Idle osmusings

vivianimbriotis | Nov. 17, 2019, 1:42 p.m.

I think I first learned about osmosis in grade seven. We cut up potatoes into long, identical strips, and placed them into three solutions - one of pure water, one with some salt, and one with a lot of salt. The first potato grew, the second stayed the same size, and the last shrank and became sad and floppy. Osmosis.

This happens whenever there is a membrane that is permeable to water, but not permeable to something that’s dissolved in the water. When you dissolve something (like salt) into something else (like water), the former is called a solute, the latter a solvent. (Actually, osmosis doesn’t require that water be the solvent). Cell membranes are highly permeable to water, but almost impermeable to charged particles like the ions of a dissolved salt.

“The water wants to equalise its concentration gradient,” my teacher at the time said, “And if the water is diluted by a solute, the water will diffuse down its concentration gradient. The lowest energy state is when there’s an equal number of water molecules on both sides of the membrane.” Well, I thought, water molecules don’t really want anything, per se, but I guess that makes sense - chemicals tend to diffuse around until they equalise their concentration.

But as I slowly learned more physics and chemistry, this explanation confused me more and more. Diffusion is just the result of random movements of particles. If you begin with a lot of particles in one space, and they move around randomly, they’ll rapidly spread out through that space:

But something is...off, here. I notice I am confused.

Diffusion is about particles in regions of high density ending up in regions of lower density, because if there’s lots of particles all in one place they’ll rapidly bounce off each other until they escape into regions with fewer particles. But in osmosis, you add particles (solute) to a region, and...more water molecules flow into that region. Maybe I’m just not imaginative enough to grasp the whole “concentration of water” thing. But check this out. Osmosis is described by the van ‘t Hoff law:

Π = icRT

Where Π is the osmotic pressure, i is the van ‘t Hoff index, c is the difference is solute concentration across the membrane, R is the ideal gas constant (8.314 joules per kilogram per mole), and T is the temperature of the solution. Most interesting are all the things that don’t matter - the charge or mass of the solute particles, for example. Note also that the concentration of water doesn’t matter at all!!

Wait, what is a "van 't Hoff index"? Well, say you dissolve 1 molecule of ammonia into water. You will get 1 molecule of ammonia floating around in the water, so the van 't Hoff index of ammonia is 1. If you dissolve one unit of table salt NaCl (one Na+ molecule and one Cl- molecule) into water, you get two molecules in the water, so the van 't Hoff index of NaCl is about 2. So how can we explain the van 't Hoff equation?

Let’s set up a thought experiment. Imagine a sphere of our semipermeable membrane. Outside the sphere is pure water. Inside the sphere is a mixture of water and salt. The molecules of water freely move through the spherical semipermeable membrane. The salt is moving around randomly within the water, and occasionally bouncing off the walls of the sphere, exerting a pressure on it.

This kind of behaviour - molecules freely travelling through a volume, interacting little with one another, and ultimately colliding with and exerting pressure on their container - recalls sharply the idea of an ideal gas. Ideal gases behave according to the ideal gas law:

PV=nRT

Which is to say, the pressure per unit volume exerted by a gas is proportional to the temperature of that gas and the number of particles present (where R is the coefficient of proportionality, the ideal gas constant). This is another thing I never really understood until I took a deep dive into this paper on a kinetic derivation of the law.

The features of an ideal gas are that particles move randomly, that the space between them is much larger than their size, that they collide elastically, and that the forces between particles are negligible. These all seem pretty close to the behavior of solute in a sufficiently dilute solution. The main difference in the case of a dissolved solute is that we don’t know the number of molecules of solute, we instead know their concentration in a given volume of solvent. But playing with the ideal gas law:

PV = nRT

P = nRT/V

P = (n/V)RT

Here, the (n/V) term represents the number of particles per unit volume, which is exactly the concentration (times the van ‘t Hoff index to account for dissociation, as mentioned earlier):

P = (ic)RT

And this is exactly the van ‘t Hoff equation! So maybe I am justified in imagining solute molecules bombarding the inside of the sphere and forcing it outwards.

This is the kind of account presented in a 1917 paper by Alfred W Porter, and also one of the interpretations of osmosis discussed by M G Bower here. It probably doesn’t hold a candle to the more sophisticated thermodynamic explanations involving the chemical potential, but it’s a lot easier to understand, and predicts the van ‘t Hoff equation nicely.

Wait, one problem! So far I’ve talked about a force from the solute particles acting on the membrane, but in my examples of osmosis the membrane is fixed in place!

Well, Newton’s third law tells us that if a body A exerts a force F on body B, then body B exerts a force -F back on body A - that is to say, after colliding with the membrane, all the solute molecules bounce off and travel away from the membrane. But those solute particles are only going to travel a tiny distance before colliding with lots of water molecules and transferring their net momentum to them, so really our membrane can be thought of as exerting a pressure on the entirety of the solute-water mixture, proportional to the concentration of solute! An osmotic pressure.

I’m not arguing that this is the actual mechanism of osmosis down in the thermodynamic weeds, but I think it has a lot more explanatory power than “water following its concentration gradient” (it predicts the van ‘t Hoff equation, for one), and I don’t think it’s more difficult to understand, either - it's old-school reification of particles as ping-pong balls. So if we have to teach biologists and medical professionals one of the two, I’d prefer the kinetic explanation.