# Osmosis, osmotic pressure, vapour pressure

Why does osmosis happens?

Does osmosis happens when we have osmotic pressure difference between the two solutions (that are connected by a membrane)?

Does osmosis happens when we have two solutions of different concentrations?

My professor told that osmosis happens because of the vapour pressure difference between the solutions. But I didn't understand what vapour pressure has to do with osmosis. Can you explain me that, please?

I've been thinking too and is the vapour pressure of a solution numerically equal to the osmotic pressure of that solution? I see this relation because I know the more concentred is the solution the bigger is it's vapour pressure... that's right?

How do I calculate vapour pressure? (Can I use p V = n R T?)

Does density of the solutions has anything to do with osmosis? And what about hidrostatic pressure?

Thank you for the help,
Rafael Andreatta

Andy Resnick
Osmotic pressure gradients arise from spatial changes in concentration of a solute, and usually it's in the context of aqueous solutions. I suppose it can be related to a vapor pressure, but I don't see what the advantage is.

http://en.wikipedia.org/wiki/Osmotic_pressure

Osmotic pressure is indeed a pressure, and can balance a hydrostatic pressure head if, for example, a semipermeable membrane is used to separate the two solutions. There's a (IIRC) Norwegian company what is trying to extract usable energy by exploiting the osmotic pressure difference between salt and fresh water, but I don't see how it's economical.

Now, if you have multiple solutes and a membrane that can (as a suggestive example) distinguish between K and Na ions, it's possible to set up directed transport using the osmotic pressure gradient of either K or Na as well as use the electric potential that is established by the imbalance of ions.

http://en.wikipedia.org/wiki/Gibbs-Donnan_effect

Mapes
Homework Helper
Gold Member
A simple way of visualizing osmosis is to consider a saline solution and pure water separated by a membrane. The concentration of water is lower on the saline side (it has to be <100% due to the dissolved salt), so there's a driving force for water to diffuse from the pure water side into the saline solution.

We can tie osmosis and vapor pressure together with the chemical potential. The chemical potential is the driving force to move matter around, just as electrical potential is the driving force to move electric charges around. Matter tends to move to where its chemical potential will be lowest.

The chemical potential often increases with increasing concentration and vice versa. Since concentration is a lot easier to visualize, mechanisms like diffusion and osmosis (which are actually driven by changes in chemical potential) are often modeled as simply being driven by concentration differences. Fick's Laws of diffusion, for example, are expressed in terms of concentration gradients.

The chemical potential $\mu$ is defined as

$$\mu=\mu_0+RT\ln a$$

where $\mu_0$ is just a reference constant for that material, R is the gas constant, T is the temperature, and a is called the activity. It's the activity that is often approximately proportional to concentration; the activity is zero when the material is absent and one when the material is pure.

Now here's the connection: if the material is in equilibrium with its own vapor, the vapor pressure p will equal the activity a. This lets us calculate the vapor pressure of any material, since the chemical potential is also equal to $H-TS$ where H is the molar enthalpy and S is the molar entropy. At equilibrium, by definition, the chemical potential of the condensed state (lets say it's a liquid) and the gas is equal:

$$\mu_\mathrm{gas}=\mu_\mathrm{liquid}$$

$$\mu_\mathrm{0,gas}+RT\ln p_\mathrm{gas}=\mu_\mathrm{0,liquid}+RT\ln a_\mathrm{liquid}$$

$$RT\ln p_\mathrm{gas}=-(\mu_\mathrm{0,gas}-\mu_\mathrm{0,liquid})$$

where we've assumed the liquid to be pure ($a=1$, as stated above). This is equivalent to

$$RT\ln p_\mathrm{gas}=-(\Delta H-T\Delta S)$$

where $\Delta H$ and $\Delta S$ are the change in enthalpy and entropy when the liquid boils. With a little algebra, we obtain

$$p\propto\exp(-\Delta H/RT)$$

and thus see a relationship between vapor pressure, heat of vaporization, and temperature. I hope this helps answer your question.

Also, I concur that vapor pressure increases with concentration, so long as the assumption holds that chemical potential increases with concentration. The density of materials comes into play because denser materials generally take more energy to melt and boil, and so $\Delta H$ is higher. Thus water, for example, has a much higher vapor pressure than aluminum, with has a much higher vapor pressure than dense osmium. Does this all make sense?

The chemical potential often increases with increasing concentration and vice versa.

I thought that chemical potential decreases, which is why water has a higher chemical potential than the solution?

Mapes
Homework Helper
Gold Member
I thought that chemical potential decreases, which is why water has a higher chemical potential than the solution?

The chemical potential of species X generally, but not always, increases with the concentration of X. I think you're referring to the change in the chemical potential of water with increasing concentration of a dissolved substance?

All the above explanations are fine but do not, I think, get to the heart of the OP's first question, which was "Why does osmosis happen?" I.e., yes, dissolving a solute in water lowers the water's chemical potential. But this leaves unanswered the question of why this happens. The answer is simple: it's entropy; more specifically, the driving force for osmotic transport is the entropy of mixing.

Consider the expression u = u_o + kT ln x_w + pV. The second term on the RHS, which tells us the chemical potential is lowered when we reduce the mole fraction of water from 1 (pure water), arises purely from the entropy of mixing. [If the solution deviates measurably from ideal behavior, then we need to replace mole fraction with activity, a; but that's a higher-order correction. When we do need to use activity, it's because energetic considerations come into play; these arise from differences between the water-water, water-solute, and solute-solute interactions.]

You can understand why the driving force is the entropy of mixing by asking yourself what would happen to the system if you removed the semi-permeable membrane. That's right, it would mix, because the number of configurations associated with the mixed state is vastly greater. So why doesn't it do that completely with the semi-permeable membrane in place -- i.e.,why doesn't all the water flow to the solution side? Well, in a typical osmotic pressure experiment, water does indeed flow from the pure to the solution side. But as it does this, the level on the solution side increases, causing the pressure on the solution side of the membrane to be higher. This causes the chemical potential of the water on that side to increase (the pV term in the above expression). Hence osmotic equilibrium is attained when the increase in pressure on the solution side causes an increase in chemical potential on that side that just balances the decrease in chemical potential from the presence of the solute. I.e., the osmotic pressure is just the pressure needed to balance the driving force from the entropy of mixing.

And, to my mind, the explanation that osmotic pressure is caused by vapor pressure lowering is not a good one. Instead, osmotic pressure and vapor pressure lowering are both colligative properties that are caused by the same phenomenon: the lowering of chemical potential by the presence of the solute, which in turn results from the entropy of mixing (plus some higher-order corrections for energetic interactions).

Finally, some may wish to say that one doesn't need to invoke entropy of mixing -- it's just water flowing down a concentration gradient (from higher to lower). But such an explanation would be incorrect. It's about the chemical potential, and thus the entropy of mixing, not the concentration. Increasing the size of the solute will increase the concentration gradient, but will not necessarily affect the osmotic pressure. Careful osmotic pressure experiments carried out by Morse* showed that dilute solutions containing equal mole fractions of glucose or sucrose have nearly the same osmotic pressure at 30°C. Yet, for equal mole fractions, the difference in water concentration across the membrane (from the pure side to the solution side) is significantly greater with sucrose than with glucose, simply because sucrose has roughly twice the molar volume of glucose.

In general, it can be shown that, if solvent compressibility is assumed to be negligible, doubling the size of the solute nearly doubles the difference in solvent concentration across the membrane (it is not exactly double because the increase in solute volume slightly increases the total volume of the solution; this deviation becomes negligible at high dilutions). Yet this does not affect the mole fraction and thus does not affect the osmotic pressure (except insofar as it affects higher-order energetic corrections). This goes back to the old statement about colligative properties: at the limit of infinite dilution, they are independent of the nature or size of the solute.

*Morse, H. N. The osmotic pressure of aqueous solutions; report on investigations made in the
Chemical Laboratory of the Johns Hopkins University during the years 1899-1913 (Carnegie
Institution of Washington, Washington, D.C., 1914).

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