Question #44291

A solution is prepared by dissolving $1.0\text{ mol}$ of a transition metal chloride, \ce{MCl3}, in exactly $50.0\text{ mol}$ of water at $300\text{ K}$. The \ce{M^{3+}} cation forms a rigid, kinetically inert octahedral hydration sphere, \ce{[M(H2O)6]^{3+}}, while the chloride ions remain unhydrated. Assuming $100\%$ dissociation of the salt into the hydrated cation and chloride ions, calculate the theoretical osmotic pressure ($\Pi$) of the solution.

Assume the remaining 'free' water behaves as an ideal solvent and use the exact thermodynamic relation for osmotic pressure: $\Pi = -\frac{RT}{V_m} \ln x_A$ where $x_A$ is the mole fraction of the free solvent.

(Given: Molar volume of pure liquid water, $V_m = 18.0\text{ mL mol}^{-1}$; $RT = 24.3\text{ L atm mol}^{-1}$ at $300\text{ K}$; $\ln(12/11) = 0.087$; $\ln(27/25) = 0.077$)