Rutherford's Model and Its Difficulties

Transcript

0:00Introduction

Hello everyone! Let us begin with a strange question. Your body is made of atoms, and in every atom, electrons are moving round a nucleus. Now, if classical physics were the whole story, every one of those atoms would collapse in about a hundred millionth of a second. So why are you still here, watching this video? Something in that picture must be wrong, right? Today we build Rutherford's planetary model of the atom, find the energy of its electron, and then see exactly where the model breaks down. Questions on this do come up in JEE, so let's begin.

0:37The planetary atom

So, here is Rutherford's picture, the planetary atom. The electrons revolve round the nucleus, the way planets go round the Sun. But anything moving in a circle needs a force pulling it towards the centre. So tell me, what supplies that pull for the electron? Yes, exactly, it is the electrostatic attraction between the positive nucleus and the negative electron. That attraction provides the centripetal force, and nothing else is needed. Okay? Now let us write it down for hydrogen, with orbit radius r. On the left, m v squared over r, the centripetal force, where m is the mass of the electron and v is its speed. On the right, the electric force, e squared, over four pi epsilon naught times r squared. Here e is the charge on the proton, and epsilon naught, the Greek letter epsilon with a small zero, is the permittivity of free space. Clear? Now multiply both sides by r, and take half. We get the kinetic energy, capital K, equal to half m v squared, which is e squared over eight pi epsilon naught r. So the smaller the orbit, the larger the kinetic energy. With me?

1:51Potential energy and total energy

Next comes capital U, the potential energy. U equals minus e squared over four pi epsilon naught r. The minus sign is there because the force is attractive, and we take zero energy with the electron at rest, infinitely far away. Okay? Now it is your turn to think. Add the kinetic energy and the potential energy. What is the total energy of the electron? Let us add them together. K is e squared over eight pi epsilon naught r, and U is minus two times that same amount. One minus two gives minus one. So the total energy, capital E, equals minus e squared over eight pi epsilon naught r. Got it? Now look at all three together. E equals minus K, which equals U over two. This is a topper's trick: if you know any one of the three energies, you know the other two immediately. Remember it, okay? Now, here is a classic trap. The total energy is negative. So does the electron have a negative kinetic energy? No, no, be careful here, because kinetic energy can never be negative. It is the potential energy that is negative, and twice as big as the kinetic energy in size. So E less than zero, with the less than sign on screen, means the electron is bound. It has less energy than an electron at rest far away, so energy must be supplied to set it free. Clear?

3:20Difficulty 1: the atom should collapse

So far the model looks beautiful, but now the trouble starts. Our electron goes round its circle at a steady speed. So tell me, is it accelerating? Yes, it is accelerating, because its direction changes all the time, so it has a centripetal acceleration towards the nucleus. So a revolving electron is accelerated. Right? And classical electromagnetism is very clear about this: an accelerated charge must radiate electromagnetic waves. Radiating means it keeps losing energy. With me? Now, losing energy makes the total energy more negative. And from our formula, a more negative total energy means a smaller orbit. So the electron spirals inwards, closer and closer, right into the nucleus. Okay? And how long would this take? About ten to the power minus eight seconds, where the small s on screen stands for seconds. That is the collapse from our opening puzzle. But real atoms are stable, so here the model fails for the first time. Got it?

4:23Difficulty 2: the wrong kind of spectrum

Now the second difficulty. As the electron spirals in, its orbit keeps shrinking, so its frequency changes continuously. And the light it radiates changes its frequency along with it. Okay? So tell me, what kind of spectrum should such an atom give out? That is correct, it should give a continuous spectrum, with all the frequencies blending smoothly into one another. That is the model's prediction. Right? But in experiments, atoms emit sharp lines, only certain wavelengths, and nothing in between. So the model fails a second time. Remember these two failures, the collapse and the continuous spectrum, because a theory question may ask you for exactly these two. Clear?

5:10Example 1: potential and total energy from K = 3.4 eV

Now, example one, an easy one. In an electron orbit, the kinetic energy is three point four electron volts, the e V on screen. Find the potential energy and the total energy. Use our trick, and pause the video to try it first. Done? Okay, let's check. Since E equals minus K, and also U over two, the potential energy U equals minus two times the kinetic energy. So minus two times three point four gives minus six point eight electron volts. Right? And the total energy is just minus K, so E equals minus three point four electron volts. Did anyone write plus three point four? That is the sign trap, so watch out for it. With me? Let us quickly check. K plus U is three point four minus six point eight, which gives minus three point four electron volts. It matches, so both answers are correct. Okay?

6:09Example 2: orbit radius from 13.6 eV ionisation energy

Now a harder one. The ionisation energy of hydrogen is thirteen point six electron volts. Using Rutherford's model, find the radius of the electron's orbit. Here is a hint: think about what the ionisation energy tells you about the total energy. Pause the video and try it first. Done? Okay, let's check. Ionisation energy is the energy we must supply to free the electron, to take it to rest far away, where the energy is zero. So what is the total energy of the electron in its orbit? Correct, the total energy is minus thirteen point six electron volts. The double arrow on screen means implies. Okay? And from our formula, the total energy is minus e squared over eight pi epsilon naught r. So we set that equal to minus thirteen point six electron volts, and the two minus signs cancel. Right? Now, a handy constant from the start of this chapter. The quantity e squared over four pi epsilon naught equals one point four four electron volt nanometre, where n m stands for nanometre. It saves you from all those powers of ten. With me? Eight pi is two times four pi, so r equals one point four four electron volt nanometre, over two times thirteen point six electron volts. Oh no, careful here, because many students forget this two and get double the radius. Clear? Two times thirteen point six is twenty seven point two. Then one point four four over twenty seven point two gives about zero point zero five three nanometres. The wavy equals sign means approximately, and the electron volts cancel, leaving nanometres. That is the radius of the orbit. Got it?

7:59Quick revision: E = -K = U/2, collapse, continuous spectrum

So, quick revision. In the planetary atom, K is e squared over eight pi epsilon naught r, and E equals minus K, which equals U over two. The accelerated electron radiates, so the atom should collapse. And the model predicts a continuous spectrum, but atoms give sharp lines. Now go and try the practice problems on this page. Give them a real try. All the best!