Part C: Concept Checks

Transcript

0:00Introduction

Hello everyone! Here is a small puzzle. Take a hydrogen atom in its ground state. Light excites it only if the photon's energy matches a level difference exactly, yet an electron with any energy above ten point two electron volts can excite it. So why is the photon so fussy, and the electron not? Keep your answer in mind; Example one settles it. Why-questions like these do come up in JEE, so let's do some quick concept checks.

0:32Check 1: probing the atom with alpha particles

So, check one. In the figure, alpha particles, alpha the Greek letter like a rounded a, fly past a nucleus of charge plus Z e, the orange dot; Z is the atomic number and e the proton's charge. The smaller the impact parameter, b, the larger the scattering angle, theta; r naught is the head-on closest approach. The formulas on the right, with K the alpha particle's kinetic energy and epsilon naught the permittivity of free space, come from the Rutherford lesson. Why use alpha particles as probes, and not electrons? Think about the mass first. The alpha particle is heavy, about seven thousand three hundred electron masses, so the atom's electrons cannot deflect it. Okay? Second, it is energetic, so it can come very close to the nucleus. Right? So any large deflection must be due to a massive, concentrated charge: the nucleus. Clear? Now, along a curved path, the linear momentum, small p, keeps changing. But the angular momentum about the nucleus, capital L, stays constant. Why the difference? Yes, it is all about the direction of the force. The force always points along the line joining the alpha particle to the nucleus: a central force. So it exerts no torque about the nucleus, and angular momentum is conserved. But the force is not zero, so linear momentum changes. Got it?

2:02Check 2: Lyman ultraviolet, Balmer visible

Now check two. In the figure, each horizontal line is an energy level of hydrogen, labelled by n, the principal quantum number, with its energy in electron volts, e V, on the right; the dashed line at zero is n equals infinity, the sideways eight. Dark blue arrows fall to level one, the Lyman series; orange ones to level two, Balmer; black ones to level three, Paschen. Why is Lyman ultraviolet, but Balmer visible? E n, with the small n below, equals minus thirteen point six over n squared electron volts, for hydrogen. The figure gives the general form for any one-electron atom, with capital Z squared on top; Z is the atomic number, and it is one for hydrogen. Beside it is the Rydberg formula, with lambda the wavelength, capital R the Rydberg constant, and n one, n two the lower and upper levels. Okay? It is all about the energy, exactly. Lyman jumps end on level one. From two to one, thirteen point six minus three point four gives ten point two electron volts, and from infinity, thirteen point six. Big energies, short wavelengths, so Lyman is ultraviolet. Right? Balmer jumps end on level two. From three to two, three point four minus one point five one gives about one point nine, and from infinity, three point four. Much smaller energies, longer wavelengths, so Balmer is visible. Clear? One small detail: Balmer's last lines, near the series limit, reach just into the near ultraviolet. So the visible label is for most of the series, not every single line. With me?

3:40Check 3: the sharp edge of the X-ray spectrum

Next, X-rays, check three. The left graph shows intensity against lambda, the wavelength: a smooth continuous spectrum, with two orange spikes, K alpha and K beta, the characteristic lines. The right graph, Moseley's plot, is for the practice set. The curve starts sharply, at the minimum wavelength marked lambda min. Why such a sharp edge? Here is the reason. Each electron in the tube is accelerated through a voltage, capital V, so it reaches the target with kinetic energy e times V, with e the electron's charge. Okay? In the target, any part of that energy can come out as a photon, but no photon can carry more than the whole e V of one electron. With me? So the photon that takes all of it has the shortest wavelength: lambda min, with min written small, equals h c over e V, where h is Planck's constant and c the speed of light. Nothing shorter is possible, hence the sharp edge, which depends only on the voltage.

4:41Check 4: one density for every nucleus

Got it? Now into the nucleus, check four. Why does nuclear matter have the same density in every nucleus, light or heavy? It all comes from one formula. Capital R, the nuclear radius, equals R naught times A to the power one third, with capital A the mass number, the number of nucleons. R naught, R with a small zero below, is about one point two femtometres, the f m; the wavy equals sign means approximately. Okay? The volume is four thirds pi R cubed, with pi the Greek letter. Cube A to the power one third and you get A, so the volume is proportional to A; the sign like a sideways fish means proportional to. Right? The mass is also proportional to the number of nucleons. So mass over volume is the same for all nuclei: each nucleon occupies the same volume. Clear?

5:34Example 1: photon vs electron exciting hydrogen at 10.2 eV

Now, Example one, our puzzle from the start. Why can a photon excite hydrogen only with an exact level difference, while any electron above ten point two electron volts can? Pause the video and try it first. Are you done? Okay, let's check. From check two, the jump from level one to level two needs exactly ten point two electron volts. Now, can a photon with a little more give just ten point two, and keep the rest? It cannot, and that is the key. The photon is absorbed whole, or not at all, so its energy must match a gap exactly. Okay? Above thirteen point six electron volts, a photon can instead ionise the atom, but that is not excitation. Right? But an electron, a particle, has kinetic energy K, and it can give up just ten point two electron volts and carry away the rest. So any electron above ten point two works, and that settles our puzzle!

6:35Example 2: why hydrogen levels crowd together as n increases

Now Example two, a little harder. Why do the energy levels of hydrogen get closer together as n increases? You can see them crowd together at the top of the figure. Pause the video and try it first. Are you done? Okay, numbers first. From level one upwards, the gaps between neighbours are ten point two, then one point eight nine, then zero point six six, and then zero point three one electron volts. Now, the energy goes as one over n squared; the slash on the badge means over. So the gap also goes as one over n squared, right? No, that is exactly the trap, so let us work it out carefully. The gap is E n plus one, minus E n, the labels written small below. That is thirteen point six times the bracket, one over n squared, minus one over n plus one, whole squared. With me? Take the common denominator: n plus one, whole squared, minus n squared, is two n plus one. So the gap is thirteen point six times two n plus one, over n squared times n plus one, whole squared. Right? For large n, two n plus one is about two n, and the bottom is about n to the power four. So the gap is about twenty seven point two over n cubed electron volts, falling as one over n cubed, one power faster than the energy: that is why the levels crowd together. Got it?

8:00Quick revision: alpha probes, Lyman, X-ray cut-off, density

So, quick revision. Alpha particles are heavy, energetic probes, and a central force keeps their angular momentum constant. Lyman photons carry ten point two to thirteen point six electron volts, ultraviolet; Balmer only one point nine to three point four; and the X-ray spectrum ends at lambda min, h c over e V, okay? R equals R naught times A to the power one third, which gives every nucleus the same density, a photon is absorbed whole or not at all, and level gaps fall as one over n cubed. Now try the practice problems on this page. All the best!