Question

Solubility product constant \(\left(K_{\mathrm{sp}}\right)\) of salts of types \(\mathrm{MX}, \mathrm{MX}_{2}\) and \(\mathrm{M}_{3} \mathrm{X}\) at temperature, \(T\) are \(4.0 \times 10^{-8}, 3.2 \times 10^{-14}\) and \(2.7 \times 10^{-15}\), respectively. Solubilities (in \(\mathrm{M}\) ) of the salts at temperature, \(T\), are in the order (a) \(\mathrm{MX}>\mathrm{MX}_{2}>\mathrm{M}_{3} \mathrm{X}\) (b) \(\mathrm{M}_{3} \mathrm{X}>\mathrm{MX}_{2}>\mathrm{MX}\) (c) \(\mathrm{MX}_{2}>\mathrm{M}_{3} \mathrm{X}>\mathrm{MX}\) (d) \(\mathrm{MX}>\mathrm{M}_{3} \mathrm{X}>\mathrm{MX}_{2}\)

Short answer

The solubilities of the salts at temperature, T, are in the order: (a) MX > MX2 > M3X

Step-by-step solution

Identify Solubility Expressions

Write the solubility product expression for each salt as per the type, assuming 's' as the solubility in mol/L.\ For salt type MX: \[ K_{\mathrm{sp}} = [M^+][X^-] = s^2 \] For salt type \(MX_2\): \[ K_{\mathrm{sp}} = [M^{2+}][X^-]^2 = 4s^3 \] For salt type \(M_3X\): \[ K_{\mathrm{sp}} = [M^{3+}][X^{3-}] = 27s^4 \]

Calculate Actual Solubility

Use the given solubility product constants to calculate solubility 's' for each salt.\ For MX with \(K_{\mathrm{sp}}=4.0 \times 10^{-8}\), \[ s = \sqrt{K_{\mathrm{sp}}} \]\[ s = \sqrt{4.0 \times 10^{-8}} \]\[ s = 2.0 \times 10^{-4} M \] For \(MX_2\) with \(K_{\mathrm{sp}}=3.2 \times 10^{-14}\), \[ s = \sqrt[3]{\frac{K_{\mathrm{sp}}}{4}} \]\[ s = \sqrt[3]{\frac{3.2 \times 10^{-14}}{4}} \]\[ s = \sqrt[3]{0.8 \times 10^{-14}} \]\[ s = 4.3 \times 10^{-5} M \] For \(M_3X\) with \(K_{\mathrm{sp}}=2.7 \times 10^{-15}\), \[ s = \sqrt[4]{\frac{K_{\mathrm{sp}}}{27}} \]\[ s = \sqrt[4]{\frac{2.7 \times 10^{-15}}{27}} \]\[ s = \sqrt[4]{0.1 \times 10^{-15}} \]\[ s = 3.2 \times 10^{-5} M \]

Compare and Order Solubilities

To determine the order, compare the calculated solubilities of each salt.\ For MX: \(2.0 \times 10^{-4} M\)\ For MX2: \(4.3 \times 10^{-5} M\)\ For M3X: \(3.2 \times 10^{-5} M\)\ By comparison, the order of solubilities is: \[ \mathrm{MX} > \mathrm{MX}_{2} > \mathrm{M}_{3} \mathrm{X} \]

Key concepts

Chemical Equilibrium

In the realm of chemistry, the concept of chemical equilibrium is a fundamental principle that describes the state of a chemical reaction in which the rate of the forward reaction equals the rate of the reverse reaction. Essentially, this means no net change in the concentration of reactants and products occurs; they have reached a balance.

This situation is a dynamic one, with reactants continually converting to products and vice versa, but because the rates are equal, you wouldn't observe any observable changes at equilibrium. It is a critical concept for understanding reactions, including those that result in the formation of a solid precipitate (an insoluble substance) in a solution.

When applying the concept of chemical equilibrium to the solubility of ionic compounds, the equilibrium exists between the solid substance and its ions in solution. Here, the solid salt dissociates into its constituent ions until the maximum concentration that can exist in equilibrium with the undissolved solid (at a given temperature and pressure) is reached. This equilibrium can be quantified using the solubility product constant, K_sp, which is unique for every substance under specific conditions.

Ionic Product

The term ionic product, often represented by the notation Q_sp, is defined as the product of the molar concentrations of the ions involved in the equilibrium, each raised to the power of its stoichiometric coefficient. Remember, the K_sp is a special case of the ionic product where the system is exactly at equilibrium.

It is worth noting that the ionic product can vary, and it is not constant like the solubility product. If Q_sp is less than the K_sp, the solution is unsaturated and more solid can dissolve. If Q_sp is greater than the K_sp, the solution is supersaturated, and a precipitate will form until Q_sp equals K_sp. To find the K_sp of a particular salt, the equilibrium concentrations of the ionic species in a saturated solution of the salt are multiplied accordingly.

Molar Solubility

The term molar solubility is the number of moles of a substance that can dissolve per liter of solution to form a saturated solution. It is quantitatively represented as the molarity, or moles per liter (M), of the salt that can be dissolved until reaching the point of saturation.

Molar solubility can be directly related to the solubility product constant of a compound; it is essentially the concentration of the salt that exists in equilibrium with the solid phase in a saturated solution. To determine the molar solubility from the solubility product constant, K_sp, you must consider the dissociation of the ionic compound into its constituent ions and apply stoichiometry to relate the concentrations of these ions back to the original compound. These calculations become more complex when dealing with salts that dissociate into multiple ions of different stoichiometries, such as those presented in the exercise with compounds MX, MX₂, and M₃X.