Question

At \(25^{\circ} \mathrm{C}, 100.0 \mathrm{~mL}\) of a \(\mathrm{Ba}(\mathrm{OH})_{2}\) solution is prepared by dissolving \(\mathrm{Ba}(\mathrm{OH})_{2}\) in an alkaline solution. At equilibrium, the solution has \(2.37 \mathrm{~g}\) of \(\mathrm{Ba}(\mathrm{OH})_{2}\) and a \(\mathrm{pH}\) of \(13.28\). Estimate \(K_{s p}\) for \(\mathrm{Ba}(\mathrm{OH})_{2}\).

Short answer

Estimate the solubility product constant (\(K_{sp}\)) for \(\mathrm{Ba}(\mathrm{OH})_{2}\) if the \(\mathrm{pH}\) of the saturated solution is 13.28. The estimated value of \(K_{sp}\) for \(\mathrm{Ba}(\mathrm{OH})_{2}\) is \(3.43 \times 10^{-3}\).

Step-by-step solution

Calculate the concentration of \(\mathrm{OH}^{-}\) ions

To find the concentration of \(\mathrm{OH}^{-}\) ions in the solution, we first need to find the \(\mathrm{pOH}\). Since \(\mathrm{pH}\) + \(\mathrm{pOH}\) = 14: \(\mathrm{pOH} = 14 - \mathrm{pH} = 14 - 13.28 = 0.72\) Now we can find the concentration of \(\mathrm{OH}^{-}\): \([\mathrm{OH}^{-}] = 10^{-\mathrm{pOH}} = 10^{-0.72} \approx 0.190 \mathrm{~M}\)

Write the dissolution equation for \(\mathrm{Ba}(\mathrm{OH})_{2}\)

When \(\mathrm{Ba}(\mathrm{OH})_{2}\) dissolves in water, it dissociates into its constituent ions: \(\mathrm{Ba}(\mathrm{OH})_{2} \leftrightharpoons \mathrm{Ba}^{2+} + 2\mathrm{OH}^{-}\)

Write the expression for \(K_{sp}\) and find the concentration of \(\mathrm{Ba}^{2+}\) ions

The solubility product constant, \(K_{sp}\), can be defined for the equilibrium reaction above as: \(K_{sp} = [\mathrm{Ba}^{2+}][\mathrm{OH}^{-}]^2\) To estimate \(K_{sp}\), we need to calculate the concentration of \(\mathrm{Ba}^{2+}\) ions in the solution. Using the relationship between the ions from the dissolution reaction: \([\mathrm{Ba}^{2+}] = \frac{1}{2}[\mathrm{OH}^{-}]\) Now substitute the concentration of \(\mathrm{OH}^{-}\) ions found in Step 1: \([\mathrm{Ba}^{2+}] = \frac{1}{2}(0.190) = 0.095 \mathrm{~M}\)

Calculate \(K_{sp}\)

Now, we substitute the concentrations of \(\mathrm{Ba}^{2+}\) and \(\mathrm{OH}^{-}\) ions into the expression for \(K_{sp}\): \(K_{sp} = [\mathrm{Ba}^{2+}][\mathrm{OH}^{-}]^2 = (0.095)(0.190)^2 = 0.095(0.0361) \approx 3.43 \times 10^{-3}\) Therefore, the estimated value of \(K_{sp}\) for \(\mathrm{Ba}(\mathrm{OH})_{2}\) is \(3.43 \times 10^{-3}\).

Key concepts

Solubility Product Constant

The solubility product constant, or Ksp, is a special type of equilibrium constant that applies to the dissolution of sparingly soluble salts. It is calculated by multiplying the concentrations of the ions produced when the salt dissolves, with each concentration raised to the power of the ion's stoichiometric coefficient from the balanced equation.

For instance, a salt AB that dissolves to produce A+ and B- in solution would have the equation AB ⇌ A+ + B-, and the Ksp expression would be Ksp = [A+][B-]. The numerical value of Ksp indicates the extent to which a compound will dissolve in water and is temperature-dependent. Ksp is crucial for predicting whether a precipitate will form when solutions of two different salts are mixed and for calculating the concentrations of ions in a saturated solution.

Barium Hydroxide Dissolution

Barium hydroxide, Ba(OH)2, is a strong base which dissociates completely in aqueous solutions. The dissolution process can be represented as: Ba(OH)2 (s) ⇌ Ba2+ (aq) + 2OH- (aq).

Upon dissolving, each formula unit of the solid releases one Ba2+ ion and two OH- ions into solution. This dissolution process is essential for understanding the behavior of barium hydroxide in different chemical environments, predicting the pH of the solution, and determining the extent to which Ba(OH)2 will dissolve at a given temperature, which directly relates to its Ksp value.

pH and pOH Calculations

The pH and pOH of a solution are measures of its acidity and alkalinity, respectively. The pH scale typically ranges from 0 to 14, with lower values being acidic, 7 being neutral, and higher values being basic or alkaline. The pOH is similarly a measure of the hydroxide ion concentration in solution. The pH and pOH are connected through the relationship pH + pOH = 14. This is fundamental to understanding the properties of solutions, especially when dealing with acid-base reactions and buffer solutions.

For example, if you know that the pH of a particular Ba(OH)2 solution is 13.28, you can find the pOH by subtracting the pH from 14. Then, using the relationship [OH-] = 10^-pOH, you can determine the molarity of hydroxide ions within that solution.

Concentration of Ions in Solution

The concentration of ions in a solution is indicative of the amount of substance that has dissolved. It can be calculated using the stoichiometry of the dissolution process and the solubility product constant (Ksp). When you understand how to calculate ion concentrations, you can predict solubility, carry out precipitation reactions, and determine the resulting solution's pH.

Let's take the dissociation of Ba(OH)2 as our example. Knowing that the stoichiometric ratio of Ba2+ to OH- is 1:2, you calculate the Ba2+ concentration by halving the OH- concentration. This relationship between the concentrations of the different ions is derived from the dissolution equation and is necessary to accurately estimate the Ksp for the substance in question.