Question

\(K_{\text {sp }}\) of \(\mathrm{Zr}_{3}\left(\mathrm{PO}_{4}\right)_{4}\) in terms of solubility-(S) is : (a) \(108 \mathrm{~s}^{7}\) (b) \(45^{3}\) (c) \(6912 \mathrm{~s}^{7}\) (d) None of these

Short answer

The solubility product constant (Ksp) in terms of solubility (S) is: \(6912 S^{7}\).

Step-by-step solution

- Write Down the Dissociation Equation

First, write down the equation for the dissociation of \(\mathrm{Zr}_{3}\left(\mathrm{PO}_{4}\right)_{4}\) in water:\[\mathrm{Zr}_{3}\left(\mathrm{PO}_{4}\right)_{4}(s) \leftrightarrow 3\mathrm{Zr}^{4+}(aq) + 4\mathrm{PO}_{4}^{3-}(aq)\]

- Express the Solubility Product Constant (Ksp)

Next, express the solubility product constant (\(K_{\text{sp}}\)) using the solubility (S) for each of the ions produced:\[K_{\text{sp}} = [\mathrm{Zr}^{4+}]^{3}[\mathrm{PO}_{4}^{3-}]^{4}\]

- Insert the Solubility Values

Insert the values for the concentrations of the ions in terms of solubility (S):\[K_{\text{sp}} = (3S)^{3}(4S)^{4}\]

- Expand the Expression

Now expand the expression to find the relationship between \(K_{\text{sp}}\) and solubility (S):\[K_{\text{sp}} = 27S^{3} \cdot 256S^{4} = 6912S^{7}\]

Key concepts

Ksp Calculation

Understanding how to calculate the solubility product constant, commonly abbreviated as Ksp, is essential for predicting the solubility of compounds in water. Ksp is a equilibrium constant that helps to determine how much of a solid compound will dissolve in a solution.

To calculate Ksp, you need to consider the degrees to which a compound dissociates into its constituent ions in a saturated solution. For a general salt, AB, which dissociates into A+ and B- ions, the Ksp expression would be \( K_{sp} = [A^+][B^-] \), where the concentrations are those at equilibrium.

For more complex salts, like \( \mathrm{Zr}_3(\mathrm{PO}_4)_4 \), the calculation involves raising the concentration of each ion to the power of its coefficient in the balanced dissociation equation. In our example exercise, this led to \( K_{sp} = (3S)^3(4S)^4 = 6912S^7 \), highlighting how the solubility of each ion affects the overall Ksp value. Calculating Ksp correctly is vital for many applications, including chemical analysis, environmental science, and pharmaceuticals.

Dissociation Equation

When a salt like \( \mathrm{Zr}_3(\mathrm{PO}_4)_4 \) dissolves in water, it dissociates into its constituent ions. Writing the dissociation equation involves expressing this breakdown process. For the compound \( \mathrm{Zr}_3(\mathrm{PO}_4)_4 \), the dissociation equation is:

\[ \mathrm{Zr}_3(\mathrm{PO}_4)_4 (s) \leftrightarrow 3\mathrm{Zr}^{4+}(aq) + 4\mathrm{PO}_4^{3-}(aq) \]

This equation indicates that one formula unit of \( \mathrm{Zr}_3(\mathrm{PO}_4)_4 \) separates into three zirconium ions \(\mathrm{Zr}^{4+}\) and four phosphate ions \(\mathrm{PO}_4^{3-}\) in water. The coefficients have a pivotal role, as they indicate the mole ratio of the dissociated ions, which are subsequently used to calculate the Ksp. Understanding how to write a correct dissociation equation is fundamental for anyone studying chemistry and is the first step towards grasping various equilibrium concepts.

Ionic Product

The term ionic product is at times used interchangeably with Ksp, but technically it represents the product of the molar concentrations of the ions of a salt at any given moment, not just at equilibrium. When the ionic product exceeds the Ksp, the solution becomes supersaturated and precipitation occurs.

In the sample exercise involving \( \mathrm{Zr}_3(\mathrm{PO}_4)_4 \) , the ionic product would be \( [\mathrm{Zr}^{4+}]^3[\mathrm{PO}_4^{3-}]^4 \), similar to the Ksp expression. By comparing the ionic product with the Ksp value, one can predict whether a precipitate will form or if the solution will remain unsaturated. In practical situations, like water treatment or medicine, controlling supersaturation and preventing unwanted precipitation is crucial, making a clear understanding of the ionic product a valuable asset.

Chemical Equilibrium

Chemical equilibrium occurs when the forward and reverse reactions in a system happen at the same rate, resulting in no net change in the concentration of reactants and products. At the macroscopic level, the system appears static, but at the microscopic level, reactions continue to occur.

The concept of chemical equilibrium is pivotal in understanding solubility. When a substance like \( \mathrm{Zr}_3(\mathrm{PO}_4)_4 \) is added to water, it begins to dissolve and dissociate into ions until the solution becomes saturated, leading to a state of equilibrium between the dissolved ions and the undissolved solid.

This state can be quantitatively described by the solubility product constant (Ksp), which only changes with temperature. Equilibrium is a fundamental concept not only in chemistry but also in various processes in biology and engineering, such as the functioning of enzymes, battery operation, and synthesizing chemical compounds.