# Osmosis and entropy/free energy and mixing.

## Main Question or Discussion Point

I sometimes have trouble with the concept of entropy being a driving force for work being done. For example, during osmosis you can raise the level of water on one side of a semipermeable membrane (i.e. do work) because the free energy of the system is minimized if there is a flow of solvent from high to low concentration. Or consider mixing two ideal gases, which results in a negative change in Gibbs Energy. I get that mixing two ideal gases is spontaneous, but I don't get how work is done because gas A and gas B are now together instead of apart.

This just doesn't make sense to me. We are talking strictly about entropy and free energy here, so don't take into account intermolecular interactions. If I have a two bags of marbles and then I combine them, I haven't changed the energy of the marbles. I can't do additional work because the marbles are now together. The marbles, or atoms or whatever don't know what is around them.

The way I understand entropy is statistical. I realize that it is not statistically likely to throw my two bags of marbles on the ground together and have them stay completely separate in neat little piles. But if for some odd chance they did, I don't see how their free energy would change. Each marble would still have the same energy regardless of whether they mixed or not.

I just can't wrap my mind around this statistical expression that is entropy leading to work being done.

## Answers and Replies

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Andrew Mason
Homework Helper
I sometimes have trouble with the concept of entropy being a driving force for work being done.
It isn't.

Entropy is not a driving force of anything. It is just a useful mathematical tool for analysing thermodynamic processes.
I get that mixing two ideal gases is spontaneous, but I don't get how work is done because gas A and gas B are now together instead of apart.
Work is not done. It is just that the reversible processes between the two states would involve work being done - if the gases are different. For example, suppose you have a 44.8 L container with a divider in the middle. There is 1 mole of He on one side and 1 mole of Ne on the other at STP. When you remove the divider, the He expands to occupy the whole container and so does the Ne. Total P is unchanged and T is unchanged. The He doubles its volume and halves its pressure, as does the Ne.

No work is done. No heat flows. There is no change in internal energy.

However, in order to calculate the change in entropy you have to determine the reversible heat flow, Qrev, in moving between the initial and final states for each gas. The reversible process for each gas is an isothermal quasi-static expansion, which necessarily involves heat flow and work. Since ΔU=0, Q = W = ∫PdV = RTln(Vf/Vi). ΔS = ∫dQ/T along this isothermal path = Q/T = Rln(Vf/Vi) = Rln(2). So the total change in entropy is 2Rln(2)

If the gases are the same, there is no change when the divider is removed because the atoms are indistinguishable. Since entropy is a state function and there is no change in state, there is no change in entropy if the mixing occurs between two identical gases in the same states.

It seems odd that entropy should depend on whether the mixing occurs between different types of atoms. This is known as Gibbs' Paradox and is the subject of much theoretical discussion, so don't feel bad if you are having trouble with it. It is just a matter of the way entropy is defined.

The way I understand entropy is statistical. I realize that it is not statistically likely to throw my two bags of marbles on the ground together and have them stay completely separate in neat little piles. But if for some odd chance they did, I don't see how their free energy would change. Each marble would still have the same energy regardless of whether they mixed or not.
Quite right. But entropy is not energy. The entropy would increase as they mixed (ie. it would be lower in the unmixed state). But this does not really have much to do with energy.
I just can't wrap my mind around this statistical expression that is entropy leading to work being done.
Refer to the calculation above. Entropy is a measure of ∫dQ/T in a reversible process between the beginning and end states.

In the mixing process no work is done and no heat flow occurs. But mixing of two different gases is not a reversible process. To calculate the entropy change between the unmixed and mixed states, you have to imagine a reversible process occurring (in which work IS done and heat flow DOES occur). However, if both gases are identical the mixing of the two sides IS perfectly reversible. There is no heatflow along that reversible path so there is no change in entropy.

AM

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Ok what you are saying makes sense. But why does osmosis happen then? Is there a reason other than entropy/the chemical potential (i.e. free energy) being minimized when the solutions are at more similar concentrations?

If osmosis can cause the water level on one side of a semipermeable membrane to rise against gravity then work is being done. So why does it happen? I have only heard explanations that hinge on entropy and free energy.

I guess it sort of makes sense if I try to justify it with chemical potential tending to be minimized. If I get over my intuition that gravity tends to pull things down and realize that if there are other forces in effect here (like the tendency to minimize chemical potential) then it could stand to reason that the water level could rise. It's just hard to do this when gravity is something I am familiar with and can observe, and chemical potential is just this nebulous concept to me that someone just happened to tell me tends to minimize itself. And it is hard to imagine this "winning" over gravity.

To me, the fact that osmosis can raise the level of water because chemical potential is minimized is equivalent to telling me that different marbles can spontaneously roll uphill if it allows them to mix at the top.

Although, if the marbles are rolling around fast enough at the bottom (and thus contain the necessary energy to roll up the hill) then they would statistically tend to mix rather than stay separate. Maybe this is analogous to the temperature term in the dG=dH-TdS equation?

I'm just kind of rambling now, thanks for the help though.

Entropy is not a "force", but it certainly is the "reason" why osmosis occurs. Suppose you have a membrane impermeable to solute partitioning a cylinder. And suppose on one side of the membrane you have .01 M NaCl solution and on the other side, you have a .03 M NaCl solution. Water will rush across the membrane to try to equalize the concentration on both sides of the cylinder. This is purely an entropically driven process. It spontaneously occurs. Processes like this can be used to do work, and many proteins in your cells use concentration gradients to drive molecular machines. ATP synthase in your mitochondria uses an H+ concentration gradient and when H+ moves across the gradient, the protein changes conformation and the energy change from the conformation is used to form ATP.

Entropy is not a "force", but it certainly is the "reason" why osmosis occurs. Suppose you have a membrane impermeable to solute partitioning a cylinder. And suppose on one side of the membrane you have .01 M NaCl solution and on the other side, you have a .03 M NaCl solution. Water will rush across the membrane to try to equalize the concentration on both sides of the cylinder. This is purely an entropically driven process. It spontaneously occurs. Processes like this can be used to do work, and many proteins in your cells use concentration gradients to drive molecular machines. ATP synthase in your mitochondria uses an H+ concentration gradient and when H+ moves across the gradient, the protein changes conformation and the energy change from the conformation is used to form ATP.
This is what I don't understand then. It can be used to do work, but it is not a force? I understand what osmosis is, I just don't understand why it can be used to do work. If this process was occurring in space with no gravity and in a vaccuum with no pressure pushing down on the water then I would say fine, the water can rise. But why would water "try" to equalize the concentration on both sides when there are forces like gravity and air pressure working against it?

Again, refer to my marbles example. To me this is all like saying marbles will spontaneously roll up hill if they can mix at the top. Yeah I get that marbles put together will normally mix, but now you have a hill in the way.

Andy Resnick
This is what I don't understand then. It can be used to do work, but it is not a force? I understand what osmosis is, I just don't understand why it can be used to do work. If this process was occurring in space with no gravity and in a vaccuum with no pressure pushing down on the water then I would say fine, the water can rise. But why would water "try" to equalize the concentration on both sides when there are forces like gravity and air pressure working against it?

Again, refer to my marbles example. To me this is all like saying marbles will spontaneously roll up hill if they can mix at the top. Yeah I get that marbles put together will normally mix, but now you have a hill in the way.
Osmosis will proceed in the absence of gravity (or when gravitational effects are unimportant- cells will swell (to bursting sometimes) when placed in a hypotonic solution and shrink when placed in a hypertonic solution. The work is 'P-V work', just the same as P-V work required to raise a column of water against gravity.

Your analogy of marbles mixing is interesting (in the absence of gravity)- entropy is maximized when the available volume for each marble is maximized, just as it is for non-interacting gas molecules. It gets interesting for colloids- maximizing the entropy can be used to drive formation of colloidal crystals in a high volume-fraction melt:

http://www.sciencemag.org/content/309/5738/1198.citation

Osmosis will proceed in the absence of gravity (or when gravitational effects are unimportant- cells will swell (to bursting sometimes) when placed in a hypotonic solution and shrink when placed in a hypertonic solution. The work is 'P-V work', just the same as P-V work required to raise a column of water against gravity.
So how can an entropic process do P-V work? This is what I don't understand. It makes sense to me that in the absence of gravity, the column could raise on one side. But add in the fact that now you have to do work to maximize entropy, why doesn't it just stop the whole process?

Entropy has been explained to me as a statistical process. If you put two gases together, it would be very odd for them to, despite their random motion, each stay in one side of the container. But now we are talking about the competition between a force acting against mixing (something the gases don't experience), and this concept that I don't quite understand called entropy which I have explained my interpretation of above.

As I understand it, entropy is not a force, it is just a word we have invented to explain spontaneous changes.

Is it that the water molecules have enough kinetic energy to do PV work or something? Thus leading to the change in osmotic pressure as temperature is increased?

Back to the marble analogy. Lets say that there are two sets of marbles separated by a hill. It takes a marble rolling around (on this frictionless surface) at 5 km/h to go over the hill. Thus if all the marbles are rolling around at 4km/h, no mixing will happen. But if they are all rolling around at 6km/h, then they will mix despite the barrier of the hill. And if some of the marbles are rolling around at 4km/h and some at 6km/h, then some will mix and some won't.

So in osmosis, moving water over to the other side of the membrane and raising the water level takes energy, but some of the water has enough energy to do this. Therefore it can do work to raise the column of water on one side because statistically, it wouldn't make sense for it not to mix if it has the energy to do so, just like it wouldn't make sense for all of a gas to stay on one side of a container. By varying different parameters like pressure and temperature, you can control how much water has enough energy to do work to raise the level on one side.

I've been thinking a lot about this and this is the only explanation I can rationalize. Let me know if I am on the right track.

EDIT: If the above is true, is it valid to also explain it in terms of brownwian motion, i.e. the particles are all moving around randomly? Since one side has more solute, there are less solvent particles moving in the direction of the less concentrated side. So at osmotic equilibrium, one side would have to be higher so that the solvent in the more concentrated side had a reason to flow over the other less concentrated side at the same rate. Yes?

Does this mean that if you have the system I have been talking about, you can never actually achieve equal concentration on both sides if the number of solute particles is different on each side?

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Andy Resnick
There's a lot in your post, let me try to untangle some of it:

First, entropy (or more accurately, the *change* in entropy associated with a specific process) does not determine if a process will spontaneously occur- it tells you if a process is irreversible or not. The Gibbs free energy (or more accurately, the *change* in the Gibbs free energy etc.) tells you if the process is spontaneous or not. A process can be spontaneous and irreversible, non-spontaneous and reversible, or any other combination.

For gases that are allowed to mix, there's also the 'Gibbs paradox'- there appears to be a difference in the change in entropy if two different gases are allowed to mix while there is not if the gases are identical. This paradox can be resolved (most clearly by Jaynes) by accounting for information: how is it that we know the gases are different?

Now for the marble analogy: your idea of the two populations possessing unequal amounts of kinetic energy can (probably) be used to model a semipermeable membrane, and hence osmosis. Something to realize is that when water moves in response to an osmotic pressure gradient, the chemical potential energy is being converted into another form: gravitational potential energy, or an increase in the pressure on one side of the membrane (Donnan equilibrium)- and the entropy *increases* because this process is irreversible.

In the case of marbles, the population that can cross the potential barrier will do so and the entropy will increase* as well because there is more available volume for each marble. Now, I've put an asterisk there because marbles have a finite size and so we must account for excluded volume effects- packing in more marbles on one side results in a decrease in the available volume per marble on that side. This is the origin of entropically-driven colloidal crystallization: the volume per particle is larger in a crystalline array than it is for a melted region.

Does this help? I found a good reference in "Water movement through lipid bilayers, proes, and plasma membranes: theory and reality" by Alan Finkelstein (Wiley & Sons, 1987).

[Entropy] is a mathematical tool...

Not really; it is a physical measure of increasing dispersion of energy.

Every child* soon learns the second law of thermodynamics:

There was a man in our town,
And he was wondrous wise;
He jumped into a bramble bush
And scratched out both his eyes.

And when he found his eyes were out,
With all his might and main,
He jumped back into another bush
And scratched them in again!

or as Lewis Caroll put it:

Humpty-Dumpty sat on a wall,
Humpty-Dumpty had a great fall;
All the King's horses and all the King's men
Cannot put Humpty together again.

* As soon as he gets to University that knowledge will
be cut out from his head with the help of "a maths tool".

I think you are right.

I sometimes have trouble with the concept of entropy being a driving force for work being done. For example, during osmosis you can raise the level of water on one side of a semipermeable membrane (i.e. do work) because the free energy of the system is minimized if there is a flow of solvent from high to low concentration. Or consider mixing two ideal gases, which results in a negative change in Gibbs Energy. I get that mixing two ideal gases is spontaneous, but I don't get how work is done because gas A and gas B are now together instead of apart.

This just doesn't make sense to me. We are talking strictly about entropy and free energy here, so don't take into account intermolecular interactions. If I have a two bags of marbles and then I combine them, I haven't changed the energy of the marbles. I can't do additional work because the marbles are now together. The marbles, or atoms or whatever don't know what is around them.

The way I understand entropy is statistical. I realize that it is not statistically likely to throw my two bags of marbles on the ground together and have them stay completely separate in neat little piles. But if for some odd chance they did, I don't see how their free energy would change. Each marble would still have the same energy regardless of whether they mixed or not.

I just can't wrap my mind around this statistical expression that is entropy leading to work being done.
I think you are absolutely right. Entropy is not energy, but it's a drive, since osmosis has one only and and exclusive cause: entropy increase. It reflects the power of probability, the obligatory step-by-step to the more probable. The macroscopic behavior (that is, what we have access) as a manifest of the microscopic world democratic will. I know it's weird, but...

What drives osmosis?

I sometimes have trouble with the concept of entropy being a driving force for work being done. For example, during osmosis you can raise the level of water on one side of a semipermeable membrane (i.e. do work) because the free energy of the system is minimized if there is a flow of solvent from high to low concentration. Or consider mixing two ideal gases, which results in a negative change in Gibbs Energy. I get that mixing two ideal gases is spontaneous, but I don't get how work is done because gas A and gas B are now together instead of apart.

This just doesn't make sense to me. We are talking strictly about entropy and free energy here, so don't take into account intermolecular interactions. If I have a two bags of marbles and then I combine them, I haven't changed the energy of the marbles. I can't do additional work because the marbles are now together. The marbles, or atoms or whatever don't know what is around them.

The way I understand entropy is statistical. I realize that it is not statistically likely to throw my two bags of marbles on the ground together and have them stay completely separate in neat little piles. But if for some odd chance they did, I don't see how their free energy would change. Each marble would still have the same energy regardless of whether they mixed or not.

I just can't wrap my mind around this statistical expression that is entropy leading to work being done.
We are not alone....

"What drives osmosis?"
John Grant Watterson
Journal of Biological Physics, 21:1-9, 1995