Composition and Size

Transcript

0:00Introduction

Hello everyone! Let us start with a crazy question. Imagine we squeeze the whole earth until it is as dense as the nucleus of an atom. How big would it be then? As big as a country? As big as a city? Keep your guess in mind, because we will calculate it today. We will learn what a nucleus is made of, how big it is, and how dense it is, a topic that comes up often in JEE, so let's begin.

0:27Inside the nucleus: protons and neutrons

So, what is inside a nucleus? Only Z protons and N neutrons, where Z is the atomic number and N is the neutron number. Together, protons and neutrons are called nucleons. Okay? Their total is the mass number, capital A, so A equals Z plus N, the sum of the two. To write a nuclide, put the mass number as a superscript at the top left of X, the element's symbol, and the atomic number as a subscript at the bottom left. Clear? Now let me ask you this. Uranium two thirty five has two hundred and thirty five on top and ninety two below. How many protons and how many neutrons does it have? It has ninety two protons, since Z is ninety two. And the neutrons are two hundred and thirty five minus ninety two, which gives one hundred and forty three. Right? But look, so many more neutrons than protons! Light stable nuclei have N about equal to Z; the wavy equals sign means approximately. Heavy ones need extra neutrons, N over Z up to about one point five, to offset the Coulomb repulsion of the protons. With me? Now here are two more facts that students often forget. There are no electrons inside the nucleus. And a free neutron is unstable, with a mean life of about fifteen minutes, written min on screen, but inside a stable nucleus it does not decay. Got it?

1:53Isotopes, isobars and isotones

Now, three names that students often mix up. Isotopes have the same Z but a different mass number, like carbon twelve and carbon fourteen. Same element, so same chemical properties. Okay? Isobars have the same A but a different Z, like carbon fourteen and nitrogen fourteen, with fourteen nucleons each. Right? Now take carbon fourteen, nitrogen fifteen and oxygen sixteen. Different atomic numbers, different mass numbers. So what do they share? Yes, exactly, they share the number of neutrons. Fourteen minus six is eight, fifteen minus seven is eight, and sixteen minus eight is also eight. Nuclei with the same N are called isotones. Clear? Here is a memory trick for the exam, so look at the bold letters. Isotopes has a p, for protons; isobars has an a, for the mass number A; and isotones has an n, for neutrons. The little arrows just mean, points to. With me?

2:54How big is a nucleus?

Now, how big is a nucleus? Scattering experiments give the radius R equals R naught, times A to the power one third. R naught, the R with a small zero below, is about one point two f m, where one f m, the femtometre, is ten to the power minus fifteen metre. Okay? Now let me test you. Two nuclei have radii in the ratio one is to two. In what ratio are their mass numbers? One is to two, right? No, that is the trap, because the radius grows only as a cube root. R is proportional to A to the power one third; the curly sign means, is proportional to. So A is proportional to R cubed, where the double arrow means, it follows. Clear? So the mass numbers are in the ratio one cubed is to two cubed, that is, one is to eight. Double the radius, eight times the nucleons. Got it? Now take the natural log, written l n, of both sides, so l n R equals l n R naught, plus one third l n A, as on screen. Plotted, l n R against l n A is a straight line with slope one third. Right?

4:01Every nucleus has the same density

Now, the density. The volume, capital V, equals four thirds pi R cubed, which becomes four thirds pi R naught cubed times A, so it is proportional to the mass number. Okay? And the mass is about A times m, where m, the small m, is the mass of one nucleon. Right? So, back to our two nuclei with radii in the ratio one is to two. The bigger one has eight times the nucleons. In what ratio are their densities? Density, rho, the Greek letter that looks like a p, is mass over volume. Divide A times m by four thirds pi R naught cubed times A, and the mass number cancels! So rho equals three m over four pi R naught cubed. Clear? So every nucleus, light or heavy, has the same density, and the ratio is one is to one. If you said one is to eight, that was the trap. Got it? Put in the numbers, and rho is about two point three times ten to the power seventeen kilograms per cubic metre, about two point three times ten to the power fourteen times the density of water. Unbelievable, right?

5:08Example 1: radii of Al-27, Zn-64 and Te-125 nuclei

Now, example one. Find the radii of three nuclei: aluminium twenty seven, zinc sixty four, and tellurium one twenty five. Take R naught as one point two femtometres. Pause the video and try it first. Done? Okay, let's check. The mass numbers twenty seven, then sixty four, then one hundred and twenty five, are all perfect cubes, with cube roots three, four and five. Nice, right? So for aluminium, R equals one point two times three, which gives three point six femtometres. Okay? For zinc, one point two times four gives four point eight femtometres, and for tellurium, one point two times five gives six point zero femtometres. The mass number grew almost five times, but the radius did not even double, because of the cube root. Clear?

5:59Example 2: radius of Earth squeezed to nuclear density

Now, example two, and this is our puzzle from the start. What would be the radius of the earth, of mass six point zero times ten to the power twenty four kilograms, if it were compressed to nuclear density? Pause the video and try it first. Done? Okay, let's go. Capital M, the mass of the earth, equals volume times density, four thirds pi R cubed, times rho. So R equals three M over four pi rho, all to the power one third. Okay? Now put in the mass and the nuclear density. The top gives one point eight times ten to the power twenty five, and the bottom about two point eight nine times ten to the power eighteen. Divide, and R cubed is about six point two times ten to the power six cubic metres. Right? Now, how do we find the cube root of this without a calculator? Here is the trick: keep the power of ten a multiple of three. The cube root of ten to the power six is ten squared, and the cube root of six point two is about one point eight four. So R is about one point eight four times ten squared metres. With me? That is about one hundred and eighty metres! Not a country, not even a city, just a small ball. That is how dense nuclear matter is. Was your guess anywhere close?

7:19Quick revision: isotopes, isobars, isotones, radius, density

So, quick revision. The mass number is Z plus N; isotopes share Z, isobars share A, and isotones share N, the neutron number. The radius is R naught times A to the power one third, so A is proportional to R cubed. And the density, three m over four pi R naught cubed, about two point three times ten to the power seventeen kilograms per cubic metre, is the same for every nucleus. Now go and try the practice problems on this page. All the best!