I dare say most of you will remember this classroom demonstration, in which water passes through a semi-permeable membrane and causes the liquid level to rise in the stem of the thistle funnel. The phenomenon is called osmosis, and at equilibrium the osmotic pressure is equal to the hydrostatic pressure.

**Historical background**

This experiment has its origins way back in the mid-18th century, when a French clergyman named Jean-Antoine Nollet tied a piece of pig’s bladder over the mouth of a jar containing alcohol and immersed the whole thing in a vat of water. What prompted him to do this is not known, but we do know the result of his experiment. The bladder swelled up and ultimately burst from the internal pressure.

Nollet published his findings in *Recherches sur les causes de Bouillonement des Liquides (1748)* in which he gave a correct interpretation of the phenomenon, which arises from the much more marked permeability of the bladder to water as compared with alcohol.

The actual measurement of osmotic pressure had to wait for over a century, until the German chemist Moritz Traube showed in 1867 that artificial semipermeable membranes could be made using gelatin tannate or copper ferrocyanide. Traube’s compatriot Wilhelm Pfeffer, a botanist, succeded in depositing the latter in the walls of a porous jar, which when filled with a sugar solution, connected to a mercury manometer and then plunged into pure water, provided a means of measuring osmotic pressures.

Following Pfeffer’s osmotic pressure measurements using sucrose solutions, on which JH van ‘t Hoff based his famously flawed gaseous theory of solutions, there were two notable teams of experimentalists – one on each side of the Atlantic – which provided high quality osmotic pressure data to test the ideas of theoreticians. In the USA, Harmon Northrop Morse and Joseph Christie Whitney Frazer led a team at Johns Hopkins University, Baltimore, Maryland from 1901 to 1923. In Britain meanwhile, the aristocrat-turned-scientist Lord Berkeley and co-worker Ernald Hartley set up a private research laboratory near Oxford which operated (with gaps due to war service) from 1904 to 1928.

While Morse and Frazer used the same principle as Pfeffer, albeit with a more advanced electrochemical method of depositing the membrane in the pores, Berkeley and Hartley reversed the arrangement of solvent and solution, applying measured pressure to the latter to attain equilibrium.

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**Theoretical development in Europe**

In Europe, the rigorous application of thermodynamics to the phenomenon of osmosis started in 1887 with Lord Rayleigh, who combined the use of the ideal gas law PV = nRT with the idea of a reversible isothermal cycle of operations in which the sum of work in the complete cycle is zero.

Being essentially an attempt to provide hypothesis-free support to van ‘t Hoff’s troubled gaseous theory of solutions, the solute in Rayleigh’s cycle was a mole of ideal gas, which was first dissolved in the solution by applied pressure and then recovered from the solution by osmotic pressure to return the system to its original state.

Rayleigh’s approach, using a zero-sum cycle of operations, was thermodynamically sound and continued to form the basis of theoretical development in its next phase, which in Europe focused on vapor pressure following the influential papers of Alfred Porter in 1907 and Hugh Callendar in 1908.

By 1928, the theoretical model in JAV Butler’s popular textbook *The Fundamentals of Chemical Thermodynamics* was close to the familiar classroom demonstration of osmosis shown at the head of this post, in which the hydrostatic pressure acting on the solution counteracts the tendency of the solvent to pass through the semi-permeable membrane. At equilibrium, the hydrostatic pressure P is equal to the osmotic pressure.

To obtain a thermodynamic relation for osmotic pressure in terms of vapor pressures, Butler uses Rayleigh’s idea of a reversible isothermal cycle of operations together with a semipermeable membrane in the form of a movable piston between the solution and the solvent:

The diagram shows a solution under hydrostatic pressure P which is equal to the osmotic pressure. Below the semi-permeable piston is pure solvent. Butler then applies the following argumentation:

1] Vaporize 1 mole of the pure solvent at its vapor pressure *p _{0}*, and expand it reversibly so that the vapor pressure falls to

*p*equal to the partial pressure of the solvent in the solution (Butler assumes that

*p*is not affected by P applied to the solution). Condense the vapor into the solution. Since the work of vaporization and condensation cancel out, the only work done is the work of expansion. Assuming the vapor obeys the ideal gas law, the work (w) done is given by the textbook isothermal expansion formula

2] Now move the semi-permeable piston up against the pressure P until a quantity of solvent equivalent to 1 mole of vapor has passed through it. If the decrease in the volume of the solution is ΔV, the work done is PΔV.

The cycle is now complete and the system has returned to its original state. The total work done is zero and we may equate the two terms

where P is the osmotic pressure, ΔV is the partial molal volume of the solvent in the solution, *p _{0}* is the vapor pressure of the pure solvent and

*p*is the vapor pressure of the solvent in the solution. This thermodynamically exact relation, which involves measured vapor pressures, is in good agreement with experimental determinations of osmotic pressure at all concentrations.

There is a great irony here, in that this equation is exactly the one that JH van ‘t Hoff found his way to in *Studies in Chemical Dynamics (1884)*, before he abandoned his good work and went completely off-track with his gaseous theory of solutions.

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**Theoretical development in America**

In the US, the theory of the semipermeable membrane and the ‘equilibrium of osmotic forces’ was the work of one supremely gifted man, Josiah Willard Gibbs, who more or less single-handedly laid the theoretical foundations of chemical thermodynamics in his milestone monograph *On the Equilibrium of Heterogeneous Substances*.

But before delving into the powerful idea he introduced, let us return to the subject of equilibrium in a system subject to osmotic pressure with a set-up that is slightly different to that used by Butler. In the diagram below, the piston supplies pressure P^{soln} to the solution which is just enough to stop solvent passing through the membrane and bring about equilibrium at constant temperature; the osmotic pressure is defined as the excess pressure P^{soln} – p^{0}_{1}.

The question can now be asked: Does the condition of osmotic equilibrium coincide with equality of a thermodynamic variable on either side of the membrane? Clearly it cannot be pressure or volume, nor can it be temperature since constant temperature does not prevent osmotic disequilibrium.

The P, V, T variables do not provide an affirmative answer, but in his monumental masterwork, Gibbs supplied one of his own invention which did – the chemical potential, symbolized μ. It is an intensive variable which acts as a ‘generalized force’, driving a system from one state to another. In the present context the force drives chemical components, capable of passing through a membrane, from a state of higher potential to a state of lower potential.

So given a membrane dividing solution from solvent and permeable only to the latter, we can understand the osmotic force driving the solvent (designated by subscript 1) through the membrane into the solution in terms of movement to a region of lower potential since

Now the difference in potential can be calculated according to the textbook formula

where *x _{1}* is the mole fraction (<1) of the solvent in the solution. To achieve equilibrium, the chemical potential of the solvent in the solution must be increased by the amount –RTlnx

_{1}(a positive quantity since lnx

_{1}is negative). This can be done by increasing the pressure on any solution exhibiting ideal behavior since

is positive (V_{1} is the partial molar volume of the solvent in the solution).

The osmotic pressure is defined as the excess pressure P^{soln} – p^{0}_{1}. As can be seen from the diagram below, this is the pressure required to raise the chemical potential of the solvent in the solution so that it becomes equal to the chemical potential of the pure solvent.

Since the slope is V_{1}, it follows that

Combining (2) and (3) and designating the osmotic pressure by P gives the desired equilibrium relation

This is exactly equivalent to equation (1) derived by Butler, since by his terminology

The two methods of proof are thus shown to be equivalent – we can regard osmotic pressure as the excess pressure required to increase either the chemical potential or the vapor pressure of the solvent in the solution. But Gibbs saw an advantage in using potentials, which he voiced in an 1897 letter to *Nature* entitled *Semi-Permeable Films and Osmotic Pressure*:

*“The advantage of using such potentials in the theory of semi-permeable diaphragms consists … in the convenient form of the condition of equilibrium, the potential for any substance to which a diaphragm is freely permeable having the same value on both sides of the diaphragm.”*

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**A closer look at the equations applied to osmotic systems**

The foregoing analysis has reached its conclusions by assuming that solutions in osmotic systems exhibit ideal behavior. On this assumption equation 2

can be interpreted as an ‘osmotic’ equation showing that excess pressure equal to –RTlnx_{1} needs to be applied to the solution to bring the system into osmotic equilibrium by stopping the passage of solvent into the solution. But there are no membrane-dependent terms in this equation, which can be re-written as

This simply shows the effect of adding a solute to a solvent. It should be noted that the equation is only valid at 1 atmosphere pressure, and this provides the clue to how the equation can properly be extended to apply to osmotic systems. If excess pressure ΔP is applied to the system an extra term is required to account for the fact that µ(solvent) is a function of pressure

We now have an equation that can be applied to osmotic systems. Since (δµ_{1}/δP)_{T} is the partial molar volume of the solvent in the solution (V_{1}) the equation becomes

If V_{1} is positive it is evident that osmotic equilibrium is obtained when the condition

is met. This equation of condition allows us to conclude that solvent passes spontaneously into the solution, and that this process can be arrested by applying to the solution an excess pressure ΔP which we call the osmotic pressure. Note that this pressure is external in origin and arises by virtue of the fact that a semi-permeable membrane exists between solvent and solution rather than through a property of the membrane itself or component interactions with it*.

But what if the partial molar volume of the solvent in the solution (V_{1}) is negative? This can in fact happen due to ionic attraction because V_{1} is defined not as a solvent volume but the *change* in volume when solvent is added to the solution. Referring to the above equation of condition, it can be seen that since RTlnx_{1} is always negative, equilibrium when V_{1} is negative can only be obtained when ΔP is also negative. Reduced pressure on the solution is equivalent in its effect to excess pressure on the solvent. This allows us to conclude that solvent passes spontaneously out of the solution and that this process can be arrested by applying to the solvent an excess pressure ΔP.

This reversal of what we commonly understand as osmosis might seem incredible, but it does happen under these conditions. It is important however not to view this as the spontaneous unmixing of a mixture. The membrane, the solvent phase and the negative partial molar volume of the solvent in the solution are part of the description of the total system, which will spontaneously move only in the direction of thermodynamic equilibrium and never away from it.

* This is consistent with Wilhelm Ostwald’s __incontrovertible proof__ that osmotic pressure is independent of the nature of the semi-permeable membrane.

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**Suggested further reading**

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P Mander March 2019, revised and extended January 2022

Science! Even when “just” fixing mistakes, it is always worth it.

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