Why does the ideal gas law exactly match the van't Hoff law for osmotic pressure? The van't Hoff law for osmotic pressure $\Pi$ is
$$\Pi V=nRT$$
which looks similar to the ideal gas law
$$PV = nRT.$$
Why is this? Also, in biology textbooks, the van't Hoff law is usually instead written as 
$$\Pi=CRT =\frac{NC_m RT}M$$
where $C_m$ is the mass concentration, $N$ the number of ions, and $R$ the ideal gas constant. Why?
 A: I'm answering your 2nd question:
It's a really easy proof :
You have $C_m=\frac{m}V$ and $C=\frac{n}V$ where $n=\frac{m}M$, thus $C=\frac{C_m}M$
Therefore :
$$\Pi=\frac{C_m R T}{M}$$
For $N$ the ions number it's related to biology, the ions are interacting and they cause this pressure, For example if it's $\mathrm{NaCl}$ the ions are $N=2$ 
Thus Van't hoff law is written in Biology and Biophysics textbooks :
$$\Pi =\frac{NC_mRT}M$$
A: The law $PV = n RT$ gives the pressure $P$ of $n$ moles of ideal gas in volume $V$. Meanwhile, the law $\Pi V = n R T$ describes the osmotic pressure $\Pi$ due to $n$ moles of solute in volume $V$. 
These are qualitatively very different situations, but there's a simple fundamental reason that they end up looking the same. Both of these laws are derived under the idealized assumption that the ideal gas/solute molecules don't interact with each other at all. So the expressions for the entropy of the ideal gas/solute are the same, and since the pressure of a system can be derived from the entropy, both situations yield the same pressure. 
The reason that you see $\Pi V = n RT$ expressed in such different units in biology textbooks is simply because they're using the units that are most convenient for them. 
