 Let us now try to solve a numerical. You can pause this video and go through this question. Experiments show that iron atoms form BCC crystal structure at room temperature with an edge length of 286 picometer. If the mass of iron is 56 cr per mole, so this is the molar mass, find the radius of iron atom and density of iron at room temperature. Okay. So in this question, it is given that iron atoms forms BCC structure and has an edge length of 286 picometer. Using this information, we are asked to calculate the radius of an iron atom and the density of iron at room temperature. Okay. So let's start. Let us first think about how we can calculate the radius. Now we know that in unit cells, the radius is very closely related to the edge length. And in BCC, the ratio of the radius to the edge length comes out to be always equal to root 3 by 4, right? This can be derived using the fact that in BCC, atoms across the body diagonal, they touch each other. So this distance comes out to be equal to 4R. And this body diagonal can also be written in terms of the edge length as root 3 of F. So root 3 of F will be equal to 4R, root 3A will be equal to 4R. So R by A in BCC will always be equal to root 3 by 4. So in this question, the radius R will be equal to root 3 by 4 into A. And since A is 286 picometer, so this will be root 3 by 4 into 286 picometer. And picometer means 10 to the power minus of 12 meters. So the answer is going to be, let me bring out my calculator, root over 3 into 286 divided by 4. So this part comes out to be 123.84. So it's 123.84 into 10 to the power minus of 12 meters. So 123.84 picometers, right? Let us now try to calculate the density of this iron sample. So what do we mean by density? Well density is a measure of heaviness of an object and is equal to the mass of the object, the weight of the object per unit volume, right? So how do we calculate the density of iron? Well, any crystal of iron or a rod of iron will not be made up of a single unit cell, but will instead be made of many, many unit cells, right? Now every single unit cell is exactly the same, right? So the mass per unit volume of this unit cell will be equal to that of this one, which will be same as this. So the mass per unit volume of the overall crystal will always be equal to the mass per unit volume of a single unit cell, right? Let me take another example to clarify my point. So a glass of water has a density of 1 gram per ml. So per unit ml of water, I'll always have 1 gram, right? This is because the composition of water is constant all throughout the sample. The molecular distribution of water out here and out here or anywhere else is exactly the same. So the overall density of water can be calculated by just calculating the density of a small section of water, right? Similarly, because the overall iron sample is made up of the same unit cells over and over again, so the density of the overall system will be equal to the density of a unit cell. So how do we calculate the density of this unit cell? So to calculate the density, we'd require the weight of the unit cell and the volume of the unit cell, right? So now if this unit cell has an edge length of A, then the volume will clearly be equal to A cube. So how do we figure out the weight of the unit cell? Well, the weight of this unit cell will be equal to the mass of these iron atoms that are present inside this unit cell, right? Now do remember that these iron atoms are shared between different unit cells. We only need the mass that is inside this unit cell. So how can we do that? How do we calculate the weight that is inside? Well, in this unit cell, the effective number of atoms that is inside the unit cell will be equal to 1 by 8 into 8 because atoms at the corners are only 1 eighth inside, plus 1 as the body-centered atom is 100% inside. So this will come out to be equal to 2. So I can say that effectively inside this unit cell, there are only 2 iron atoms. So the weight that is going to be inside will be 2 times the mass of iron atom, right? So I can write the weight inside to be equal to 2 times the mass of iron atom, right? So what is the mass of iron atom? Well in the question, it is given as 56 grams per mole. So per mole, which means 6.022 into 10 to the power 23. So per mole of iron atoms weighs 56 grams. However, out here, we need the weight of a single iron atom because we are calculating the density of a single unit cell. And a single unit cell will have 2 iron atoms inside. So it's 2 into mass of a single iron atom, right? So if 1 mole weighs 56 grams, then the mass of a single iron atom will be 56 divided by 6.022 into 10 to the power 23, right? So now if I plug in the values, so it's going to be 2 into 56 divided by 6.022 into 10 to the power 23, 56 grams divided by a cube. So what is a? A is 286 bicometer. So it's 286 into 10 to the power minus of 12 meters, whole thing cube, right? So I can write this as 2 into 56. Let me keep this 6.022 into 10 to the power 23 out here. divided by 286 whole thing cube into 10 to the power minus of 12 cube. So 10 to the power minus of 36. So this will be in terms of gram per meter cube, right? So this can be written as 2 into 56 divided by 286 cube into 6.022 into 1 divided by 10 to the power minus of 36 into 10 to the power 23, which will come out to be 10 to the power minus of 13, right? So now if I do the math, let me bring back my calculator again. So 2 into 56 divided by 286 whole thing cubed divided by 6.022, right? So this value, this value comes out to be equal to 7.95 into 10 to the power minus of 7. So this will be equal to 7.95 into 10 to the power minus of 7 divided by 10 to the power minus of 13 gram per meter cube, right? So this will be equal to 7.95 into 10 to the power minus of 7 plus 13 gram per meter cube, which will be equal to 7.95 into 10 to the power 6 gram per meter cube, right? So I can write this as 7.95 into 10 to the power 3 into 10 to the power 3 grams per meter cube and 10 to the power 3 grams is equal to 1 kg. So I can write the density as 7.95 into 10 to the power 3 kg per meter cube.