Excitation, Collisions and Recoil

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. Two particles hit a hydrogen atom in its ground state: a photon carrying eleven electron volts, and an electron also carrying eleven electron volts. The atom needs only ten point two electron volts to reach its first excited level. So both of them excite it, right? Hold on to your answer; we will check it today. We will also see why a neutron needs double that energy, and why an atom recoils when it gives out light. Questions like these do appear in JEE, so let's begin.

0:37Exciting an atom with a photon

So, first, the photon. Let me ask you this: when a photon hits an atom, can it give only a part of its energy, and fly away with the rest? The answer is no. The photon is absorbed only if its energy equals the gap between two levels exactly; not a little more, and not a little less. Clear? There is one more way in. If its energy exceeds the ionisation energy, the photon is absorbed and the electron is freed completely. For hydrogen with n equal to one, where n is the principal quantum number, that is thirteen point six electron volts; the e V on screen stands for electron volts. With me? And the excess, the energy beyond the ionisation energy, goes to the freed electron as kinetic energy. Got it?

1:23Exciting an atom with an electron

Now we fire an electron instead. The electron is different: it can give up any part of its kinetic energy, capital K, and keep the rest. Right? So it excites the atom whenever K is at least the excitation energy; the sign, greater than or equal to, means at least. No exact match is needed. Okay? Hydrogen needs ten point two electron volts to reach n equal to two. So what if the electron has less than that? Then it cannot excite the atom at all. As the line says, K less than ten point two electron volts gives elastic collisions only; the double arrow means it follows. The atom stays in its ground state.

2:07Heavy particles: momentum takes its share

Clear? Alright, now a heavy particle, of mass small m and kinetic energy K, strikes an atom of mass capital M at rest. With me? The moving particle carries momentum. So, can all of its kinetic energy go into exciting the atom? The answer is no, because momentum is conserved. After the collision the bodies must still be moving, and that motion keeps some kinetic energy. Okay? So the greatest loss of kinetic energy is capital M, over capital M plus small m, times K, and only this much can go into exciting the atom. Right? Now a hydrogen atom, or a neutron, hits a hydrogen atom at rest, so the two masses are practically equal. What fraction of K can be lost now? Yes, it is half, because capital M over two capital M is one half. So the loss is at most K over two; the sign, less than or equal to, means at most. Got it? To excite hydrogen, K over two must be at least ten point two electron volts, so K must be at least twenty point four, double the gap. So even a neutron with twenty electron volts cannot excite hydrogen. Remember this one. Okay?

3:21Recoil: the atom gets a small kick

Now the last idea, recoil. When an atom emits a photon of energy capital E, the photon carries momentum E over c, where c is the speed of light. Got it? The atom was at rest, so to keep the total momentum zero it must move the other way. M times v equals E over c, so the recoil speed v is E, over M times c. Clear? Its kinetic energy, K recoil, the K with the small word recoil written below it, is one half times M times v squared, which gives E squared, over two times M times c squared. With me? And that is tiny, about ten to the power minus nine of the photon energy, because M times c squared is huge compared with a few electron volts. So we usually ignore it, unless a question asks for it. Right?

4:08Example 1: 11 eV photon vs 11 eV electron on hydrogen

Now, Example one, our opening puzzle. Photons of eleven point zero electron volts fall on hydrogen atoms in the ground state. What happens? And what if electrons of eleven point zero electron volts are used instead? Pause the video and try it first. Done? Okay, let's check the photons first. Is eleven point zero electron volts the gap to any level? From the energy levels of hydrogen, the gaps from n equal to one are ten point two, twelve point zero nine, twelve point seven five, and so on; the three dots mean the list continues. None of them is eleven point zero. Okay? And eleven point zero is less than thirteen point six, so there is no ionisation either. So the photons are not absorbed at all, even though they carry more than ten point two. That is the trap: for a photon, more energy is not enough, it must match. Clear? Now the electrons, with the same eleven point zero electron volts. What happens to them? Very good, an electron can give up any part of its energy. Eleven point zero is at least ten point two, so it can lift the atom to n equal to two, but not to n equal to three, which needs twelve point zero nine. Right? And the electron keeps eleven point zero minus ten point two, which gives zero point eight electron volts. So the puzzle is solved: the photon fails, and the electron succeeds. Got it?

5:44Example 2: least kinetic energy of alpha to excite hydrogen

Now Example two, a harder one. What is the least kinetic energy of an alpha particle, alpha being the Greek letter on screen, that can excite a hydrogen atom at rest in its ground state? Pause the video and try it first. Done? Okay, let's check. Many students say: the alpha is heavier than a hydrogen atom, so it hits harder, so it needs less than twenty point four electron volts. Is that right? That guess is the trap, so let us use the formula instead. The alpha particle has about four times the mass of a hydrogen atom, so small m is four times capital M, and the hydrogen atom at rest has mass capital M, okay? The greatest loss is capital M, over capital M plus four capital M, times K, which is just K over five. Only one fifth of the energy can be lost! With me? This must be at least ten point two electron volts. So K min, the K with a small min below, is five times ten point two, which gives fifty one electron volts, far more than twenty point four. The heavier the moving particle, the more energy stays locked in the motion.

6:54Quick revision: photon, electron, collision and recoil rules

Clear? So, quick revision. For a photon, the energy must equal a level gap exactly, or exceed the ionisation energy. An electron can give any part of its kinetic energy, so K at least the excitation energy is enough. For a heavy particle, at most the fraction capital M, over capital M plus small m, of K can be lost, so hydrogen on hydrogen needs twenty point four electron volts. And in recoil, v is E, over M times c, and the recoil energy, E squared over two times M times c squared, is tiny. Now go and try the practice problems on this page. All the best!