Nuclear Fission

Transcript

0:00Introduction

Hello everyone! Here is a puzzle. One neutron makes a uranium nucleus split, that fission makes two more split, and each of those makes two more. Each generation takes ten to the power minus eight seconds. So how long until ten to the power twenty four nuclei split in a single generation? Keep your guess in mind; we will work it out by the end. Fission comes up often in JEE, so let's begin.

0:28Fission: a heavy nucleus splits

So, what is fission? Let a heavy nucleus absorb a neutron, the small n on screen. It splits into two fragments of comparable mass, and two or three neutrons come out; the arrow means, becomes. The left part of the figure shows exactly this. Okay? Here is the standard example: uranium two thirty five absorbs a neutron and becomes uranium two thirty six. The number on top is the mass number, the ninety two below is the atomic number, and the star means this nucleus is excited. Right? It then splits into barium one forty one and krypton ninety two, the red circles B a and K r in the figure. Let me ask you this: on screen, two thirty six equals one forty one plus ninety two plus what, so how many neutrons come out? Yes, exactly, three neutrons come out. One hundred and forty one plus ninety two gives two hundred and thirty three, which is three short of two hundred and thirty six. The atomic numbers balance too, since fifty six plus thirty six gives ninety two. So we write plus three n, the black dots in the figure.

1:33Where does the energy come from?

Clear? Each fission releases about two hundred mega electron volts, written M e V on screen. But splitting a nucleus means pulling nucleons apart, which should cost energy, right? So where does this energy come from? The answer is binding energy. Look at E b n, the binding energy per nucleon, the E with a small b n below it. It rises from seven point six to eight point five mega electron volts, as the arrow shows, so the fragments are more tightly bound than the uranium nucleus. Okay? So each nucleon gains about zero point nine mega electron volts of binding energy, and that energy is released. It appears mostly as kinetic energy of the two fragments. Got it? One kilogram, the k g on screen, of uranium two thirty five releases about eight point two times ten to the power thirteen joules, the J on screen. Clear?

2:28Chain reaction: the multiplication factor

Now look at the figure. The three neutrons make three more uranium nuclei split; that is the second generation. The neutrons from one fission cause further fissions, and this is a chain reaction. With me? We describe the chain with the multiplication factor, small k, the ratio of the numbers of fissions in successive generations. Now tell me, for steady power, what value should k have? Yes, exactly, k equals one. Each generation has as many fissions as the one before, so the power stays steady, and we call this critical. Right? If k is greater than one, the fissions multiply; that is supercritical, and the figure says the chain grows, as in a bomb. If k is less than one, the chain dies out, and that is subcritical. Okay?

3:16Inside a nuclear reactor

Now, inside a reactor, the fuel is enriched uranium. And here is a trap: fission neutrons are fast. Which is better at making uranium two thirty five split, a fast neutron or a slow one? The answer is slow, and many students get this wrong by saying fast. So the reactor has a moderator, such as water, heavy water or graphite, to slow the fast neutrons. Got it? They are slowed to thermal energies, about zero point zero two five electron volts, the e V on screen, and uranium two thirty five captures these neutrons best, because a slow neutron spends longer near the nucleus. With me? Then the control rods, made of cadmium or boron, absorb neutrons. Add a coolant and shielding, and the reactor is complete. Clear?

4:06Why light nuclei make good moderators

So why do these materials make good moderators? Say the neutron hits a nucleus of mass capital M, at rest, head-on, in an elastic collision. Okay? The neutron, of mass small m, loses the fraction four times small m times capital M, over small m plus capital M in brackets, the whole squared. For hydrogen, capital M equals small m, so what fraction is lost? It is four times m squared over four times m squared, that is one, so the neutron can lose all its energy at once. For deuterium it is zero point eight nine, and for carbon only zero point two eight, so lighter nuclei slow neutrons better. Right?

4:47Example 1: energy released when a 240 nucleus splits in two

Now, here is an example from the course: a nucleus of mass number two hundred and forty, with E b n equal to seven point six mega electron volts, splits into two fragments of mass number one hundred and twenty, with E b n equal to eight point five mega electron volts. Find the energy released. Pause the video and try it first. Done? Okay, let's check. Many students write eight point five minus seven point six, which gives zero point nine mega electron volts. Is that the answer? It is not, and that is the trap, because zero point nine is per nucleon. So find the total binding energy: before the split, two hundred and forty times seven point six gives one thousand eight hundred and twenty four mega electron volts. Okay? After the split, two fragments of one hundred and twenty nucleons each, so two times one hundred and twenty times eight point five gives two thousand and forty mega electron volts. Right? The rise is released: two thousand and forty minus one thousand eight hundred and twenty four, equals two hundred and sixteen mega electron volts. Or in one line, two hundred and forty nucleons times zero point nine. With me?

6:01Example 2: time for 10 to the 24 fissions in one generation

Now our puzzle from the start, a harder one. In an uncontrolled chain reaction, each fission causes two more, and one generation takes ten to the power minus eight seconds, the s on screen. Starting from one fission, how long does it take for ten to the power twenty four nuclei to undergo fission in one generation? Pause the video and try it first. Done? Okay, let's check. Some students multiply ten to the power twenty four by ten to the power minus eight, and get ten to the power sixteen seconds. Is that right? It is not, because the fissions do not happen one by one. The number doubles every generation: one, two, four, eight, and the three dots mean, and so on. So generation n has two to the power n fissions. Okay? So we need two to the power n equals ten to the power twenty four. Here is a topper's trick: two to the power ten is one thousand and twenty four, about ten to the power three, and the wavy sign means approximately. Right? So ten to the power twenty four is ten cubed, in brackets, to the power eight, about two to the power ten, to the power eight, which is two to the power eighty. The double arrow means, it follows that n equals eighty. Clear? So the time is eighty times ten to the power minus eight seconds, or eight times ten to the power minus seven seconds. That is zero point eight microseconds; the Greek letter mu means micro, a millionth. So our puzzle's answer is less than a millionth of a second! Got it?

7:36Quick revision: 200 MeV per fission, k = 1, moderator, rods

So, quick revision. In fission, uranium two thirty five absorbs a neutron and splits into two fragments, two or three neutrons, and about two hundred mega electron volts. The energy comes from the rise in E b n, about zero point nine mega electron volts per nucleon. And in a reactor, k equals one is critical, the moderator slows neutrons, and control rods absorb them. Now try the practice problems on this page. Great work today, and all the best!