Atomic Radius

Transcript

0:00Introduction

Hello everyone! One small puzzle before we start. Sodium has eleven electrons, and chlorine, in the same period, has seventeen. So chlorine must be the bigger atom, right? Well, the numbers say no. Sodium's radius is one hundred and eighty six picometres, and chlorine's is only ninety nine. More electrons, and yet a smaller atom? Size trends come up often in JEE, so let's solve this mystery together.

0:31Three kinds of atomic radius

First, we cannot put a ruler on one atom, so we measure the distance between two nuclei and take half. For two identical atoms joined by a covalent bond, half the bond length is the covalent radius. One half times the bond length just means half of it. Okay? Next, in a metal, half the distance between neighbouring nuclei is the metallic radius. It is the same idea, right? And the third kind is the van der Waals radius, for atoms that are not bonded, sitting in neighbouring molecules. Again, take half the distance between them. Now tell me, which of the three radii is the largest? The van der Waals radius, yes, exactly! Non-bonded atoms are not pulled together by a bond, so they sit farther apart. We use this radius for the noble gases, whose atoms do not bond to each other. Clear? Here is a small trick. Some tables show a noble gas bigger than the halogen just before it. The trend is not broken: the noble gas value is a van der Waals radius, and the halogen value is covalent. So compare only radii of the same kind. Got it?

1:43Across a period: the radius decreases

Look at this graph of atomic radius in picometres, written p m, for periods two and three. Across each period the radius keeps falling, from sodium at one hundred and eighty six down to chlorine at ninety nine. With me? To see why, we need the effective nuclear charge. Z effective, the capital Z with the small e f f below it, equals Z minus S, okay? Capital Z is the nuclear charge, the atomic number, and capital S is the screening by the other electrons. So Z effective is the pull an electron really feels. Across a period, the atomic number goes up by one at each step, so the nuclear charge keeps rising. Right? And each extra electron goes into the same outer shell. So let me ask you this: do electrons in the same shell screen each other well? They screen poorly, very good! So the screening grows only a little while the nuclear charge rises by a whole unit, and Z effective rises across the period. Clear? So the outer electrons are pulled in harder, and the atom shrinks. That answers our puzzle: within a period, the pull decides the size, not the number of electrons. Example two will put numbers on it. Got it?

3:00Down a group: the radius increases

Alright, now let us go down a group. Each step down adds a whole new shell, but the nuclear charge also rises. So which wins, do you think: the new shell, or the extra charge? Normally, the new shell wins. The outer electrons sit in a shell farther from the nucleus, so the radius increases down a group. Okay? Look at the graph again. The red triangles at the top are potassium, K, and calcium, C a, from period four. One extra shell, and they sit far above sodium and magnesium. With me?

3:36Exceptions: poor screening by d and f electrons

Now let us test our rule. Gallium, G a, sits just below aluminium, A l, in group thirteen. One more shell, so gallium must be bigger, right? Oh no, careful, it is the other way round! Gallium is one hundred and thirty five picometres, and aluminium is one hundred and forty three. The sign between them means less than, so gallium is slightly smaller. This is a classic trap. Clear? Just before gallium come the ten three d elements, and three d electrons screen the nucleus poorly. So gallium feels its extra nuclear charge strongly, and that more than cancels the new shell. Right? Now look at the f electrons. Hafnium, H f, sits just below zirconium, Z r, and between them lie the fourteen lanthanoids, filling the four f subshell. And four f electrons also screen poorly. Okay? This steady shrinking across the lanthanoids is the lanthanoid contraction. It cancels the extra shell, so hafnium is one hundred and fifty nine picometres, and zirconium is one hundred and sixty. The wavy equals sign means almost equal. That is why these two are so hard to separate. Got it?

4:50Example 1: covalent radius of Cl and metallic radius of Cu

Now, example one, an easy warm-up. The bond between two chlorine atoms, written C l dash C l, is one hundred and ninety eight picometres long, and the C u to C u distance in copper metal is two hundred and fifty six picometres. Find the covalent radius of chlorine and the metallic radius of copper. Pause the video and try it first. Okay, let's check your answer. So the radius of chlorine is one hundred and ninety eight picometres, right? No, no, wait, that is the whole bond length, from one nucleus to the other, so it holds two radii! One hundred and ninety eight over two gives ninety nine picometres, the same chlorine size as in our puzzle. Okay? Copper is a metal, so we need the metallic radius. Two hundred and fifty six over two gives one hundred and twenty eight picometres. Simple, right? Always take half the distance between the nuclei.

5:49Example 2: Slater's rules, Z eff of Na and Cl

Now example two, which solves our puzzle with numbers. In Slater's rules, small n is the shell number. Each other electron in the same n s, n p group screens zero point three five, each electron in shell n minus one screens zero point eight five, and each deeper electron screens one point zero zero. Estimate Z effective for sodium's valence electron, and for a three p electron of chlorine. Pause the video and try it first. Done? Okay, let's check, starting with sodium, written N a on screen. Its configuration is one s two, two s two, two p six, three s one, and the small raised numbers count the electrons. The valence three s electron is alone in its group, with eight electrons in shell two, and two in shell one. With me? The bracket is the screening: eight times zero point eight five, plus two times one, which gives eight point eight. Z effective is eleven minus eight point eight, which gives two point two. Clear? Now chlorine, whose configuration is one s two, two s two, two p six, three s two, three p five. Take one three p electron. How many other electrons share its three s, three p group? Seven, right? Careful here, because the answer is not seven! The rule counts each other electron, so we leave out our own, which makes six. So, six times zero point three five, plus eight times zero point eight five, plus two times one. Okay? Now add them up. Two point one, plus six point eight, plus two, gives ten point nine. So Z effective is seventeen minus ten point nine, which gives six point one. Got it? Chlorine's outer electrons feel a pull of six point one, almost three times sodium's two point two! That is why chlorine, at ninety nine picometres, is far smaller than sodium, at one hundred and eighty six. Puzzle solved: more electrons, but a much stronger pull.

7:53Quick revision: radius types, trends, Ga and Hf exceptions

So, quick revision. First, a radius is half the distance between two nuclei, and van der Waals is the largest. Second, across a period Z effective rises and the radius falls, while down a group the radius grows. And third, poor screening by d and f electrons makes gallium smaller than aluminium, and hafnium almost the size of zirconium. Now try the practice problems on this page, and all the best!