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R edu ce Confu s ion Abou t Diffu s ionDiffusion, a fundamental process in

the movement of biological materials,is one of the most commonly misun-derstood processes taught in biologyRecent articles in this journal (Marek etal. 1994; Vogel 1994) attest to the fre-quency and nature of these misunder-standings. Furthermore, while nm',tphysiological processes rely on diffu-sion in whole or in part, few studentscan explain why diffusion alone israrely sufficient for long-distance trans-port of liquids or gases.

Diffusion as a Rando mWalkThe underlying principle of dif-

fusion is the continuous, randommotion of molecules. This motionis driven by heat, and so it occurs atany temperature above absolutezero (-273 C). With all the mole-cules of the gas or liquid moving atthe same time, collisions are fre-quent and molecules bounce offone another. If you could followthe route of a single molecule, thepath would look like a randomwalk through three-dimensionalspace. To help advanced highschool students understand theidea of molecular motion, I havethem work in teams of three tosimulate such a random (but two-dimensional) walk. Before starting,however, the class as a whole mustagree on rules that the student mol-ecules, or "walker," will follow.

First it is necessary to establishthe directions that the walkers can

Mary R . Hebrank is an Instructor at theDuke School for Children MiddleSchool, 3716 Old Erwin Rd., Durham,NC 27705.

Mary R. Hebrank

take. For simplicity, students usu-ally restrict the possibilities to thefour orthogonal directions: for-ward, backward, right and left.Next, they must decide how thesefour directions can be randomlychosen. Since many students willhave coins in their pockets, theycan easily work out a simple codebased on a pair of coin tosses (e.g.heads-heads = forward, tails-tails = backward, etc.). Finally, stu-dents must agree on a uniform steplength, generally the length of asize 8-10 athletic shoe.

I supply each team with paper, apencil, a roll of masking tape, ameter stick, and a penny before wemove from the classroom to theschool gym or parking lot. Thereeach team determines the roles ofits members: one person acts ascoin flipper, another keeps track ofthe number of steps taken, and thethird is the walker. Students beginthe simulation by choosing a start-ing point on the floor or groundand marking it with a piece of tape.Then the coin tosser flips the pennyto determine the direction of thewalker's first step. The walkertakes one step in that direction,while the tracker marks one step ona tally sheet. A second step is takenin the direction determined by asecond pair of coin tosses, and soon, until 25 steps have been taken.(It is important to make sure stu-dents understand that each step isdetermined by a new pair of cointosses.) After the 25tlrctep has beentaken, the walker uses tape to markhis or heu final location. Then theteam measures and records the

straight-line distance between thestart and end points. A schematicdiagram of the path a walker mighttake using this method is shown inFigure 1.

Next, each team repeats the pro-cedure, this time taking 50 steps.After the students have recordedthese distances, I ask if the walkerstraveled twice as far in 50 steps asthey did in 25 steps. While such aresult is possible, it rarely happens.Usually the two distances are quitesimilar, and sometimes the seconddistance is slightly less than thefirst.

Computer Simulation ofDiffusionStudents next are challenged tofigure out the relationship between

the time elapsed in a diffusing sys-tem (analogous to the number ofsteps taken) and the distance trav-eled by an individual molecule. Tospeed the process up, each team isgiven a computer program that al-lows them to watch an animatedmolecule undergo a random walkon the computer's video display(Figure 2). Students can choose thenumber of steps to be taken, andafter displaying the path itself, theprogram provides numerical dataon the distance traveled. For eachtime increment (number of steps)selected, the program repeats for20 trials. It displays the results ofeach trial as Nvell as an average ofthe 20 trial "walks."

Student teams are advised to try160 THE AMERICAN BIOLOGY TEACHER, VOLUME 59, NO. 3, MARCH 1997

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Figure 1. Examples of random walks to simulate the movement of molecules. Inthe example on the left, the walker has taken 25 steps; in the example on the right,the walker has taken 50 steps. In both cases the directions of all steps taken havebeen determined by independent coin tosses. The letters B and E denote thelocation of the walker at the beginning and the end of each walk.a total of 6-10 different step incre-ments, starting with 100 and rang-ing up to 2500 or more. The dataanalysis will be easier if the class asa whole agrees on the numbers tobe used by each team as step incre-ments. It is also a good idea tosurreptitiously suggest that theyuse some square numbers, or num-bers that are close to square.

After all groups have completedthis portion, I put a table on theblackboard summarizing the aver-age distances obtained at each stepincrement (Table 1). At this pointstudents are asked to look for a

Table 1. Class results of computer-simulated diffusion, in s hichmolecules take 100, 200, 400, SOO,1600, and 2500 steps on randomwalks. Numbers in the cells indicatethe average straight-line distances (instep-units) between the starting andending points of the molecules, for 2Otrials. Average distances areapproximately the square roots of thenumber of steps taken.I 'can, # 10 0 20 0 -100 800 1600 'SUU

1 9. 8 14.1 23.3 28.4 40.o 57.1)2 75 15.9 19.2 35.0 41.6 54.13 9. 6 15.5 23.3 30.0 45.2 52.04 10.1 14 .5 24.2 35.9 41.9 45.25 10.8 13.0 23.5 36.2 36.7 59.-16 10 .3 14 .9 20.9 39.0 39.0 51.2

pattern, or mathematical relation-ship between time (number ofsteps taken) and actual distancetraveled. Usually some studentsare able to notice that the distancetraveled is, on average, approxi-mately equal to the square root ofthe number of steps taken.Diffusion's Limitations

Once students realize that in thesimulation the number of stepstaken is analogous to time of diffu-sion, they are in a good position tounderstand why diffusion is notnecessarily a fast process. Our con-ventional ways of thinking lead usto believe that if it takes an objectone minute to travel some knowndistance, it should take 10 times aslong to travel 10 times as far. How-ever, in diffusion it takes 100 timeslonger for a molecule to travel 10times as far. This is a simple conse-quence of the fact that distancetraveled is proportional to thesquare root of time. Therefore, dif-fusion max' work well for movingmolecules small distances, but it isless effective for moving them overlarge distances.

If you think about where inphvsiologica*'stems diffusion oc-curs, it isn't hard to verify that the

distances involved are usuallysmall. For example, neurotransmit-ters diffuse across synapses, whichtypically measure about 20 mano-meters across. The time required isapproximately a thousandth of asecond (Vogel 1902). Likewise, ittakes about a hundredth of a sec-ond for oxygen to diffuse the one-hundredth of a millimeter betweena cell and a capillary. During stren-uous exercise our muscle cells maycry out for oxygen; and as long asthe heart and lungs keep bringingoxygenated blood, diffusion will al-low it to get to the muscle cellspromptly.

This brings up an importantpoint, which is that diffusion aloneis not adequate for moving mole-cules of oxvgen over long dis-tances. If there was no heart topump the oxygenated blood, orany other mechanism to bringoxygen to the muscles, we wouldhave to rely on diffusion alone. Inthat case, it would take about 100million seconds, or about threeyears, for oxvgen to travel the one-meter distance between a humanlung and hand or foot (Schmidt-Nielsen 1900). Clearly exercise (oranything else) would he out of thequestion under these circum-stances.

Diffusion V ersusConvectionA confusing aspect of diffusion is

the concept of movement of thediffusing substance from an area ofhigh concentration to one of lowconcentration. Many teachers try todemonstrate this phenomenon byopening a bottle of perfume at thefront of the room and asking stu-dents to raise their hands when thesmell reaches them. Unfortunately,as Vogel (1988, 1004) points out,what is being demonstrated is amixture of diffusion and convec-tion. Convection is the movementof air within the room due to vari-ations in density, which are in turndue to small differences in air tem-perature. The air in a room may

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feel like still air, but the presence ofwarm bodies alone is enough to setup small currents, strong enoughto move lightweight gas moleculesabout the room. Of course the nor-mal body movements of studentsand teachers alike, whether fidget-ing, note-taking, or simply inhalingand exhaling, help speed the mol-ecules on their way. So, althoughthe volatile components of the per-fume evaporate into and then dif-fuse throughout the air, convectioncarries these molecules to the nosesof the students much faster thandiffusion alone can.

Nevertheless, if diffusion is arandom process, why does it seemto carry molecules from areas ofhigh concentration to low concen-tration? The reason is purely prob-abilistic. If you can imagine theopen bottle of perfume in abso-lutely still air (i.e. in the absence ofconvection), at any time there willhe more molecules of the diffusinggas near the perfume surface thanfar from it. However, when thesemolecules are near the surface,there are more directions leadingaway from it than toward it. Withmore possibilities for movementaway, more molecules (on average)will move away. The result is netmovement from the area of highconcentration to the area of lowconcentration. Vogel (1488, 1994)has suggested clear demonstra-tions of diffusion in the absence ofconvection.Benefits to Students

I find that the random walk ac-tivities described above allow stu-dents to explore a fundamental,but poorly understood topic by ap-pealing to their kinesthetic sensesfirst, and then challenging their an-alvtical skills as they try to deducethe mathematical principal in-volved. It also allows them to seefor themselves how a computer canintroduce an element of "time com-pression" into the mathematicalmodeling of a physical system,since the computer does essentially

Diffusion Simulation Programc l scall introdoclearcisrandomize tinterscreen 2clsprint "How many steps do you wantthe molecule to take on each trip?":

input stepsprint "Do you want to see the pathsof the molecules? (y/n)"dopathflagS=inkey$loop until pathtlag'#""cisline (350,30)-(600,170),,bfor j =1 to 2(1boundflag=0line (350,30)-(600,170),,bx =475y-100for i=I to steps

pause-inkevSit: pause$="p" then call pauserandomnum =int (rnd`8)if (pathflagS="y" then call pstselect case randomnumcase 0x=x -Icase 1v=-v-1case 2x-x+Icase 3y==y -sIcase 4x--x- IY- Y-Icase 5x1-x-1y =y-Icase 6x-x*1y -v- Icase 7X==x-Iv -=y+ Iend selectif (x,350) or (x>600) or (y,0)or (y>199) then call bynext ipset (x,v)call distavgdistance=av distance t distanceprint "Distance between points ";print using "###.###; distance: line(350,0)-(600,199),0,bfnext]avgdistance=avgdistance/20printprint "The average distance themolecules traveled isprint using "###.###";avgdistanceprint "DOI,JE!"

beepprint "Would you like to run the

simulation again? (v/n)"dorunagainS inkevSloop until runagain5-

loop until runagain5 "vendsub pausedoaS - inkevSloop until aSrod subsuh pstshared x,ypst (x,y)end subsub dirtshared x,v , xdist,rdist,distancexdist=475-xvdist-100-)distance-sgr(xdist 'ydist' vdist"vdist)end subsub by

;hared boundflagif boundflag 0 then beepIhound flag h 'u nd I lagend sub

sub introprint phis program simulates amolecule on a random walk, takingone step at"print "a time. The molecule will take

as many steps as you instruct it to,but the"print "direction of each step will berandomly determined by the computer,"print "For the number of steps chosen,the molecule will take 20 random walks."

print "At the end of each walk, thedistance between the start and end pointswill

print "be displayed. After all 20 walkshave been taken, the average distanceprint "traveled will be displayed."print ""Chen von will he asked if you want

to run the sin elation again."printprint "You may pause the simulation at

any time by pressing 'p'."printprintprint " Press any key to continue."doa$==inkevSloop untilend sub

Figure 2. Computer program in BASIC that models diffusion as the random walkof a molecule. The simulation is similar to the random walk that students take, butthe simulated molecules can move in any of eight directions ( the four compassdirections plus the four intermediate directions) instead of the four directionsstudents take.

what they themselves did. Finally,it helps them appreciate the subtlesophisttion of living machineswhen they realize that diffusion

alone cannot always get a physio-logical job done, but more elabo-rate transport systems may be re-quired.

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AcknowledgmentsThanks are due to my friend and

former teacher, Steve Vogel, for intro-ducing me to the subject of biomechan-ics in general and diffusion in particu-lar. I also thank my son, AndrewHebrank, for taking a brief outline of aprogram that would simulate a ran-dom walk of a molecule, and turning itinto an elegant and effective computermodel of diffusion.

ReferencesMarek, E.A., Cowan, C.C. & Cavallo, A.M.L.

(1994). Students' misconceptions aboutdiffusion: How can they be eliminated?I he A tncric an Biology Teacher, 65, 74-77.

Schmidt-Nielsen, K. (1990). Animal Phys-iology: Adaptation and Etrrironrnent, 4thed. New York: Cambridge UniversityPress.

Vogel, S. (1988). Life's Devices: The PhysicalWorld of A nimals and Plants. Princeton,Ni: Princeton University Press.Vogel, S. (1992). Vital Circuits: On Pumps,Pipes, and the Workings of CirculatorySustrrrls. New York: Oxford UniversityPress.Vogel, S. (1994). Dealing honestly withdiffusion. The American Biology Teacher,56, 405-407.

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