A solution saturated with a salt of the type \(\mathrm{M}_{3} \mathrm{X}_{2}\) has an osmotic pressure of \(2.64 \times 10^{-2}\) atm at \(25^{\circ} \mathrm{C}\). Calculate the \(K_{s p}\) value for the salt, assuming ideal behavior.

The van 't Hoff equation relates osmotic pressure, temperature, and molar concentration for an ideal solution. It is given by: \[Π = \mathrm{cRT}\] Here, Π: osmotic pressure (atm) c: molar concentration (mol/L) R: the gas constant in L.atm/mol.K (0.0821 L.atm/mol.K) T: temperature in Kelvin

Given temperature is \(25^{\circ} \mathrm{C}\), we can convert it to Kelvin: \[T(K) = T(^\circ \mathrm{C}) + 273.15\] \[T = 25 + 273.15\] \[T = 298.15\, K\]

Now, we can use the van 't Hoff equation to find the molar concentration (c) of the solution. We are given the osmotic pressure Π = \(2.64 \times 10^{-2}\) atm and temperature T = 298.15 K. The gas constant R = 0.0821 L.atm/mol.K. Rearranging the equation for c: \[c = \frac{Π}{\mathrm{RT}}\] Substitute the values into the equation: \[c = \frac{2.64 \times 10^{-2}}{(0.0821)(298.15)}\] \[c ≈ 1.07 \times 10^{-3} \, \mathrm{mol/ L}\]

The balanced dissolution reaction for the salt \(\mathrm{M}_{3}\mathrm{X}_{2}\) is: \[\mathrm{M}_{3}\mathrm{X}_{2} \, (s) \rightleftharpoons \, 3\mathrm{M}^{+} \, (aq) + 2\mathrm{X}^{-} \, (aq)\] Now, let's calculate the \(K_{sp}\) for the salt. Since the molar concentration (c) = 1.07 × 10⁻³ mol/L, the concentration of \(\mathrm{M}^{+}\) ions = 3(\(1.07 \times 10^{-3}\)) and the concentration of \(\mathrm{X}^{-}\) ions = 2(\(1.07 \times 10^{-3}\)). So, the \(K_{sp}\) expression for the salt is: \[K_{sp} = [\mathrm{M}^{+}]^{3} [\mathrm{X}^{-}]^{2}\] Now, plug in the concentrations into the equation: \[K_{sp} = (3 \times 1.07 \times 10^{-3})^{3} (2 \times 1.07 \times 10^{-3})^{2}\] \[K_{sp} ≈ 1.23 \times 10^{-16}\] So, the \(K_{sp}\) value for the salt \(\mathrm{M}_{3}\mathrm{X}_{2}\) is approximately \(1.23 \times 10^{-16}\).