Why is the van ’t Hoff relation an approximation?

Question

Lately, I've been looking and thinking about the fact that a professor told us in an advanced physiology lecture. When it came to mentioning osmotic pressure and the van ’t Hoff relationship

$$ \pi = c \cdot R \cdot T\tag{1} $$

he said it was just an approximation, and if we ever got bored, we should try to prove it mathematically. I remember this relationship from when I took physical chemistry classes but I never looked into it further. Well, since I was bored, I started looking at the derivation of this relationship and possible clues to why it is just an approximation, and it kind of messed with my head.

Is it because in the equation shown in Wikipedia — Osmotic pressure: Derivation of the van ’t Hoff formula

$$ \pi = -\frac{RT}{V_\mathrm m}\ln(1 - x_\mathrm s) \tag{2} $$

we can apply Taylor series for $\ln(1 - x_\mathrm s)$ in the case of very dilute solutions? Is the result of Taylor’s development a relationship that we can also sometimes see, when instead of concentration $c$ we calculate the sum of the product of concentration and activity for each osmotically active substance in dilute solutions?

Does this part of the relation tell us that it is a simple approximation? And how can I somehow mathematically prove it?

Answer 1

It uses the approximation that the solution is very dilute, so the mole fraction of the solvent $x_a$ is bigger than the mole fraction of the substance $x_b$ which is dissolved. So for small $x_b$:

$ln(x_a) = ln(1) - x_b \approx -x_b$

$-RTx_b \approx -V_m\cdot p_{osm}$

$p_{osm}\frac{V}{n_A} = RT x_b$

Also we approximate that

$x_b = \frac{n_b}{n_a + n_b} \approx \frac{n_b}{n_a}$

$p_{osm}\frac{V}{n_A} = RT \frac{n_b}{n_a}$

$p_{osm}V = RT n_b$