Question #44537

A biochemist is investigating the osmotic pressure ($\pi$) of a tetrameric protein solution at $300\text{ K}$. In its native folded state, the protein behaves as an ideal solute with a molar mass $M_{native} = 1.0 \times 10^5\text{ g mol}^{-1}$. Upon the addition of a chemical denaturant, the protein completely dissociates into four identical unfolded monomeric chains. The osmotic pressure is measured against a dialysate containing the exact same concentration of denaturant, ensuring only the protein contributes to the measured $\pi$.

Due to the massive increase in the hydrodynamic radius and excluded volume upon unfolding, the denatured monomers exhibit non-ideal behavior. The osmotic pressure of such macromolecular solutions follows the virial expansion: $\frac{\pi}{C} = RT \left( \frac{1}{M} + BC \right)$ where $C$ is the protein concentration in $\text{g L}^{-1}$, $M$ is the molar mass of the kinetic solute particle, and $B$ is the second virial coefficient. For the denatured state, $B = 3.6 \times 10^{-5}\text{ L mol g}^{-2}$.

Calculate the ratio of the osmotic pressure of the denatured state to the native state ($\frac{\pi_{den}}{\pi_{nat}}$) at a protein concentration of $C = 10\text{ g L}^{-1}$.