JEE Advanced and IIT-JEE Questions (2001–2015): Nuclear Energy

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. Uranium two thirty six splits into iodine one thirty seven, yttrium ninety seven and two neutrons. One hundred and thirty seven, plus ninety seven, plus two, gives two hundred and thirty six, so the nucleons match. So is the rest mass after the split the same as before? It looks like it should be, right? Today we solve JEE Advanced and IITJEE questions on nuclear energy, and this puzzle, from IITJEE two thousand seven, is one of them. So, let's begin.

0:36Q value: lost mass becomes energy

So, first the idea behind every question today. In a nuclear reaction, small a plus capital X goes to capital Y plus small b, as the arrow shows, and the charge number and the nucleon number are conserved. Okay? But the mass is not conserved. The energy released, the Q value, is the mass of small a plus the mass of capital X, minus the masses of capital Y and small b, all times c squared. Each m with a small label below is a mass, and c is the speed of light. Right? In numbers, one atomic mass unit, u, of mass corresponds to nine hundred and thirty one point five mega electron volts, M e V on screen; the double-headed arrow means, corresponds to. Clear? Let me ask you this: when a reaction gives out energy, does the total mass go up, or down? It goes down, yes, exactly. Q greater than zero means the mass decreases and energy is released, and we call the reaction exothermic. With me? Q is also the total binding energy of the products, minus that of the reactants. More tightly bound products means energy comes out. Got it?

1:48Our puzzle, from IIT-JEE 2007

Now, back to our puzzle. Each nucleus is written with its chemical symbol, the mass number on top and the atomic number below, and two n means two neutrons. The nucleons match, so is the rest-mass energy the same on both sides? It is not the same, and that is the trap. Fission releases energy, so Q is positive, and the rest mass must fall. Right? So, with capital E for rest-mass energy, E of uranium two thirty six is greater than E of iodine one thirty seven, plus E of yttrium ninety seven, plus two times E of a neutron. That greater-than sign answers our puzzle. Okay?

2:28Fission and fusion: both climb the curve

Now look at this graph: the binding energy per nucleon, plotted against capital A, the mass number. In fission, a heavy nucleus like uranium, at the right end, splits in two; see the arrow marked fission. Right? Fusion is the opposite: two light nuclei, at the left end, join into one, along the arrow marked fusion. Clear? Now tell me, how can two opposite processes both release energy? Yes, exactly, both arrows climb the curve. The products have a larger binding energy per nucleon, E b n on screen, the E with a small b n below it. They are nearer the peak at iron fifty six, more tightly bound, so energy comes out. With me? But fusion has a catch: both nuclei are positive, and the Coulomb repulsion pushes them apart. So fusion needs ten to the power seven to ten to the power nine kelvin, the capital K on screen. Got it?

3:23Sharing the energy: the lighter piece gets more

Now, one more tool first. When a nucleus at rest breaks into two products, the total momentum stays zero, so they get equal and opposite momenta. Which one gets more kinetic energy, the light piece or the heavy one? The light one, though many students guess the heavy one. Kinetic energy, capital K, is p squared over two m, with p the momentum and m the mass. The same p for both, so K is proportional to one over m; the curly sign means, is proportional to. Okay? So the energy is shared in the inverse ratio of the masses. Right?

4:01Example 1: match fusion, fission, beta decay (IIT-JEE 2006)

Now, example one, a matching question from IITJEE two thousand six. Option A is nuclear fusion, B is fission, C is beta decay, with the Greek letter beta, and D is an exothermic nuclear reaction. Then, p says, converts some matter into energy; q, low atomic number; r, higher atomic number; and s, proceeds by weak nuclear forces. Pause the video and try it first. Done? Okay, let's check. Which of the four convert some matter into energy? All four of them, yes, exactly. Each one releases energy, so in each the mass decreases. So p goes with every option. Okay? Fusion joins light nuclei, so it takes q, low atomic number. Fission splits heavy nuclei, so it takes r, higher atomic number. So option A is p and q, and option B is p and r. Right? Beta decay proceeds by the weak nuclear force, so option C takes p and s. Clear? Now option D, the exothermic reaction. Many students write only p and move on. Is that the full answer? It is not complete, and that is the trap. Fusion and fission are both exothermic nuclear reactions, so D matches p, q and r. Very good, that is the full answer.

5:27Example 2: KE of Xe and Sr fission fragments (JEE Adv 2015)

Now a harder one, example two, from JEE Advanced two thousand fifteen. Uranium two thirty six, at rest, splits into xenon one forty, written X e, strontium ninety four, written S r, and two unknown particles; x and y each have two mega electron volts of kinetic energy. The binding energies per nucleon are seven point five for uranium, and eight point five for xenon and for strontium. Identify x and y, and find the kinetic energies of xenon and strontium. Pause the video and try it first. Done? Okay, let's check. For the nucleons, two hundred and thirty six, minus one hundred and forty, minus ninety four, leaves two. For the charge, ninety two, minus fifty four, minus thirty eight, leaves zero. Okay? Together, x and y carry two nucleons and no charge, which means they are two neutrons. Right? Now Q, from binding energies. For the products, many students multiply eight point five by two hundred and thirty six. Is that right? That is exactly the trap: free neutrons have no binding energy, so only two hundred and thirty four nucleons are bound. So Q equals two hundred and thirty four times eight point five, minus two hundred and thirty six times seven point five. One thousand nine hundred and eighty nine, minus one thousand seven hundred and seventy, gives two hundred and nineteen mega electron volts. Clear? The neutrons take two plus two, that is four. So the fragments share two hundred and nineteen minus four, which is two hundred and fifteen mega electron volts. With me? Now, which fragment gets more of this energy, xenon or strontium? Strontium, the lighter one, yes, exactly. The neutrons are light, so their momenta are small, and we neglect them. Then, just as we saw, the two fragments have nearly equal and opposite momenta, and they share this energy in the inverse ratio of their masses. So K with a small S r below, the kinetic energy of strontium, is two hundred and fifteen times one hundred and forty over two hundred and thirty four; strontium gets the xenon mass number over the total. That is about one hundred and twenty nine mega electron volts; the wavy equals sign means approximately. Okay? And xenon gets two hundred and fifteen times ninety four over two hundred and thirty four, about eighty six mega electron volts. So the lighter strontium really does get more. Got it?

7:59Quick revision: Q from mass lost, binding energy, KE sharing

So, quick revision. Energy released is the mass lost times nine hundred and thirty one point five mega electron volts per atomic mass unit. In fission and fusion, the products have a larger binding energy per nucleon. And from rest, the energy is shared in the inverse ratio of the masses. Now try the other questions on this page. You have already done the hardest one, so all the best!