Nanoparticles: size and the surface area to volume ratio (triple)

Why does a Roman glass cup look green in daylight, but glow red when a lamp shines through it? It is the Lycurgus Cup, made about sixteen hundred years ago, and there is no paint or dye on it. The colour comes from specks of gold and silver spread through the glass, each far too small to see. A lump of the same metals would never do that. The cup's answer is a number, and the number starts with a cube.

The rest of this chapter reads properties off structures. This video keeps the structure the same and changes only the size. It is one of two videos in the chapter for separate chemistry only. Nanoscience is the science of particles between one and one hundred nanometres across, called nanoparticles. A nanometre is one billionth of a metre, which is one times ten to the minus nine metres. An atom is roughly a tenth of a nanometre across. So a nanoparticle is anything from a few atoms to a few hundred atoms across. Foundation and Higher students both need this scale. Fine particles, the size band next up the scale, run from one hundred to two thousand five hundred nanometres across. A nanoparticle sits below them. Now compare two particles of the same metal. A nanoparticle is twenty nanometres across, and a fine particle is two thousand nanometres across. How many times wider is the two thousand nanometre particle than the twenty? One hundred times. Two thousand divided by twenty is one hundred. In metres, in standard form, twenty nanometres is two times ten to the minus eight metres. Two thousand nanometres is two times ten to the minus six metres. The powers differ by two, which is the same factor of one hundred.

Size changes behaviour through one number: surface area to volume ratio. The name gives the order: surface area first, volume second. A cube has six faces. Unfold one into a flat net and count them: one, two, three, four, five, six. Each face is a square, so its area is side times side. The surface area of a cube is six times side times side. Its volume is side times side times side. Keep the six written in the working, every time. Take a cube with sides of two centimetres. Surface area first: six times two times two. Two times two is four. Six times four is twenty-four. So the surface area is twenty-four centimetres squared. Then the volume: two times two times two, which is eight. The volume is eight centimetres cubed. Now the ratio, surface area first: twenty-four to eight. Divide both sides by eight, the smaller number. That gives three to one. A ratio is not finished until it is in its simplest whole numbers, here with a one on the volume side. Now shrink the cube to sides of one centimetre. The six faces, the order and the simplifying all stay the same. Pause and work out the ratio for the one centimetre cube, in all three lines. Surface area: six times one times one, which is six centimetres squared. Volume: one times one times one, which is one centimetre cubed. Ratio: six to one. Halving the side doubled the ratio, from three to one up to six to one. A smaller cube has more surface for every bit of volume. The six is the step one examiner's report found missing most: "Volume was often correctly calculated while many gave the surface area of one face as one hundred forgetting that the cube had six surfaces." One face is not the surface. The fix is to write six times side times side as the first line, before any number is multiplied out. Here's an exam-style version. A question tells you a cube's volume is eight centimetres cubed, and the area of one face is four centimetres squared. What two traps are hidden in the volume and the face area given in that question, before any working starts? First trap: working out the volume again, when it is given. Second trap: using four as the surface area, when that is one face. Surface area is six times four, twenty-four, so the ratio is twenty-four to eight, three to one. One examiner's report on a question like this says: "Many also recalculated the volume of the cube, despite being given that value, and often disagreeing with the provided value." The same report says: "This question discriminated very well; about two fifths of students scored all four marks and two thirds scored two or more." This calculation is very winnable. The habit is to box what the question gives you before you start. Anything in the box is done, and if your own answer disagrees with it, the question is right.

The pattern has a rule. When the side of a cube is divided by ten, its surface area to volume ratio is multiplied by ten. A cube with sides of ten centimetres has a surface area to volume ratio of nought point six to one. Without calculating, what is the ratio for a cube with sides of one centimetre? Six to one. The side went down by a factor of ten, so the ratio went up by a factor of ten. Picture one big cube, then the same material cut into a thousand small cubes, each with a tenth of the side. The amount of material is the same, and the total surface is ten times bigger. A nanoparticle has a far higher share of its atoms on its surface than a lump does. Chemistry happens at surfaces, so a nanoparticle can behave differently from the same material in bulk. A much smaller quantity can also do the same job. Be precise about what is bigger. One nanoparticle has a tiny surface area. What it has is a far higher surface area for its volume than a lump of the same material. That is why the full phrase, surface area to volume ratio, is the one to use. The comparison word matters too. One examiner's report notes: "Some students did not give answers in comparative terms, for example referring to a high, rather than higher, surface area to volume ratio." High compared with what is the missing half. The fix is to finish the sentence every time: a higher surface area to volume ratio than the same material in bulk. Now, of the twenty and two thousand nanometre particles, which has the higher ratio? The twenty nanometre particle. It is one hundred times narrower, so its ratio is one hundred times bigger. One more report line on these calculations: "Many candidates did not show their working and so incorrect responses scored zero as error carried forward marks could not be awarded." The fix is three lines, every time: surface area, volume, ratio.

Pencil and paper for this part: a calculation first, then the words. A new cube has sides of three centimetres. What is its surface area to volume ratio? Surface area: six times nine, fifty-four centimetres squared. Volume: twenty-seven centimetres cubed. Ratio: fifty-four to twenty-seven, which is two to one. Now the rule: a cube's side is divided by ten. What happens to the ratio? It is multiplied by ten. Something different now. What is missing from this sentence: nanoparticles have a high surface area to volume ratio? Compared with the same material in bulk, nanoparticles have a higher surface area to volume ratio. One more: why do the specks in the Roman cup behave so differently from a lump of gold and silver? They are nanoparticles, with a far higher surface area to volume ratio than a lump, so a much bigger share of their atoms sit at the surface.

A thumbs-up on this one means you have got this down, and it saves you sitting through the cube arithmetic again before an exam. Without a thumb, it stays on your list, and saving it makes it easy to find. Try three cube questions of your own; three questions in, it starts to feel normal.

Next in the chapter: nanoparticles, what they are used for and the risks. Hopefully that's next in your feed, but if not, search for ScholaFly C H oh five, ten.

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