Osmosis: Diffusive or Something Else?

[McCoy13]

So I've been trying to cook up a homework assignment for a class that guides them through deriving osmotic flow from diffusion. I've been working on it for a month, and running into all kinds of stumbling blocks, so I thought I'd ask the simple question here: is osmotic flow fundamentally diffusive or fundamentally something different? Rather than lay out my own arguments now (which I'm a little tired of going over in my head), I'd like to respond to arguments perhaps presented by someone who is a better authority on the matter than myself (I'm just a physics undergrad who hasn't taken biology courses, though has read quite a bit about osmosis at this point) and get some dialogue going.

Thanks.

P.S. For the record, I am not doing a homework, I am designing a homework for my job.

 

[Fewmet]

I teach high school chemistry and physics. Students acquire a conception from bio classes that there is some sort of natural force driving diffusion. For those that have also been through physics, I nudge them to question the origin of such a force. After some struggle, they usually realize there is no repulsion specifically causing molecules of similar composition to separate and work out the random nature of it.

It that what you are wanting your students to discover? I can see how that would be hard to do in a homework assignment because you are not present to lead them away from the misconceptions they hold.

How about giving qualitative arguments for both sides and asking them to assess the validity of each? Would that address your goal?

 

[McCoy13]

Teaching diffusion will actually be done separately from this homework. When the students receive this homework, they will hopefully already have a good understanding of how diffusion arises from Brownian motion including Fick's law.

My question is more along the lines of how to appropriately apply to diffusion to arrive at the phenomenon of osmosis. Consider for example two compartments of equal volume containing equal amounts of water separated by a semipermeable membrane. In compartment A let there be double the solute as in compartment B. We expect osmotic flow from B into A. However, Fick's law frames the flow of water along its concentration gradient, and since the compartments are equal volume with equal amounts of water, there is no concentration gradient. I've tried out arguments using partial volume (which I think has lots of problems) and omitting some amount of water in each compartment as "bound" to the solute, but both of these are some what wishywashy, handwaving explanations.

Now it's possible I'm making some elementary mistake in the above scenario, but I've thought about many many scenarios that don't necessarily hinge on having equal volume and equal amounts of water initially, and I've still encountered difficulty kind of just getting diffusion to make osmosis "go" as it were.

 

[McCoy13]

Furthermore, the most convincingly rigorous mathematical derivation I've seen started with (phenomenological) osmotic pressure and hydrostatic pressure and analyzed the pressure gradients at a pore in the membrane and derived the flow as a result of the extra thermal energy carried by the solute (if I remember correctly, I can check again tomorrow). However, I don't know if this is sort of the fundamental underpinning of the phenomenon or if it's a case of one derivation vs. another (ala matrix mechanics vs. partial diff eq derivations of quantum mechanics).

 

[Pythagorean]

If you have double the solute in one compartment I'd think you'd have double the concentration.

 

[Fewmet]

I understand what you are looking for. At the level I teach I usually go for a more qualitative approach.

Let me ask a fundamental question. You are starting with A and B being of equal volume and assert there are equal numbers of water molecules in A and B. Is that really true if A has more solute in it? I can see that you would get a slightly higher density of water around each solute particle...can it be shown that decrease in water volume necessarily equals the extra solute volume?

In other words, suppose you start with 1 mole of water in A and B and add 0.2 mol solute to A and 0.1 mol solute to B. Will the volumes of A and B still be the same? If not, doesn't each unit volume of A have less water in ti that the same volume in B?

(It's about 1 am here. I'll not be checking for a response for about 5 hours.)

 

[McCoy13]

I understand what you are looking for. At the level I teach I usually go for a more qualitative approach.

Let me ask a fundamental question. You are starting with A and B being of equal volume and assert there are equal numbers of water molecules in A and B. Is that really true if A has more solute in it? I can see that you would get a slightly higher density of water around each solute particle...can it be shown that decrease in water volume necessarily equals the extra solute volume?

In other words, suppose you start with 1 mole of water in A and B and add 0.2 mol solute to A and 0.1 mol solute to B. Will the volumes of A and B still be the same? If not, doesn't each unit volume of A have less water in ti that the same volume in B?

(It's about 1 am here. I'll not be checking for a response for about 5 hours.)

So this is essentially the argument I have resorted to already. However, the professor I'm working with believes that that we should be able to present it in a way that does not invoke the electric forces between water and solute.

A quick counter-question about such an explanation. Osmosis is a colligative phenomenon, yes? So in theory it should not matter if we have a binary salt that ionizes in solution with +2 and -2 charge or a salt that ionizes with +1 and -1 charge, right? But if the charge is stronger, the electric force between ions and water molecules should be stronger, so the local density change around ions should be more extreme, making the osmotic effect more extreme, but this is a contradiction with the assumption that osmosis is colligative.

The merit of such a volume argument is obvious, but it's not quite at the level of rigor my boss would want. He wants me to come up with a model that is a bit more simple minded, that preferably doesn't involve intermolecular interactions (e.g. electric force or collision). Of course, my whole belief at this point is that such cannot be done. A simple counterexample would be that ignoring electric forces, there is nothing special about having solute in water compared to having bulk material in water, which obviously does not generate osmosis. Similarly, if you replace the solute with oil, you'll observe hydrophobicity rather than osmosis. I just want to kind of double check with others before I try to explain to my boss again that osmosis is more than just diffusion with two species of particle.

P.S. I'll also consider this more tomorrow and see if I can work out some more math behind it to build a complete picture.

 

[Mapes]

is osmotic flow fundamentally diffusive or fundamentally something different?

Osmosis and diffusion both arise from minimization of total free energy through random motion. They are both mechanisms whereby gradients in chemical potential are eliminated, if possible, via thermal motion of atoms or molecules. Osmosis is a special case in which a barrier blocks the transfer of one or more components. Diffusion is a special case in which the enthalpy of mixing is exothermic or not too endothermic (equivalently, the activity of each component increases monotonically with concentration). (If this requirement isn't met, components don't mix but rather phase separate like oil and water.)

This is a graduate level definition. At the undergraduate level, I don't think anyone would object to defining and modeling osmosis as "diffusion through a semipermeable membrane."

 

[Fewmet]

A quick counter-question about such an explanation. Osmosis is a colligative phenomenon, yes? So in theory it should not matter if we have a binary salt that ionizes in solution with +2 and -2 charge or a salt that ionizes with +1 and -1 charge, right? But if the charge is stronger, the electric force between ions and water molecules should be stronger, so the local density change around ions should be more extreme, making the osmotic effect more extreme, but this is a contradiction with the assumption that osmosis is colligative.

How interesting...I hadn't known (or had forgotten) that osmosis is colligative. Could it be that osmosis is colligative to a good approximation? Looking around a little online, I see that osmotic pressure is colligative. It seems like that means that the phenomenon of diffusion is colligative, but I'm not taking the time to think though it at the moment.

This is just speculation: osmotic pressure is http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch15/colligative.php#osmotic". That strongly implies that intermolecular attractions come into play in temperature.

Different speculation: can you address your problem by arguing that the solvent molecules in A (where the concentration of the solute is greater) have fewer free paths that take them to the semipermeable barrier than the solvent molecules in B? That gives more frequent "effective collisions" for the solvent molecules in B.

 

[McCoy13]

So I've been thinking about your suggestion from before and ran through some math just now.

Let compartment A and B contain a volume of water V corresponding with some amount of moles $N_{w}$. Let some moles of solute $N_{s}$ be added to compartment A and $2N_{s}$ be added to compartment B. We hypothesize that the addition of such solute will generate a volume change, and since the mechanism of this volume change should be identical, let compartment A undergo volume change $\Delta V$ and compartment B $2 \Delta V$. Let's examine the concentrations of water to see how the diffusion will work.

$$C_{A}=\frac{N_{w}}{V+\Delta V} > C_{B}=\frac{N_{w}}{V+2\Delta V}$$

Therefore we expect diffusive flow from compartment A to compartment B. This agrees with what we expect in osmotic flow. Suppose $\delta$ amount of water flows along the concentration gradient.

$$\frac{N_{w}-\delta}{V+\Delta V}=\frac{N_{w}+\delta}{V+2\Delta V}$$

$$\delta=\frac{N_{w}\Delta V}{2V+3\Delta V}$$

This looks okay, and the dependencies all seem to make sense. However, let's quickly consider where the volume change $\Delta V$ comes from. In general, we expect polar water molecules to be attracted to polar or ionic solute, forming pockets of high density around the solute, as you suggested. However, this means that in fact $\Delta V$ should be negative, so $\delta$ is negative, and the flow goes from compartment B to compartment A (high solute concentration to low solute concentration). Furthermore, (according to Wikipedia, granted, not a very scholarly source) some solutions will expand while others will contract when solute is added, therefore in general we cannot predict the sign of the volume change a priori, and osmosis should not depend on it. We might alternatively propose a simple displacement mechanism for generating the volume change (ignoring the claim from Wikipedia), but this contradicts the colligative nature of osmosis, as displacement should be linked to molecule size.

 

[McCoy13]

How interesting...I hadn't known (or had forgotten) that osmosis is colligative. Could it be that osmosis is colligative to a good approximation? Looking around a little online, I see that osmotic pressure is colligative. It seems like that means that the phenomenon of diffusion is colligative, but I'm not taking the time to think though it at the moment.

It seems to me that if osmotic pressure is colligative, and we characterize sort of the "strength" of osmotic flow by the osmotic pressure, that the osmotic phenomenon in general is colligative. That is to say, if dissociating ions of charge 2 generates the same osmotic pressure as ions of charge 1, then the osmotic flows should not be any different. Nor should the flow depend on the volume displacement or mass of the particles.

This is just speculation: osmotic pressure is http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch15/colligative.php#osmotic". That strongly implies that intermolecular attractions come into play in temperature.

I agree, though one could perhaps counter-argue that this is because diffusion occurs more quickly at higher temperature due to increased thermal motion if one doesn't accept that the intermolecular forces are relevant to begin with. Again, I think they are, but I need to build the case that they are relevant to a diffusion-based model to the professor.

Different speculation: can you address your problem by arguing that the solvent molecules in A (where the concentration of the solute is greater) have fewer free paths that take them to the semipermeable barrier than the solvent molecules in B? That gives more frequent "effective collisions" for the solvent molecules in B.

Perhaps, but again I don't see how one can then avoid the size of the solute coming into play. Furthermore, I could see a student arguing that such an argument should work equally well in reverse, that the greater concentration of solute in A should interfere with solvent moving into A rather than out of A, and honestly I'm not sure how I'd counter such an argument.

 

[McCoy13]

At the undergraduate level, I don't think anyone would object to defining and modeling osmosis as "diffusion through a semipermeable membrane."

Part of the challenge of this design problem is the who I'm designing for. There is a great deal of need for physical accessibility, a sort of intuitive tangibility. I personally object to a claim that osmosis is diffusion through a semipermeable membrane because I think it's misleading in terms of how students will be thinking about diffusion. Diffusion is usually taught using an ideal gas model, which stipulates that particles don't interact attractively and don't even have to scatter, and I believe this is how we will be teaching it. Therefore, such interactions, which I believe are critical to making osmosis "go" so to speak, will fall outside of our students' diffusion model and will have to be incorporated into osmosis either from electricity (and I'm not sure how much time we're spending on electricity in this course) or by axiom (which I'd like to avoid).

 

[Fewmet]

A minor point:

$$C_{A}=\frac{N_{w}}{V+\Delta V} > C_{B}=\frac{N_{w}}{V+2\Delta V}$$

Your original premise was that A has a greater concentration of solute. If adding the solute increases the volume of A more than B, doesn't the concentration of water in A decrease more than it does in B? That would reverse your equality, but I don't see how that invalidates anything that follows.

All else here seems sound.

 

[McCoy13]

Sorry, in this example B has twice the solute as A. But as my last few posts indicate, I still have unresolved difficulties, though the math works out to the osmotic flow we expect.

 

[Fewmet]

I brainstormed this with my high school students. We came up with another potential approach.

Focus on A and B having the same average temperature. Since A has more solute, the solute particles has a greater proportion of the thermal energy than in B. Conversely, the water molecules in B will have a greater portion of the thermal energy than the water molecules in A. (This ignores any change in temperature due to the solution process). That means the water molecules in B have a higher average velocity and will have more frequent collisions with the membrane between A and B.

(I want to acknowledge that I am just throwing out possibilities that might be helpful. You're doing the hard work of thinking though the consequences. If this is not seeming fruitful to you, feel free to ignore the suggested mechanism).

 

[McCoy13]

This seems like a possible answer. I'm going to reference back to the text I was using, which seemed to suggest this to see if I can tease apart its argument into some as simple as what you suggest. Thank you very much for your help.

 

[Mapes]

In general, we expect polar water molecules to be attracted to polar or ionic solute, forming pockets of high density around the solute, as you suggested. However, this means that in fact $\Delta V$ should be negative, so $\delta$ is negative, and the flow goes from compartment B to compartment A (high solute concentration to low solute concentration).

No, it won't. Now that you've assumed some affinity between the water and the solute, you can't still use water concentration alone to predict the direction of net water flow. In dilute solutions, the solvent (here, water) follows Raoultian, or ideal, behavior; its activity coefficient is one, independent of the solute's activity concentration. That's why osmosis is colligative in dilute solutions.

 

[McCoy13]

No, it won't. Now that you've assumed some affinity between the water and the solute, you can't still use water concentration alone to predict the direction of net water flow. In dilute solutions, the solvent (here, water) follows Raoultian, or ideal, behavior; its activity coefficient is one, independent of the solute's activity concentration. That's why osmosis is colligative in dilute solutions.

Okay, but do you agree with the explanation presented before that the solute generates a volume change $\Delta V$ and that this volume change is responsible for the change in water concentration? If so, how do you suggest this volume change comes about?

 

[Mapes]

Okay, but do you agree with the explanation presented before that the solute generates a volume change $\Delta V$ and that this volume change is responsible for the change in water concentration? If so, how do you suggest this volume change comes about?

Sure, I'm on board with a negative volume of mixing. The problem I have is with using water concentration alone as a predictor of diffusion, since you've now added the assumption that the water has an affinity for the solute that exists on one side only and can't pass through the barrier. This affinity is going to suppress water diffusion away from the solute. In the end, the water still moves to where its chemical potential is lower, which is the side containing solute and a lower molar concentration of water.

EDIT: Corrected "solute-free."

 

[McCoy13]

Sure, I'm on board with a negative volume of mixing. The problem I have is with using water concentration alone as a predictor of diffusion, since you've now added the assumption that the water has an affinity for the solute that exists on one side only and can't pass through the barrier. This affinity is going to suppress water diffusion away from the solute. In the end, the water still moves to where its chemical potential is lower, which is the solute-free side.

Do you mean the solute side? It seems like you made contradictory statements. The affinity between water and solute will suppress diffusion away from the solute, but the water will flow to the solute-free side? It seems to me if the solute and water have affinity, then the side with solute would have lower chemical potential.

 

[Mapes]

Whoops; yes, I meant the solute side.

 

[McCoy13]

Okay, thank you both. I think I feel prepared to make an argument to my professor/boss now.

 

[DavidMcC]

McCoy13: (according to Wikipedia, granted, not a very scholarly source)

If you check the page notes on Wikipedia, you should find that it often is scholarly these days. At first, it was notoriously un-scholarly, but that has changed, and any dubious pages are flagged by Wikipedia themselves.