\(A\) basic solution has a pH of \(9.8\). What is its molar \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration?

Understanding the relationship between pH and hydronium ion concentration is essential for students of chemistry. The pH scale is a measure of the acidity or basicity of a solution. It ranges from 0 to 14, with 7 representing a neutral solution—neither acidic nor basic.

The term 'pH' stands for 'potential of hydrogen' or 'power of hydrogen' and is a logarithmic scale. This means that each whole pH value below 7 is ten times more acidic than the next higher value, and each whole pH value above 7 is ten times less acidic (more basic) than the one before it.

For a solution with a pH of 9.8, which is basic as it's greater than 7, the concentration of hydronium ions can be calculated using the formula: \[ [H_3O^+] = 10^{-pH} \] This indicates a low concentration of hydronium ions, reflecting the basicity of the solution. Specifically, a pH of 9.8 corresponds to a hydronium ion concentration of approximately \(1.58 \times 10^{-10} \) M.

Logarithms play a pivotal role in chemistry, particularly when dealing with pH calculations. A logarithm answers the question: to what exponent must we raise a base number (often 10 in chemistry) to obtain a certain number?

In the context of pH, the logarithmic relationship is inverse; as the hydronium ion concentration increases, the pH decreases, and vice versa. This is due to the pH formula: \[ pH = -\log_{10} [H_3O^+] \] Logarithmic scales allow chemists to work with the very large range of hydronium ion concentrations without having to write out lengthy decimal numbers. For example, the difference in concentration between pH 2 and pH 3 is a factor of 10, which is relatively easy to comprehend compared to the actual concentrations, which might be \(1 \times 10^{-2} \) M and \(1 \times 10^{-3} \) M, respectively.

Acidic and basic solutions are distinguished by their pH values. An acidic solution has a pH less than 7 and contains a higher concentration of hydronium ions (\(H_3O^+\)). Basic solutions, on the other hand, have a pH greater than 7 and contain lower concentrations of hydronium ions.

Acids donate protons (hydrogen ions) in a solution, increasing the concentration of hydronium ions, while bases accept protons, decreasing the hydronium ion concentration. This interchange of protons is what chemists refer to as the Bronsted-Lowry acid-base theory.

A solution with a pH of 9.8, as in the original exercise, is basic, and this reflects a lower concentration of hydronium ions, as also demonstrated by the calculation \(1.58 \times 10^{-10} \) M. The ability to determine the nature of a solution - whether it is acidic or basic - is crucial in various fields such as biology, environmental science, medicine, and many industrial applications.