diffusion is a key subject in physiology. By means of diffusional processes, substances can be efficiently transferred at short distances. For that reason, it is a very important mechanism of exchange between the cells comprising a plant or an animal and the surrounding compartments with which they interact as well as between the compartments themselves (such as the interstitial and capillary compartments, for example). However, diffusion is too a slow process when matter has to be transported along the large distances found in the bodies of multicellular organisms. Consequently, as organisms became larger, diffusion was no longer capable of providing for all their needs, and convective transport was made necessary.

Explaining to undergraduate students why diffusion works so badly at larger distances while at the same time being a process of paramount relevance at the cellular level is not a trouble-free task (4). Although students easily grasp the notion that random processes are capable of giving rise to emergent behavior, such as the net directional movement of molecules, they frequently confound a macroscopic ballistic view of movement with a microscopic diffusive one. It appears to arise from a deep-seated misconception about random processes (2). In the present article, we describe an inexpensive and simple way to make students intuitively experience the probabilistic nature and nonorientated motion of diffusing particles. This understanding allows students to realize why diffusion works so well over short distances and becomes increasingly and rapidly less effective as the distances involved become progressively larger. We believe that this activity is in agreement with the urge for the adoption of teaching approaches that more actively involve the student in the learning process, focus on problem solving, and lead to more meaningful learning (3).

The only required materials are some coins. In a classroom, the instructor has to assemble three groups of, say, five students each. The first group represents the “particles,” and the second and third groups are composed, respectively, by the “dealers” and “secretaries.” Each particle is assigned to its own dealer and secretary. To start the experiment, the students representing the particles must stand in line above the same reference mark located centrally at the front of the classroom (see Fig. 1). At the signal of the instructor, each dealer, by tossing a coin, will inform his/her respective particle to which side to move, according to a rule previously established by the instructor: for instance, heads means one step to the right and tails means one step to the left (the instructor should alert the students representing the dealers to pay attention at the process of coin flipping, so that the coin, once tossed up, should rotate several times to reduce outcome biases; this is extremely important, as the coin is to be used as a random number generator). Upon being informed about the respective outcome (left or right), each particle should walk one step in the informed direction, which is accordingly registered by the team's secretary. The demonstration finishes after 16, 25, or 36 trials, as decided by the instructor. After that, the instructor should ask the students to appreciate where each particle has finished its “random walk” and how far it is from the reference mark signaling the initial position. The students are instructed to consider each trial as an arbitrary unit of time of equal duration and each step as spanning an arbitrary unit of space with constant length. The expected average result of this demonstration should follow this equation:

Consequently, after 16, 25, or 36 trials, the average net displacement (distance) from the initial mark is expected to be 4, 5, or 6, respectively (the agreement between the expected and observed averages increases with the number of students playing the particles). The instructor should then ask the students why this happened, that is, why do some particles remain so close to their original positions after so many steps (time units)? It is quite possible that after all trials one or more particles may finish the demonstration at their starting position, whereas some of them may hit the wall of the room. At this time, some of the students have probably realized the nonoriented nature of this process: a unidimensional random walk. The instructor should stimulate the imagination of the students by asking them to calculate how many trials (or how long) would be necessary for a given particle to cover the distance between the starting central mark and the classroom's door or beyond.

To make the experiment even clearer, the instructor should ask the students to calculate the theoretical average distances for a single trial, two trials, and three trials, which can be easily shown in a simple table (Table 1). From these results (Table 1), the students should be able to appreciate a few important features:

After one trial, the ending position is always (and obviously) at a distance = 1 (taking into account the absolute magnitude of the displacement).

After two trials, the average distance for a pool of many particles is distance = 1, although each particle, individually, will always end up with equal probability at either distance = 2 or distance = 0.

After three trials, among eight possible outcomes, only two of them correspond to a distance = 3 (the maximum allowed); the six other possible outcomes (75% of the total) will lead to a final net displacement distance = 1.

A graphing exercise can be performed before the actual data collection. Based on the above theory, one can propose to students to plot distance versus time as well as time versus distance, accordingly to *Eq. 1*. The shape of the first graph will show that even after long periods of time, a given particle only covers short distances. The second plot will show how long a particle takes to cover long distances. The comparison between theoretical and observed data helps students capture the way in which the process unfolds.

After the students have fully comprehended the random walk and diffusion in one dimension, the instructor can use the same concepts to explain and extrapolate the situation to three dimensions.

Internal transport systems in multicellular organisms must assure that nutrients, gases, and other vital substances move from sites of production or uptake to sites of use; furthermore, these systems have to provide for the excretion of byproducts of cell metabolism. Over short distances of hundreds of micrometers, materials can diffuse through the body in short times. For example, the time that it takes for oxygen to diffuse in water, the major constituent of living bodies, through the 5 μm of a cell is ∼0.2 ms (1). Therefore, diffusion takes place in microscopic body structures maintained along the evolutionary course: inside the cell, across the plasma, capillary, nephron, and alveoli membranes, over the synapse cleft, and over many other microscopic structures. Nevertheless, this demonstration shows that due to the probabilistic and nonoriented nature of the diffusion process, a particle can take minutes to diffuse even over the tiny distance of 1 in. As body size increases, the time it takes for materials to move from one end of the organism to the other by diffusion increases greatly. The time needed for oxygen diffusion through the body of a small animal of 6 cm is ∼7 h, through a human body of 2 m is ∼10 mo, and over the 15 m of some big animals is ∼48 yr (1)! This fact was the foremost selective pressure to the phylogenetic development of some internal transport systems by convection, like the respiratory, circulatory, and urinary systems–all convective transport systems.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

## ACKNOWLEDGMENTS

The authors thank Nestor Caticha for helpful discussions on the central topic of this article.

- Copyright © 2010 the American Physiological Society