't really change anything to the proportionality. You can ignore that. We can write this right away as r cube proportional to a. It's four by three pi is just a constant, doesn't really change the proportionality or affect the proportionality in any way. So now when we take cube roots of both the sides, we can write this as r, this is proportional to a to the power one by three. And when we remove this proportionality, we need to add a constant, that is r naught, that is multiplied with a to the power one by three. This expression gives us the radius of a nucleus. Here the constant r naught, this was again verified experimentally. This is somewhere around 1.2 into 10 to the power minus 15 meters. And, and there's a way of writing this, we can write this as 1.2 femto femtometers or fermi meters. And this one, one FM, this is equal to 10 to the power minus 15 meters. So this is the expression for the radius of a nucleus. We can also think about the density. If we add more protons and more nucleons, does it make, does it make the nucleus more dense or does it not affect the density? For that we can try and recall, how did we figure out the density in the first place? We know that density, density was given as rho, rho that was equal to mass divided by volume. And if we take, let's say if we take this helium atom, what, what could be the mass of this helium atoms? It's equal to the mass of this proton plus this proton plus this neutron plus this neutron. It's equal to the mass of all of these nucleons. So mass of any general nucleus, we can write this as the number of nucleons that is a number of protons and neutrons multiplied by the mass that each of them carry and turns out they carry almost the same mass that is one atomic mass unit. So this is one AMU. And one AMU can be expressed in kilograms. We'll do that in a while. But volume is 4 by 3 pi r cube. Volume is 4 by 3 pi r cube. Now in place of r, we can put r naught into a to the power 1 by 3. So when we do that, this becomes equal to a multiplied by 1 AMU. And this is 1.6 into 10 to the power minus 27 kilograms. And this is divided by, this is divided by 4 by 3 pi r cube. So 4 by 3 pi into r naught cube multiplied with a because a is already to the power of 1 by 3. So when you cube it, it just becomes a. One interesting thing that we see here is that a just gets cancelled. A just gets cancelled. Everything else is a constant. r naught is a constant. This is a number. This is a number. So it turns out that the density of a nucleus does not depend on the number of protons and neutrons in the nucleus. Density for every nucleus is constant. It's one fixed number. And turns out the value of that number, when you solve all of this, when you put in r naught, when you work out this calculation, this comes out to be equal to 2.3 into 10 to the power 17 kilogram per meters cube. Insanely high number. But the interesting bit is that it's constant for all the nucleus. It's just like a water drop. If you have a small water drop, it has some density. But if you get a bigger water drop, the density still is the same. Density doesn't change. Kind of like how the density for nucleus is behaving over here. It's the same no matter the size. So that's about it. This is an expression for the radius of the nucleus. And this is the density of a nucleus, which is the same for all nuclei.