Find the \(\mathrm{pH}\) of a solution consisting of 0.29 mol of \(\mathrm{HBr}\) in 1.0 \(\mathrm{L}\) of water.

Understanding strong acid ionization is crucial when calculating the pH of a solution. Strong acids, such as hydrobromic acid (HBr), sulfuric acid (H₂SO₄), and hydrochloric acid (HCl), dissociate completely in water. This means that in an aqueous solution, strong acids release all their hydrogen ions (H⁺) into the solution, resulting in a highly acidic environment.

When we say an acid ionizes 'completely', it signifies that the acid's molecules dissociate to form hydrogen ions and their respective anions to an extent of nearly 100%. In terms of chemical reactions:
\( \text{HA} \rightarrow \text{H}^+ + \text{A}^- \)
For \( \text{HBr} \), this would be:
\( \text{HBr} \rightarrow \text{H}^+ + \text{Br}^- \)
The concentration of \( \text{H}^+ \) ions is thus equal to the initial concentration of the strong acid if we assume that the volume does not change significantly. This direct relationship simplifies the process of calculating the pH of strong acid solutions and forms the basis for the subsequent steps in pH calculations.

Focusing on hydrobromic acid (HBr), classified as a strong acid, is instrumental to understanding the concept of pH in acid solutions. HBr is a diatomic molecule consisting of one hydrogen atom and one bromine atom.

When dissolved in water, HBr undergoes ionization:
\( \text{HBr(aq)} \rightarrow \text{H}^+ (\text{aq}) + \text{Br}^- (\text{aq}) \)
Given the exercise, if we have a 0.29 mol/L concentration of HBr, it means that \( \text{H}^+ \) ions will also be at 0.29 mol/L since each molecule of HBr releases one hydrogen ion into the solution. This concentration will directly impact the solution's pH, a measure of its acidity. Due to the complete ionization of HBr, the solution's pH can be predicted accurately from the molarity of the original acid.

Logarithmic functions play a fundamental role in the calculation of pH. The pH scale is a logarithmic scale used to specify the acidity or basicity of an aqueous solution. The 'p' in pH stands for 'potenz' in German, meaning power or exponent, reflecting its basis in logarithms.

pH is defined as the negative base-10 logarithm of the activity of hydrogen ions in a solution:
\( \text{pH} = -\log [\text{H}^+] \)
For a 0.29 mol/L hydrogen ion concentration, the pH is calculated using this logarithmic relationship. Logarithms are the inverse operations of exponentiation, meaning:
\( \log_{10} (10^x) = x \)
The logarithm asks the question 'to what power must 10 be raised to obtain a certain number?'. In the context of our pH example, we use the base-10 logarithm of the concentration of hydrogen ions and then take the negative of this value to determine the pH. This calculation reflects how much more acidic the solution is compared to pure water, which has a pH of 7, the neutral point on the pH scale.