# The connection of the “R” in the ideal gas law and osmotic pressure [closed]

I am studying chemistry and biology at the same time, and have encountered a problem. In the ideal gas law, the $$R$$ is the ideal gas constant, and its value is $$\pu{0.0821 L atm mol-1 K-1}.$$

However, in the osmotic pressure formula $$\Psi_s = -iCRT,$$ the $$R$$ is called the pressure constant, and its value is $$\pu{0.0831 L bar mol-1 K-1}.$$ Is it a coincidence that their values are identical (after adjusting for the $$\pu{atm}$$ and $$\pu{bar}$$ conversion)?

• For R in all sorts of units see en.wikipedia.org/wiki/Gas_constant – MaxW May 8 '20 at 1:59
• No coincidence, and you’ll encounter it in other contexts as well. A bit like pi and e in maths. – Karsten Theis May 8 '20 at 2:29
• R exists because we measure temperature in different units than energy. If you measure temperature in joules/mole, R disappears. – WaterMolecule May 8 '20 at 14:15

## 1 Answer

It's not a coincidence at all! If you do an online search for "derivation of osmotic pressure", you'll see how $$R$$ enters into the derivation.

Indeed, that's one of the beauties of the van 't Hoff equation for osmotic pressure – it reveals that (under the modest simplifying assumptions of the van 't Hoff equation derivation) the osmotic pressure created by a dilute solute is equal to the pressure that solute would exert if it instead acted as an ideal gas!:

$$\text{van 't Hoff equation: } \Pi_i = \frac{n_i}{V} R T = C_i R T$$

$$\text{ideal gas equation: } p_i = \frac{n_i}{V} R T = C_i R T$$

Here the "$$i$$" refers to species $$i$$.