Question

Calculate \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) and \(\mathrm{pH}\) in saturated \(\mathrm{Ba}(\mathrm{OH})_{2}(\mathrm{aq})\) which contains \(3.9 \mathrm{g} \mathrm{Ba}(\mathrm{OH})_{2} \cdot 8 \mathrm{H}_{2} \mathrm{O}\) per \(100 \mathrm{mL}\) of solution.

Short answer

The concentration of the \(\mathrm{H}_{3} \mathrm{O}^{+}\) ions is \(4.03 \times 10^{-16} \: M\) and the pH of the solution is 15.4.

Step-by-step solution

Calculate number of moles

Calculate the number of moles of \(\mathrm{Ba}(\mathrm{OH})_{2}(\mathrm{aq})\) in 100 mL of the solution using the molar mass (formula weight \(\mathrm{Ba}(\mathrm{OH})_{2} \cdot 8 \mathrm{H}_{2} O = 315.46 \: g/mol\)). You do this by dividing the mass of solute by the molar mass. \(n = \frac{3.9 \: g}{315.46 \: g/mol} = 0.0124 \: mol\).

Calculate concentration of hydroxide ions

Calculate the concentration of \(\mathrm{OH}^-\) ions in the solution. Since each formula unit of \(\mathrm{Ba}(\mathrm{OH})_{2}\) contains 2 \(\mathrm{OH}^-\) ions, the concentration of \(\mathrm{OH}^-\) ions will be twice the number of moles of \(\mathrm{Ba}(\mathrm{OH})_{2}\) since it dissociates fully in solution in 1L, but since it's in 100mL the concentration \([\mathrm{OH}^-]\) will be \(\frac{(2*0.0124mol)}{100 mL} * 1000 = 0.248 M\)

Calculate concentration of hydronium ions

Use the ion-product of water (\(K_w = [\mathrm{H}_{3} \mathrm{O}^{+}][\mathrm{OH}^{-}]\) = \(1.0 \times 10^{-14}\)) to find concentration of \(\mathrm{H}_{3} \mathrm{O}^{+}\) ions. Rearrange the formula to find \([\mathrm{H}_{3} \mathrm{O}^{+}]\), which equals to \(K_w / [\mathrm{OH}^{-}] = \frac{1.0 \times 10^{-14}}{0.248} = 4.03 \times 10^{-16} M\).

Calculate pH

Finally find the pH using the formula pH = -\(\log[\mathrm{H}_{3} \mathrm{O}^{+}]\), so the pH = \( -\log(4.03 \times 10^{-16}) = 15.4\).

Key concepts

pH Calculation

Understanding how to calculate the pH of a solution is an essential skill in chemistry. pH is a measure of the acidity or basicity of an aqueous solution. On the pH scale, which ranges from 0 to 14, solutions with a pH less than 7 are acidic, and solutions with a pH greater than 7 are basic.

The pH is calculated using the concentration of hydronium ions \(\mathrm{H}_{3}\mathrm{O}^{+}\) in the solution. The relationship between pH and hydronium ion concentration is given by the formula:
\[\text{pH} = -\log\left([\mathrm{H}_{3}\mathrm{O}^{+}]\right)\]
This means to find the pH, you take the negative logarithm (base 10) of the molar concentration of hydronium ions. A key point to remember is that a higher concentration of hydronium ions corresponds to a lower pH (more acidic), and a lower concentration corresponds to a higher pH (more basic).

Molar Concentration

Molar concentration, often simply called concentration, refers to the amount of a substance (solute) dissolved in a given volume of solvent. It's denoted by the unit molarity (M), which is moles per liter (mol/L).

To calculate the molar concentration, you'll need to:

• Determine the number of moles of solute present.
• Measure the volume of the solution in liters.
• Divide the moles of solute by the volume of the solution.

For solutions of strong electrolytes like \(\mathrm{Ba}(\mathrm{OH})_{2}\), which dissociate completely in water, the concentration of ions is directly related to the molarity of the solute. With this in mind, you can easily calculate the concentrations of resulting ions in a solution, which is crucial for further calculations like determining the pH.

Ion-Product of Water

The ion-product of water \(K_w\) is a constant value that represents the product of the concentrations of hydronium \(\mathrm{H}_{3}\mathrm{O}^{+}\) and hydroxide \(\mathrm{OH}^{-}\) ions in water at a given temperature. At room temperature, \(K_w\) is \(1.0 \times 10^{-14}\).\[K_w = [\mathrm{H}_{3} \mathrm{O}^{+}][\mathrm{OH}^{-}] = 1.0 \times 10^{-14}\]
This constant is an important concept when solving for unknown concentrations in acid-base chemistry. If you know the concentration of one of the ions, you can easily solve for the other using this constant. This relationship is what allows us to find the \(\mathrm{H}_{3}\mathrm{O}^{+}\) concentration if the \(\mathrm{OH}^{-}\) concentration is known, providing a link between pH and the basicity of a solution.