# What is the mathematical reason behind the gradual increase in osmotic pressure?

Question:

Arrange the following in increasing order of osmotic pressure, $0.1~\mathrm{M}$ cane sugar, $0.1~\mathrm{M}$ $\ce{NaCl}$ solution, $0.1~\mathrm{M}$ $\ce{H2SO4}$.

Solution: As the number of particles/effective concentration of the solutions increase in the following order:

Cane sugar $(0.1~\mathrm{M})$ > $\ce{NaCl}$ $(0.2~\mathrm{M})$ > $\ce{H2SO4}$ $(0.3~\mathrm{M})$

So the osmotic pressure which is related to concentration by the relation $P=cRT$ varies as:

Cane sugar > $\ce{NaCl}$ > $\ce{H2SO4}$

Is the following reasoning correct?

That formula for the osmotic pressure is known as the van 't Hoff law. It is described in some detail at the Wikipedia page for osmotic pressure, which also includes a brief (albeit incomplete and rather sloppy) derivation. To reach the van't Hoff law

$$\Pi = cRT$$

you have to approximate $\ln x_v$ by a Taylor series $(\ln x_v = \ln (1 - x_\mathrm{solute}) \approx -x_\mathrm{solute}$ which holds for small values of $x_\mathrm{solute})$ to get

$$\Pi = \frac{x_\mathrm{solute}RT}{V_\mathrm{m}}$$

The Wikipedia page uses $V$ to refer to the molar volume of the solvent (which should really be represented by $V_\mathrm{m}$).[1] In any case, $x_\mathrm{solute} = n_\mathrm{solute}/n_\mathrm{tot}$ and $V_\mathrm{m} \approx V/n_\mathrm{tot}$ such that

$$\frac{x_\mathrm{solute}}{V_\mathrm{m}} \approx \frac{n_\mathrm{solute}}{V} = c_\mathrm{solute}$$

and if the solute is a 1:1 electrolyte, such as $\ce{NaCl}$, then you have to multiply by two to account for the total number of solute particles.

I am not sure about the term "effective concentration" (I personally have never seen it, and if I had to, I would probably write something like the "total concentration of solute particles") but the logic used is correct as long as you are in the regime where the van 't Hoff law is applicable i.e. very dilute solvent.

Lastly you should probably use $<$ instead of $>$ in your ordering to be clear.

[1] In the actual derivation this should strictly be the partial molar volume of the solvent

$$\overline{V}_{\!\!\mathrm{m,solvent}} = \left(\frac{\partial V}{\partial n_\mathrm{solvent}}\right)_{\!n_\mathrm{solute}}$$

but this can to a good extent be approximated by the molar volume of the pure solvent.