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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)?

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    $\begingroup$ For R in all sorts of units see en.wikipedia.org/wiki/Gas_constant $\endgroup$
    – MaxW
    May 8, 2020 at 1:59
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    $\begingroup$ No coincidence, and you’ll encounter it in other contexts as well. A bit like pi and e in maths. $\endgroup$
    – Karsten ♦
    May 8, 2020 at 2:29
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    $\begingroup$ R exists because we measure temperature in different units than energy. If you measure temperature in joules/mole, R disappears. $\endgroup$
    – WaterMolecule
    May 8, 2020 at 14:15

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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 $i$.

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