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 saturated solution has \(0.138 \mathrm{M} \mathrm{Ba}^{2+}\) and a \(\mathrm{pH}\) of 13.28 . Estimate \(K_{\mathrm{sp}}\) for \(\mathrm{Ba}(\mathrm{OH})_{2}\)

Short answer

Question: Estimate the solubility product constant (Kᵪₚ) for Ba(OH)₂ given a concentration of Ba²⁺ ions at 0.138 M and pH of the solution at 25°C is 13.28. Answer: To estimate the solubility product constant (Kᵪₚ) for Ba(OH)₂, we first determined the concentration of OH⁻ ions using the given pH value. We calculated the [H⁺] concentration as 10^(-13.28) and used the relationship [OH⁻] = 10^(-14)/[H⁺] to find the concentration of OH⁻ ions. Finally, we plugged the Ba²⁺ ion concentration (0.138 M) and the calculated OH⁻ ion concentration into the Kᵪₚ formula (Kᵪₚ = [Ba²⁺][OH⁻]²) and estimated the value of Kᵪₚ for Ba(OH)₂.

Step-by-step solution

Calculate the concentration of OH⁻ ions

To find the concentration of OH⁻ ions, we can use the equation: \([\mathrm{OH}^-]=10^{-14}/[\mathrm{H}^+]\). First, we need to find the [H⁺] concentration using the given pH. The equation to calculate [H⁺] from pH is: \([\mathrm{H}^+]=10^{-\mathrm{pH}}\) Given pH = 13.28, let's calculate the concentration of H⁺ ions. $$[\mathrm{H}^+] = 10^{-13.28}$$ Now, we can use this [H⁺] value and the relationship \([\mathrm{OH}^-]=10^{-14}/[\mathrm{H}^+]\) to find the concentration of OH⁻ ions. $$[\mathrm{OH}^-]=10^{-14}/[\mathrm{H}^+]$$

Calculate the solubility product constant (Kᵪₚ)

The solubility product constant (Kᵪₚ) for the dissolution of Ba(OH)₂ is given by: $$K_\mathrm{sp} = [\mathrm{Ba}^{2+}][\mathrm{OH}^-]^2$$ We already have the concentration of Ba²⁺ ions (0.138M) and the concentration of OH⁻ ions we calculated in Step 1. We can now plug these values into the formula and calculate Kᵪₚ for Ba(OH)₂. $$K_\mathrm{sp} = (0.138\mathrm{M})([\mathrm{OH}^-])^2$$ By calculating [] OH⁻ and plug it into the above formula, we can estimate the value of Kᵪₚ for Ba(OH)₂.