Question

What is the \(\mathrm{H}_{3} \mathrm{O}^{+}\)concentration in each of the following solutions? (a) lake water, \(\mathrm{pOH}=6.00\) (b) coffee, \(\mathrm{pOH}=8.90\) (c) borax, pOH \(=4.50\)

Short answer

The concentration of \(\mathrm{H}_{3} \mathrm{O}^{+}\) ions in lake water is \(10^{-8}\) M, in coffee is \(10^{-5.1}\) M, and in borax is \(10^{-9.5}\) M.

Step-by-step solution

Calculating the \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration for lake water

The \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration can be found using the formula \[ [\mathrm{H}_{3} \mathrm{O}^{+}] = 10^{-(14- \mathrm{pOH})} \] So for lake water with a \(\mathrm{pOH}\) of 6.00, a computation of \[ [\mathrm{H}_{3} \mathrm{O}^{+}] = 10^{-(14-6)} = 10^{-8} \] will give the concentration value for \(\mathrm{H}_{3} \mathrm{O}^{+}\) ions.

Calculating the \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration for coffee

The \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration for coffee can be found in the same way using its \(\mathrm{pOH}\) value. For coffee, where \(\mathrm{pOH}\) = 8.90, a computation of \[ [\mathrm{H}_{3} \mathrm{O}^{+}] = 10^{-(14-8.90)} = 10^{-5.10} \] will give the concentration value for \(\mathrm{H}_{3} \mathrm{O}^{+}\) ions.

Calculating the \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration for borax

The same process we used earlier can be repeated for borax to find the \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration. For borax with a \(\mathrm{pOH}\) value of 4.50, the computation would be \[ [\mathrm{H}_{3} \mathrm{O}^{+}] = 10^{-(14-4.50)} = 10^{-9.5} \] This will give the concentration value for \(\mathrm{H}_{3} \mathrm{O}^{+}\) ions in the borax solution.

Key concepts

pOH

Understanding the pOH scale is fundamental when discussing the acidity or basicity of a solution. The pOH is a measure of the hydroxide ion (OH^-) concentration and is calculated as the negative logarithm to the base 10 of the hydroxide ion concentration. Hence, a solution with a higher OH^- concentration will have a lower pOH value, indicative of a more basic solution. Conversely, a lower OH^- concentration results in a higher pOH, suggesting a more acidic environment. It is important to remember that the pOH can range from 0 to 14, where lower values are basic, and higher ones are acidic. This intuitive scale helps students easily categorize a solution’s basicity by just knowing the pOH value.

For instance, a pOH of 6.00, typical for lake water, indicates the presence of a base, albeit one that is not strongly basic. By understanding the pOH value alone, one can already infer the acidity or basicity of a solution before even embarking on any concentration calculations.

pH and pOH relationship

The connection between pH and pOH is vital in understanding the holistic nature of a solution's acidity or basicity. The pH scale measures the hydrogen ion concentration (H^+ or H3O^+), while the pOH scale measures hydroxide ion concentration (OH^-). These two values are inversely related and summed together always equal to 14 in pure water and aqueous solutions at 25°C, known as the neutral point where H3O^+ equals OH^-. This relationship is expressed by the equation: pH + pOH = 14 . This formula allows us to solve for one value when the other is given, and it serves as a bridge between the pH and pOH scales. For example, with a pOH of 8.90, as you would find in coffee, you can subtract this value from 14 to find the corresponding pH, helping you to quantify the solution's overall acidity.

Acid and base concentration

Delving into the realm of acid and base concentration involves understanding the amount of hydrogen ions (H3O^+) for acids and hydroxide ions (OH^-) for bases present in a solution. The strength or weakness of an acid or base is determined by its dissociation in water – strong acids and bases fully dissociate, releasing a greater concentration of ions, whereas weak acids and bases partially dissociate. By determining the concentration of these ions, we gain insight into the solution’s pH or pOH. For instance, in the context of borax with a pOH of 4.50, we learn that it contains a certain hydroxide ion concentration directly correlated to its basic strength. Being able to calculate the ion concentrations from pOH or pH expands on our understanding of chemical properties and reactivity of solutions.

Logarithmic concentration calculations

The logarithmic aspect in calculating concentrations, specifically with the pH and pOH scales, is a critical mathematical concept in chemistry. pH and pOH are examples of how logarithms simplify handling very small or large numbers, such as ion concentrations in solution, into manageable figures. A logarithm is the power to which a number must be raised to obtain some other number. For example, if 10 is raised to the power of -7 to get 0.0000001, then the logarithm of 0.0000001 is -7. When applied to the pH or pOH, this concept allows us to convert between the ion concentration and its corresponding pH or pOH value. Logarithmic calculations are used because the H3O^+ and OH^- ion concentrations often span several orders of magnitude, so a logarithmic scale provides a concise and uniform way to express these values. As illustrated in the exercise with lake water and other solutions, we use the formulas pH = -log(H3O^+) and pOH = -log(OH^-) to move from ion concentration to their respective p scales.