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The hyperphysics article defines osmosis as a process driven by internal energy and internal energy is defined as energy associated with random,disordered motion of molecules. Osmosis develops pressure and lifts weight so it has the output of a machine.

But what drives osmosis? Is osmosis driven at the expense of thermal energy although there is no temperature gradient?

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Perhaps surprisingly the osmotic pressure is related to the vapour pressure of the solvent. To see why, consider this though experiment:

Vapour pressure

The vapour pressure of the pure solvent, $P$, is greater than the vapour pressure of the solvent with some solute added, $P'$. This is simply because the mole fraction of the solvent is higher when it's pure than when something else is present - see Raoult's Law for more details. So if we put both beakers inside a sealed box there will be a net transfer of solvent from the pure beaker into the beaker with the solute.

Obviously there is no vapour present in an osmosis cell, but the theromdynamics are the same i.e. the change in free energy of the solvent moving from the pure side to the solution side is the same as if we evaporated some of the pure solvent then condensed it into the solution.

The molar change in free energy in evaporating then condensing the solvent is:

$$ \Delta G = -RT \ln\frac{P'}{P} $$

and the work done in moving one mole of solvent against an osmotic pressure $\Pi$ is:

$$ W = \int_0^\Pi VdP = V\Pi $$

where $V$ is the molar volume and we assume $V$ is constant i.e. not dependant on the pressure. At equilbrium the free energy change will be equal to the work done, so we equate the two equations above to get:

$$ \Pi = \frac{RT}{V} \ln \frac{P}{P'} $$

and since $P \gt P'$ that means $\Pi \gt 0$.

All very well, but none have this has addressed the microscopic origin of the pressure and I would guess that's what you're really after i.e. how do we explain the pressure by considering single solvent molecules. Well, the simplest case is when the solvent and solution are ideal fluids so the only effects are due to entropy. There are more solvent molecules per cubic metre in the pure solvent than in the solution, so assuming the solvent molecules move randomly it is more probable that a molecule will move from the pure side to the solution side than vice versa.

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