Diffusion and Osmosis Simulator

What diffusion and osmosis are

Diffusion is the net movement of particles from where they are more concentrated to where they are less, driven by nothing but their own random motion. Osmosis is the net movement of water through a partially permeable membrane into the more concentrated solution. Fick’s first law gives the rate of diffusion, J = −D dc/dx, and van ’t Hoff’s law gives the osmotic pressure behind osmosis, Π = cRT.

This simulator puts both on one membrane. Let the solute through and it diffuses from the stronger side to the weaker until the two match, while the water, on balance, stays where it is. Hold the solute back and it cannot move, so the water does: it crosses into the stronger solution, the membrane slides to make room, and the two concentrations close from both ends. The only thing that changes between the two is the membrane, which is why it is the first control.

How to read the simulator

The tank is half a millimetre wide. The larger dots are the solute, each standing for 2 mmol/L on its side, and the small faint dots are water. Every one of them takes its own random step many times a second, so what you are watching is Brownian motion, sampled: each dot is one molecule followed out of the vast number a real tank would hold. The strip under the tank counts the solute in twenty slices, each bar straight below the slice it counts, so the concentration gradient in the picture and the gradient in the numbers are the same thing seen twice.

With the membrane letting water and solute through, the readouts put the concentration counted on each side beside what Fick’s law predicts, and the line over the strip is Fick’s second law at that moment. With it letting only water through, they give each side’s concentration and water potential and where the two will settle, the tank labels each side hypotonic or hypertonic, and the strip’s dashed lines are each side’s mean. Choosing a membrane restarts the run at a speed that suits it, 8x for osmosis and 2x for diffusion, because across this distance osmosis through a red blood cell’s membrane is several times slower than diffusion.

Fick’s law, and why the counts scatter about it

Fick’s first law says the flux of solute is proportional to the concentration gradient, J = −D dc/dx, with D the diffusion coefficient. Written for a membrane it becomes the form school biology uses: the rate of diffusion is proportional to the surface area times the concentration difference, divided by the thickness. Fick’s second law, dc/dt = D d²c/dx², says how a gradient flattens, and for a tank split down the middle it has an exact solution. The difference between the halves falls to half its starting value at t½ = 0.0492 L²/D, where L is the width of the tank.

Two things follow that are easy to get wrong. The half-way time does not depend on the size of the difference: 250 against 260 mmol/L and 0 against 500 mmol/L of sucrose both close half their gap in 23.6 s, because a bigger difference drives a proportionally bigger flow. And it goes as the square of the distance, so twice the width takes four times as long, which is why cells are small and why large organisms need a circulation to carry things any distance.

The counts do not sit on the curve, and they should not. Fick’s law is what random molecular motion averages to, in the way the exponential of the radioactive decay simulator is what random decays average to. With the 200 solute molecules the defaults draw, each side’s count scatters about the law by about 14 mmol/L once they have mixed, seven molecules either way, and the scatter shrinks only as the square root of the number counted. Press Restart twice and the two runs differ; the curve does not.

Worked example: 100 and 300 mmol/L of sucrose at 25 °C

These are the settings the simulator opens with. Van ’t Hoff’s law gives each side’s osmotic pressure, with 1 mmol/L equal to 1 mol/m³: on the left, Π = 100 × 8.314 × 298.15 = 247.9 kPa, and on the right 743.7 kPa. Water potential is minus that while nothing presses on either side, so the left is at −248 kPa and the right at −744 kPa, and water moves from the left to the right.

The membrane has the water permeability of a human red blood cell’s, 200 µm/s. The osmotic flow law, Jv = Pf Vw Δc, then gives a starting flow of 200 µm/s × 18.07 cm³/mol × 200 mmol/L = 0.72 µm/s, or 0.040 mol of water through each square metre every second. The membrane slides left at that speed, slowing as the two sides converge, and covers half its journey in 117 s, about 15 s on screen at 8x. It comes to rest a quarter of the way across, where the left side has shrunk to half its width and the right has grown by half, so both hold 200 mmol/L. Timing a membrane like this is a real measurement: the moving membrane method reads a lipid bilayer’s water permeability off the speed at which it moves between two solutions (Yano and colleagues, 2021).

Now switch the membrane to let water and solute through. The same starting solutions now even out by diffusion instead, with the membrane standing still, and Fick’s law has the halves at 150 and 250 mmol/L after 23.6 s. Sucrose diffuses with D = 520 µm²/s at 25 °C, so in one second a molecule wanders about 32 µm along each axis, √(2Dt), and a typical one takes about four minutes to wander the width of the tank.

Water potential, and the pressure that stops osmosis

Water potential measures how readily water leaves a place: Ψ = Ψs + Ψp, the solute potential plus the pressure potential, with pure water at atmospheric pressure set at zero. The solute potential is Ψs = −iCRT, minus the osmotic pressure, and water always moves from the higher water potential to the lower. AP Biology’s formula sheet writes it with R = 0.0831 L bar/(mol K) and the temperature as °C + 273, which gives −7.35 bar for 0.3 mol/L of sucrose at 22 °C. Set both sides to 300 mmol/L at 22 °C, so that nothing moves, and the simulator shows −736 kPa, the same to within about 0.1 percent. A-level biology usually quotes water potential in kilopascals, and plant physiologists in megapascals.

Set the left side to 0 for pure water and watch what nothing can balance. The water keeps crossing until the membrane meets its stop a fifth of the way across, after about 180 s. The right side is then at 187.5 mmol/L, and the stop holds 465 kPa: the pressure it takes to stop osmosis, which is what osmotic pressure means. That pressure appears as the right side’s pressure potential, bringing both water potentials to zero, and the water stops. It is large: 465 kPa would hold up a column of water about 47.5 m tall, and the 744 kPa of 300 mmol/L of sucrose about 76 m. Squeezing a solution harder than its osmotic pressure pushes water the other way, which is reverse osmosis, used to make drinking water from sea water.

Hypotonic, isotonic and hypertonic

The three words compare two solutions across a membrane by the solute it holds back. The hypotonic one has less and the hypertonic one more, so water moves from the hypotonic solution into the hypertonic one; isotonic solutions have the same and exchange no water on balance. A red blood cell in a hypotonic solution swells, and in a hypertonic one it shrinks, which is why a drip is made up with 0.9 percent sodium chloride, the concentration isotonic with it. A plant cell swells until its wall pushes back hard enough to stop the water, which is the same thing the stop does here.

A solute the membrane lets through does not count, however concentrated it is, because it evens itself out instead of drawing water. That is why the simulator gives no tonicity when its membrane lets the solute through, and it is the reason the serum osmolality calculator reports effective osmolality with urea left out: urea crosses cell membranes, so it raises osmolality without moving water. The kidney puts osmosis to work: it builds a salty, hypertonic medulla so that water leaves the urine wherever hormones have opened water channels, which the nephron explorer follows along the tubule.

Why temperature and the size of the molecule matter

The diffusion coefficient follows the Stokes-Einstein relation, D = kT/(6πηr): it rises with temperature, falls with the viscosity η of the water, and falls with the size r of the molecule. Urea, the smallest solute offered, diffuses 2.65 times as fast as sucrose, so it closes half the gap in 8.9 s where sucrose takes 23.6 s. Warming the water from 5 °C to 45 °C lowers its viscosity from 1.52 to 0.60 mPa s and raises every diffusion coefficient 2.9 times, taking sucrose’s half-way time from 43.2 s to 14.8 s.

Osmotic pressure responds far less, because it goes as the absolute temperature: the same warming raises it by only 14 percent. The random motion behind both is the same motion the kinetic theory gas simulator follows atom by atom in a gas, where molecules fly freely between collisions instead of jostling in a liquid.

Common mistakes

• Explaining osmosis as water diffusing down its own concentration gradient. It is driven by the membrane turning back the solute, not by the solution holding less water (Kramer and Myers, 2013), which is why the water dots here are drawn at one density on both sides.
• Thinking the molecules stop at equilibrium. They keep crossing in both directions at the same rate. What stops is the net movement.
• Thinking water moves only one way in osmosis. It crosses both ways all the time; osmosis is the imbalance between the two.
• Expecting a bigger difference to even out sooner. It flows faster, in proportion, so the half-way time is the same.
• Assuming osmosis always ends with equal concentrations. Only when nothing pushes back. A cell wall, or the stop here, ends it at a pressure instead.
• Calling a solution hypertonic because it holds more of a solute the membrane lets through. Tonicity only counts solutes the membrane holds back.
• Counting a salt once. Sodium chloride gives two particles, so 150 mmol/L of it is close to 300 mmol/L of particles, which is what osmosis responds to.

What this model leaves out

Real membranes are neither perfectly selective nor perfectly open. Most let a solute through slowly, and a solute that leaks through, as urea does across a cell membrane, draws water at first and then lets it return as it spreads. The two settings here are the two extremes, and most real membranes sit somewhere between them.

The solutions are ideal and dilute, so the osmotic pressure is exactly cRT, where real solutions depart from it as they get more concentrated, and every solute stays as whole molecules. When the water flow is worked out, each side is treated as well mixed, but in a real tank the solute piles up against the membrane on the shrinking side and thins out on the growing one, which slows osmosis a little. The dots here do pile up, and the strip under the tank shows it; the flow is worked out as though they did not. The water permeability is held at its set value at every temperature, the membrane slides freely rather than bulging, the tank lies flat so gravity plays no part, and the molecules neither attract nor repel each other. Charged solutes add a voltage to the story, which the resting membrane potential simulator takes up.