Transitions, Ionisation and Excitation

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Take hydrogen atoms in the ground state, and hit them with electrons of energy twelve point five electron volts. But pulling the electron out of hydrogen needs thirteen point six electron volts, so that is not enough. Does anything happen at all? And how many spectral lines come out afterwards? Hmm, think about it. By the end of this class, you will answer it in one minute. And look, this is a very JEE style question. So, let's begin.

0:35Jumping between levels

So, first a quick recall from our last class. The energy of level n, E with a small n below it, is minus thirteen point six, times Z squared over n squared, in electron volts, the e V on screen. Capital Z is the nuclear charge. Okay? Now the electron jumps from a lower level, n one, up to a higher level, n two: n with a small one, and n with a small two, below it. Tell me, how much energy does this jump need? Simple, just subtract! E of n two, minus E of n one. Take thirteen point six Z squared common: delta E equals thirteen point six Z squared, into one over n one squared, minus one over n two squared, electron volts. Delta, this Greek triangle, means change in. And n one is the lower level, so the bracket comes out positive. Got it? When the electron falls down, the atom emits delta E as a photon. When it jumps up, the atom absorbs delta E, the same amount. Same formula, both directions. That's it! Next, the wavelength of that photon. Lambda, the Greek letter for wavelength, equals h c over delta E; h is Planck's constant, and c the speed of light. And the handy form from Planck's class: lambda in nanometres equals twelve hundred and forty, divided by delta E in electron volts. No powers of ten. Super quick!

2:02Ionisation energy: pulling the electron out

Alright, now ionisation: pulling the electron right out. Remember this picture: hydrogen on the left in blue, the helium ion on the right in orange, energies in electron volts. To remove the electron, we lift it all the way up to the dashed zero line. From the ground state, that takes thirteen point six Z squared electron volts. For hydrogen, thirteen point six. Now the helium ion, H e plus, with the small plus up top: a one electron species with Z equal to two. So two times thirteen point six, twenty seven point two electron volts, right? No, no, careful! This is a very common trap. It is Z squared, not Z! Thirteen point six times four: fifty four point four electron volts. See the orange ground level at minus fifty four point four? Four times deeper than hydrogen's. And L i two plus, the lithium ion, with Z equal to three: thirteen point six times nine, one hundred twenty two point four electron volts. Simple as that. Now, if the electron is already up in level n, it is closer to the zero line, so we need less: thirteen point six Z squared, over n squared. Bigger n, smaller ionisation energy. Clear? And for one mole of hydrogen atoms, the ionisation energy is one thousand three hundred and twelve kilojoules, the k J on screen. Write this number down, okay?

3:27Topper's trick: same jump, scale by Z squared

Now a trick that toppers use. Take the same jump, same n one and n two, but in the helium ion instead of hydrogen. How does the energy gap compare? Only Z squared changes; the bracket stays the same. So delta E is proportional to Z squared, this stretched alpha sign means proportional to, and lambda goes as one over Z squared. For the helium ion, Z squared is four, so four times the hydrogen gap. Yes, just like its ionisation energy! Don't recalculate, just scale. That's it.

4:04Excitation energy and potentials

Now excitation. The electron is not removed; it only moves up, from the ground state to level n. So the excitation energy is E n minus E one, okay? The first excitation energy of hydrogen: from level one up to level two. In the picture, hydrogen's level two sits at minus three point four. So the answer is three point four electron volts, right? Hold on, wait a second! Three point four is only the energy of level two, not the jump! One arrow two means the jump from level one to level two: minus three point four, minus of minus thirteen point six, ten point two electron volts. Remember ten point two; you will meet it again and again! Sometimes a question says potential instead of energy: ionisation potential, excitation potential. Don't panic. Same numbers, in volts. So for hydrogen, thirteen point six volts and ten point two volts; the capital V means volts. Really simple, right?

5:05Example 1: photon from n = 3 to n = 2 in hydrogen

Now, example one. Find the energy and wavelength of the photon emitted when the electron in hydrogen falls from n equal to three, to n equal to two. Pause the video, and try it first. Done? Okay, let's check. The electron goes from three to two. So which one is n one here? Two, exactly! n one is always the lower level, whichever way the electron moves. So n one is two, n two is three, and Z is one. Delta E equals thirteen point six, into one by four minus one by nine: thirteen point six times five by thirty six, one point eight nine electron volts. And lambda, with our handy form: twelve hundred and forty over one point eight nine, six hundred and fifty six nanometres. No powers of ten. See how fast that was? With me?

5:58Example 2: spectral lines after 12.5 eV electron impact

Now, our puzzle from the start, JEE level. Hydrogen atoms in the ground state are bombarded with electrons of energy twelve point five electron volts. Which spectral lines can be emitted afterwards? Pause the video, and try it. Hint: excitation energies first. Done? Okay, let's check. To level two: thirteen point six, into one minus one by four, ten point two electron volts. Twelve point five is more, so level two is reachable. To level three: thirteen point six, into one minus one by nine, twelve point zero nine electron volts. Still less than twelve point five. Reachable too! Alright, and level four? Can our electrons push the atoms up there? Thirteen point six, into one minus one by sixteen: twelve point seven five electron volts. More than twelve point five. So no, level four is out of reach. So twelve point zero nine is less than twelve point five, less than twelve point seven five; this sign means less than. The atoms reach at most n equal to three. So yes, something does happen: excitation, up to level three! Now the lines. The electron comes back down from level three. So just one line, three to one, right? No, no, careful! It can also come down in steps: three to two, then two to one. So three lines: three to two, three to one, and two to one. Puzzle solved. Perfect!

7:31Quick revision: transitions, ionisation and excitation

So, quick revision. One: delta E equals thirteen point six Z squared, into one over n one squared minus one over n two squared, and lambda in nanometres is twelve hundred and forty over delta E in electron volts. Two: ionisation energy from level n is thirteen point six Z squared over n squared; excitation energy is E n minus E one. Three: potentials are the same numbers, in volts. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!