Question

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.

Short answer

The solubility product (\(K_{sp}\)) for the salt \(\mathrm{M}_{3}\mathrm{X}_{2}\) is approximately \(1.23 \times 10^{-16}\).

Step-by-step solution

Identify the van 't Hoff equation

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

Convert given temperature to Kelvin

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

Calculate the molar concentration

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: \[c = \frac{2.64 \times 10^{-2}}{(0.0821)(298.15)}\] \[c ≈ 1.07 \times 10^{-3} \, \mathrm{mol/ L}\]

Write the balanced dissolution reaction and calculate \(K_{sp}\)

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 \(1.23 \times 10^{-16}\).

Key concepts

Understanding Osmotic Pressure

Osmotic pressure is a fundamental concept in chemistry that helps us understand how solutions behave. It occurs when there is a difference in solute concentration on either side of a semi-permeable membrane. The pressure is necessary to prevent water from flowing into the solution through the membrane from pure solvent. This process is essential in biological systems and many industrial applications.
The formula to calculate osmotic pressure is based on the van 't Hoff equation:

• \( Π = ext{cRT} \)
• \( Π \) is the osmotic pressure,
• \( c \) is the molar concentration,
• \( R \) is the gas constant, and
• \( T \) is temperature in Kelvin.

Understanding osmotic pressure is crucial for calculating various properties of solutions, especially when dealing with solubility and equilibrium.

Molar Concentration

Molar concentration refers to the amount of solute, in moles, present in one liter of solution. It's a way to express how "concentrated" or "diluted" a solution is. Calculating molar concentration helps to determine how much of a compound will dissolve in a given solvent.
The formula used is:

• \( ext{c} = rac{n}{V} \)
• where \( n \) is the number of moles,
• and \( V \) is the volume in liters.

Learning to calculate molar concentration can help in predicting the outcomes of chemical reactions, purity of solutions, and even guide the experimenter's next steps.

The van 't Hoff Equation

The van 't Hoff equation is an essential tool in chemistry for understanding how temperature affects the properties of solutions. It ties together several factors: osmotic pressure, molar concentration, and temperature.
This equation is written as:

• \( Π = ext{cRT} \)
• \( Π \) reflects osmotic pressure,
• \( c \) is the concentration of the solute,
• \( R \) represents the gas constant (0.0821 L.atm/mol.K),
• \( T \) is the temperature in Kelvin.

By applying this equation, you can derive the concentration of a substance in a solution if you know the osmotic pressure and temperature. This is a powerful technique for chemists studying solution behavior under various conditions.

Dissolution Reaction and Solubility Product

A dissolution reaction occurs when a solute dissolves in a solvent, breaking into its constituent ions or molecules. Understanding this process is crucial for calculating the solubility product constant \( K_{sp} \).
Let’s take the salt \( ext{M}_3 ext{X}_2 \) as an example. When it dissolves, it separates into its ions as:

• \( ext{M}_3 ext{X}_2 \leftrightharpoons 3 ext{M}^+ + 2 ext{X}^- \)

The solubility product \( K_{sp} \) reflects the equilibrium condition of a dissolution reaction. It is given by:

• \( K_{sp} = [ ext{M}^+]^3 [ ext{X}^-]^2 \)

Calculating the \( K_{sp} \) helps you understand how much of the salt will dissolve in the solution, essentially determining the solution's saturation point. This type of calculation is essential in fields like pharmacology, environmental science, and chemical engineering. Understanding dissolution reactions can aid in predicting product formation and solubility issues in both lab and industrial settings.