Exchange surfaces and the surface area to volume ratio

You have far more skin than an earthworm, so why can't you breathe through it? An earthworm takes in all the oxygen it needs straight through its skin. A flatworm, as flat as a sheet of paper, does exactly the same thing. You have nearly two square metres of skin, far more than either of them. Yet if skin were all you had to breathe through, you would get nowhere near enough oxygen. More skin than either of them, and still not enough. The way into that starts with a cube of beetroot.

Surface area is the total area of the outside of an object. Volume is how much space it takes up inside. Divide one by the other and you get a ratio. That is what surface area to volume ratio means: how much outside there is for each bit of inside. Every bit of the inside of a living thing needs oxygen, and all of it arrives by diffusion, spreading in through the outside. Which matters for supply: surface area alone, or surface area per bit of volume? Surface area for each bit of volume. A big surface helps little if there is far more inside to supply. A cube has six square faces, so its surface area is six times the side squared. Its volume is the side cubed. Surface area comes out in centimetres squared, and volume in centimetres cubed. The ratio itself has no unit. Three cubes of beetroot have sides of one centimetre, two centimetres and forty millimetres. Convert first, on its own line. There are ten millimetres in a centimetre, so forty millimetres is four centimetres. The one centimetre cube first. Six times one squared is six centimetres squared. One cubed is one centimetre cubed. That makes the ratio six to one. Now the two centimetre cube, with the same three steps. What is the two centimetre cube's ratio, as something to one? Its ratio comes out at three to one, once you simplify. Two squared is four, and six times four is twenty-four centimetres squared. Two cubed is eight centimetres cubed. Twenty-four to eight, with both sides divided by eight, is three to one. Now the four centimetre cube, converted from millimetres. Four squared is sixteen, and six times sixteen is ninety-six centimetres squared. Four cubed is sixty-four centimetres cubed. Ninety-six to sixty-four, with both sides divided by sixty-four, is one point five to one. Read down the ratios: six, then three, then one and a half. Each time the side doubles, the ratio halves. That means the bigger the object, the smaller its ratio. Now carry that rule from cubes to animals, with an elephant and a mouse. Which has the larger surface area to volume ratio, the elephant or the mouse? The mouse. It is much smaller, so it has much more outside for each bit of inside. One exam specification warns teachers about this very point. Learners commonly show some confusion regarding surface area to volume ratio, particularly how larger animals have a smaller surface area to volume ratio. The way to keep it straight is the beetroot cubes: as the side doubled, the ratio halved. Here's the second half of the beetroot question. The three cubes are dropped into water, and the red colour leaks out of them. Which cube goes colourless right to its middle first, and why? The one centimetre cube. It has the largest ratio, and the distance from its surface to its middle is the shortest. A linked answer puts a so after the ratio. The small cube has the largest ratio, so more of it is close to the surface, so the colour escapes from its middle sooner. The particles diffuse just as fast in every cube. The small cube's middle is closer to the outside, and that is all. One examiner's report on a Higher paper, about cubes turning colourless, records this. Most candidates scored well by recognising that cube A was quicker because it was smaller or that it had a large surface area to volume ratio. The same report adds a warning. No credit was given to the idea of a faster rate of diffusion, as the linked explanation related to diffusion distance and not rate. The habit, then, is distance and ratio for a cube question, never faster diffusion.

Shape changes the ratio too, even at the same volume. Three shapes each have a volume of eight centimetres cubed: a cube two by two by two, a rod one by one by eight, and a flat sheet four by four by nought point five. The cube is the one already done: twenty-four to eight, which is three to one. The rod has no single side to square, so add up its faces. Its two square ends are one by one, which makes two. Its four long sides are one by eight, which makes thirty-two. That is thirty-four centimetres squared in all. Thirty-four to eight. Divide both sides by eight, and the rod's ratio is four point two five to one. Now the flat sheet, with two big faces and four thin edges. What is the sheet's ratio, once you add up all six faces? The sheet's ratio comes to five to one, the largest of the three. The two big faces are four by four, which makes thirty-two. The four edges are four by nought point five, which makes eight. Forty centimetres squared, over eight, is five to one. Here's the point of it. A tiny animal with no blood system gets all its oxygen by diffusion through its skin. Which shape suits it best: the cube, the rod or the flat sheet, and why? The flat sheet. It has the largest ratio, and no cell is far from the surface, so diffusion alone reaches every cell. That is why a flatworm is flat. Its shape does the job a blood system would otherwise have to do.

Now scale that up to a whole animal, one link at a time. A single-celled organism has a large ratio, and nothing inside it is far from the outside. Diffusion across its surface supplies all of it. A large animal has a small ratio. Most of its cells sit deep inside, so the diffusion distance, how far the particles must travel, is long. That makes diffusion through the outer surface alone far too slow to supply them. So a large organism needs exchange surfaces, specialised places where substances get in and out, and a transport system to carry them to every cell. Now picture a small animal that grows much bigger, but keeps exactly the same shape. What happens to its ratio, and what does it come to need? Its ratio gets smaller. Its deepest cells end up far from the skin, so it needs an exchange surface and a transport system. Through exchange surfaces pass oxygen, carbon dioxide, water, dissolved food molecules, mineral ions and urea. A second report, this time about two model cells shaped as cubes, records two slips. Many realised that the surface area of cube Y is larger than the surface area of cube X, but did not relate this to the volume. Quite a lot said active transport would have to be used by the multicellular organism. The fix for both: never give a surface area without its volume beside it. And the answer to a small ratio is an exchange surface, not a different way across a membrane.

Your lungs are an exchange surface. Air goes down the trachea, the windpipe, into the bronchi, its two branches, and ends in millions of tiny air sacs called alveoli. Each alveolus is wrapped in a dense network of capillaries, the narrowest blood vessels. Oxygen, between the alveolus air and the blood: which way does it diffuse? From the air in the alveolus into the blood, because the air holds the higher concentration of oxygen. Carbon dioxide goes the other way, from the blood into the air. Feature one: millions of alveoli give a huge surface area, so more oxygen can diffuse across at once. Feature two: the alveolus wall is one cell thick, and so is the capillary wall beside it. That gives a short diffusion path. Now a job, and you predict the feature. Once oxygen is in, it must not build up next to the alveolus. What feature keeps oxygen from building up in the blood beside the alveolus? A good blood supply. The dense capillaries carry oxygen away as soon as it arrives, so the concentration gradient stays steep. A report on a Foundation paper, about the adaptations of an alveolus, notes the one that went missing. Few candidates mentioned a good blood supply, with the capillary network having a large surface area seen just once. The habit that fixes it: count the features to four, and say what the blood does for the gradient. Feature four is ventilation, the breathing in and out that keeps replacing the air in the alveoli. Fresh air keeps the gradient steep too. That is four of four: a large surface area, a short diffusion path, a good blood supply and ventilation. Those four make any exchange surface effective, though the blood supply belongs to animals, and ventilation to animals exchanging gases. Here are three loose ways of describing feature two. Thin alveoli, a thin membrane, thin cell walls: what is wrong with each? The alveoli are not thin; their wall is one cell thick. A thin membrane is too vague. And animal cells have no cell wall at all. Another report, on a Higher paper, records those same slips. Having thin walls or having walls only one cell thick was often poorly expressed, with answers implying that the alveoli themselves were thin or that they have a thin membrane; neither of which are correct. The fix is one phrase, said every time: the wall is one cell thick.

The same features turn up wherever living things exchange substances. In the small intestine, villi, tiny finger-like folds of the lining, give a large surface area and a good blood supply. Plant roots and leaves are exchange surfaces too, each with a large surface area, and the chapter on plant tissues covers how they are built. Now fish gills. Gills are many thin filaments covered in tiny plates. The surface is a cell or two thick, blood flows through it, and water is kept flowing over it. Explain how gills are adapted for gas exchange, and for every feature, say what it does for diffusion. Can you link each gill feature to diffusion before the reveal? The gill's many filaments and plates give a very large surface area, so more oxygen enters the blood at once. A surface a cell or two thick gives a short diffusion path. The blood carries oxygen away as soon as it arrives, so the gradient from water to blood stays steep. Water kept flowing over the gills brings fresh oxygen, which keeps the gradient steep as well. Every feature is linked to diffusion, and none is only listed. On a Foundation paper about how the breathing system is adapted, the examiner's report records this. Very few students were able to make the link between gas exchange and diffusion. Many students wrote lengthy responses about the journey of oxygen or the mechanism of breathing, neither of which were asked about in the question. The fix: link every feature to diffusion with a so, and leave the oxygen's journey round the body, and how breathing works, out of it.

Four checks before the close, two on ratios and two on surfaces. A cube has sides of three centimetres. What is its ratio, as something to one? Two to one. Six times nine is fifty-four centimetres squared, and three cubed is twenty-seven centimetres cubed. Now, why does a large animal need exchange surfaces at all? Its ratio is small and most of its cells are far from the outside, so diffusion through its skin is too slow to supply them. Now a damaged wall: an alveolus wall thickens to three cells. Which feature is lost? The short diffusion path. Oxygen would have further to travel to reach the blood. One more, back to the start: why can an earthworm breathe through its skin, and you can't? The earthworm is thin, with a large ratio and short distances. You have more skin but a far smaller ratio, so you breathe through millions of alveoli folded up inside you.

If you're comfortable with the ratio and the four features, hit the thumbs-up, and you won't need this one again. Still shaky on it, then save it, and write the four features out once. Written down once, they tend to stay.

Next in the chapter: What sets the rate of diffusion, and Fick's law. Hopefully that's next in your feed, but if not, search for ScholaFly B I oh four, oh four.

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