Question

The molar solubility of barium phosphate in water at \(25^{\circ} \mathrm{C}\) is \(1.4 \times 10^{-8} \mathrm{~mol} \mathrm{~L}^{-1}\). What is the value of \(K_{\mathrm{sp}}\) for this salt?

Short answer

\(K_{\text{sp}} = 4.70 \times 10^{-46}\)

Step-by-step solution

Write the Dissolution Equation

Write down the balanced chemical equation for the dissolution of barium phosphate in water: \( \text{Ba}_{3}(\text{PO}_{4})_{2}(s) \rightarrow 3\text{Ba}^{2+}(aq) + 2\text{PO}_{4}^{3-}(aq) \).

Write the Expression for the Solubility Product Constant (Ksp)

The expression for the solubility product, \(K_{\text{sp}}\), is given by the concentrations of the ions raised to the power of their coefficients in the balanced equation: \(K_{\text{sp}} = [\text{Ba}^{2+}]^{3}[\text{PO}_{4}^{3-}]^{2}\).

Calculate Ions Concentration From Molar Solubility

For every mole of \(\text{Ba}_{3}(\text{PO}_{4})_{2}\) that dissolves, 3 moles of \(\text{Ba}^{2+}\) and 2 moles of \(\text{PO}_{4}^{3-}\) are produced. Therefore, their concentrations are \([\text{Ba}^{2+}] = 3 \times 1.4 \times 10^{-8}\) and \([\text{PO}_{4}^{3-}] = 2 \times 1.4 \times 10^{-8}\).

Substitute Concentration Values into Ksp Expression

Substitute the calculated concentrations into the \(K_{\text{sp}}\) expression: \(K_{\text{sp}} = (3 \times 1.4 \times 10^{-8})^{3}(2 \times 1.4 \times 10^{-8})^{2}\).

Calculate the Solubility Product Constant (Ksp)

Calculate \(K_{\text{sp}}\) by evaluating the expression: \(K_{\text{sp}} = (3 \times 1.4 \times 10^{-8})^{3}(2 \times 1.4 \times 10^{-8})^{2} = 470.292 \times 10^{-48}\).

Simplify the Ksp Value

To simplify the \(K_{\text{sp}}\) value, rewrite it to a standard scientific notation: \(K_{\text{sp}} = 4.70 \times 10^{-46}\).

Key concepts

Molar Solubility

Molar solubility is a term used in chemistry to describe the number of moles of a substance that can be dissolved in a liter of solution before the substance stops dissolving and the solution reaches saturation. It is typically expressed in units of moles per liter (\text{mol/L}). In the case of sparingly soluble salts like barium phosphate, the molar solubility is quite low, which means only a small amount of the salt can dissolve in water.

Understanding the concept of molar solubility is critical for predicting the behavior of salts in solution, especially when it involves reactions that depend on the concentrations of dissolved ions. For instance, knowing the molar solubility of barium phosphate, which is given as \(1.4 \times 10^{-8}\) \(\text{mol/L}\), allows us to figure out how much of the compound can be in a saturated solution before precipitation occurs.

Dissolution Equation

The dissolution equation represents the chemical process where a solid compound, such as a salt, breaks apart into its constituent ions in a solvent like water. For barium phosphate, the dissolution equation is written as \(\text{Ba}_3(\text{PO}_4)_2(s) \rightarrow 3\text{Ba}^{2+}(aq) + 2\text{PO}_4^{3-}(aq)\).

In this equation, the solid barium phosphate (\text{Ba}_3(\text{PO}_4)_2) dissociates to form three barium ions (\text{Ba}^{2+}) and two phosphate ions (\text{PO}_4^{3-}) in the aqueous phase. To understand any solubility problem, it is important to begin with the correct dissolution equation since it guides further calculations involving the solubility product constant and ion concentration.

Ions Concentration

When discussing ions concentration, we're referring to the amount of ions present in a solution. During the dissociation of compounds like barium phosphate, the concentration of ions is directly linked to the molar solubility of the compound. As per the dissolution equation provided, three moles of \(\text{Ba}^{2+}\) and two moles of \(\text{PO}_4^{3-}\) ions are produced per mole of the dissolved compound.

Using the molar solubility of barium phosphate, we can calculate the concentrations of these ions to be \( [\text{Ba}^{2+}] = 3 \times 1.4 \times 10^{-8} \) and \( [\text{PO}_4^{3-}] = 2 \times 1.4 \times 10^{-8} \). This step is crucial for finding the value of the solubility product constant, as the concentrations will be used in the mathematical formula to calculate \(K_{\text{sp}}\).

Chemical Equilibrium

Chemical equilibrium is a state in a chemical reaction where the rates of the forward and reverse reactions are equal, resulting in no overall change in the concentration of reactants and products. With regard to solubility, chemical equilibrium occurs in the context of a saturated solution, where the solid salt is in dynamic balance with its dissolved ions.

At this point, the solution cannot dissolve any more of the salt, and the ions' activities are defined by the solubility product constant (\(K_{\text{sp}}\)). This constant is a unique value for a given temperature and represents the extent to which a compound dissociates in water. The lower the \(K_{\text{sp}}\), the less soluble the compound is. Using the calculated ion concentrations, we can determine the value of \(K_{\text{sp}}\) for barium phosphate, which helps in predicting the solubility behavior under various conditions.