Agar Cubes, a Flatworm and a Thick Worm
A student cuts three agar cubes containing Universal Indicator, with sides of 1 cm, 2 cm and 3 cm, and places them in dilute hydrochloric acid. The time for each cube to change colour completely is recorded: 1 cm, 8 min; 2 cm, 32 min; 3 cm, 72 min. She then examines two worms from a rock pool. Worm X is a flatworm about 0.4 mm thick and 15 mm long with no circulatory system. Worm Y is a cylindrical worm 6 mm in diameter and 10 cm long with a simple circulatory system. She predicts that Worm Y will exchange gases faster than Worm X because it is much longer, and concludes that the agar experiment shows diffusion is fast enough for any animal as long as the animal is thin somewhere.
1. Calculate the surface-area-to-volume ratio of each cube and explain the pattern in the colour-change times.
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Using SA:V = 6a divided by a cubed, which simplifies to 6/a:
- 1 cm cube: 6/1 = 6.0 per cm
- 2 cm cube: 6/2 = 3.0 per cm
- 3 cm cube: 6/3 = 2.0 per cm
The times are 8, 32 and 72 min. Doubling the side from 1 to 2 cm multiplies the time by 4; tripling it multiplies the time by 9. That is, time scales approximately with the square of the linear dimension.
The reasoning has two parts that students often merge. First, the volume that must change colour rises with the cube of the side, while the surface through which acid enters rises only with the square, so the volume-to-surface ratio worsens as size increases. Second, the acid must diffuse further to reach the centre, and random-walk diffusion time increases with the square of distance. Both effects push in the same direction, giving the observed roughly fourfold and ninefold increases. Quoting only the SA:V ratio without linking it to diffusion distance and volume would earn limited credit.
2. Evaluate the prediction that Worm Y exchanges gases faster than Worm X because it is longer.
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The prediction is wrong, and the stated reason is wrong. Length does not rescue a thick body: for a cylinder, SA:V equals 2/r, so making it longer adds surface and volume in the same proportion and leaves the ratio unchanged. Thickness is what matters.
Worm X is 0.4 mm thick, so no cell is more than about 0.2 mm from the surface, well within the range over which diffusion is effective; its large SA:V and flattened body allow sufficient gas exchange without a circulatory system.
Worm Y is 6 mm in diameter, so cells at the centre are about 3 mm from the surface. Diffusion over that distance is far too slow to supply an active animal, regardless of the worm's length. Its circulatory system does not speed up diffusion across the body surface; it maintains a steep concentration gradient at the surface and transports oxygen and carbon dioxide between the surface and deep tissues. The correct comparison is surface area per unit volume and diffusion distance, not total length.
3. The student concludes that the agar experiment shows diffusion is adequate for any animal provided it is thin. Explain two limitations of the agar model that undermine this conclusion.
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The model captures the geometry of diffusion but not the physiology of an animal.
No metabolism and no convection. Agar is inert: it does not consume oxygen, produce carbon dioxide, or move fluid. A living animal uses oxygen, which steepens the gradient and can increase net uptake, but it also has a high demand that the model does not represent. Agar also lacks a blood system that could carry gases away, which is exactly the adaptation that lets a thick animal survive.
A single diffusing substance and no membranes. The acid simply reacts with indicator; there is no selectively permeable membrane, no transport proteins and no active transport. Real exchange surfaces are thin, moist and richly supplied with blood, and their permeability is controlled.
A further point is scale: the cubes change colour completely within 72 minutes, which says nothing about whether the rate of supply meets a metabolic demand. The model supports the general principle that diffusion is only fast enough over short distances, but it cannot be used to set a universal thickness limit for animals, because animals can add ventilation and circulation.