# 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?

• What law is that? May 5 '20 at 18:37
• @BobD Van't Hoff law, the other one is the Ideal gas law !
– user257533
May 5 '20 at 18:38
• What's the signification of $C_m$ and $N$ ?
– user257151
May 5 '20 at 18:41
• I know about the ideal gas law. Just never hear of Van't Hoff law. May 5 '20 at 18:42
• @Electroelf I added the signification
– user257533
May 5 '20 at 18:48

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.

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 : $$\Pi =\frac{NC_mRT}M$$